Cardassian Expansion: a Model in Which the Universe Is Flat, Matter

Physics Letters B 540 (2002) 1–8 www.elsevier.com/locate/npe

Cardassian expansion: a model in which the universe is ﬂat, matter dominated, and accelerating

Katherine Freese, Matthew Lewis

Michigan Center for Theoretical Physics, University of Michigan, Ann Arbor, MI 48109, USA Received 15 January 2002; received in revised form 30 May 2002; accepted 30 May 2002 Editor: J. Frieman

Abstract A modification to the Friedmann–Robertson–Walker equation is proposed in which the universe is flat, matter dominated, and accelerating. An additional term, which contains only matter or radiation (no vacuum contribution), becomes the dominant driver of expansion at a late epoch of the universe. During the epoch when the new term dominates, the universe accelerates; we call this period of acceleration the Cardassian era. The universe can be flat and yet consist of only matter and radiation, and still be compatible with observations. The energy density required to close the universe is much smaller than in a standard cosmology, so that matter can be sufficient to provide a flat geometry. The new term required may arise, e.g., as a consequence of our observable universe living as a 3-dimensional brane in a higher-dimensional universe. The Cardassian model survives several observational tests, including the cosmic background radiation, the age of the universe, the cluster baryon fraction, and structure formation.  2002 Elsevier Science B.V. All rights reserved.

1. Introduction radiation. Pure matter (or radiation) alone can drive an accelerated expansion if the Friedmann–Robertson– Recent observations of type IA Supernovae [1, Walker (FRW) equation is modiﬁed by the addition of 2] as well as concordance with other observations a new term on the right-hand side as follows: (including the microwave background and galaxy H 2 = Aρ + Bρn, (1) power spectra) indicate that the universe is accel- ˙ erating. Many authors have explored a cosmologi- where H = R/R is the Hubble constant (as a function cal constant, a decaying vacuum energy [3,4], and of time), R is the scale factor of the universe, the quintessence [5–7] as possible explanations for such energy density ρ contains only ordinary matter and an acceleration. radiation, and we will take Here we propose an alternative which invokes n<2/3. (2) no vacuum energy whatsoever. In our model the universe is ﬂat and yet consists only of matter and In the usual FRW equation B = 0. To be consistent with the usual FRW result, we take 8π E-mail address: [email protected] A = . (M. Lewis). 2 3mpl 0370-2693/02/$ – see front matter  2002 Elsevier Science B.V. All rights reserved. PII:S0370-2693(02)02122-6 2 K. Freese, M. Lewis / Physics Letters B 540 (2002) 1–8

We note here that the geometry is flat, as required The second term starts to dominate at a redshift zeq n by measurements of the cosmic background radiation when Aρ(zeq) = Bρ (zeq), i.e., when [8], so that there are no curvature terms in the = 1−n + 3(1−n) equation. There is no vacuum term in the equation. B/A ρ0 (1 zeq) . (5) This Letter does not address the cosmological constant From evaluating Eq. (1) today, we have (Λ) problem; we simply set Λ = 0. In this Letter, we first study the phenomenology of 2 n H = Aρ0 + Bρ (6) the ansatz in Eq. (1), and then turn to a discussion 0 0 of the origin of this equation.1 Directions for a future so that search for a fundamental theory will be discussed. = 2 − n−1 A H0 /ρ0 Bρ0 . (7) From Eqs. (5) and (7), we have 2. The role of the Cardassian term2 2 3(1−n) H (1 + zeq) B = 0 . (8) The new term in the equation (the second term n 3(1−n) ρ [1 + (1 + zeq) ] on the right-hand side) is initially negligible. It only 0 comes to dominate recently, at the redshift zeq ∼ We have two parameters in the model: B and n, O(1) indicated by the supernovae observations. Once or, equivalently, zeq and n. Note that B here is the second term dominates, it causes the universe chosen to make the second term kick in at the right to accelerate. We can consider the contribution of time to explain the observations. As yet we have ordinary matter, with no explanation of the coincidence problem; i.e., we have no explanation for the timing of zeq.Suchan −3 ρ = ρ0(R/R0) (3) explanation would arise if we had a reason for the to this new term. Once the new term dominates the required mass scale of B. The parameter B has units 2−4n right-hand side of the equation, we have accelerated of mass . Later, we will discuss the origin of the expansion. When the new term is so large that the Cardassian term in terms of extra dimensions, and ordinary first term can be neglected, we find discuss the origin of the mass scale of B. As discussed below, to match the CMB and supernovae data we take 2 R ∝ t 3n (4) 0.3 zeq 1, but this value can easily be refined to better fit upcoming observations. so that the expansion is superluminal (accelerated) for n<2/3. As examples, for n = 2/3wehaveR ∼ t; for n = 1/3wehaveR ∼ t2;andforn = 1/6wehave 3. What is the current energy density of the R ∼ t4. The case of n = 2/3 produces a term in the universe? FRW equation H 2 ∝ R−2; such a term looks similar to a curvature term but is generated here by matter in a Observations of the cosmic background radiation universe with a flat geometry. Note that for n = 1/3the show that the geometry of the universe is flat with acceleration is constant, for n>1/3 the acceleration is Ω = 1. In the Cardassian model we need to revisit diminishing in time, while for n<1/3 the acceleration 0 the question of what value of energy density today, ρ , is increasing (the cosmic jerk). 0 corresponds to a flat geometry. We will show that the energy density required to close the universe is much 1 As discussed below, we were motivated to study an equation smaller than in a standard cosmology, so that matter of this form by work of Chung and Freese [10] who showed that can be sufficient to provide a flat geometry. terms of the form ρn can generically appear in the FRW equation as The energy density ρ0 that satisfies Eq. (6) is, by a consequence of embedding our observable universe as a brane in extra dimensions. definition, the critical density. From Eqs. (1) and (5), 2 The name Cardassian refers to a humanoid race in Star Trek we can write whose goal is to take over the universe, i.e., accelerated expansion. 2 = + 1−n + 3(1−n) n This race looks foreign to us and yet is made entirely of matter. H A ρ ρ0 (1 zeq) ρ . (9) K. Freese, M. Lewis / Physics Letters B 540 (2002) 1–8 3

Fig. 1. The ratio F(n,zeq) = ρc/ρc,old as given by Eq. (13). The contour labeled 0.3 corresponds to parameters n and zeq roughly consistent with present observations.

= 2 3 Evaluating this equation today with A 8π/(3mpl), above, i.e., we have = × × −29 2 3 ρ0 (1/3, 1/5, 0.15) 1.88 10 h0 gm/cm 2 = 8π + + 3(1−n) H0 ρ0 1 (1 zeq) . (10) 2 for n = (2/3, 1/3, 1/6) and zeq = 1. (16) 3mpl

Defining ρ0 = Ω0ρc we find that the critical density For larger values of zeq, the modification to the ρc has been modified from its usual value, i.e., the value of ρc can be even larger. Note the amusing number has changed. We find result that for zeq = 2andn = 1/12, we have ρc = 2 2 0.046ρc,old so that baryons would close the universe 3H0 mpl (not a universe we advocate). ρ = . (11) c 3(1−n) 8π[1 + (1 + zeq) ] Thus 4. Cluster baryon fraction ρc = ρc,old × F(n), (12) where For the past ten years, a multitude of observations has pointed towards a value of the matter density = + + 3(1−n) −1 F(n) 1 (1 zeq) (13) ρ0 ∼ 0.3ρc,old. The cluster baryon fraction [11,12] as and well as the observed galaxy power spectrum are best fit if the matter density is 0.3 of the old critical density. = × −29 2 −3 ρc,old 1.88 10 h0 gm/cm (14) Recent results from the CMB [8,9] also obtain this value. In the standard cosmology this result implied and h0 is the Hubble constant today in units of 100 km s−1 Mpc−1. In Fig. 1, we have plotted the new critical density ρc as a function of the two 3 An alternate possible definition would be to keep the standard parameters n and zeq. For example, if we take zeq = 1, value of ρc and discuss the contribution to it from the two terms on we find the right-hand side of the modified FRW equation. Then there would be a contribution to Ω from the ρ term and another contribution F = (1/3, 1/5, 0.15) from the ρn term, with the two terms adding to 1. This is the approach taken when one discusses a cosmological constant in lieu for n = (2/3, 1/3, 1/6), respectively. (15) of our second term. However, the situation here is different in that we have only matter in the equation. The disadvantage of this second We see that the value of the critical density can be choice of definitions would be that a value of the energy density = much lower than previously estimated. Since Ω0 1, today equal to ρc according to this second definition would not we have today’s energy density as ρ0 = ρc as given correspond to a flat geometry. 4 K. Freese, M. Lewis / Physics Letters B 540 (2002) 1–8

Table 1 fected. Below we discuss the impact on late structure Values of zeq for various values of n corresponding to a universe formation during the era where the Cardassian term = with ρ0 0.3ρc,old. The age of the universe today t0 corresponding is important. This term accelerates the expansion of to the two parameters n and z is listed in the last column, where eq the universe, and freezes out perturbation growth once H0 is the value of the Hubble constant today it dominates (much like when a curvature term dom- nz H t eq 0 0 inates); this freeze out happens late enough that it is 0.60 1.00 0.73 relatively unimportant. To obtain an idea of the type 0.50 0.76 0.78 0.40 0.60 0.83 of effects that we may find, instead of analyzing the 0.30 0.50 0.87 exact perturbation equations with metric perturbations 0.20 0.42 0.92 included, we will merely modify the time dependence 0.10 0.37 0.95 of the scale factor in the usual Jeans analysis equation. 0.00 0.33 0.99 For now we take the standard equation for perturbation growth; as a caveat, we warn that recent structure that matter could not provide the entire closure density. formation may be further modified due to a change in Here, on the other hand, the value of the critical Poisson’s equations as described below. For we now density can be much lower than previously estimated. we take Hence the cluster motivated value for ρ0 is now compatible with a closure density of matter, Ω = 1, 0 δ¨ + 2(R/R)˙ δ˙ = 4πρδ/m2 , (17) all in the form of matter. For example, if n = 0.6 with pl z = 1, or if n = 0.2 with z = 0.4, a critical density eq eq where δ = (ρ −¯ρ)/ρ¯ is the fluid overdensity. Now of matter corresponds to ρ ∼ 0.3ρ , as required by 0 c,old one must substitute Eq. (1) for R/R˙ . In the standard the cluster baryon fraction and other data. In Fig. 1, FRW cosmology with matter domination, R ∼ t2/3, one can see which combination of values of n and z eq and there is one growing solution to δ with δ ∼ R ∼ produce the required factor of 0.3. If we assume that t2/3. This standard result still applies throughout most the value ρ = 0.3ρ is correct, for a given value 0 c,old of the (matter dominated) history of the universe in our of n (that is constant in time) we can compute the new model, so that structure forms in the usual way. value of z for our model from Eq. (13). Table 1 lists eq Modifications set in once the new Cardassian term these values of n and z . Henceforth, we shall focus eq becomes important. When R ∼ tp, Eq. (17) can be on these combinations of parameters. written

2 5. Age of the universe 2p 3p − δ (x) + δ (x) − x 3pδ = 0, (18) x 2 In the Cardassian model, the universe is older due to the presence of the second term. In Table 1, we where x ≡ t/t0 with t0 denoting the time today and present the age of the universe for various values of n superscript prime refers to d/dx. This equation can (under the assumption that ρ0 = 0.3ρc,old). generally be solved in terms of Bessel functions for If one takes t0 > 10 Gyr as the lower bound on constant p (such as is the case once the Cardassian globular cluster ages, then one requires t0H0 > 0.66 term completely overrides the old term). A simple = = for h0 = 0.65. If one requires globular cluster ages example is the case of n 2/3andp 1; in greater than 11 Gyr [23], then t0H0 > 0.73 for h0 = the limit x 3/4, the last term in Eq. (18) can 0.65. All values in Table 1 satisfy these bounds. be dropped and the equation is solved as δ(t) = −1 a1 + a2t . Perturbations cease growing and become frozen in. This result agrees with the expectation 6. Structure formation that in a universe that is expanding more rapidly, the overdensity will grow more slowly with the scale Since the new (Cardassian) term becomes impor- factor. As mentioned at the outset, as long as the tant only at zeq ∼ 1, early structure formation is not af- Cardassian term becomes important only very late K. Freese, M. Lewis / Physics Letters B 540 (2002) 1–8 5 in the history of the universe, much of the structure with F given in Eq. (13). As discussed previously, we see will have already formed and be unaffected. as our standard case we will take F ≡ ρc/ρc,old = Further comments on late structure formation (e.g., 0.3. With this assumption, and by using expression cluster abundances) follow below. Eq. (24) in Eq. (22), we find that d changes by a factor of (1.47, 1.88, 2.04 and 2.23) for n = (0.6, 0.3, 0.2 and 0.1), respectively, compared to the usual 7. Doppler peak in cosmic background radiation (nonCardassian) FRW universe with ρ0 = ρc,old.In addition Here we argue that the location of the first Doppler √ √ 1 1 + R∗ + R∗ + r∗R∗ peak is only mildly affected by the new Cardassian s∗ ∝ √ ln √ , cosmology. We need to calculate the angle subtended F 1 + r∗R∗ by the sound horizon at recombination. In the standard − where r∗ = 0.042(F h2) 1 and R∗ = 30Ω h2 and we FRW cosmology with flat geometry, this value corre- b use h = 0.7andΩ = 0.04. We find that s∗ changes sponds to a spherical harmonic with l = 200. A peak b by a factor of (1.44, 1.62, 1.67, 1.29) for n = (0.6, at this angular scale has indeed been confirmed [8]. In 0.3, 0.2 and 0.1), respectively, compared to the usual the Cardassian cosmology we still have a flat geome- FRW universe with ρ = ρ . The angle subtended try. Hence, we can still write 0 c,old by the sound horizon on the surface of last scattering θ = s∗/d, (19) decreases and the location (l) of the first Doppler peak increases by roughly a factor of where s∗ is the sound horizon at the time of recombination tr and d is the distance a light ray trav- (1.02, 1.11, 1.12, 1.13) els from recombination to today. To calculate these n = , , lengths, we use the fact that for a light ray ds2 = 0 = for (0.6, 0.3, 0.2, 0.1) respectively (25) − 2 + 2 2 dt a dx to write compared to the usual FRW universe with ρ0 = ρc,old. t0 This shift still lies within the experimental uncertainty d = dt/a. (20) on measurements of the location of the Doppler peak. We note the following: in the same way that a t r nonzero Λ may make the current CBR observations Following the notation of Peebles [13], we define the compatible with a small but nonzero curvature, indeed redshift dependence of H as a nonzero Cardassian term could also allow for a nonzero curvature in the data. A more accurate study H(z)= H E(z) (21) 0 of the effects of Cardassian expansion on the cosmic so that Eq. (20) can be written background radiation (including the first and higher

zr peaks) is the subject of a future study. 1 dz d = . (22) H0R0 E(z) 0 8. The cutoff energy density Similarly, the sound horizon at recombination is ∞ An alternate way to write Eq. (1) is = s∗ dt/a. (23) 2 n−1 H = Aρ 1 + (ρ/ρcutoff) , (26) zr ≡ = In standard matter dominated FRW cosmology where ρcutoff ρ(zeq) A/B. This notation of- 3/2 fers a new interpretation; it indicates that the sec- with Ωm,0 = 1, E(z) = (1 + z) in Eq. (22) and ond term only becomes important once the energy d = 2/H0R0. For the cosmology of Eq. (1), we have density of the universe drops below ρcutoff,which has a value a few times the critical density. Hence, E(z)2 = F × (1 + z)3 + (1 − F)× (1 + z)3n (24) regions of the universe where the density exceeds 6 K. Freese, M. Lewis / Physics Letters B 540 (2002) 1–8 this cutoff density will not experience effects asso- 10. Best fit of parameters to current data ciated with the Cardassian term. In particular, we can be reassured that the new term will not affect We can find the best fit of the Cardassian parame- gravity on the Earth or the Solar System. The den- ters n and zeq to current CMB and Supernova data. 3 sity of water on the Earth is 1 gm/cm ,whichis The current best fit is obtained for ρ0 = 0.3ρc,old (as 28 orders of magnitude higher than the critical den- we have discussed above) and n<0.4 (equivalently, sity. w<−0.6) [20,21]. In Table 1 one can see the values of zeq compatible with this bound, as well as the resultant age of the universe. As an example, for n = 0.2 (equivalently, w =−0.8), we find that z = 0.42. 9. Comparing to quintessence eq Then the position of the first Doppler peak is shifted by a factor of 1.12. The age of the universe is 13 Gyr. We note that, with regard to observational tests, The cutoff energy density is ρcutoff = 2.7ρc,sothat [14–18], one can make a correspondence between the new term is important only for ρ<ρcutoff = 2.7ρc. the Cardassian and quintessence models; we stress, Hence, as mentioned above, the Cardassian term will however, that the two models are entirely different. not affect the physics of the Earth or Solar System in Quintessence requires a dark energy component with a any way. specific equation of state (p = wρ), whereas the only We note the enormous uncertainty in the current ingredients in the Cardassian model are ordinary mat- data; future experiments (such as SNAP [22]) will ter (p = 0) and radiation (p = 1/3). However, as far constrain these parameters further. as any observation that involves only R(t), or equivalently H(z), the two models predict the same effects on the observation. Regarding such observations, we 11. Extra dimensions can make the following identifications between the Cardassian and quintessence models: n ⇒ w+1, F ⇒ A Cardassian term may arise as a consequence + Ωm,and1− F ⇒ ΩQ,wherew is the quintessence of embedding our observable universe as a (3 1)- equation of state parameter, Ωm = ρm/ρc,old is the ra- dimensional brane in extra dimensions. Chung and tio of matter density to the (old) critical density in the Freese [10] showed that, in a 5-dimensional universe standard FRW cosmology appropriate to quintessence, with metric = ΩQ ρQ/ρc,old is the ratio of quintessence energy ds2 =−q2(τ, u) dτ 2 + a2(τ, u) dx2 density to the (old) critical density, and F is given by + 2 2 Eq. (13). In this way, the Cardassian model can make b (τ, u) du , (27) contact with quintessence with regard to observational where u is the coordinate of the fifth dimension, one tests. may obtain a modified FRW equation on our observ- All observational constraints on quintessence that able brane with H 2 ∼ ρn for any n (see also [19]). depend only on the scale factor, R(t) (or, equivalently, This result was obtained with 5-dimensional Einstein H(z)) can also be used to constrain the Cardassian equations plus the Israel boundary conditions relating model. However, because the Cardassian model re- the energy–momentum on our brane to the derivatives quires modified Einstein equations (see below), the ofthemetricinthebulk. gravitational Poisson’s equations and consequently We do not yet have a fundamental higher-dimen- late-time structure formation may be changed; e.g., the sional theory, i.e., a higher-dimensional Tµν ,which redshift dependence of cluster abundance should be we believe describes our universe. Once we have this, different in the two models. These effects (and others, we can write down the modified four-dimensional Ein- such as the fact that quintessence clumps) may serve stein’s equations and compute the modified Poisson’s to distinguish the Cardassian and quintessence mod- equations, as would be required, e.g., to fully under- els. The correspondence with quintessence, as well as stand latetime structure formation. discussion of distinguishing tests will be the subject of There is no unique 5-dimensional energy–momen- a future paper. tum tensor Tµν that gives rise to Eq. (1) on our brane. K. Freese, M. Lewis / Physics Letters B 540 (2002) 1–8 7

Hence, in this Letter we construct an example which 10−101, which cancels other small numbers in such a is easy to find but is clearly not our universe, simply way as to again require roughly Eq. (30) to be satis- as a proof that such an example can be written down. fied. The form of Tµν given in Eq. (28) is by no means Following [10] (see Eqs. (24) and (25) there with unique and has been presented merely as an existence F(u)= u), we have constructed an example of a bulk proof; we hope a more elegant Tµν may be found, per- 2 n Tµν for arbitrary n in H ∼ ρ , matter on the brane as haps with a motivation for the required value of B. in Eq. (3), and with q = b in Eq. (27). We display only 0 T0 here (the other components will be published in a future paper): 12. Discussion − 4+n − 2 3 n B n , κ2T 0 =− We have presented Eq. (1) as a modification to 5 0 n2τ 2 the FRW equations in order to suggest an explana- 4 n tion of the recent acceleration of the universe. In the 1 2 1 4 1 2 2 2 × 4 · 81 n B n − 16 n κ n τ + u , 5 nτ Cardassian model, the universe can be flat and yet matter dominated. We have found that the new Car- (28) dassian term can dominate the expansion of the uni- where verse after zeq = O(1) and can drive an acceleration. 2 + n We have found that matter alone can be responsible 2 n − 1 2 1 , = exp −(2/3) n B n κ u (29) for this behavior (but see the comments below). The 5 nτ current value of the energy density of the universe is and the constant κ5 is related to the 5-dimensional then smaller than in the standard model and yet is at Newton’s constant G5 and 5D reduced Planck mass the critical value for a flat geometry. Structure forma- 2 = = −3 M5 by the relation κ5 8πG5 M5 .Thisis tion is unaffected before zeq. The age of the universe merely one (inelegant) example of many bulk Tµν that is somewhat longer. The first Doppler peak of the cos- produce Cardassian expansion. mic background radiation is shifted only slightly and We may now investigate the meaning of the values remains consistent with experimental results. Such a of B(n) required by Eq. (8), where B(n) is the modified FRW equation may result from the existence parameter in front of the new Cardassian term in of extra dimensions. Further work is required to find a Eq. (1). As mentioned previously, the mass scale of B simple fundamental theory responsible for Eq. (1). − n has units of m2 4 . We find that the corresponding Questions of interpretation remain. We have said mass scale is very small for n<1/2, is singular at n = that matter alone is responsible for the accelerated 1/2, and then goes over to a very large value for n> behavior. However, if the Cardassian behavior results 1/2. Specifically, for n = 2/3andzeq = 1, we obtain from integrating out extra dimensions, then one may − − B ∼ 10 52 GeV 2/3 which corresponds to a mass ask what behavior of the radii of the extra dimensions scale of 1078 GeV. In the context of extra dimensions, is required. The Israel conditions connect the energy this large mass scale turns out to cancel against other density on the brane to fields in the bulk. The required large numbers in such a way that it corresponds to behavior of bulk fields is not transparent when one reasonable values of the energy–momentum tensor in writes the modified FRW equation. We have found the bulk. We find that τ is roughly the age of the a large or small mass scale to be required, which universe and we have , ∼ 1forallu.Thenwehave must result from the extra dimensions. In principle − one would like to have a complete 5-dimensional T 0 ∼ 10 5 GeV 5. (30) 0 theory so as to perform post-Newtonian tests on the Although this value is not motivated, it is not unrea- model and also to check other consequences. For sonable. In other words, reasonable bulk values can example, with a 5-dimensional model, one would like generate the required parameters in Eq. (1). Numeri- to compare with limits from fifth force experiments cal values for other components of Tµν are the same and to check that none of the higher-dimensional order of magnitude, with the exception that T04 ∼ 0. fields are overcontributing to the energy density of the For the case of n = 1/3, we obtain a mass scale of universe at any point (the moduli problem). 8 K. Freese, M. Lewis / Physics Letters B 540 (2002) 1–8

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Black hole entropy without brick walls

Li Xiang a,b

a CCAST (World Lab.), P.O. Box 8730, Beijing 100080, PR China b Institute of Theoretical Physics, Chinese Academy of Sciences, P.O. Box 2735, Beijing 100080, PR China 1 Received 12 April 2002; received in revised form 5 May 2002; accepted 5 May 2002 Editor: J. Frieman

Abstract The properties of the thermal radiation are discussed by using the new equation of state density motivated by the generalized uncertainty relation in the quantum gravity. There is no burst at the last stage of the emission of a Schwarzschild black hole. When the new equation of state density is utilized to investigate the entropy of a scalar ﬁeld outside the horizon of a static black hole, the divergence appearing in the brick wall model is removed, without any cutoff. The entropy proportional to the horizon area is derived from the contribution of the vicinity of the horizon.  2002 Elsevier Science B.V. All rights reserved.

PACS: 04.70.Dy; 04.62.+v; 97.60.Lf

The title is the same as Ref. [1] where Demers et by introducing some regulators. These fictitious fields al. show that the divergence appearing in the brick are especially designated in the number, statistics and wall model [2] can be absorbed into the renormal- masses. To my surprise, the entropy expressed by the ized Newton’s constant. By using the WKB approx- masses of the regulators can be precisely renormalized imation, ’t Hooft investigates the statistical properties to the Bekenstein–Hawking formula, S = A/(4GR), of a scalar field outside the horizon of a Schwarzschild GR is the renormalized Newton’s constant. However, black hole. The entropy proportional to the horizon it is hard to understand the introduction of the “bare area is obtained, but with a cutoff utilized to remove entropy” in Ref. [1]. The “bare entropy” seems to be the divergence of the density of states. The cutoff is negative and its meaning is unclear.2 Is there a better introduced by hand and looks unnatural. Susskind and method can remove the divergence appearing in the Uglum suggest that the explosive free energy and en- brick wall model? tropy in the model of ’t Hooft are related to the diver- Recently, many efforts have been devoted to the gence of the one-loop effective action of the quantum generalized uncertainty relation field theory in curved space [3]. Their conjecture is confirmed by [1]. The authors of [1] remove the cutoff λ x p h¯ + ( p)2, (1) and regularize the divergent free energy and entropy h¯

E-mail address: [email protected] (L. Xiang). 2 Dr. Fursaev told me, this difﬁculty can be overcome in the 1 This is the mailing address. Sakharov’s induced gravity [4].

0370-2693/02/$ – see front matter  2002 Elsevier Science B.V. All rights reserved. PII:S0370-2693(02)02123-8 10 L. Xiang / Physics Letters B 540 (2002) 9–13 and its consequences [5–11], especially the effect on then the density of states [10,11]. Here h¯ is the Planck −4 constant, λ is of order of the Planck length. Eq. (1) u = β G(0) + G (0)a √ 4 2 means that there is a minimal length, 2 λ. As well- π − 40π λ = β 4 1 − . (6) known, the number of quantum states in the integrals 15 7β2 d3x d3p is given by In the usual case, above equation does not essentially d3x d3p , (2) change the well-known conclusion for the black body (2πh)¯ 3 radiation because the correction is very slight. For ex- which can be understood as follows: since the uncer- ample, the temperature of the center of the neutron star 9 32 2 tainty relation x p ∼ 2πh¯ , one quantum state cor- is 10 K, but the Planck temperature is 10 K,λ/β ∼ −46 responds to a “cell” of volume (2πh)¯ 3 in the phase- 10 . However, Eq. (6) is no longer valid for the case 2 space. Based on the Liouville theorem, the authors λ/β 1, that is higher than the Planck temperature. of Ref. [11] argue that the number of quantum states We calculate the upper bound of energy density, that is should be changed to the following ∞ 2 − x dx 3 3 4 d x d p u<β 2 , (3) 1 + λx 3 (2πh)¯ 3(1 + λp 2)3 0 β2 −3/2 2 = i = − π λ where p pi p ,i 1, 2, 3. Eq. (3) seriously de- = β 4 · forms the Planckian spectrum of the black√ body ra- 16 β2 diation at the Planck temperature, T = 1/λ (see π − λ = β 1, (7) Ref. [11], Fig. 2). 16λ3/2 Let us discuss the more details than Ref. [11]. This where the inequality is due to ex − 1 >x.This will benefit the following investigation of the black means that when the temperature is higher than the hole entropy. From Eq. (3), we directly write down the Planck temperature the state equation of the thermal density of internal energy of the thermal radiation radiation is essentially different from the well-known ∞ conclusion, u ∼ β−4. This will influence the emission ω3 dω u = of the black hole. According to the Stefan–Boltzmann βω − + 2 3 (e 1)(1 λω ) law, the loss mass rate of a Schwarzschild black hole 0 ∞ reads 3 −4 x dx = β dM −4 1 x 2 3 ∼ β A ∼ , (8) (e − 1)(1 + ax ) 2 0 dt M − = β 4G(a), (4) where M the mass of the hole. At the last stage of emission, M → 0, so the emission rate becomes 2 where a = λ/β ,x = βω. We take the units G = c = divergent. However, from Eq. (7), at the last stage, the h¯ = kB = 1. The above integral cannot be expressed as rate will be changed to a simple formula, but we can investigate its asymptotic dM − behavior in the two different conditions. We first con- ∼ β 1A ∼ M → 0, (9) sider the case a 1. This means that the temperature dt is much less than the Planck temperature. We have here is no burst. ∞ We turn to the problem of black hole entropy. Re- x3 dx π4 G(0) = = , calling the brick wall model, the number of quantum ex − 1 15 states less than energy ω is given by [2,12,13] 0 ∞ L 5 6 3 2 x dx 24π 2ω r dr G (0) =−3 =− , (5) Γ(ω)= , (10) ex − 1 63 3π f 2 0 r0+" L. Xiang / Physics Letters B 540 (2002) 9–13 11 which is for a massless scalar field in a spherical and From Eq. (3), the number of quantum states with static space–time as follows energy less than ω is given by 2 =− 2 + −1 2 + 2 2 + 2 2 2 1 dr dθ dϕdpr dpθ dpϕ ds fdt f dr r dθ r sin θdφ , g(ω) = 3 + 2 3 (11) (2π) (1 λω /f ) 1 dr dθ dϕ where f = f(r). The horizon is located by f(r0) = 0. = (2π)3 (1 + λω2/f )3 " is the cutoff near the horizon. Obviously, the number of states is divergent if " = 0. We carefully check 2 ω2 1 1 1/2 × − p2 − p2 the derivation of Eq. (10) and find that it agrees with f 1/2 f r2 θ r2 sin2 θ ϕ Eq. (2), not (3). The former leads to the following × dpθ dpϕ formula 4πω3 r2 dr 8π3 r2 dr = sin θdθdϕ S = , (12) 3(2π)3 f 2(1 + λω2/f )3 45β3 f 2 2ω3 r2 dr = , (18) which is analogous with√ the usual state equation of the 2 + 2 3 −1 3π f (1 λω /f ) thermal radiation:√ (β f) is the local temperature, 4πr2 dr/ f is the element of the spatial volume of where the integration goes over those values of pθ ,pϕ the spherical shell. The divergent entropy means the for which the argument of the square root is positive = invalidity of the usual state equation near the black (please refer to Refs. [2,12,13]). When λ 0, Eq. (18) = hole horizon. If we take Eq. (3), the situation may be naturally returns to (10). However, in the case λ 0, essentially different. Why not have a try? Substituting Eq. (18) is essentially different from (10): it is conver- the wave function Φ = exp(−iωt)ψ(r,θ,ϕ) into the gent at the horizon without any cutoff! By using the usual method, the free energy is given by equation of massless scalar field 1 √ = 1 − −βω √ ∂ −ggµν ∂ Φ = 0, (13) F(β) dg(ω)ln 1 e −g µ ν β ∞ we obtain =− g(ω)dω βω − ∂2ψ f 2 ∂ψ e 1 + + 0 ∂r2 f r ∂r ∞ 2 r2 dr ω3 dω 1 ω2 1 ∂2 ∂ 1 ∂2 =− . + + + cotθ + ψ 2 βω − + 2 3 2 2 2 2 3π f (e 1)(1 λω /f ) f f r ∂θ ∂θ sin θ ∂ϕ r0 0 (19) = 0. (14) The entropy reads By using the WKB approximation with ∂F S = β2 ψ ∼ exp[iS(r,θ,φ)], ∂β ∞ we have 2β2 r2 dr eβωω4 dω = 2 3π f 2 (eβω − 1)2(1 + λω2/f )3 2 1 ω 1 2 1 2 = − − r0 0 pr 2 pθ 2 pϕ , (15) f f r r2 sin θ ∞ 2β−3 r2 dr x4 dx where = 2 , 3π f 2 (1 − e−x )(ex − 1) 1 + λx 3 ∂S ∂S ∂S β2f p = ,p= ,p= . (16) r0 0 r ∂r θ ∂θ ϕ ∂ϕ (20) where x = βω. Taking into account the following We also obtain the square module of momentum inequalities ω2 2 = i = rr 2 + θθ 2 + ϕϕ 2 = −x x p pi p g pr g p g pϕ . (17) 1 − e > , θ f 1 + x 12 L. Xiang / Physics Letters B 540 (2002) 9–13

ex − 1 >x. (21) near the horizon. This convergency is due to the effect of the generalized uncertainty relation on the quantum We obtain states. This provides an evidence for the idea of Li ∞ 2β−3 r2 dr (x3 + x2)dx and Liu. The more details between the Li–Liu equa- S< tion and the generalized uncertainty relation will be 3π f 2 + λx2 3 1 2 r0 0 β f investigated in the future. − − 2β−3 r2 dr 1 λ 2 π λ 3/2 As pointed by Ref. [6], the generalized uncertainty = + relation (1) may have a dynamical origin since it π 2 2 2 3 f 4 β f 16 β f contains a dimensional coupling constant λ.IfEq.(1) r0 − is indeed due to the string theory, λ should be β λ 3/2 r2 dr = 2 + associated with the stringy scale l2. This implies 2 r dr 1/2 . (22) s 6πλ 24 f that one takes into account the contribution from r0 r0 the stringy excitation when calculating the density of We are only interested in the contribution from the quantum states. The convergent property should be [ + ] vicinity near the horizon, r0,r0 " , which corre- reexamined. There are some evidences (or arguments) sponds to√ a proper distance of order of the minimal for the convergence of the density of states even if length, 2 λ. This is because the entropy closes to the considering the stringy excitation: firstly, the bosonic upper bound only in this vicinity. Furthermore, it is string can be described by a discrete field theory, then just the vicinity neglected by brick wall model. We the number of degrees of freedom of it is smaller have than that of the usual field theory [16]. Secondly, the + + r0 " r0 " entropy of a string is proportional to its mass since √ dr dr 2 λ = √ ≈ √ the degeneracy increases exponentially with the mass − f 2κ(r r0) level. However, the massive string cannot be excited r0 r0 in the low energy effective theory (such as general 2" = , (23) relativity) [17]. Therefore, the contribution from the κ stringy excitation is ignored in the case of the massive where κ is the surface gravity at the horizon of black hole where the semi-classical approximation is black hole and it is identified as κ = 2πβ−1. Thus still valid. As to the black hole at the Planck scale we naturally derive the entropy proportional to the the usual quantum field theory is no longer valid. The horizon area entropies of the black hole and the excited string states β λ−3/2 √ are matched in the correspondence principle [18]. S ∼ r2" + · 2r2 λ 6πλ2 0 24 0 3A = , (24) Acknowledgements 16λπ = 2 where A 4πr0 is the surface area of the black hole. The author thanks X.J. Wang, Z.Q. Bai and H.G. As early as 1992, Li and Liu phenomenally pro- Luo for their zealous help during the research. The posed that the state equations of the thermal radiation author also thank Profs. Y.Z. Zhang, R.G. Cai and near the horizon should be changed to a series of new C.G. Huang for their comments on this research. This formulae rather than Eq. (12), in order to maintain the work is supported by the Post Doctor Foundation of validity of the generalized second law of thermody- China and K.C. Wong Education Foundation, Hong namics [14]. Using the Li–Liu equation, Wang inves- Kong. tigates the entropy of a self-gravitational radiation system and obtains the Bekenstein’s entropy bound [15]. Here, Parallel to the brick wall model, the scalar field References near the horizon of a static black hole is investigated again, we obtain the entropy proportional to the hori- [1] J.G. Demers, R. Lafrance, R.C. Myers, Phys. Rev. D 52 (1995) zon area. There is no divergence without any cutoff 2245. L. Xiang / Physics Letters B 540 (2002) 9–13 13

[2] G. ’t Hooft, Nucl. Phys. B 256 (1985) 727. [10] S.K. Rama, Phys. Lett. B 519 (2001) 103. [3] L. Susskind, J. Uglum, Phys. Rev. D 50 (1995) 2700. [11] L.N. Chang et al., hep-th/0201017. [4] V.P. Frolov, D.V. Fusaev, A.I. Zelnikov, Nucl. Phys. B 486 [12] T. Padmanabhan, Phys. Lett. B 173 (1986) 43. (1997) 339, hep-th/9607104; [13] J.L. Jing, Int. J. Theor. Phys. 37 (1998) 1441. V.P. Frolov, D.V. Fursaev, Phys. Rev. D 56 (1997) 2212, hep- [14] L.X. Li, L. Liu, Phys. Rev. D 46 (1992) 3296. th/9703178; [15] D.X. Wang, Phys. Rev. D 50 (1994) 7385. V. Frolov, D. Fursaev, J. Gegenberg, G. Kunstatter, Phys. Rev. [16] I. Klebanov, L. Susskind, Nucl. Phys. B 309 (1988) 175. D 60 (1999) 024016. [17] A.W. Peet, Class. Quantum Grav. 15 (1998) 329. [5] A. Kempf, G. Mangano, R.B. Mann, Phys. Rev. D 52 (1995) [18] L. Susskind, hep-th/9309145; 1108. E. Halyo, A. Rajaraman, L. Susskind, Phys. Lett. B 392 (1997) [6] L.J. Garay, Int. J. Mod. Phys. A 10 (1995) 145. 319; [7] H.A. Kastrup, Phys. Lett. B 413 (1997) 267. G.T. Horowitz, J. Polchinski, Phys. Rev. D 55 (1997) 6189. [8] D.V. Ahluwalia, Phys. Lett. A 275 (2000) 31, gr-qc/0002005. [9] R.J. Adler et al., gr-qc/0106080. Physics Letters B 540 (2002) 14–19 www.elsevier.com/locate/npe

Implications of the ﬁrst neutral current data from SNO for solar neutrino oscillation

Abhijit Bandyopadhyay a, Sandhya Choubey b, Srubabati Goswami c,D.P.Royd,e

a Saha Institute of Nuclear Physics, Bidhannagar, Kolkata 700 064, India b Department of Physics and Astronomy, University of Southampton, Highﬁeld, Southampton S017 1BJ, UK c Harish-Chandra Research Institute, Chhatnag Road, Jhusi, Allahabad 211-019, India d Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400 005, India e Physics Department, University of California, Riverside, CA 92521, USA Received 1 May 2002; received in revised form 13 June 2002; accepted 15 June 2002 Editor: P.V. Landshoff

Abstract We perform model independent and model dependent analyses of solar neutrino data including the neutral current event rate from SNO. The inclusion of the first SNO NC data in the model independent analysis determines the allowed ranges of 8Bflux normalisation and the νe survival probability more precisely than what was possible from the SK and SNO CC combination. We perform global νe–νactive oscillation analyses of solar neutrino data using the NC rate instead of the SSM prediction for the 8B flux, in view of the large uncertainty in the latter. The LMA gives the best solution, while the LOW solution is allowed only at the 3σ level.  2002 Published by Elsevier Science B.V.

= NC CC+NC The neutral current results from the Sudbury Neu- where r σνµ,τ /σνe 0.157 for a threshold en- trino Observatory measures for the first time the to- ergy of 5 MeV (including the detector resolutions tal flux of 8B neutrinos coming from the Sun [1]. In and the radiative corrections to ν–e scattering cross- a recent paper [2] we had examined the role of the sections). All the rates are defined with respect to anticipated NC data from SNO in enhancing our un- the BBP2000 Standard Solar Model (SSM) [4]. We derstanding of the solar neutrino problem. The SNO showed in [2] that because SNO has a greater sensi- NC rate can be expressed in terms of SNO CC and SK tivity to the NC scattering rate as compared to SK, the elastic scattering rates as [3] SNO NC measurement will be more precise and hence incorporation of this can be more predictive than the NC = CC + el − CC SNO CC and SK combination. We took three repre- RSNO RSNO RSK RSNO /r, (1) SNO = ± sentative NC rates, RNC 0.8, 1.0 and 1.2 ( 0.08) and showed that

E-mail addresses: [email protected] ν (A. Bandyopadhyay), [email protected] (1) For a general transition of e into a mixture of (S. Choubey), [email protected] (S. Goswami), active and sterile neutrinos the size of the sterile [email protected] (D.P. Roy). component can be better constrained than before.

0370-2693/02/$ – see front matter  2002 Published by Elsevier Science B.V. PII:S0370-2693(02)02138-X A. Bandyopadhyay et al. / Physics Letters B 540 (2002) 14Ð19 15

(2) For transition to a purely active neutrino the where Pee and Pea denote the probabilities folded with 8B neutrino flux normalisation and the survival the detector response function [5] and averaged over probability Pee are determined more precisely. energy. To extract a model independent bound on Pee (3) We had also performed global two flavour oscilla- one has to ensure an equality of the response functions tion analysis of the solar neutrino data for the νe– which amounts to slight adjustment of the SK thresh- CC νactive case, where instead of RSK and RSNO we old energy and the rate [5,6]. Our approach is slightly el NC CC NC different. We treat P to be effectively energy inde- used the quantities RSK/RSNO and RSNO/RSNO. ee These ratios are independent of the 8B flux nor- pendent. The SK spectrum data indicates a flat proba- malisation and hence of the SSM uncertainty. We bility down to 5 MeV [7]. This is corroborated by SNO showed that use of these ratios can result in drastic [8,9] which now has a threshold of 5 MeV for kinetic reduction of the allowed parameter regions spe- energy of the observed electron. Hence we consider cially in the LOW-QVO area depending on the this assumption as justified and expect the results to be value of the NC rate. insensitive to the differences in the response functions. It should be noted here that in contrast to the SNO CC We now have the actual experimental result events their NC events correspond to a neutrino energy threshold of 2.2 MeV. However it is clear from RNC = 1.01 ± 0.12 (2) Eq. (5) that for a νe to νa transition there is no rea- SNO NC son to expect any energy dependence in RSNO.Onthe while Eq. (1) gives 1.05 ± 0.15. Thus in 306 live days (577 days) the SNO NC measurement has achieved a precision, which is already better than that obtained from the SK and SNO CC combination. This Letter follows closely the analysis that we have done in [2] but incorporating the actual data. In addition we also perform an alternative global analysis for νe–νactive 8 oscillation by letting the B normalisation factor fB NC = vary freely, where the inclusion of RSNO ( fB) in the fit serves to control this parameter. As we shall see below the two methods of global analysis give very similar results. In Section 1 we discuss the constraints on the electron neutrino survival probability, the 8B normalisation factor fB and the fraction of sterile component without assuming any particular model for the probabilities. In Section 2 we perform the global analyses assuming two flavour νe–νactive oscillation.

1. Model independent analysis

For the general case of νe transition into a combination of νactive (νa)andνsterile (νs ) states one can write the SK, SNO CC and SNO NC rates as

el = + RSK fBPee fBrPea, (3) CC = Fig. 1. The SNO CC and NC rates shown relative to their SSM RSNO fBPee, (4) predictions. The dashed line is the prediction of the pure νe to νs NC = + transition. The pure sterile solution is seen to be disfavored at 5.3 σ. RSNO fB(Pee Pea), (5) 16 A. Bandyopadhyay et al. / Physics Letters B 540 (2002) 14Ð19

8 Fig. 2. Best fit value of the B neutrino flux fB shown along with its 8 1σ and 2σ limits against the model parameter sin2 α, representing Fig. 3. The 1σ and 2σ contours of solutions to the B neutrino flux fB and the νe survival probability Pee assuming νe to νa transition. νe transition into a mixed state (νa sin α + νs cos α). The dashed line denote the ±2σ limits of the SSM. The 1σ SSM error bar for fB is indicated on the right. other hand for the general case of νe transition into a of fB from this fit with the 2σ upper limit from the combination of νa and νs our approach effectively as- SSM (vertical lines) gives a lower limit of sin2 α> sumes Pes to be energy independent down to 2.2 MeV. 0.45, i.e., the probability of the active component CC NC A comparison of the current values RSNO with RSNO is is > 45%. Note that there is no upper limit on this shown in Fig. 1. It constitutes a 5.3 σ signal for transi- quantity, i.e., the data is perfectly compatible with νe tion to a state containing an active neutrino component transition into purely active neutrinos. or alternatively a 5.3 σ signal against a pure sterile so- Assuming transition into purely active neutrinos lution. (Pea = 1 − Pee) we show in Fig. 3 the 1σ and 2σ Next we consider the general case where νe goes to contours in the fB–Pee plane from the combinations a mixed state = νa sin α + νs cosα. Then one can write SK + SNOCC and SK + SNOCC + SNONC. The 2 Pea = sin α(1 − Pee). Substituting this in Eqs. (3) inclusion of the NC rate narrows down the ranges of and (5) and eliminating Pee using Eq. (4) one gets the fB and Pee. The error in fB after the inclusion of NC 2 following set of equations for fB and sin α [2] data is about half the size of the corresponding error sin2 α f − RCC = Rel − RCC /r, (6) from SSM as is seen from Fig. 3. B SNO SK SNO 2 − CC = NC − CC sin α fB RSNO RSNO RSNO. (7) We treat sin2 α as a model parameter. And for different 2. Model dependent analysis input values of sin2 α we determine the central value 2 and the 1σ and 2σ ranges of fB by taking a weighted In this section we present the results of our χ average of Eqs. (6) and (7). The corresponding curves analysis of solar neutrino rates and SK spectrum data are presented in Fig. 2. Combining the 2σ lower limit in the framework of two flavour oscillation of νe A. Bandyopadhyay et al. / Physics Letters B 540 (2002) 14Ð19 17

Fig. 4. The νe → νa oscillation solutions to the global solar neutrino data using (a) Ga rate, the SK zenith angle energy spectra and the SK and SNO (CC) rates, both normalised to the SNO (NC) rate and (b) total Ga, Cl, SK, SNO (CC) and SNO (NC) rates along with the SK zenith angle 8 8 energy spectra, keeping the B flux normalisation fB free. In both cases we use the SNO (NC) error as the error in the B flux. (c) is similar to (b), but without using the SNO (NC) rate. to an active flavour. We use the standard techniques SK spectrum data. The SNO CC and NC rates have a described in our earlier papers [10,11] excepting for large anticorrelation. We have taken into account this el CC the fact that instead of the quantities RSK and RSNO correlation between the measured SNO rates in our el NC CC NC global analyses. Further details of this fitting method we now fit the ratios RSK/RSNO and RSNO/RSNO. The 8B flux normalisation gets cancelled from these can be found in [2]. In Table 2 we present the best-fit 2 ratios and the analysis becomes independent of the parameters, χmin and goodness of fit (GOF). The best- large (16–20%) SSM uncertainty associated with this. fit comes in the HIGH(LMA) region as before [11,12]. We include in our global analyses the 1496 day SK However as is seen from Fig. 4(a) the incorporation of zenith angle spectra [15]. Since we use both SK rate the NC data narrows down the allowed regions, and in and SK spectrum data we keep a free normalisation particular the LOW region becomes much smaller. 2 factor for the SK spectrum. This amounts to taking We have also performed an alternative χ fit to the the information on total rates from the SK rates data rates of Table 1 [1,7,13,14] along with the 1496 day and the information of the spectral shape from the SK spectra [15], keeping fB as a free parameter. Even 18 A. Bandyopadhyay et al. / Physics Letters B 540 (2002) 14Ð19

Table 1 The observed solar neutrino rates relative to the SSM predictions (BP2000) are shown along with their compositions for different experiments. For the SK experiment the νe contribution to the rate R is shown in parentheses assuming νe → νa transition. In the combined Ga rate we have included the latest data from SAGE and GNO Experiment R Composition Ga 0.553 ± 0.034 pp(55%), Be(25%), B(10%) Cl 0.337 ± 0.030 B(75%), Be(15%) SK 0.465 ± 0.014 (0.363 ± 0.014) B(100%) SNO(CC) 0.349 ± 0.021 B(100%) SNO(NC) 1.008 ± 0.123 B(100%)

Table 2 2 The χmin, the goodness of fit and the best-fit values of the oscillation parameters obtained for the analysis of the global solar neutrino data 2 2 2 2 Data used Nature of solution m in eV tan θχmin Goodness of fit (%) − Ga + LMA 9.66 × 10 5 0.41 35.95 80.08 − SK/NC + LOW 1.04 × 10 7 0.61 46.73 36.09 − CC/NC + VO 4.48×10 10 0.99 54.25 13.84 − − SKspec SMA 6.66 × 10 6 1.35 × 10 3 67.06 1.41 − Cl + Ga+ LMA 6.07 × 10 5 0.41 40.57 65.99 − SK + CC+ LOW 1.02 × 10 7 0.60 50.62 26.14 − NC + SKspec VO 4.43×10 10 1.156.11 12.39 −6 −3 +fB free SMA 5.05 × 10 1.68 × 10 70.97 0.81

though we allow fB to vary freely the NC data serves Letter we have discussed two useful strategies, of 2 to control fB within a range determined by its error. incorporating the NC data in the global χ analysis of As we see from Table 2 and Fig. 4(b) the results of rates and spectrum data, by which one can avoid the this fit are very similar to the previous case. The best fit large 8B flux uncertainty from the SSM. comes from the HIGH(LMA) region, while no allowed • We fit the ratios of the SK elastic and SNO CC region is obtained for the LOW solution at the 99% rates w.r.t. the NC rate, from which the f cancels CL level. Maximal mixing is seen to be disallowed B out. at the 3σ level. To illustrate the impact of the NC • We fit the rates by keeping f as a free parameter, rate on the oscillation solutions we have repeated the B where the inclusion of the SNO NC rate (= f ) free f fit without this rate. The results are shown in B B serves to control this parameter. Fig 4(c). Evidently the NC data plays a pivotal role in constraining the oscillation solutions, particularly in Both the analyses give very similar results. They the LOW/QVO region, which is allowed only at the 3σ clearly favour the HIGH(LMA) solution, while a level. It puts an upper bound on the m2 in the LMA limited region of the LOW solution is also acceptable region and rules out maximal mixing. at the 3σ level. The maximal mixing solution is disfavoured at the 3σ level. As more data accumulate one expects a substantial reduction in the error bar of 3. Summary and conclusions the SNO NC rate, resulting in further tightening of the allowed regions of neutrino mass and mixing. The first SNO NC data constitutes a 5.3 σ signal for transition into a state containing an active neutrino component. The inclusion of this data puts much Note added tighter constraints on fB and Pee from a model independent analysis involving active neutrinos as The paper [16] appeared on the net after completion compared to the SNO CC/SK combination. In this of our work. In the region of overlap our results agree A. Bandyopadhyay et al. / Physics Letters B 540 (2002) 14Ð19 19 with theirs as well as with the updated version of [17]. S. Goswami, D. Majumdar, A. Raychaudhuri, hep- It may be added here that the SNO CC and NC rates ph/9909453; given in Table 1 are obtained assuming undistorted S. Choubey, S. Goswami, N. Gupta, D.P. Roy, Phys. Rev. D 64 (2001) 053002. energy spectra above 5 MeV, which for transitions [11] A. Bandyopadhyay, S. Choubey, S. Goswami, K. Kar, Phys. to active neutrinos has good empirical justification as Lett. B 519 (2001) 83; mentioned above. We thank Prof. Mark Chen of SNO S. Choubey, S. Goswami, K. Kar, H.M. Antia, S.M. Chitre, Collaboration for communication on this point. Phys. Rev. D 64 (2001) 113001; S. Choubey, S. Goswami, D.P. Roy, Phys. Rev. D 65 (2002) 073001; A. Bandyopadhyay, S. Choubey, S. Goswami, K. Kar, Phys. References Rev. D 65 (2002) 073031. [12] G.L. Fogli, E. Lisi, D. Montanino, A. Palazzo, Phys. Rev. D 64 [1] SNO Collaboration, Q.R. Ahmad et al., nucl-ex/0204008, to (2001) 093007; appear in Phys. Rev. Lett. J.N. Bahcall, M.C. Gonzalez-Garcia, C. Pena-Garay, [2] A. Bandyopadhyay, S. Choubey, S. Goswami, D.P. Roy, hep- JHEP 0108 (2001) 014; ph/0203169, to appear in Mod. Phys. Lett. A. P.I. Krastev, A.Yu. Smirnov, hep-ph/0108177; [3] See, e.g., V. Barger, D. Marfatia, K. Whisnant, Phys. Rev. M.V. Garzelli, C. Giunti, JHEP 0112 (2001) 017. Lett. 88 (2002) 011302. [13] J.N. Abduratshitov et al., SAGE Collaboration, astro- [4] J.N. Bahcall, M.H. Pinsonneault, S. Basu, Astrophys. J. 555 ph/0204245; (2001) 990. W. Hampel et al., GALLEX Collaboration, Phys. Lett. B 447 [5] F.L. Villante, G. Fiorentini, E. Lisi, Phys. Rev. D 59 (1999) (1999) 127; 013006. M. Altman et al., GNO Collaboration, Phys. Lett. B 490 (2000) [6] G.L. Fogli, E. Lisi, D. Montanino, A. Palazzo, Phys. Rev. D 64 16; (2001) 093007. E. Belloti, Talk at Gran Sasso National Laboratories, May 17, [7] S. Fukuda et al., Super-Kamiokande Collaboration, Phys. Rev. 2002; Lett. 86 (2001) 5651. T. Kirsten, Talk at Neutrino 2002, Munich. [8] SNO Collaboration, Q.R. Ahmad et al., Phys. Rev. Lett. 87 [14] B. Cleveland et al., Astrophys. J. 496 (1998) 505. (2001) 071301. [15] M.B. Smy, Talk at noon2001, hep-ex/0202020. [9] SNO Collaboration, Q.R. Ahmad et al., nucl-ex/0204009, to [16] V. Barger, D. Marfatia, K. Whisnant, B.P. Wood, hep- appear in Phys. Rev. Lett. ph/0204253. [10] S. Goswami, D. Majumdar, A. Raychaudhuri, Phys. Rev. D 63 [17] P. Creminelli, G. Signorelli, A. Strumia, hep-ph/0102234. (2001) 013003; Physics Letters B 540 (2002) 20–24 www.elsevier.com/locate/npe

Non-standard contributions to ν–e elastic scattering from solar neutrinos observations and LSND measurement

A. Ianni

INFN-Laboratori Nazionali del Gran Sasso, S.S. 17bis Km 18+910, I-67010 Assergi (Aquila), Italy Received 16 May 2002; received in revised form 6 June 2002; accepted 10 June 2002 Editor: L. Rolandi

Abstract We calculate bounds on neutrino magnetic moment and charge radius squared from solar neutrinos observations after Super- Kamiokande and SNO results. We use LSND νe–e elastic scattering measurement to constrain further the charge radius. −10 −10 2 −32 2 With few assumptions, we derive µν 1 × 10 µB (µν 1.2 × 10 µB ), when r =0, and −1.2 × 10 r −32 2 −32 2 −32 2 2.7 × 10 cm (−0.5 × 10 r 3.5 × 10 cm ), when µν = 0, at 90% C.L. for Dirac (Majorana) neutrinos. We −10 also show that µν cannot be larger than 1.2 × 10 µB for any allowed value of the charge radius. Moreover, we show that from our ﬁt a fraction of about 1.3% of νxR (x = µ,τ) could be present in the solar neutrinos ﬂux on Earth.  2002 Published by Elsevier Science B.V.

1. Introduction Dirac neutrino species of the order of [1] −19 mν µν ∼ 3.2 × 10 µB . (1) In the Standard Model neutrinos are massless and 1eV do not have electromagnetic dipole and transition mo- Among the non-standard processes allowed by a non- ments [1], i.e., they do not couple to the electromag- zero µν ,theν–e elastic scattering is of primary netic field. However, there are many possible exten- importance because can be studied extensively in sions of the Standard Model which allow neutrinos different energy ranges. In this exotic scenario the to interact directly to the electromagnetic field [2]. differential cross-section for ν–e elastic scattering is This possibility is of great interest because such a written as [4] new coupling would allow a variety of non-standard 2 dσ(Eν,T) GF me 2 processes [1]. The strong evidence of neutrino flavor = (CV + X + CA) dT 2π transformation reported by the SNO Collaboration [3] shows that neutrino have a non-zero mass and as a con- 2 + − 2 − T sequence they should have a non-zero magnetic mo- (CV CA) 1 Eν ment. In particular, a minimal extension of the Stan- 2 2 meT dard Model predicts a magnetic moment for massive + C − (CV + X) A E2 ν 2 + 2 πα 1 − 1 E-mail address: [email protected] (A. Ianni). µν 2 . (2) me T Eν 0370-2693/02/$ – see front matter  2002 Published by Elsevier Science B.V. PII:S0370-2693(02)02126-3 A. Ianni / Physics Letters B 540 (2002) 20–24 21

2 1 In Eq. (2) CV = 2sin θW + 1/2, CA = 1/2forνe, take into account an energy-dependent deficit. With 2 CV = 2sin θW − 1/2, CA =−1/2forνx (x = µ,τ). this in mind we should consider the possibility of Moreover, dealing with anitneutrinos, we must make both flavor oscillations (MSW) and, as we are deal- the substitution CA →−CA. In Eq. (2) the magnetic ing with a non-zero µν , of resonant spin-flavor pre- moment, µν ,√ is given in units of µB (Bohr magne- cession (RSFP) [10,11,14]. For this latter we distin- 2 παr2 → →¯ ton) and X = ,wherer2 is the mean-square guish νeL νeR,νxR and νeL νxR transitions for 3GF 3 charge radius. With this definition we can see that a Dirac and Majorana neutrinos, respectively, where as − = charge radius of ∼ 6.5 × 10 16 cm2 gives X = 1and above x µ,τ. this corresponds to a significant change in the stan- Both processes (MSW and RSFP) could be used dard cross-section. We notice that Eq. (2) is given as a to fit solar neutrinos data [15–17]. Hence, following function of the neutrino energy, Eν , the recoil electron a reasoning similar to the one presented in [18], kinetic energy, T , and contains two phenomenologi- we call α the survival probability at Earth of νeL’s 2 β cal parameters, namely µν and r . For this reason produced at the core of the Sun and the fraction of a correct approach to study non-standard electromag- νxL’s which have not changed chirality because of an netic interactions should take into account both para- electromagnetic interaction with the Sun’s magnetic 4 meters as they are strongly correlated [5]. Finally, we field. With this model in mind, on Earth, we will also point out that the interaction cross-section propor- measure νeL’s with a reduction factor (with respect to tional to µ2 changes the chirality of the incoming neu- the SSM prediction) α and, as an example, a fraction ν ¯ − − trino. This could play an important role if we study of νxR equal to (1 α)(1 β). electromagnetic processes involving solar neutrinos In the most general scenario both α and β are en- which have been travelling through the magnetic field ergy dependent. In the following, however, as a first inside the Sun before reaching a detector on Earth. approximation we consider α and β to be constant. As 2 a matter of fact in the LMA5 scenario for solar neutri- Existing laboratory limits on µν and r come from ν–e elastic scattering [6–8]. These are of the nos, the survival probability is with good approxima- −10 −9 E order of few ×10 µB for νe and few ×10 µB for tion constant [16] for ν 5 MeV. So, in the energy range of Super-Kamiokande and SNO this is a good νµ. In particular, a study using 825-days data from 2 −10 W(T) 6 Super-Kamiokande gives µν 1.5 × 10 µB at assumption. In this case, if we call, , the de- 90% C.L. [7] On the other hand astrophysical limits tector energy resolution function, the effective cross- −13 −11 ν e are of the order of 10 –10 µB [9–11]. section for solar neutrinos from – elastic scattering, Best limits on r2 are derived using accelerator + and beam dump experiments as the sensitivity on this Eν /(1me/2Eν ) dσ(Eν,T) parameter increases with the neutrino energy. For νe σ(Eν ) = dT W(T) , (3) − × −32 2 × −32 2 dT it is found [12] 3 10 r 4 10 cm , 0 −16 while for νµ [13] the limit is | r | 0.7 × 10 cm.

3 Here we are not considering the possibility to have electron 2. Searching for neutrino non-standard antineutrinos in the final state due to a conversion of the kind νeL →¯νxR →¯νeL as these could be easily identified through the electromagnetic interactions with solar neutrinos inverse β-decay reaction. 4 observations For Dirac neutrinos the final state, νxR, is sterile with respect to weak interactions. 5 In this Letter we analyse data from solar neutri- The Large Mixing Angle scenario is strongly favored by a 2 global analysis of solar neutrinos data [16]. nos observations to search for µν and r . A real- 6 Here, using a gaussian energy resolution function, istic analysis of solar neutrinos observations should Tup − T T − T W(T)= 0.5 Erf √ − Erf low√ , 2σ 2σ 1 2 T T In this Letter we use sin θW = 0.2315. 2 This can be considered a laboratory limit as the solar neutrino where Erf is the error function, Tup and Tlow are the experimental flux is now experimentally established. recoil electron kinetic energy upper and lower limit, respectively. 22 A. Ianni / Physics Letters B 540 (2002) 20–24 can be split, using Eq. (2), into terms proportional to Table 1 2 2 µν and r . Doing this we can write σ from Eq. (3) Parameters for the SNO χSNO function. See text for details as ρx (ρx¯ )ηem η1e η1x (η1x¯ )η2 − . ( . ) . × −4 σ = ασweak + β(1 − α)σweakµ2σ 0.15(0.12) 0.011 0.033 0 011 0 009 2 7 10 e x ν em 2 + r ασ1e + β(1 − α)σ1x + 2 SK + − SK 2 2 r αη1e β(1 α)η1x + r α + β(1 − α) σ2, (4) + r2 2 α + β(1 − α) η2SK − data . (8) and Dirac neutrinos7 and i for Dirac neutrinos, and σ = ασweak + β(1 − α)σweak e x weak 2 SK = + − SK + (1 − α)(1 − β)σ + µ σ ai α β(1 α)ρx x¯ ν em 2 + − − SK + 2 SK + r ασ1e + β(1 − α)σ1x (1 α)(1 β)ρx¯ µν ηem + − − + 2 2 + r2 αη1SK + β(1 − α)η1SK (1 α)(1 β)σ1x¯ r σ2, (5) e x + − − SK weak (1 α)(1 β)η1x¯ for Majorana neutrinos. In Eqs. (4) and (5) σe weak weak 2 2 SK and σx (σx¯ ) are the Standard Model cross- + r η2 − datai (9) sections for ν–e elastic scattering for νe and νµ,τ for Majorana neutrinos. (ν¯µ,τ ), respectively, σem is the cross-section due to the In Eqs. (8) and (9) spin-flip term in Eq. (2). The cross-sections σ1e, σ1x (σ1 ¯ )andσ2 are from the contributions proportional x ρSK ρSK = σ weak/σ weak(σ weak/σ weak), to r2 and r22 for ν (ν¯), respectively. x x¯ x e x¯ e SK = weak Moreover, to derive Eqs. (4) and (5) we have ηem σem/σe , assumed that µν is an effective magnetic moment and SK weak η1 ¯ = σ1e,x,x¯ /σ , is the same for different neutrino flavors, the same e,x,x e SK = weak being true for the charge radius. η2 σ2/σe In this Letter, in particular, we take into account and data is the Super-Kamiokande relative measure- the Super-Kamiokande measurement of the recoil i ment in bin i. In evaluating Eq. (7) we have used electron spectrum [19] for 1258-days data taking and 8Bandhep spectra according to [20], a gaussian en- the SNO measurement of ν–e elastic scattering [3]. √ ergy resolution with σ = 0.47 T/1 MeV and data For the Super-Kamiokande data we calculate the error T from [19]. Cross-sections are folded over the neutri- covariance matrix according to data reported in [19] as nos energy spectra. = 2 + For SNO the χ2 function is written Sij σi δij si sj , (6) where σ 2 is the sum in quadrature of the statistical aSNO − rSNO 2 i χ2 = ES , (10) and uncorrelated errors and si are the correlated ones. SNO SNO σr With Eq. (6) we can define a χ2 function for the Super- SNO SK SNO = SNO SSM Kamionade data as where a is similar to a , rES φES /φ SNO SNO SNO = ± 2 2 2 SK −1 SK and σr is the error on rES .HereφES (2.39 χ α, β, µ , r = a S a , (7) − − − SK ν i ij j 0.27) × 10 6 cm 2 s 1 is the measured elastic scat- ij tering neutrinos flux [3], φSSM = (5.05 ± 1.01) × where 10−6 cm−2 s−1 is the SSM 8B flux [20]. In evaluating Eq. (10) we have used a gaussian energy√ resolution aSK = α + β(1 − α)ρSK + µ2ηSK i x ν em function with σT =−0.0684 + 0.331 T = 0.0425T as in [3] with a lower threshold at 5 MeV for the re- 7 Eq. 4 holds for both νeL → νeR and νeL → νxR with the coil electron kinetic energy. Table 1 reports the values assumptions made in the Letter (see below). of terms used in Eq. (10). A. Ianni / Physics Letters B 540 (2002) 20–24 23

3. The LSND contribution

To improve the sensitivity on r2 we used the measurement of νe–e cross-section performed by the LSND Collaboration [12]. Using for νe a spectrum shape of the form ∝ 2 2 2 − −1 −1 dΓ/dE mµc Eν (mµc 2Eν) MeV s , 2 with mµc muon rest energy, and detector information from [12], we have calculated 167 expected events in the Standard Model scenario against the 191 ± 22 measured [12]. Hence in 1 d.o.f. analysis we derive × −9 µνe 1.1 10 µB at 90% C.L. in agreement with what reported in [12]. However, in a proper 2 d.o.f. 2 2 analysis we will write a χ function similar to χSNO with α = β = 1. Fig. 1. Allowed regions (90–99% C.L.) from Super-Kamiokande, In order to combine results from Super-Kamiok- SNO and LSND data on ν–e elastic scattering electromagnetic phenomenological parameters for Dirac neutrinos. See text for ande, SNO and LSND we add Eqs. (7), (10) and the details. χ2 function for LSND. The SNO measurement of neutral current [3] al- SNO SNO = ± lows us to measure φCC /φNC 0.35 0.02, where SNO SNO φCC and φNC are the measured charge and neutral current fluxes, respectively. Therefore, to minimize the combined χ2 function we search for a mini- 2 2 mum in µν and r with 0.29 α 0.41 (3σ range) and β in [0.5, 1] for Majorana neutrinos, for in this SNO SNO = case φCC /φNC α. On the other hand, for Dirac SNO SNO = + − neutrinos φCC /φNC α/(α β(1 α)). Hence, for any β in [0.5, 1], we search for a minimum with (β + 0.155)/3.85 α (β + 0.28)/3.125, which corresponds to a slide in the α–β plane where we can constrain these parameters on the basis of the 3σ range SNO SNO measured value for the φCC /φNC ratio. Result of minimization for both kind of neutrinos 2 ≈ { = gives χmin 20.24 and best-fit parameters are α = = × −4 2= } Fig. 2. Allowed regions (90–99% C.L.) from Super-Kamiokande, 0.35,β 0.98,µν 3.0 10 , r 1.50 for SNO and LSND data on ν–e elastic scattering electromagnetic Dirac neutrinos and {α = 0.342,β = 0.995,µν = phenomenological parameters for Majorana neutrinos. See text for − 3.4 × 10 5, r2=1.47} for Majorana ones.8 Allowed details. 2 2 regions, in the (r –µν ) plane with α and β equal 2 = to the best fit values, at 90% and 99% C.L. (χ 10−32 cm2 (−0.5 × 10−32 r2 3.5 × 10−32 cm2) 2 − 2 = χ χmin 4.61, 9.21) are reported in Figs. 1 and 2, when µ = 0 for Dirac (Majorana) neutrinos. In order ν respectively. From these figures we can see that µν to have a better idea of how the fit result combine × −10 × −10 1.0 10 µB (µν 1.2 10 µB ) at 90% C.L. with the experimental data in Fig. 3 we show the 2= − × −32 2 × when r 0, and 1.2 10 r 2.7 measured Super-Kamiokande relative spectrum for solar neutrinos [19] with the predicted spectrum with 8 −10 2 Here as above µν is given in units of 10 µB and r in the best fit parameters reported above. In particular, − units of 10 32 cm2. the solid line is for Dirac neutrinos. 24 A. Ianni / Physics Letters B 540 (2002) 20–24

including for this latter a possible calibration test with a neutrino source [25], could improve further the sensitivity to study lower values of µν ,forthese experiments will probe lower energy regions around 1 MeV. Moreover, a possible combined analysis using CHARM II data [8] could help improving limits on r2.

Acknowledgements

The author thanks J.F. Beacom, D. Montanino and F. Vissani for discussions.

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+ Measurements of relative branching ratios of c decays into states containing

FOCUS Collaboration J.M. Link a,M.Reyesa,P.M.Yagera,J.C.Anjosb,I.Bediagab,C.Göbelb,J.Magninb, A. Massafferri b,J.M.deMirandab,I.M.Pepeb,A.C.dosReisb,S.Carrilloc, E. Casimiro c,E.Cuautlec, A. Sánchez-Hernández c,C.Uribec,F.Vázquezc, L. Agostino d, L. Cinquini d,J.P.Cumalatd, B. O’Reilly d,J.E.Ramirezd, I. Segoni d, J.N. Butler e, H.W.K. Cheung e, G. Chiodini e, I. Gaines e, P.H. Garbincius e, L.A. Garren e,E.Gottschalke,P.H.Kaspere,A.E.Kreymere,R.Kutschkee, L. Benussi f,S.Biancof,F.L.Fabbrif,A.Zallof,C.Cawlﬁeldg,D.Y.Kimg,K.S.Parkg, A. Rahimi g,J.Wissg,R.Gardnerh, A. Kryemadhi h,K.H.Changi, Y.S. Chung i, J.S. Kang i,B.R.Koi,J.W.Kwaki,K.B.Leei,K.Choj,H.Parkj, G. Alimonti k, S. Barberis k,A.Ceruttik,M.Boschinik,P.D’Angelok,M.DiCoratok,P.Dinik, L. Edera k,S.Erbak, M. Giammarchi k,P.Inzanik,F.Leverarok, S. Malvezzi k, D. Menasce k, M. Mezzadri k, L. Moroni k,D.Pedrinik, C. Pontoglio k,F.Prelzk, M. Rovere k,S.Salak, T.F. Davenport III l,V.Arenam,G.Bocam, G. Bonomi m, G. Gianini m, G. Liguori m,M.M.Merlom, D. Pantea m,S.P.Rattim, C. Riccardi m, P. Vitulo m, H. Hernandez n,A.M.Lopezn,H.Mendezn,L.Mendezn,E.Montieln, D. Olaya n,A.Parisn, J. Quinones n,C.Riveran,W.Xiongn,Y.Zhangn,J.R.Wilsono, T. Handler p, R. Mitchell p,D.Enghq,M.Hosackq, W.E. Johns q,M.Nehringq, P.D. Sheldon q,K.Stensonq, E.W. Vaandering q,M.Websterq,M.Sheaffr

a University of California, Davis, CA 95616, USA b Centro Brasileiro de Pesquisas Físicas, Rio de Janeiro, RJ, Brazil c CINVESTAV, 07000 Mexico City, DF, Mexico d University of Colorado, Boulder, CO 80309, USA e Fermi National Accelerator Laboratory, Batavia, IL 60510, USA f Laboratori Nazionali di Frascati dell’INFN, Frascati I-00044, Italy g University of Illinois, Urbana-Champaign, IL 61801, USA h Indiana University, Bloomington, IN 47405, USA i Korea University, Seoul 136-701, South Korea j Kyungpook National University, Taegu 702-701, South Korea k INFN and University of Milano, Milano, Italy l University of North Carolina, Asheville, NC 28804, USA m Dipartimento di Fisica Nucleare e Teorica and INFN, Pavia, Italy n University of Puerto Rico, Mayaguez, PR 00681, USA

o University of South Carolina, Columbia, SC 29208, USA p University of Tennessee, Knoxville, TN 37996, USA q Vanderbilt University, Nashville, TN 37235, USA r University of Wisconsin, Madison, WI 53706, USA Received 8 June 2002; accepted 10 June 2002 Editor: L. Montanet

Abstract + + ∗0 + + + − We have studied the Cabibbo suppressed decay c → K (892) and the Cabibbo favored decays c → K K , + + + ∗0 + − + + + + − c → φ and c → ( K )K and measured their branching ratios relative to c → π π to be (7.8 ± 1.8 ± 1.3)%, (7.1 ± 1.1 ± 1.1)%, (8.7 ± 1.6 ± 0.6)%and(2.2 ± 0.6 ± 0.6)%, respectively. The ﬁrst error is statistical and the + − + + + + ∗0 second is systematic. We also report two 90% conﬁdence level limits (c → K π )/ (c → K (892)) < 35% + + + − + + + − and (c → K K )NR/ (c → π π )<2.8%.  2002 Elsevier Science B.V. All rights reserved.

1. Introduction complete the charged particle tracking and momentum measurement system. Three multi-cell, thresh- Past experiments have reported results on non- old Cerenkovˇ counters discriminate between different leptonic branching fractions of the lowest lying particle hypotheses, namely, electrons, pions, kaons + charmed baryon c [1,2]. In this Letter we report on and protons. The FOCUS apparatus also contains one + several c decay channels containing a baryon in hadronic and two electromagnetic calorimeters as well the final state. These measurements may be useful in as two muon detectors. testing theoretical predictions of the contributions to Events are selected using a candidate driven ver- inclusive decay amplitudes. For instance, as pointed texing algorithm where the vector components of the out by Guberina and Stefancic [3], direct measure- reconstructed decay particles define the charm flight + ments of c singly Cabibbo suppressed decay rates direction. This is used as a seed track to find the pro- + can improve our theoretical understanding of the c duction vertex [6]. Using this algorithm we determine lifetime, which can then be compared to recent high the confidence level of the decay and production ver- statistics measurements [4]. tices, and the significance of their separation. For each of the decay modes analyzed, we require the primary vertex to have a confidence level greater than 1% and 2. Reconstruction to contain at least two tracks other than the charm seed track. − → − + → + + → 0 This analysis uses data collected by the FOCUS The nπ , nπ and pπ 1 experiment at Fermilab during the 1996–1997 fixed- decays are reconstructed using a kink algorithm [7] target run and is based on a topological sample of where the properties of the neutral particle in the decay events with a charged Sigma hyperon plus two other are not detected, but rather inferred, with a two-fold + ambiguity in the momentum solution for some decays. charged particles emerging from the c decay vertex. FOCUS is a photo-production experiment equipped Systematic effects due to this ambiguity are reduced + → + + − with very precise vertexing and particle identification by normalizing to the decay mode c π π , detectors. The vertexing system is composed of a sili- where the same effect exists. To aid in fitting the con microstrip detector (TS) [5] interleaved with seg- mass distribution, for channels containing a ,we ments of the BeO target and a second system of twelve microstrip planes (SSD) downstream of the target. Be- 1 Throughout this Letter the charge conjugate state is implied − yond the SSD, five stations of multi-wire proportional unless explicitly stated. Note that the is not the charge conjugate + chambers plus two large aperture dipole magnets partner of the . FOCUS Collaboration / Physics Letters B 540 (2002) 25–32 27 implement a double Gaussian to determine the yield of signal events. By double Gaussian we mean two Gaussian shapes with separate amplitudes, means and widths. In this Letter we will discuss the +π+π−, +K+π−, −K+π+ and +K+K− final states. In + → + + − the c π π mode we let all the parameters float while in the other lower statistics modes we fix some of the parameters to their Monte Carlo values.

+ → + + − 3. c π π normalization mode + → + + − The c π π mode is our highest statistics decay containing a + particle. The events are selected requiring a detachment between primary and secondary vertex divided by its error (l/σl ) greater + + − then 5.5. A minimum + momentum cut of 50 GeV/c Fig. 1. π π invariant mass distribution fit with a double c Gaussian for the signal and a linear background. is imposed, as is a minimum secondary vertex confidence level of 10%. We also apply a cut on the lifetime + → + ∗0 + → − + + resolution, σt < 120 fs for the run period where we 4. c K (892) and c K π had a silicon detector (TS) in the target region (about decay modes 2/3 of the events) and σt < 150 fs otherwise (NoTS). + Further, we reject events which have a lifetime greater The c is reconstructed in the decay channel + + + − than six times the c lifetime. K π . We fit the invariant mass distribution We identify charged tracks using information from with and without a mass cut (832 −6. ondary vertex detachment cut, l/σl > 5.5, which re- Further, pions must not be identified as muons by the jects much of the combinatoric hadronic background. muon detector. For + → pπ0 decays, the proton is A minimum cut of 50 GeV/c on the momentum of − − + → + required to satisfy Wπ Wp > 3 while for the c candidate is also applied. The secondary decay + nπ , the pion must satisfy Wp − Wπ > −3and vertex must have a confidence level greater than 10%. PICON > −6 and have a momentum greater than Cerenkovˇ identification cuts are applied to the kaon + 5GeV/c. andpionfromthec as well as on the charged + + − + The π π mass distribution (Fig. 1) is fit with daughter of the . In particular, we require Wπ − + + a double Gaussian and a linear background. We obtain WK > 3.5ontheK while for the pion (from the c a yield of 1706 ± 88 events. decay) we require PICON > −6. In the + → pπ0 28 FOCUS Collaboration / Physics Letters B 540 (2002) 25–32

+ ∗ ∗ + − ∗ Fig. 2. (a) The K 0 invariant mass distribution where the K 0(892) is reconstructed in the K π channel. Only events in the K (892) − + + signal region are selected. (b) Invariant mass distribution for the ﬁnal state K π . The distributions are ﬁt with a double Gaussian for the signal and a linear background.

Systematic uncertainties were determined by fit variations on binning, fitting range, and counting sideband subtracted events in the K+π− invariant mass distribution. In order to test for possible biases in the fitting procedure, we also performed the fit of the +K+π− mass distribution using the Monte Carlo shape. We measured the relative branching ratio for statistically independent sub-samples divided by momentum, particle-antiparticle, run period, and + + + ∗0 Fig. 3. Spectator diagram for c → K (892). In this case no decay modes. Using a technique modeled on the PDG quark pair needs to be created. S-factor method we evaluated a systematic uncertainty from these split samples. After a correction for the case we apply a soft pion–proton separation cut of branching ratio of K∗0(892) → K+π−, our final − − + → + + → + ∗0 Wπ Wp > 3, while for nπ the pion must result for the branching ratio of c K (892) + + − satisfy Wp − Wπ > −3andPICON> −6. The pion with respect to π π is: from the + is also required to have a minimum momentum of 5 GeV/c. We remove much of the + → + ∗0 (c K (892)) remaining background by requiring σt < 120 fs for + + + − (c → π π ) the TS run period and σt < 150 fs for the NoTS run period. We also require the lifetime to be less than six = 7.8 ± 1.8(stat.) ± 1.3(syst.) %, + times the c lifetime. The mass distribution is fit using a double Gaussian where the systematic error is obtained by adding in plus a linear background. We fixed the widths, the quadrature the contributions from fit variants and split relative yield ratio and the shift between the means samples. of the two Gaussians to the Monte Carlo values. The We searched for the similar Cabibbo suppressed ± + → − + + resulting yield is 49 10 events. decay channel c K π . The event selection FOCUS Collaboration / Physics Letters B 540 (2002) 25–32 29

+ → + + − is identical to that of the c K π selection where the + is reconstructed in a neutron and a charged pion. The invariant mass distribution is shown in Fig. 2. To set an upper limit on the branching ratio for this decay we fit the data using a double Gaussian with a linear background where all the parameters of the Gaussians, except for the total yield, are fixed to the Monte Carlo values. The fit returns a yield of 10 ± 11 events. The systematic uncertainty is computed by varying the range, binning and the fitting shape function. As we do not observe a signal, after correcting by the branching ratio of K∗0(892) → K+π− we determine an upper limit of:

+ → − + + (c K π ) 35% + + − + + ∗0 Fig. 4. The fit to the K K invariant mass distribution fit with (c → K (892)) a double Gaussian plus a first degree polynomial. with a 90% confidence level, where we have combined the statistical and systematic errors in quadrature. +π+π− to be: + → + + − (c K K ) + ( → +π+π−) + + + − c 5. → K K decay mode c = 7.1 ± 1.1(stat.) ± 1.1(syst.) %.

The reconstruction of +K+K−events requires a + + → + detachment cut of l/σl > 3, the candidate c must 6. c φ decay mode have a minimum momentum of 30 GeV/c and a secondary vertex with a minimum confidence level For the +K+K− final state we also measured the + + + of 1%. The kaons from the c must be favored resonant φ contribution. The φ events are se- + + − with respect to the pion hypothesis, Wπ − WK > 1. lected using the same cuts used for the K K in- + The identification of the charged pion from the clusive mode, except: the l/σl cut is lowered to 2.5, (in the nπ + decay mode) is achieved by requiring the K+K− invariant mass is required to be within 3σ 2 Wp − Wπ > −3andPICON> −6. A soft separation (10 MeV/c )oftheφ mass and the absolute differ- + − ∗ cut of Wπ − Wp > −3, is applied to the proton ence between the K invariant mass and the for the decay + → pπ0. Further, we require the nominal value (1.690 GeV/c2) must be greater than 2 lifetime resolution σt < 110 fs for the TS period and 20 MeV/c . This last cut is applied to suppress the + → ∗0 + σt < 140 fs for the NoTS period. The invariant mass contamination from c (1690)K where the distribution is plotted in Fig. 4. ∗0 decays to +K−. A sideband subtraction is per- We fit the distribution using a double Gaussian for formed to remove a possible non-resonant contribu- the signal region plus a linear background and find tion. The +φ invariant mass distribution, with the 103±15 events. The widths of the Gaussians, the yield ∗0(1690) exclusion cut, is shown in Fig. 5. ratio and the shift between the two means are all fixed The fitting procedure follows the strategy applied to Monte Carlo values. Systematic studies were per- in the +K+K− inclusive mode and give a yield of formed in a manner similar to that for the +K∗(892) 57 ± 10 events. To assess the final systematic uncer- state. Adding in quadrature the two uncertainties ob- tainty on this measurement, we follow similar crite- tained by fit variations and split samples we quote ria to those described previously. We also investigated + → + + − the branching ratio for c K K relative to possible systematic contributions due to our choice 30 FOCUS Collaboration / Physics Letters B 540 (2002) 25–32

Fig. 5. The figure shows the sideband subtracted invariant mass + + + Fig. 6. Invariant mass distribution for the c decay to → ∗ + − + distribution for c φ fit using a double Gaussian for the 0(1690)( K )K fit with a double Gaussian for the signal signal and a linear background. and a linear background. of sidebands. double Gaussian and linear background We apply the same fitting procedure used in the + + − have been used for the fit giving. inclusive decay mode K K . The signal yield is Adding in quadrature the contribution from fit 34 ± 8 events. Fit variations and split samples were variations and split samples and correcting for the used to evaluate the systematic uncertainty. We quote branching fraction of φ to K+K− and the fraction of a final result of: + − events lost from our K mass cut, we quote the + ∗0 + ( → (1690)K ) ∗ + − + → + c ∗ B 0(1690) → K final result for the branching ratio of c φ with + → + + − + → + + − (c π π ) respect to c π π to be: = 2.2 ± 0.6(stat.) ± 0.6(syst.) %. + → + (c φ) + From the measurements of the resonant decays in ( → +π+π−) + + − c the K K final state we infer that almost all of + → + + − = 8.7 ± 1.6(stat.) ± 0.6(syst.) %, c K K occurs through the resonant modes +φ and ∗0(1690)K+. where unseen decay modes of the φ are included.

+ → + + − 8. Non-resonant decay mode c K K + → ∗0 + 7. c (1690)K decay mode We also looked for a non-resonant contribution + → ∗0 × + → + + − We also searched for the decay c (1690) to the decay c K K . We studied the K+, with the ∗0 reconstructed in +K−.Thecuts region M(K+K−)>1.03 GeV/c2 and M(+K−)> used in the selection of these events are the same 1.71 GeV/c2 applying the same selection cuts used in as for the inclusive +K+K− mode, except that we the inclusive mode. + + − lowered the detachment cut to l/σl > 2.5andweap- In Fig. 7 we show the K K invariant mass plied two mass cuts. The first cut excludes the φ re- distribution where the double Gaussian fitting proce- gion by requiring M(K+K−)>1.03 GeV/c2.The dure has been applied. All the parameters of the dou- second cut requires the +K− invariant mass to ble Gaussian, except the total yield, are fixed to their be within 20 MeV/c2 of the ∗0 nominal mass of Monte Carlo values. The yield from the fit is 14 ± 8 1.690 GeV/c2. A clean signal is shown in Fig. 6. events. After further corrections due to possible conta- FOCUS Collaboration / Physics Letters B 540 (2002) 25–32 31

+ → + + → ∗0 + mination from c φ and c (1690)K determined by varying the fit conditions in a manner decays we find a yield of 8 ± 8 events. With no evi- similar to that previously described. dence of a signal, we quote the 90% confidence level limit for the non-resonant component of the decay + → + + − + → + + − c K K with respect to c π π 9. Conclusions to be: + + + + − We have measured the branching ratio of four c ( → K K )NR + c < 2.5%. decay modes containing a particle reconstructed + → + + − + (c π π ) in both the pπ0 and nπ channels. These modes + → + ∗0 + → + + − Our systematic error, added in quadrature to the are c K (892), c K K inclu- + → + + → ∗0 + − + statistical error and included in the upper limit, is sive, c φ and c ( K )K .For these last two modes our measurements are consistent with the recent results reported by the Belle Collabo- ration [1]. We also set an upper limit for the Cabibbo + → − + + suppressed decay mode c K π and the + → + + − non-resonant contribution to c K K .Our final results, the relative efficiency of each mode with respect to the normalization mode as well as comparisons to previous measurements, are summarized in Table 1.

Acknowledgements

We wish to acknowledge the assistance of the staffs of Fermi National Accelerator Laboratory, the INFN of Italy, and the physics departments of the collabo- rating institutions. This research was supported in part + + + − by the US National Science Foundation, the US De- Fig. 7. c → K K invariant mass distribution for + − + − M(K K )>1.03 GeV/c2 and M( K )>1.710 GeV/c2. partment of Energy, the Italian Istituto Nazionale di The ﬁt is performed using a double Gaussian for the signal plus Fisica Nucleare and Ministero dell’Università e della a linear background. Ricerca Scientiﬁca e Tecnologica, the Brazilian Con-

Table 1 FOCUS results compared to previous measurements [1,2] where applicable. The relative efficiencies are computed with respect to the normalization mode Efficiency FOCUS results Belle results CLEO results ratio +→ + ∗ (c K (892)) ± ± + + + − 0.35 (7.8 1.8 1.3)%– – (c → π π ) + − + + (c → K π ) + + ∗ 1.49 < 35% @ 90% CL – – (c → K (892)) +→ + + − (c K K ) ± ± ± ± ± ± + + + − 0.85 (7.1 1.1 1.1)% (7.6 0.7 0.9)% (9.5 1.7 1.9)% (c → π π ) +→ + (c φ) ± ± ± ± ± ± + + + − 0.39 (8.7 1.6 0.6)% (8.5 1.2 1.2)% (9.3 3.2 2.4)% (c → π π ) +→ ∗ + − + (c ( K )K ) ± ± ± ± + + + − 0.92 (2.2 0.6 0.6)% (2.3 0.5 0.5)%– (c → π π ) + + − + (c → K K )NR + + + − 0.44 < 2.8% @ 90% CL < 1.8% @ 90% CL – (c → π π ) 32 FOCUS Collaboration / Physics Letters B 540 (2002) 25–32 selho Nacional de Desenvolvimento Científico e Tec- [3] B. Guberina, H. Stefancic, hep-ph/0202080. nológico, CONACyT-México, the Korean Ministry of [4] J.M. Link et al., FOCUS Collaboration, Phys. Lett. B 523 Education, and the Korean Science and Engineering (2001) 53. [5] J.M. Link et al., FOCUS Collaboration, hep-ex/0204023, Foundation. submitted to Nucl. Instrum. Methods A. [6] P.L. Frabetti et al., E687 Collaboration, Nucl. Instrum. Meth- ods A 320 (1992) 519. References [7] J.M. Link et al., FOCUS Collaboration, Nucl. Instrum. Meth- ods A 484 (2002) 174. [8] J.M. Link et al., FOCUS Collaboration, Nucl. Instrum. Meth- [1] K. Abe et al., Belle Collaboration, Phys. Lett. B 524 (2002) 33. ods A 484 (2002) 270. [2] P. Avery et al., CLEO Collaboration, Phys. Rev. Lett. 71 (1993) 2391. Physics Letters B 540 (2002) 33–42 www.elsevier.com/locate/npe

Measurement of χc2 production in two-photon collisions

Belle Collaboration K. Abe h,K.Abean,R.Abeab,T.Abeao,I.Adachih, Byoung Sup Ahn o,H.Aiharaap, M. Akatsu u,Y.Asanoau,T.Asoat,V.Aulchenkob,T.Aushevl,A.M.Bakichak, Y. Ban af,E.Banasz,A.Bayr,P.K.Beheraav, A. Bondar b,A.Bozekz,M.Brackoˇ s,m, J. Brodzicka z, B.C.K. Casey g,P.Changy,Y.Chaoy,B.G.Cheonaj,R.Chistovl, S.-K. Choi f,Y.Choiaj, M. Danilov l,L.Y.Dongj,A.Drutskoyl,S.Eidelmanb, V. Eiges l,Y.Enariu, C. Fukunaga ar, N. Gabyshev h, A. Garmash b,h, T. Gershon h, R. Guo w,F.Handaao,T.Haraad, Y. Harada ab, H. Hayashii v,M.Hazumih, E.M. Heenan t, I. Higuchi ao,T.Hojoad, T. Hokuue u, Y. Hoshi an, K. Hoshina as, S.R. Hou y,W.-S.Houy, H.-C. Huang y,T.Igakiu,Y.Igarashih,K.Inamiu, A. Ishikawa u,R.Itohh,M.Iwamotoc,H.Iwasakih,Y.Iwasakih,H.K.Jangai, J. Kaneko aq,J.H.Kangay,J.S.Kango,P.Kapustaz, S.U. Kataoka v, N. Katayama h, H. Kawai c,Y.Kawakamiu,N.Kawamuraa, T. Kawasaki ab,H.Kichimih,D.W.Kimaj, Heejong Kim ay,H.J.Kimay,H.O.Kimaj, Hyunwoo Kim o,S.K.Kimai,T.H.Kimay, P. Krokovny b,R.Kulasirie,S.Kumarae,A.Kuzminb,Y.-J.Kwonay,G.Lederk, S.H. Lee ai,J.Liah,D.Liventsevl,R.-S.Luy, J. MacNaughton k, G. Majumder al, F. Mandl k, S. Matsumoto d, T. Matsumoto ar, K. Miyabayashi v,H.Miyakead, H. Miyata ab, G.R. Moloney t,T.Morid, T. Nagamine ao, Y. Nagasaka i, E. Nakano ac, M. Nakao h,J.W.Namaj, Z. Natkaniec z, K. Neichi an,S.Nishidap,O.Nitohas, S. Noguchi v, T. Nozaki h,S.Ogawaam, F. Ohno aq, T. Ohshima u,T.Okabeu, S. Okuno n, S.L. Olsen g, Y. Onuki ab,W.Ostrowiczz, H. Ozaki h,P.Pakhlovl,H.Palkaz, C.W. Park o,H.Parkq,K.S.Parkaj,L.S.Peakak, J.-P. Perroud r,M.Petersg, L.E. Piilonen aw,N.Rootb,K.Rybickiz,H.Sagawah,S.Saitohh,Y.Sakaih, M. Satapathy av,O.Schneiderr,S.Schrenke,C.Schwandah,k,S.Semenovl,K.Senyou, R. Seuster g,M.E.Seviort, H. Shibuya am,B.Shwartzb,V.Sidorovb,J.B.Singhae, S. Stanicˇ au,1,M.Staricˇ m, A. Sugiyama u,K.Sumisawah, T. Sumiyoshi ar, S. Suzuki ax, S.K. Swain g, T. Takahashi ac, F. Takasaki h,K.Tamaih,N.Tamuraab,M.Tanakah, G.N. Taylor t,Y.Teramotoac, S. Tokuda u, T. Tomura ap,S.N.Toveyt, T. Tsuboyama h, T. Tsukamoto h, S. Uehara h, K. Ueno y,S.Unoh,S.E.Vahsenag,G.Varnerg, K.E. Varvell ak,C.C.Wangy,C.H.Wangx,J.G.Wangaw, M.-Z. Wang y, Y. Watanabe aq,

E. Won o,B.D.Yabsleyaw,Y.Yamadah, A. Yamaguchi ao,Y.Yamashitaaa, M. Yamauchi h, H. Yanai ab,J.Yashimah,Y.Yuanj,Y.Yusaao,J.Zhangau,Z.P.Zhangah, V. Zhilich b,D.Žontarau

a Aomori University, Aomori, Japan b Budker Institute of Nuclear Physics, Novosibirsk, Russia c Chiba University, Chiba, Japan d Chuo University, Tokyo, Japan e University of Cincinnati, Cincinnati, OH, USA f Gyeongsang National University, Chinju, Republic of Korea g University of Hawaii, Honolulu, HI, USA h High Energy Accelerator Research Organization (KEK), Tsukuba, Japan i Hiroshima Institute of Technology, Hiroshima, Japan j Institute of High Energy Physics, Chinese Academy of Sciences, Beijing, PR China k Institute of High Energy Physics, Vienna, Austria l Institute for Theoretical and Experimental Physics, Moscow, Russia m J. Stefan Institute, Ljubljana, Slovenia n Kanagawa University, Yokohama, Japan o Korea University, Seoul, Republic of Korea p Kyoto University, Kyoto, Japan q Kyungpook National University, Taegu, Republic of Korea r Institut de Physique des Hautes Énergies, Université de Lausanne, Lausanne, Switzerland s University of Maribor, Maribor, Slovenia t University of Melbourne, Melbourne, Victoria, Australia u Nagoya University, Nagoya, Japan v Nara Women’s University, Nara, Japan w National Kaohsiung Normal University, Kaohsiung, Taiwan, ROC x National Lien-Ho Institute of Technology, Miao Li, Taiwan, ROC y National Taiwan University, Taipei, Taiwan, ROC z H. Niewodniczanski Institute of Nuclear Physics, Krakow, Poland aa Nihon Dental College, Niigata, Japan ab Niigata University, Niigata, Japan ac Osaka City University, Osaka, Japan ad Osaka University, Osaka, Japan ae Panjab University, Chandigarh, India af Peking University, Beijing, PR China ag Princeton University, Princeton, NJ, USA ah University of Science and Technology of China, Hefei, PR China ai Seoul National University, Seoul, Republic of Korea aj Sungkyunkwan University, Suwon, Republic of Korea ak University of Sydney, Sydney, NSW, Australia al Tata Institute of Fundamental Research, Bombay, India am Toho University, Funabashi, Japan an Tohoku Gakuin University, Tagajo, Japan ao Tohoku University, Sendai, Japan ap University of Tokyo, Tokyo, Japan aq Tokyo Institute of Technology, Tokyo, Japan ar Tokyo Metropolitan University, Tokyo, Japan as Tokyo University of Agriculture and Technology, Tokyo, Japan at Toyama National College of Maritime Technology, Toyama, Japan au University of Tsukuba, Tsukuba, Japan av Utkal University, Bhubaneswer, India aw Virginia Polytechnic Institute and State University, Blacksburg, VA, USA ax Yokkaichi University, Yokkaichi, Japan ay Yonsei University, Seoul, Republic of Korea Belle Collaboration / Physics Letters B 540 (2002) 33–42 35

Received 31 May 2002; accepted 10 June 2002 Editor: L. Montanet

Abstract

The production of the χc2 charmonium state in two-photon collisions has been measured with the Belle detector at the + − + − −1 KEKB e e collider. A clear signal for χc2 → γJ/ψ, J/ψ → is observed in a 32.6 fb data sample accumulated at center-of-mass energies near 10.6 GeV, and the product of its two-photon decay width and branching fraction is determined to + − be Γγγ(χc2)B(χc2 → γJ/ψ)B(J/ψ → ) = 13.5 ± 1.3(stat.) ± 1.1(syst.) eV.  2002 Elsevier Science B.V. All rights reserved. PACS: 13.20.Gd; 13.40.Hq; 13.65.+i; 14.40.Gx

Keywords: Two-photon collisions; Charmonium; χc2; Partial decay width

1. Introduction = e or µ [5]. This channel is suitable for the experimental determination of Γγγ(χc2), since the de- The two-photon decay widths (Γ )oftheeven cay branching fraction is known with a relatively small γγ + − charge-parity charmonium states provide valuable in- error, B(χc2 → γJ/ψ)B(J/ψ → ) = (1.59 ± formation for testing models describing the nature of 0.13)% for both lepton families [3]. However, the two- heavy quarkonia. Various theoretical calculations de- photon decay width results obtained from previous scribing the quark–antiquark system predict the value measurements with this process seem to be system- ¯ → → of Γγγ(χc2) to be within the range 0.2–0.8 keV [1]. atically larger than those from pp χc2 γγ ex- A precise experimental determination of Γγγ(χc2) periments [6]. Further studies with high statistics data will provide a strong constraint on these models. The samples are needed to clarify the situation. ratio of the two-gluon decay width to the two-photon Recently, CLEO Collaboration has reported a mea- + − + − decay width Γgg(χc2)/Γγγ(χc2) has been calculated surement of Γγγ(χc2) in the π π π π final state within the framework of perturbative QCD with first- [7]. Although the branching fraction of this decay + − + − order correction [2] and the result gives Γ (χ ) = mode, B(χc2 → π π π π ) = (1.2 ± 0.5)%[3], γγ c2 + − 2 → 0.47 ± 0.13 keV. A comparison of Γγγ(χc2) with the is comparable to that of the γJ/ψ γ mode, two-gluon width will provide a way to study the va- its large error precludes a precise determination of lidity of perturbative QCD corrections for quarkonium Γγγ(χc2). decays. We have measured χc2 production in two-photon Measurements of two-photon decay widths for processes using the decay channel χc2 → γJ/ψ, + − −1 charmonium states are difficult because of their small J/ψ → . The results are based on a 32.6 fb production cross section and small detection efficien- data sample collected with the Belle detector. cies. To date, several experiments have reported the observation of two-photon production of the χc2 in the + − decay channel χc2 → γJ/ψ, J/ψ → ,where 2. Experimental data and detector system

The experiment was performed with the Belle E-mail address: [email protected] (S. Uehara). detector [8] at the asymmetric e+e− collider KEKB, 1 On leave from Nova Gorica Polytechnic, Slovenia. where an 8.0 GeV e− beam collides with a 3.5 GeV 2 This value is derived from measurements [3] using the as- + sumption that Γ (χ ) is given by Γ(χ → hadrons) − Γ(χ → e beam with a crossing angle of 22 mrad. We use a gg c2 c2 c1 −1 hadrons) [4]. Here the strong coupling constant is set to be 29.6 fb sample of data collected at the c.m. energy αs (mc) = 0.29 ± 0.02. corresponding to the peak of the Υ(4S) resonance 36 Belle Collaboration / Physics Letters B 540 (2002) 33–42

(10.58 GeV) and a 3.0 fb−1 sample collected 60 MeV The main backgrounds are leptonic final states from below the peak. QED processes such as e+e− → e+e− + −γ .The The basic topology of events that we select is E/p information, which is the ratio of the energy two tracks of opposite charge and a photon. The deposit in the ECL to the track’s momentum, is used to recoiling e+ and e− are not tagged in order to select identify the leptons and eliminate small backgrounds quasi-real two-photon collisions with high efficiency. that contain hadron tracks. Events induced by highly virtual photons (i.e., photons with high Q2) are effectively rejected by a strict transverse momentum (pt ) requirement applied to the 3. Event selection χc2 daughter particles, as described in the following section.3 The event selection criteria are as follows. (1) Ex- The charged track momenta are measured with a actly two oppositely charged tracks reconstructed by cylindrical drift chamber (CDC) located in a uniform the CDC, where both tracks satisfy the following lab- 1.5 T magnetic field. Track trajectory coordinates near oratory frame conditions: −0.47 cosθ +0.82, the collision point are provided by a silicon vertex where θ is the polar angle; pt 0.4GeV/c; |dr| detector (SVD). Photon detection and energy mea- 1cm,|dz| 3cm,where(|dr|,dz) are the cylin- surements are performed with a CsI electromagnetic drical coordinates of the track’s point of closest ap- calorimeter (ECL). The resolutions of track momen- proach to the nominal collision point in the rϕ plane; tum and photon energy measurements are 0.4% for |dz| 1cm,wheredz is the difference between the leptons, which, for the signal process, have typi- the dz’s of the two tracks; and no other well recon- cal pt values of 1.4 GeV/c, and 2.0% for the photons, structed tracks with pt higher than 0.1 GeV/c.(2)The with typical energies of 0.4 GeV. The magnet return opening angle (α) of the two tracks satisfies cosα> iron is instrumented to form the KL and muon detec- −0.997. (3) There is just one electromagnetic clus- tor (KLM), which detects muon tracks and provides ter in the ECL with an energy Eγ 0.2 GeV and trigger signals. isolated from the nearest charged track by an angle The majority of the signal events are triggered by greater than 18◦. (4) The scalar sum of the momenta two-track triggers that require at least two tracks with of the two charged tracks is less than 6 GeV/c,and transverse momenta larger than 0.2 GeV/c detected the invariant mass of the two tracks is between 1.5 in the CDC in coincidence with matching signals and 4.5 GeV/c2. (5) The total energy deposited in the from TOF counters and trigger scintillation counters, ECL is less than 6 GeV. (6) The absolute value of the isolated cluster or energy-sum signals from the ECL, total transverse momentum vector in the c.m. frame + − | ∗tot|=| ∗+ + ∗− + ∗γ | or muon tracks in the KLM. A constraint on the of the e e beams, pt pt pt pt ,is opening angle in the plane transverse to the e+ less than 0.15 GeV/c, while that for the two tracks ◦ | ∗+ + ∗−| beam axis (rϕ plane), ϕ > 135 , is applied at the only, pt pt , is larger than 0.10 GeV/c,where ∗+ ∗− ∗γ trigger level. The signal inefficiency due to this trigger pt , pt and pt are measured transverse momen- constraint is negligibly small because the final-state tum vectors (defined as two-dimensional momentum leptons tend to be back-to-back in the rϕ plane due vectors projected onto the plane perpendicular to the to the kinematic properties of our selected final states, beam axis in the e+e− c.m. system) for the positive namely, the strict pt requirement on the χc2 and the track, the negative track and the photon, respectively. small mass difference between the χc2 and J/ψ.The (7) For electron pairs, both tracks are required to have events from the J/ψ → e+e− mode are efficiently E/p 0.8; for muon pairs, both tracks are required to triggered by a total energy trigger derived from the have E/p 0.4. ECL with a threshold set at 1.0 GeV. Selection criterion (2) rejects cosmic-ray backgrounds. Criterion (6) rejects two-photonic lepton- pair production events with radiation from a recoil 3 2 electron or with a fake photon, which tend to pop- Q is defined as the negative of the invariant mass squared of ∗+ ∗− 2 | + |≈ a virtual incident photon. It is approximately equal to |pt | of the ulate the region pt pt 0. The scatter plot + − | ∗+ + ∗−| | ∗tot| virtual photon with respect to the e e beam axis. in the pt pt – pt plane before the applica- Belle Collaboration / Physics Letters B 540 (2002) 33–42 37

Fig. 2. Scatter plot of the invariant mass M+− of the two-track system versus the mass difference M = M+−γ − M+−.The arrows indicate the nominal J/ψ mass and the M value expected for χc2 → γJ/ψ.

tive for partially compensating for the radiative-decay events of J/ψ, J/ψ → e+e−γ .

4. Derivation of the number of signal events Fig. 1. Scatter plots for the absolute values of the two kinds of vector sums of the transverse momenta: the horizontal axis for the sum of Fig. 2 shows a scatter plot of the invariant mass the two charged particles, and the vertical axis of all three particles of the two tracks (M+−) versus the invariant-mass including the photon: (a) for real data and (b) for Monte Carlo events difference M = M+−γ − M+− for the selected of the signal process. Straight lines show the selection requirements. events, where M+−γ is the invariant mass of all three particles. A clear concentration is observed around M+− = 3.097 GeV/c2 and M = 0.459 GeV/c2,the tion of requirement (6) is shown in Fig. 1(a), where signal region for χc2 → γJ/ψ. | ∗+ + ∗−|≈ two separate clusters of events at pt pt 0 The mass difference distribution is shown in | ∗tot|≈ and pt 0 are apparent. The latter cluster corre- Fig. 3(a) for the events falling within the J/ψ signal + − sponds to exclusive γ ﬁnal states produced by mass region 3.06 M+− 3.13 GeV/c2.Aftertheﬁ- two-photon collisions. The distribution from the signal selection requirement for signal candidate events, nal events obtained from the Monte Carlo (MC) simu- 0.42 M 0.49 GeV/c2, is applied, 176 events lation (described in Section 5) is shown in Fig. 1(b). remain. Of these, 82 events have electron pairs and We correct the absolute momenta of detected elec- 94 have muon pairs. The contribution of χc1 produc- trons or positrons for bremsstrahlung in e+e−γ event tion, which would peak at M = 0.42 GeV/c2,ises- candidates. If photons of energy between 0.02 and timated to be less than one event, as expected from the 0.2 GeV are present within a cone of half-angle 3◦ suppression of spin-1 meson production in quasi-real around the electron direction, the energy of the most two-photon collisions [9]. energetic photon in the cone is added to the absolute The M distribution of events in the J/ψ-mass momentum of the track. This correction is also effec- sideband regions (2.65

from the χc0 [3,7], we avoid that mass region. This expected χc0 yield is consistent with the small excess of events seen near M = 0.32 GeV/c2 in Fig. 3(a). We determine the number of χc2 signal events to be 136.0 ± 13.3 after subtracting the number of background events from the total in the signal region. We find that the events in M sideband regions are dominated by non-J/ψ backgrounds since the quantity and shape of the distribution agree with that of the J/ψ sideband. The 368 events in the 0.42 M 0.49 GeV/c2 region in Fig. 3(b) consist of 176 electron pairs and 192 muon pairs. These backgrounds are consistent with higher-order QED events such as e+e− → e+e− + −γ , which would give comparable numbers of events with electron and muon pairs. In contrast, hadron production (in which pions would fake muons) would give primarily events containing muon pairs. Since a complete calculation of this process that takes interference effects into account is not available, we cannot estimate the background yield theoretically. | ∗tot| Fig. 4 shows distributions for ϕ, pt , +− Fig. 3. The mass difference distributions for (a) events in the γ +− | cosθγ | and cosθ− for the final candidate events, J/ψ-mass region (closed circles with error bars) and (b) sideband where ϕ is the azimuthal angle difference between events. The curves in (a) and (b) indicate the results of the fits that are used to determine the background contribution in the signal the two lepton’s momenta in the laboratory frame, and | ∗tot| + − region. The histogram in (a) shows the distribution of the signal pt is the transverse momentum of the γ sys- + − +−γ MC events normalized to the observed signal. The arrows show the tem in the c.m. frame of the e e beams. θγ and signal region (a) and the background control region (b). +− + − θ− are the polar angles of the photon in the γ c.m. frame and of the negatively charged lepton in the + − c.m. frame, respectively, where the polar angles 2 3.15

Fig. 4. Comparison of the ﬁnal samples (closed circles with error bars, backgrounds included) with the sum of the signal MC events (open ∗tot +−γ +− histogram) and estimated background contributions (hatched region) for: (a) ϕ;(b)|p |;(c)| cos θγ |; and (d) cos θ− .Thereareno ◦ t entries in the ϕ < 135 region in (a). The dashed histograms in (c) and (d) show the distribution for the pure helicity = 0 production case (see the text in Section 5).

5. Monte Carlo calculations the polar-angle distribution of the photon shown in Fig. 4(c) can be used to evaluate the possible contri- = = + − bution from a λ 0 component. The λ 2 compo- We used Monte Carlo (MC) simulated e e → +−γ + − + − nent produces a cosθγ distribution that is propor- e e χc2, χc2 → γJ/ψ, J/ψ → ( = e or 2 +−γ tional to [1 + cos θγ ], whereas the λ = 0 compo- µ) events to calculate the efficiency for the signal +− [ − 2 γ ] process. The TREPS MC program [10] is used for nent is proportional to 5 3cos θγ . The dashed the event generation. The effects of J/ψ radiative histograms in Figs. 4(c) and (d) show the expected distributions from the pure λ = 0 state. The present ex- decays are modeled with the PHOTOS [11] simulation +−γ code, which generates photon radiation from a final- perimental cosθγ distribution has better agreement = 2 = state lepton generated by TREPS with a probability with a pure λ 2 hypothesis (χ /dof 12.0/9) than = 2 = determined by a QED calculation. All of the final-state that for pure λ 0(χ /dof 45.0/9), or any other mixture of the two helicity states. Here, each χ2/dof particles in the MC events are processed by the full +− | γ | detector simulation program. is calculated from the cosθγ distribution divided into ten bins, where the total event number in the ex- We assume that the χc2 decay to the γJ/ψ final state is an E1 transition, since experimental observa- pected signal distribution is fixed to the number of ob- tions indicate that this transition dominates the de- served signal events. The trigger efficiency is experimentally determined cay [12]. Since the helicity state of χc2 produced in + − two-photon collisions is not known, we assume a pure using Bhabha and µ µ events that are collected with two or more different redundant triggers, as described λ = 2 state [13], where λ is the helicity of χc2 with respect to the γγ-incident axis. The measurement of above in Section 2. We estimate the probability for 40 Belle Collaboration / Physics Letters B 540 (2002) 33–42

2 the signal-process events that survive the selection mass scale of the vector mesons. The Qmax value does criteria to pass the trigger conditions to be (99 ± 1)% not affect the product of the luminosity function and ((94 ± 3)%) for eeγ (µµγ )events;and(96 ± 2)% the detection efficiency, since it is chosen to be large | ∗tot| in average. This estimation agrees with the results enough to cover the acceptance of our pt require- of a trigger simulation that is applied to the signal ment. The value of the TREPS luminosity function is MC events. The trigger efficiency is also confirmed compared with that obtained from a full-diagram cal- by the experimental yield of two-photonic lepton-pair culation of the e+e− → e+e−µ+µ− process [14] in + − → + − + − | ∗tot| events, e e e e , which agrees with the the small pt region. From the difference of the two expectation from a QED calculation [14]. results, the systematic error for the luminosity func- The ECL photon energy resolution was studied by tion is estimated to be 5%, which includes ambiguities | ∗tot| comparing experimental and MC mass-difference dis- from the choice of the form factor and the finite pt tributions for D∗0 → D0γ decays in e+e− annihila- requirement. tion events. We find that the photon energy resolution is 1.3 times the MC prediction. The same tendency is confirmed in the χc1 → γJ/ψ samples from B-meson 6. Results and discussion decays and η → γγ samples from two-photon collisions. If we use a correspondingly wider M distrib- The two-photon decay width of the χ is related to → c2 ution for the χc2 γJ/ψ, it decreases the efficiency the signal event yield as of the M selection by 3.8% from the MC-determined L (m )η value. We take this effect into account as a correction Yield 2 γγ χc2 = 20π Γγγ(χc2) for the efficiency and assign a systematic error of the L dt (c/h)¯ 2m2 χc2 same size (±3.8%). We confirm that the observed po- × B(χ → γJ/ψ)B(J/ψ → + −) sition and width of the signal peak in the data are con- c2 sistent with their expected values. = (0.309 fb/eV)Γγγ(χc2) When the trigger effects and the difference between + − × B(χc2 → γJ/ψ)B(J/ψ → ), the data and MC photon energy resolutions are taken into account, the overall efficiency is found to be 6.6%. where L dt is the integrated luminosity, η is the + − + − = 2 This is the average of the e e and µ µ decay efficiency, mχc2 ( 3.556 GeV/c ) is the χc2 mass and = × −4 −1 channels; the ratio of the efficiencies for the two lepton Lγγ(mχc2 )( 7.75 10 GeV ) is the two-photon species is ee/µµ = 0.70 ± 0.03. The lower efficiency luminosity function at the χc2 mass. The total width + − for e e is due to the occasional presence of extra of χc2 (2.00 ± 0.18 MeV [3]) is much smaller than the high-energy photons from radiative J/ψ decays and present M resolution (∼ 9 MeV), and does not affect electron bremsstrahlung. When the ee/µµ ratio for the present measurement. the background component determined from the J/ψ The observed number of events, 136.0 ± 13.3, im- sideband events is taken into account, we expect the plies the result Γγγ(χc2)B(χc2 → γJ/ψ)B(J/ψ → event yields in the final samples to have a ratio + −) = 13.5 ± 1.3 ± 1.1 eV, where the first and ee/µµ = 0.74 ± 0.04. This value is consistent with second errors are statistical and systematic, respec- the observed ratio for the final experimental samples: tively. This result corresponds to Γγγ(χc2)B(χc2 → 0.87 ± 0.13. γJ/ψ)= 114 ± 11(stat.) ± 9(syst.) ± 2(B.R.) eV or The two-photon luminosity function is also calcu- Γγγ(χc2) = 0.85 ± 0.08(stat.) ± 0.07(syst.) ± 0.07 lated by the TREPS program [10]. For consistency, the (B.R.) keV, where the last errors correspond to the un- 2 2 = B → = same upper cutoff value of the photon Q , Qmax certainties of the branching ratios, (χc2 γJ/ψ) 1.0GeV2, and the same vector-meson pole effect are (13.5 ± 1.1)%andB(J/ψ → + −) = (11.81 ± used in the calculation of the luminosity function and 0.20)% [3]. The systematic error has contributions in the event generation. The uncertainty in the lumi- from the trigger efficiency (2%), lepton identifica- nosity function due to the vector-meson pole effect tion efficiency (1.5%), photon detection efficiency (we adopt the J/ψ mass) is small, about 2%, since we (2%), inefficiency due to fake photons (less than 2%), | ∗tot| apply a strict pt requirement that is well below the J/ψ detection efficiency (2%), the M cut efficiency Belle Collaboration / Physics Letters B 540 (2002) 33–42 41

(3.8%), background subtraction (2.3%), the luminos- as has been indicated by a measurement, M2/E1 − +0.039 = ity function (5%) and other sources (less than 3%); 0.093−0.041( a2(χc2)) in amplitude [12], gives these total 8% when combined in quadrature. only a 2% effect on the efficiency. The error in the background subtraction is derived The Belle result for Γγγ(χc2) is compared with from the difference in signal yields in the M sig- those from previous experiments [5–7] in Fig. 5. The nal region between the present method (counting the present result has the smallest statistical and system- events in the signal region and subtracting the back- atic errors of all the two-photon measurements and is ground contribution) and an alternative method in consistent with the previous two-photon results. How- which the signal and the background components are ever, it is larger than the pp¯ results. A review of simultaneously fitted to the M distribution with all the experimental results of various branching ratios the shape and size parameters for the background and of ψ(2S) and χc decays [16] suggests that this dis- signal distributions allowed to float, with the Crystal- crepancy may come from incorrect values of B(χc2 → Ball line shape [15] used for the signal distribution. γJ/ψ)and B(χc2 → pp)¯ that are used for the deriva- The error of the background normalization is also tion of Γγγ(χc2) in these experiments. combined with this error. The inefficiency due to an extra (fake) photon with E>0.2 GeV is estimated to be less than 2% from 7. Conclusion + − an experimental study of the pt -balanced e e → + − + − e e µ µ process. We have measured χc2 production from two-photon −1 The uncertainty due to the assumption that the χc2’s collisions with a 32.6 fb data sample collected are produced in a pure λ = 2 state and decay via pure with the Belle detector at the KEKB e+e− col- E1 transitions is not included in the systematic error. lider, using the decay mode χc2 → γJ/ψ, J/ψ → Production in the λ = 0 state at the 10% level would + −. We find 136.0 ± 13.3 signal events after back- increase the detection efficiency by 7% and decrease ground subtraction. The observed polar-angle dis- the measured Γγγ(χc2) value by the same amount. tributions of the photon and leptons are consistent Meanwhile, a small mixture of the M2 transitions with those expected from the production of χc2 in the pure helicity 2 state. The product of the two- photon decay width of χc2 and branching fractions, + − Γγγ(χc2)B(χc2 → γJ/ψ)B(J/ψ → ) = 13.5± 1.3(stat.) ± 1.1(syst.) eV, is obtained. This result corresponds to Γγγ(χc2) = 0.85 ± 0.08(stat.) ± 0.07 (syst.) ± 0.07(B.R.) keV, where the product of the branching fractions, B(χc2 → γJ/ψ)B(J/ψ → + −) = (1.59 ± 0.13)% [3] is used.

Acknowledgements

We wish to thank the KEKB Accelerator Group for the excellent operation of the KEKB accelerator. We acknowledge support from the Ministry of Education, Culture, Sports, Science, and Technology of Japan and the Japan Society for the Promotion of Science; the Australian Research Council and the Australian Fig. 5. Comparison of the Belle result for Γγγ(χc2) value with those from previous measurements [5–7]. The solid error bars show Department of Industry, Science and Resources; the the statistical errors. The length of the dashed part in each error National Science Foundation of China under contract bar corresponds to the size of the systematic error, including the No. 10175071; the Department of Science and Tech- branching ratio uncertainty. nology of India; the BK21 program of the Ministry 42 Belle Collaboration / Physics Letters B 540 (2002) 33–42 of Education of Korea and the CHEP SRC program L3 Collaboration, M. Acciarri et al., Phys. Lett. B 453 (1999) of the Korea Science and Engineering Foundation; the 73. Polish State Committee for Scientiﬁc Research under [6] E760 Collaboration, T.A. Armstrong et al., Phys. Rev. Lett. 70 (1993) 2988; contract No. 2P03B 17017; the Ministry of Science E835 Collaboration, M. Ambrogiani et al., Phys. Rev. D 62 and Technology of the Russian Federation; the Min- (2000) 052002. istry of Education, Science and Sport of the Repub- [7] CLEO Collaboration, B.I. Eisenstein et al., Phys. Rev. Lett. 87 lic of Slovenia; the National Science Council and the (2001) 061801. Ministry of Education of Taiwan; and the US Depart- [8] Belle Collaboration, A. Abashian et al., Nucl. Instrum. Meth- ods A 479 (2002) 117. ment of Energy. [9] G.A. Schuler, F.A. Berends, R. van Gulik, Nucl. Phys. B 523 (1998) 423. [10] S. Uehara, KEK Report 96-11 (1996). References [11] E. Barberio, Z. Was, Comput. Phys. Commun. 79 (1994) 291. [12] E835 Collaboration, M. Ambrogiani et al., Phys. Rev. D 65 (2002) 052002; [1] C.R. Münz, Nucl. Phys. A 609 (1996) 364; E760 Collaboration, T.A. Armstrong et al., Phys. Rev. D 48 H.-W. Huang, C.-F. Qiao, K.-T. Chao, Phys. Rev. D 54 (1996) (1993) 3037; 2123. M. Oreglia et al., Phys. Rev. D 25 (1982) 2259; [2] W. Kwong, P.B. Mackenzie, R. Rosenfeld, J.L. Rosner, Phys. L.S. Brown, R.N. Cahn, Phys. Rev. D 13 (1976) 1195. Rev. D 37 (1988) 3210. [13] M. Poppe, Int. J. Mod. Phys. A 1 (1986) 545; [3] D.E. Groom et al., Eur. Phys. J. C 15 (2000) 1, and 2001 off- H. Krasemann, J.A.M. Vermaseren, Nucl. Phys. B 184 (1981) year partial update for the 2002 edition available on the PDG 269. WWW pages (URL: http://pdg.lbl.gov/). [14] F.A. Berends, P.H. Daverveldt, R. Kleiss, Nucl. Phys. B 253 [4] G.T. Bodwin, E. Braaten, G.P. Lepage, Phys. Rev. D 46 (1992) (1985) 441; R1914. F.A. Berends, P.H. Daverveldt, R. Kleiss, Comput. Phys. [5] TPC/Two-Gamma Collaboration, D.A. Bauer et al., Phys. Lett. Commun. 40 (1986) 285. B 302 (1993) 345; [15] T. Skwarnicki, Ph.D. Thesis, Institute for Nuclear Physics, CLEO Collaboration, J. Dominick et al., Phys. Rev. D 50 Krakow, 1986; DESY Internal Report, DESY F31-86-02 (1994) 4265; (1986). OPAL Collaboration, K. Ackerstaff, Phys. Lett. B 439 (1998) [16] C. Patrignani, Phys. Rev. D 64 (2001) 034017. 197; Physics Letters B 540 (2002) 43–51 www.elsevier.com/locate/npe

The e+e− → Zγγ → qq¯γγ reaction at LEP and constraints on anomalous quartic gauge boson couplings

L3 Collaboration

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a I Physikalisches Institut, RWTH, D-52056 Aachen, Germany 1 III Physikalisches Institut, RWTH, D-52056 Aachen, Germany 1 b National Institute for High Energy Physics, NIKHEF, and University of Amsterdam, NL-1009 DB Amsterdam, The Netherlands c University of Michigan, Ann Arbor, MI 48109, USA d Laboratoire d’Annecy-le-Vieux de Physique des Particules, LAPP, IN2P3-CNRS, BP 110, F-74941 Annecy-le-Vieux cedex, France e Institute of Physics, University of Basel, CH-4056 Basel, Switzerland f Louisiana State University, Baton Rouge, LA 70803, USA g Institute of High Energy Physics, IHEP, 100039 Beijing, PR China 6 h Humboldt University, D-10099 Berlin, Germany 1 i University of Bologna and INFN-Sezione di Bologna, I-40126 Bologna, Italy j Tata Institute of Fundamental Research, Mumbai (Bombay) 400 005, India k Northeastern University, Boston, MA 02115, USA l Institute of Atomic Physics and University of Bucharest, R-76900 Bucharest, Romania m Central Research Institute for Physics of the Hungarian Academy of Sciences, H-1525 Budapest 114, Hungary 2 n Massachusetts Institute of Technology, Cambridge, MA 02139, USA o Panjab University, Chandigarh 160 014, India p KLTE-ATOMKI, H-4010 Debrecen, Hungary 3 q Department of Experimental Physics, University College Dublin, Belﬁeld, Dublin 4, Ireland r INFN Sezione di Firenze and University of Florence, I-50125 Florence, Italy s European Laboratory for Particle Physics, CERN, CH-1211 Geneva 23, Switzerland t World Laboratory, FBLJA Project, CH-1211 Geneva 23, Switzerland u University of Geneva, CH-1211 Geneva 4, Switzerland v Chinese University of Science and Technology, USTC, Hefei, Anhui 230 029, PR China 6 w University of Lausanne, CH-1015 Lausanne, Switzerland x Institut de Physique Nucléaire de Lyon, IN2P3-CNRS,Université Claude Bernard, F-69622 Villeurbanne, France y Centro de Investigaciones Energéticas, Medioambientales y Tecnológicas, CIEMAT, E-28040 Madrid, Spain 4 z Florida Institute of Technology, Melbourne, FL 32901, USA aa INFN, Sezione di Milano, I-20133 Milan, Italy ab Institute of Theoretical and Experimental Physics, ITEP, Moscow, Russia ac INFN, Sezione di Napoli, and University of Naples, I-80125 Naples, Italy ad Department of Physics, University of Cyprus, Nicosia, Cyprus ae University of Nijmegen and NIKHEF, NL-6525 ED Nijmegen, The Netherlands af California Institute of Technology, Pasadena, CA 91125, USA ag INFN, Sezione di Perugia, and Università Degli Studi di Perugia, I-06100 Perugia, Italy ah Nuclear Physics Institute, St. Petersburg, Russia ai Carnegie Mellon University, Pittsburgh, PA 15213, USA aj INFN, Sezione di Napoli, and University of Potenza, I-85100 Potenza, Italy ak Princeton University, Princeton, NJ 08544, USA al University of Californa, Riverside, CA 92521, USA am INFN, Sezione di Roma, and University of Rome “La Sapienza”, I-00185 Rome, Italy an University and INFN, Salerno, I-84100 Salerno, Italy ao University of California, San Diego, CA 92093, USA ap Bulgarian Academy of Sciences, Central Laboratory of Mechatronics and Instrumentation, BU-1113 Soﬁa, Bulgaria aq The Center for High Energy Physics, Kyungpook National University, 702-701 Taegu, South Korea ar Purdue University, West Lafayette, IN 47907, USA as Paul Scherrer Institut, PSI, CH-5232 Villigen, Switzerland at DESY, D-15738 Zeuthen, Germany au Eidgenössische Technische Hochschule, ETH Zürich, CH-8093 Zürich, Switzerland av University of Hamburg, D-22761 Hamburg, Germany aw National Central University, Chung-Li, Taiwan, ROC ax Department of Physics, National Tsing Hua University, Hsinchu, Taiwan, ROC

Received 15 May 2002; received in revised form 9 June 2002; accepted 10 June 2002

Editor: L. Rolandi 46 L3 Collaboration / Physics Letters B 540 (2002) 43–51

Abstract + − − The cross section of the process e e → Zγγ → qq¯γγ is measured with 215 pb 1 of data collected with the L3 detector during the ﬁnal LEP run at centre-of-mass energies around 205 GeV and 207 GeV. No deviation from the Standard Model − expectation is observed. The full data sample of 713 pb 1, collected above the Z resonance, is used to constrain the coefﬁcients of anomalous quartic gauge boson couplings to:

−2 2 −2 −2 2 −2 −0.02 GeV

1. Introduction nearest quark: E > 5GeV, | cosθ | < 0.97, √γ γ + − High energy e e collisions offer a unique envi- s − mZ < 2ΓZ, cos θγ q < 0.98, (1) ronment to unveil the structure of the couplings be- where m and Γ are the Z boson mass and width. tween gauge bosons. Extensive studies of boson pair- Z Z Events with hadrons and initial state photons falling production are performed to probe triple vertices of outside the signal definition cuts are referred to as neutral and charged bosons. Results were recently re- “non-resonant” background. ported on the investigation of triple boson produc- + − + − A single initial state radiation photon can also lower tion through the reactions e e → W W γ [1,2] and + − + − the effective centre-of-mass energy of the e e colli- e e → Zγγ [3,4]. These processes give access to sion to around m . This photon can be mistaken for the possible anomalous Quartic Gauge boson Couplings Z most energetic photon of the signal and two sources (QGCs). can then mimic the least energetic photon: the direct Figs. 1(a)–(c) display three of the six Standard + − radiation of photons from the quarks, or photons orig- Model diagrams that describe the e e → Zγγ proc- inating from hadronic decays, misidentified electrons ess with the radiation of photons from the incoming or unresolved π0’s. These background processes are electrons. This process is studied exploiting the high depicted in Figs. 1(d) and (e), respectively. branching fraction of the Z boson decay into hadrons. + − In the Standard Model, the Zγγ production via The e e → Zγγ → qq¯γγ signal is defined [4] by QGCs is forbidden at tree level. Possible contributions phase-space requirements on the energies E and γ of anomalous QGCs, through the diagram sketched in angles θ of the two photons, on the propagator mass √ γ Fig. 1(f), are described by two terms of dimension-six s and on the angle θ between each photon and the γ q in an effective Lagrangian [5,6]:

0 πα µν ρ L =− a Fµν F Wρ · W , 6 4Λ2 0 1 Supported by the German Bundesministerium für Bildung, c πα µσ ρ L =− acFµρ F W · Wσ , Wissenschaft, Forschung und Technologie. 6 4Λ2 2 Supported by the Hungarian OTKA fund under contract numbers T019181, F023259 and T037350. where α is the ﬁne structure constant, Fµν is the

3 Also supported by the Hungarian OTKA fund under contract photon ﬁeld and Wσ is the weak boson ﬁeld. The number T026178. parameters a0 and ac describe the strength of the 4 Supported also by the Comisión Interministerial de Ciencia y QGCs and Λ represents the scale of the New Physics Tecnología. responsible for these anomalous contributions. In the 5 Also supported by CONICET and Universidad Nacional de La Standard Model, a0 = ac = 0. Experimental limits Plata, CC 67, 1900 La Plata, Argentina. + − 6 Supported by the National Natural Science Foundation of on QGCs were derived from studies of the e e → + − China. W W γ process [1,2]. However, the a0 and ac L3 Collaboration / Physics Letters B 540 (2002) 43–51 47

(a) (b) (c)

(d) (e) (f)

+ − Fig. 1. Representative diagrams of (a)–c) the Standard Model contribution to the e e → Zγγ signal and the “non-resonant” background, (d) the background from direct radiation of a photon from the quarks, (e) the background from photons, misidentified electrons or unresolved π0’s originating from hadrons and (f) the anomalous QGC diagram. couplings might be different in the e+e− → Zγγ nal states, generated with EXCALIBUR [16]. The L3 case. Alternative parametrisations can be found in detector response is simulated using the GEANT [17] Refs. [7,8]. Indirect bounds on QGCs were extracted and GHEISHA [18] programs, which model the ef- in Ref. [9] using Z pole data. fects of energy loss, multiple scattering and shower- ing in the detector. Time-dependent detector ineffi- ciencies, as monitored during data taking periods, are 2. Data analysis also simulated Candidates for the e+e− → Zγγ → qq¯γγ process Ref. [4] describes the analysis of the e+e− → are longitudinally and transversely balanced hadronic Zγγ → qq¯γγ process with 497.6pb−1 of data col- events with two isolated photons with reconstructed lected by the L3√ detector [10] at LEP at centre-of- energy above 5 GeV, detected in a polar angle range mass energies, s, between 130 and 202 GeV. This | cosθ| < 0.97. The invariant mass of the reconstructed Letter details the equivalent findings from the√ final hadronic system, Mqq¯ , is required to be consistent with LEP run, when the machine was operated at s = mZ:74GeV

Fig. 2. Distributions of (a) the relativistic velocity√ βZ of the Z boson reconstructed from the measured photons, (b) the invariant mass Mqq¯ of the hadronic system, (c) the scaled energy Eγ 1/ s of the most energetic photon and (d) the angle ω between the least energetic photon and the nearest jet. Data, signal and background Monte Carlo samples are shown. Monte Carlo predictions are normalised to the integrated luminosity of the data. The arrows show the positions of the ﬁnal selection cuts. In each plot, cuts on all other variables have been applied.

Table 1 + − Results of the e e → Zγγ → qq¯γγ selection. The signal efficiencies, ε, are given, together with the observed and expected numbers of qq¯ Other events. Expectations for signal, Ns , hadronic processes with photons, Nb , and other backgrounds, Nb , are listed. Uncertainties are due to Monte Carlo statistics √ qq¯ Other s (GeV) ε(%) Data Monte Carlo Ns Nb Nb 204.8 51 17 14.7 ± 0.511.3 ± 0.53.09 ± 0.02 0.31 ± 0.03 206.6 50 23 24.7 ± 0.519.5 ± 0.54.53 ± 0.04 0.67 ± 0.03 L3 Collaboration / Physics Letters B 540 (2002) 43–51 49 that of the closest jet is also imposed. Data and Monte Carlo distributions of these selection variables are presented in Fig. 2. Good agreement is observed. Table 1 lists the signal efficiencies and the numbers of events selected in the data and Monte Carlo samples. A signal purity around 75% is obtained. The dominant background consists of hadronic events with photons. Half of these are “non-resonant” events, the other half being events with final state radiation or fake photons.

3. Cross section measurement

A clear Z signal is observed in the spectrum of the recoil mass to the two photons, as presented in Fig. 3(a). The e+e− → Zγγ → qq¯γγ cross section, σ , is determined in the kinematical√ region defined by Eq. (1) at each average s by a fit to the recoil mass spectrum. The background predictions and the signal shape are fixed, while the signal normalisation is fitted. The results are:7 = +0.11 ± σ(204.8GeV) 0.30−0.09 0.03 pb (σSM = 0.287 ± 0.003 pb), = +0.07 ± σ(206.6GeV) 0.25−0.06 0.03 pb (σSM = 0.281 ± 0.003 pb). Here and below, the first quoted uncertainties are statistical and the second ones systematic. The systematic uncertainties on the cross section measurement are of the order of 10% [4], mainly due to the limited Monte Carlo statistics and the uncertainty on the energy scale of the detector. The measurements are in good agreement with the theoretical predictions, σSM, as calculated with the KK2f Monte Carlo program. The uncertainty on the predictions (1.5%) is the quadratic sum of the theory uncertainty [11] and the statistical uncertainty of the Fig. 3. Mass recoiling from photon pairs in data, signal and background Monte Carlo for (a) the data sample analysed in this Monte Carlo sample used for the calculation. These Letter and (b) the total sample collected above the Z resonance. results and those obtained at lower centre-of-mass Monte Carlo predictions are normalised to the integrated luminosity of the data.

7 The cross section is also measured in the more restrictive phase space deﬁned by tightening the bounds on θγ and θγ q to | | −1 energies [4] are compared in Fig. 4 to the expected√ cos θγ < 0.95 and√ cos θγ q < 0.9. For the full 215 pb at the combined average s of 205.9 GeV, the result is: σ(205.9GeV) = Standard Model cross section as a function of s. 0.18±0.06±0.02 pb, with a Standard Model expectation of σ = Fig. 3(b) shows the recoil mass spectrum for SM − 0.172 ± 0.003 pb. the total data sample of 712.9pb1 collected at 50 L3 Collaboration / Physics Letters B 540 (2002) 43–51

Fig. 5. Energy spectrum of the least energetic photon in data, signal √and background Monte Carlo. The full integrated luminosity at s = 130–209 GeV is considered. Monte Carlo predictions are normalised to the integrated luminosity of the data. Examples of anomalous QGC predictions are also given.

+ − → → ¯ Fig. 4. The cross√ section of the process e e Zγγ qqγγ as a function of s. The signal is deﬁned by the phase-space cuts of Eq. (1). The width of the band corresponds to the statistical and theoretical uncertainties of the predictions of the KK2f Monte Carlo. Dashed and dotted lines represent anomalous QGC predic- 2 −2 2 −2 tions for a0/Λ = 0.05 GeV and ac/Λ = 0.10 GeV , respec- − tively. The inset presents three combined samples: 231.6pb 1 at √ − √ s = 182.7–188.7 GeV, 232.9pb 1 at s = 191.6–201.7 GeV and the data described in this Letter.

LEP above the Z resonance, comprising the data discussed in this Letter and those at lower centre-of- mass energies [4]. A fit to this spectrum determines the ratio R between all the observed data and the Zγγ Fig. 6. Two-dimensional confidence level contours for the fitted signal expectations as: 2 2 QGC parameters a0/Λ and ac/Λ . The fit result is shown together with the Standard Model (SM) predictions. σ R = = 0.86 ± 0.09 ± 0.06, Zγγ σ SM 4. Constraints on quartic gauge boson couplings in agreement with the Standard Model. The correlation of systematic uncertainties between the different data Anomalous values of QGCs would manifest them- + − samples amounts to 50% and is taken into account in selves as deviations in the√ total e e → Zγγ cross the fit. section as a function of s, as presented in Fig. 4. L3 Collaboration / Physics Letters B 540 (2002) 43–51 51

A harder energy spectrum for the least energetic pho- [2] L3 Collaboration, M. Acciarri et al., Phys. Lett. B 490 (2000) ton [6] constitutes a further powerful experimental sig- 187; L3 Collaboration, M. Acciarri et al., Phys. Lett. B 527 (2002) nature, as√ shown in Fig. 5 for the full data sample col- = 29. lected at s 130–209 GeV. QGC predictions for [3] L3 Collaboration, M. Acciarri et al., Phys. Lett. B 478 (2000) the cross section and this spectrum are obtained by 39. reweighting the Standard Model signal Monte Carlo [4] L3 Collaboration, M. Acciarri et al., Phys. Lett. B 505 (2001) events. A modified version of the WRAP [19] Monte 47. Carlo program, that includes the QGC matrix element, [5] G. Bélanger, F. Boudjema, Phys. Lett. B 288 (1992) 201. is used. [6] W.J. Stirling, A. Werthenbach, Eur. Phys. J. C 14 (2000) 103. [7] G. Bélanger et al., Eur. Phys. J. C 13 (2000) 283. The energy spectra√ of the least energetic photon are [8] A. Denner et al., Eur. Phys. J. C 20 (2001). fitted for the two s values√ discussed in this Letter [9] A. Brunstein, O.J.P. Éboli, M.C. Gonzales-Garcia, Phys. Lett. and the eight values of s of Ref. [4]. Each of the two B 375 (1996) 233. parameters describing the QGCs is left free in turn, [10] L3 Collaboration, B. Adeva et al., Nucl. Instrum. Methods the other being fixed to zero. The fits yield the 68% A 289 (1990) 35; L3 Collaboration, O. Adriani et al., Phys. Rep. 236 (1993) 1; confidence level results: I.C. Brock et al., Nucl. Instrum. Methods A 381 (1996) 236; a0 +0.02 −2 ac +0.01 −2 M. Chemarin et al., Nucl. Instrum. Methods A 349 (1994) 345; = 0.00− GeV , = 0.03− GeV , Λ2 0.01 Λ2 0.02 M. Acciarri et al., Nucl. Instrum. Methods A 351 (1994) 300; in agreement with the expected Standard Model values A. Adam et al., Nucl. Instrum. Methods A 383 (1996) 342; of zero. A simultaneous fit to both parameters yields G. Basti et al., Nucl. Instrum. Methods A 374 (1996) 293. [11] KK2f version 4.13 is used; the 95% confidence level limits: S. Jadach, B.F.L. Ward, Z. W¸as, Comput. Phys. Commun. 130 − a − −0.02 GeV 2 < 0 < 0.03 GeV 2, (2000) 260. Λ2 [12] PYTHIA version 5.772 and JETSET version 7.4 are used; − a − T. Sjöstrand, Preprint CERN-TH/7112/93 (1993), revised −0.07 GeV 2 < c < 0.05 GeV 2, Λ2 1995; − T. Sjöstrand, Comput. Phys. Commun. 82 (1994) 74. as shown in Fig. 6. A correlation coefficient of 16% [13] KORALZ version 4.03 is used; is observed. Experimental systematic uncertainties as S. Jadach, B.F.L. Ward, Z. W¸as, Comput. Phys. Commun. 79 + − well as those on the Standard Model e e → Zγγ → (1994) 503. qq¯γγ cross section predictions are taken into account [14] PHOJET version 1.05 is used; in the fit. These results supersede those previously R. Engel, Z. Phys. C 66 (1995) 203; √ R. Engel, J. Ranft, Phys. Rev. D 54 (1996) 4244. obtained at lower s [4], as they are based on the [15] KORALW version 1.33 is used; full data sample and an improved modelling of QGC M. Skrzypek et al., Comput. Phys. Commun. 94 (1996) 216; effects. M. Skrzypek et al., Phys. Lett. B 372 (1996) 289. In conclusion, the e+e− → Zγγ → qq¯γγ process [16] R. Kleiss, R. Pittau, Comput. Phys. Commun. 85 (1995) 447; is found to be well described by the Standard Model R. Pittau, Phys. Lett. B 335 (1994) 490. [17] GEANT version 3.15 is used; predictions [11], with no evidence for anomalous R. Brun et al., Preprint CERN-DD/EE/84-1 (1984), revised values of QGCs. 1987. [18] H. Fesefeldt, Report RWTH Aachen PITHA 85/02 (1985). [19] G. Montagna et al., Phys. Lett. B 515 (2001) 197. We are References indebited to G. Montagna, M. Moretti, O. Nicrosini, M. Osmo and F. Piccinini for having provided us with the WRAP reweighting function. [1] OPAL Collaboration, G. Abbiendi et al., Phys. Lett. B 471 (1999) 293. Physics Letters B 540 (2002) 52–61 www.elsevier.com/locate/npe

Analysis of ﬂuctuation of ﬂuctuations in 32S–AgBr interactions at 200 A GeV

Dipak Ghosh, Argha Deb, Mitali Mondal, Jayita Ghosh

Nuclear and Particle Physics Research Centre, Department of Physics, Jadavpur University, Kolkata 700032, India Received 20 November 2001; received in revised form 13 May 2002; accepted 6 June 2002 Editor: J.P. Schiffer

Abstract The possible signature of chaotic behavior of multiparticle production in nucleus–nucleus collision has been investigated with the help of a new parameter, the entropy index µq . This index describes the event-to-event ﬂuctuation of factorial moments which measures the spatial ﬂuctuation of the multiplicity distribution. An analysis of 32S–AgBr interaction data at 200 A GeV reveals a large value for the entropy index signifying chaotic behavior of the multipion production process in ultrarelativistic nuclear interactions.  2002 Elsevier Science B.V. All rights reserved.

1. Introduction mechanism for particle production has a self-similar property. Unusually large density fluctuations in (pseudo) The method of factorial moments Fq does not rapidity have been observed in cosmic events [1] fully account for all the fluctuations that the system as well as in hadron–hadron [2,3], hadron–nucleus exhibits. In case of vertically averaged horizontal and nucleus–nucleus [4] collisions. These fluctuations moments, only the spatial fluctuation is taken into have led to interpretations in terms of possible ev- account neglecting the event space fluctuation. On the idence for a hadronic phase transition [5], hadronic other hand, horizontally averaged vertical moments Cherenkov radiation [6] and recent years have wit- lose information about spatial fluctuation and only nessed a remarkably intense experimental and theoret- measure the fluctuations from event-to-event. Thus, ical activity in search of scale invariant fluctuations in the common methods lead to the loss of information multihadron production processes, commonly known on the chaotic nature of the multiparticle production as “intermittency” [7,8]. In particle physics, intermit- processes. tency refers to the power-law behavior of the nor- Recently, new moments Cp,q have been introduced [10] which are the moments of factorial mo- malized factorial moments Fq , on the bin size. The observation of such a behavior [9] suggests that the ment distributions and take into account the spatial fluctuation as well as the event space fluctuation. From these moment, an effective parameter, the entropy in- E-mail address: [email protected] (D. Ghosh). dex, µq , is defined by the author [10] to characterize

0370-2693/02/$ – see front matter  2002 Elsevier Science B.V. All rights reserved. PII:S0370-2693(02)02088-9 D. Ghosh et al. / Physics Letters B 540 (2002) 52–61 53 the chaotic behavior of the multiparticle production (i) The incident beam track should not exceed 3◦ process. from the main beam direction in the pellicle. This In the history of particle physics, there is no appro- is done to ensure that we have taken the real priate measure for chaos in multiparticle production projectile beam. till date. The possibility of chaotic behavior of particle (ii) Events showing interactions within 20 µm from production in branching processes has recently been the top and bottom surface of the pellicle were investigated [10,11]. rejected. This is done to reduce the loss of In order to extract the entropy indices, one should tracks as well as to reduce the error in angle study the phase space and the event space simultane- measurement. ously. For each event, one can calculate the factorial (iii) The incident particle tracks which induced inter- moments in pseudorapidity η space. Event by event, actions were followed in the backward direction the values of factorial moments fluctuate greatly. The to ensure that they indeed were projectile beam entropy index, µq describes the degree of such fluctu- starting from the beginning of the pellicle. ation from event-to-event. According to the emulsion terminology [19], the Recently, Wang et al. [12,13] have studied chaotic- particles emitted from interactions are classified ity with the help of this parameter in case of p–p colli- as: sion at 400 GeV/c. Numerous experimental investiga- (a) Black particles. They are target fragments tions on intermittent type of fluctuations for nucleus– with ionization greater or equal to 10I0, nucleus collisions at relativistic and ultra-relativistic I0 being the minimum ionization of a singly energy have been performed in last twenty years, e.g., charged particle. The range of them is less [9,14–17]. But no rigorous study on chaoticity has than 3 mm, the velocity less than 0.3c and been reported so far on nucleus–nucleus collisions at the energy less than 30 MeV, where c is the relativistic and ultra-relativistic energy. In this Letter velocity of light in vacuum. we have analyzed the experimental data of multiparti- (b) Grey particles. They are mainly fast target cle production in 32S–AgBr interactions at 200 A GeV recoil protons with energy up to 400 MeV. in terms of entropy indices in order to shed light on They have ionization 1.4I0 I 10I0.Their possible chaotic behavior of multiparticle production ranges are greater than 3 mm and they have in ultra-relativistic nuclear collisions. velocities (ν),0.7c ν 0.3c. (c) Shower particles. The relativistic shower tracks with ionization I less than or equal to 1.4I0 are mainly produced by pions and 2. Experimental data are not generally confined within the emulsion pellicle. (d) The projectile fragments are a different class Stacks of G5 nuclear emulsion plates were hori- of tracks with constant ionization, long range zontally exposed to a 32S beam at 200 A GeV from and small emission angle. CERN SPS at CERN. Details about the data sets may be found in Ref. [18]. A Leitz Metaloplan microscope To ensure that the targets in the emulsion are silver with a 10X objective and 10X ocular lens provided or bromine nuclei, we have chosen only the events with a semi-automatic scanning stage was used to scan with at least eight heavily ionizing tracks (black + the plates. Each plate was scanned by two independent grey). observers to increase the scanning efficiency. The fi- According to the above selection procedure, we nal measurements were done using an oil-immersion have chosen 140 events of 32S–AgBr interactions at 100X objective. The measuring system fitted with it 200 A GeV. The emission angle (θ) was measured has 1 µm resolution along the X and Y axes and 0.5 µm for each track by taking the coordinates (X0,Y0,Z0) resolution along the Z-axis. of the interaction point, coordinates (X1,Y1,Z1) at After scanning, the events were chosen according the end of the linear portion of each secondary track to the following criteria: and coordinate (Xi,Yi ,Zi) of a point on the incident 54 D. Ghosh et al. / Physics Letters B 540 (2002) 52–61 beam. The accuracy in pseudorapidity in the region Where p is a positive real number. If Cp,q(M) has a of interest (−2 η 2) is of the order of 0.1 power law behavior as the division number M goes to pseudorapidity units. infinity It is worthwhile to mention that the emulsion tech- Ψq (p) nique possesses very high spatial resolution, which Cp,q(M) ∝ M ,M→∞, (3) makes it a very effective detector for studying the then the phenomenon is referred to as erraticity, and is chaotic behavior of multiparticle production. characterized by the slope µq of Ψq (p) at p = 1which is called entropy index defined by [10] 3. Entropy and entropy index d µ = Ψ (p)| = (4) q dp q p 1 The single-particle density distribution in pseudorapidity space is non-flat. As the shape of this distri- and describes the width of the fluctuation. This is a bution influences the scaling behavior of the factorial new parameter defined in the event space and is related moments, we have used the “cumulative” variable [20] to the entropy as X(η) instead of η. The cumulative variable X(η) is − S = ln NM µq . (5) given by q

η η2 From (5) we can say that as µq increases, i.e., the event-to-event fluctuation of the factorial moments X(η) = ρ η ∂η ρ η ∂η , increases, Sq decreases. The meaning of (5) can better η1 η1 be understood following a very simple but illustrative where η1 and η2 are two extreme points in the distrib- example sited by Hwa [10]. One can consider two e ution ρ(η). The corresponding region of investigation extreme cases: (a) If Fq is the same for every event, = e = for X(η) then becomes (0, 1). then Sq ln N; (b) If only one event has Fq 0, e = = One can consider a two-dimensional space-like and Fq 0 in all others, then Sq 0. Thus case (b) pseudorapidity (η) space as the horizontal axis which is more ordered in the event space than (a); that is, is divided into M bins and the vertical axis has N it is more disordered to spread out an observable e sites corresponding to the N events in the event space. (Fq in this case) over all events than to confine it to For each event, the factorial moments are calculated a few events having non-zero values (analogous to the according to the following formula increase of entropy of an expanding gas). Thus, Sq decreases when there are more events with F e = 0, 1 M q f e(M) = n (n − 1) ···(n − q + 1), signifying more order in the event space. From (5) it q M i i i i=1 is now obvious that µq is a measure of that decrease which in turn implies more fluctuation in F e. where ni is the number of particles in the ith bin for q At large M, only large spikes in small bins con- eth event. The normalized factorial moments for the e eth event are then defined as tribute to Fq , especially when q is large. Events with large spikes are rare. Consequently, the fluctuation e = e e q e Fq (M) fq (M) f1 (M) . (1) in Fq from event-to-event becomes more pronounced e with increasing q. That behavior is now quantified Since Fq (M) fluctuates from event to event, one e by µq . We may therefore use µq to characterize the obtains a distribution P(Fq ) for the whole event e ‘spatial’ properties of the chaotic behavior of multi- sample. Let the average of Fq (M) determined from e e particle production processes. P(Fq ) be denoted by Fq (M) ,and We have divided the X(η) region into M bins and = e e e Φq (M) Fq (M) Fq (M) . have calculated the factorial moments Fq (M) for each event using Eq. (1) with M = 5, 6, 7,...,25. Here q, In order to quantify the degree of that fluctuation, the order for spatial fluctuation, is varied from 2 to 5 in a new normalized moment is defined as [10] e steps of 1. We have plotted ln Fq (M) against ln M for p Cp,q(M) = Φq (M) . (2) q = 2, 3, 4 and 5 in Figs. 1(a)–(d). It shows power law D. Ghosh et al. / Physics Letters B 540 (2002) 52–61 55

(a) (b)

e Fig. 1. The dependence of the logarithm of the event-average factorial moment Fq (M) on the logarithm of phase-space partition number M for experimental and Monte Carlo simulated data of 32S–AgBr interactions at 200 A GeV with (a) q = 2, (b) q = 3, (c) q = 4and(d)q = 5. For clarity of presentation the error bars are shown at some representative points. behavior of factorial moments with bin size, signifying (ii) The multiplicity distribution and rapidity distribu- intermittent behavior. tion of Monte Carlo simulated events reproduces To check whether the observation is non-statistical those of the real ensemble. in nature, we have done Monte Carlo simulation based on the following assumptions: With this Monte Carlo simulated events we have performed the same analysis and the variation of e (i) Pions are emitted independently of each other. ln Fq (M) with ln M is shown in Fig. 1 with the corre- 56 D. Ghosh et al. / Physics Letters B 540 (2002) 52–61

(a) (b)

Fig. 2. The dependence of the logarithm of the factorial moment Fq (M) for a typical event on the logarithm of phase-space partition number M for experimental and Monte Carlo simulated data of 32S–AgBr interactions at 200 A GeV with (a) q = 2, (b) q = 3, (c) q = 4and(d)q = 5. sponding experimental data. Non-intermittent behav- typical event is shown in Fig. 2(a)–(d), respectively, ior is observed for the Monte Carlo simulated data. for q = 2, 3, 4 and 5. The intermittency analysis has also been done indi- The above analysis provides a strong evidence in vidually for few randomly chosen events. There also favour of our data to be dynamically important. So we we see that Monte Carlo simulated data show non- can use this data for studying the delicate fluctuation intermittent behavior, whereas experimental data man- of fluctuations analysis. ifest intermittent type of fluctuation. Analysis for a D. Ghosh et al. / Physics Letters B 540 (2002) 52–61 57

(a) (b)

Fig. 3. The dependence of the logarithm of Cp,q(M) on the logarithm of phase-space partition number M for p = 0.7, 0.9, 1.0, 1.1, 1.3 and 1.5 for 32S–AgBr interactions at 200 A GeV with (a) q = 2, (b) q = 3, (c) q = 4and(d)q = 5. For clarity of presentation the error bars are shown at some representative points.

e To probe the event-to-event fluctuation of Fq (M) Cp,q(M) shows power law behavior with M in the we have calculated Cp,q(M), the moment of factorial neighborhood of p = 1 for the entire range of M. moments, using Eq. (2). Here p is the order for event- The best linear fit of ln Cp,q(M) versus ln M has been to-event fluctuation. For each q we have calculated performed in the region of investigation for p = 0.9 the values of Cp,q(M) for p = 0.7, 0.9, 1.0, 1.1, 1.3 and 1.1 for all orders (q). The confidence levels and 1.5. To check whether Cp,q(M) follows scaling for the best fits never fall below 90%. According to behavior with M,lnCp,q(M) is plotted against ln M Eq. (3) the slopes of plots give Ψq (p). The slopes are in Fig. 3(a)–(d), respectively, for q = 2, 3, 4 and 5. given in Table 1. To quantify the degree of fluctuation 58 D. Ghosh et al. / Physics Letters B 540 (2002) 52–61

e Table 1 of Fq (M) from event-to-event the values of entropy = + The slopes of the linear fits (y a bx) to the experimental data of index, µq has been calculated using these slopes 32 200 A GeV S–AgBr interactions following the definition given in Eq. (4). The values qp bof the entropy index, µq are tabulated in Table 2 for 20.9 −0.0021 ± 0.0002 different orders and Fig. 4 represents the variation of ± 1.10.0040 0.0004 µq with order q. It is evident from the Fig. 4 that 30.9 −0.0330 ± 0.0025 the values of entropy indices, µq increase with q 1.10.0480 ± 0.0041 40.9 −0.1291 ± 0.0100 which signifies that the chaoticity increases with q, 1.10.1641 ± 0.0117 the order for spatial fluctuation, or, in other words, 50.9 −0.2204 ± 0.0113 the event-to-event fluctuations become more erratic 1.10.2386 ± 0.0094 with the increase of q implying the decrease in event space entropy Sq . However, as there is no quantitative criterion on how small µq must be in order to indicate no chaotic behavior, it is not justified to comment about the exact characteristics of this chaotic behavior of particle production. It is worthwhile to mention that Fu et al. [21] and Liu et al. [22] have demonstrated that in a low- multiplicity sample, erraticity analysis is dominated by the statistical fluctuations. This dominance will disappear and the observed erraticity will deviate from that of pure statistical fluctuations only if the events of the studied samples are coming from some new kind of physical processes, because within the framework of traditional high energy nuclear physics the dominance of statistical fluctuations in a given physical process does not depend on the concrete conditions, e.g., the collision energy, the mass of colliding nuclei, the cut of phase space etc. So the dominance of dynamical fluctuation over statistical fluctuation should be checked. In order to make a faithful comparison between the results from experimental data and the pure-statistical-fluctuation case, the authors [22] have demonstrated how to construct a model of pure statis- Fig. 4. The dependence of µ on q for experimental and Monte q tical fluctuations. Following them [22] we have taken Carlo simulated data of 32S–AgBr interactions at 200 A GeV and for 400 GeV/c p–p collision [13]. one random number distributed uniformly in the re-

Table 2 Entropy indices and relative parameters for experimental and Monte Carlo simulated data of 200 A GeV 32S–AgBr interactions and for 400 GeV/c p–p collision [13]

Interaction Reference Average multiplicity µ2 µ3 µ4 µ5 200 A GeV This work 95.70.030 ± 0.002 0.405 ± 0.024 1.466 ± 0.077 2.295 ± 0.093 32S–AgBr Monte Carlo simulated This work 95.70.004 ± 0.0007 0.027 ± 0.005 0.093 ± 0.030 0.237 ± 0.060 200 A GeV 32S–AgBr 400 GeV/c [13] multiplicity 10.27 0.203 ± 0.009 0.909 ± 0.021 1.752 ± 0.054 – p–p 4 D. Ghosh et al. / Physics Letters B 540 (2002) 52–61 59 gion [0, 1] as the value of cumulative variable, X(η) It is obvious from Eqs. (7) and (8) that for each particle in an event. Repeating n times, the 1 X(η) values of all the n particles in the event are deter- Γ0 = 1andΓ1 = . n + 1 mined and a Monte Carlo event, containing only statistical fluctuations, has been generated. Constructing At higher q, Γq are progressively smaller but in this way N events, we have calculated the moment, are increasingly more dominated by the large xi Cst (M) using Eq. (2) and entropy index, µst using components in Se, which in turn emphasize on large p,q q rapidity gaps. The moment Γ fluctuates from event- Eq. (4). We have included the values of µst in Table 2 q q to-event. This event-to-event fluctuation of Γ can be and potted in Fig. 4. It can clearly be seen that the sta- q quantified by the erraticity measure tistical values of entropy indices lie far below the experimental points. Therefore, we can safely conclude sq =−Γq ln Γq , (9) that our erraticity analysis is not dominated by statis- where stands for average over all events. tical fluctuations. The moment Γq does not filter out statistical fluctuations. However, one can estimate how much sq stands out above the statistical fluctuation by first 4. Rapidity-gap analysis calculating st =− st st The use of rapidity gaps is proposed by Hwa sq Γq ln Γq , (10) et al. [23] to measure the fluctuations of spatial st where Γq is determined by using only the statistical patterns from event-to-event when event multiplicity distribution of the gaps, i.e., when all n particles in an is low. In our case, the average multiplicity of data event are distributed randomly in X(η) space and then sample is not too high, for that we are now revisiting taking the ratio this problem adopting the procedure proposed by Hwa = st et al. [23]. Sq sq /sq . (11) One can consider an event with n particles, labeled If S deviates from 1 then it will be the erraticity = q by i 1, 2,...,n, located in the X(η) space at Xi , measure of the spatial pattern. ordered from the left to the right. The distances The q dependence of Sq is to be determined from between neighboring particles are given by the analysis. However, the specific way in which Sq depends on q have no physical significance [23]. xi = Xi+1 − Xi,i= 0, 1,...,n, (6) We have calculated Γq moment for each event using Eq. (8). Here q the order for spatial fluctuations, with X0 = 0andXn+1 = 1 being the boundaries of the X(η) space. Every event e is thus characterized by is varied from 2 to 5 in steps of 1. For each q, Γ moments fluctuate from event-to-event. To probe asetSe of n + 1 numbers: Se ={xi | i = 0,...,n}, q which clearly satisfy this event-to-event fluctuation we have calculated the erraticity measure sq using Eq. (9). To eliminate the n st statistical part of this measure we have calculated sq xi = 1, (7) by randomly distributing the particles of each event i=0 = st in X(η) space and have taken the ratio Sq sq /sq . and these numbers are referred as ‘rapidity gaps’. To The ln Sq values are plotted against ln q in Fig. 5. It is evident from the figure that S deviates significantly study the fluctuation of Se from event-to-event, the q from 1. It implies that it is a statistically significant moment of xi for each event is defined as measure of erraticity of rapidity gaps in multiparticle n production. The S values increase with the increase 1 q q Γq = x , (8) n + 1 i of q putting more weight on large gaps. Evidently, the i=0 result indicates a power-law behavior in q for q>2 where q is the order for spatial fluctuation. Since a Sq ∝ q , with a = 3.89 ± 0.43. xi < 1, Γq are usually 1. 60 D. Ghosh et al. / Physics Letters B 540 (2002) 52–61

ment, because this behavior is still seen in the unlike- sign charged particles [24,25]. If it exists in the unlike- sign sector, then it must also exist in the like-sign sector, though small in comparison to the BE effect. We emphasize on the fact that an effect that is small is not necessarily unimportant. Thus the study of event- to-event fluctuation of normalized factorial moment will certainly provide important informations of the dynamics of particle production. For comparison of erratic behavior, the results obtained by Wang et al. for p–p collisions at 400 GeV/c [12,13] are also included in Table 2 and Fig. 4. It is observed that the values of the entropy index for 32S–AgBr interactions at 200 A GeV are smaller than those for p–p collisions at 400 GeV/c.Itisworth- while to mention here that the values of intermittency exponent for nucleus–nucleus collisions are less than that for hadron–hadron collisions [9]. This property Fig. 5. The dependence of logarithm of S on logarithm of q for q is also reflected in our analysis. This may be due to 32S–AgBr interactions at 200 A GeV. the fact that the nucleus is composed of many nucleons and nucleus–nucleus collision at a particular im- 5. Discussion pact parameter involves a number of participants. Our fluctuation of fluctuation analysis is therefore sensitive Fluctuations of spatial patterns from event-to-event to two types of fluctuations: one due to the particle can reveal greater details about the underlying dy- production of an elementary collision between nucle- namics of particle production than average properties. ons, and the other due to the fluctuation in the sources. Erraticity analysis of multiparticle production data In p–p collision there is only one source. Spectator extracts the maximum amount of information on self- quarks in a p–p collision can also fragment and con- similar fluctuations. In this Letter, we have used two tribute to particle production, where as the spectator methods to describe erraticity: one is based on bin nucleons in a nucleus–nucleus collision do not con- multiplicities and the other on rapidity gaps. It is intu- tribute to particle production. itively obvious that the two quantities are complemen- This is the fundamental difference between our data tary: the latter measures how far apart neighboring par- and Wangs’ [12,13]. According to the sense of the ticles are, while the former measures how many parti- definition of the entropy index, one may then conclude cles fall into the same bin. Detailed study of these two that the particle production process in case of nucleus– methods gives positive evidence of chaos in particle nucleus collisions is chaotic but is less chaotic than production in relativistic nucleus–nucleus collisions. that of hadron–hadron collisions. This is the first ever Regarding the origin of dynamical fluctuation pres- rigorous evidence of chaotic particle production of ent in heavy ion interactions, it is believed that a main nucleus–nucleus collisions at ultra-relativistic energy. part is dominated by the Bose–Einstein correlation effect of identical pions. The bunching of particles in small bins cannot be distinguished from the interfer- Acknowledgements ence effect due to the coherent emission of same type particles from an extended source. The rise of nor- The authors express their gratitude to the referees malized factorial moment with decreasing bin size for whose valuable suggestions have enriched the Letter like-sign charged particles is believed to be due to BE a lot. We further express our gratitude to Prof. P.L. effect. However, one should not let the BE effect com- Jain, Buffalo State University, USA for providing us pletely obscure the rise of normalized factorial mo- with the exposed and developed emulsion plates used D. Ghosh et al. / Physics Letters B 540 (2002) 52–61 61 for this analysis. We also gratefully acknowledge the [12] W. Shaoshun et al., Phys. Lett. B 416 (1998) 216. financial help from the University Grants Commission [13] W. Shaoshun et al., Phys. Rev. D 57 (1998) 3036. (India) under the COSIST programme. [14] EMU-01 Collaboration, M.I. Adamovich et al., Phys. Rev. Lett. 65 (1990) 412. [15] EMU-01 Collaboration, M.I. Adamovich et al., Z. Phys. C 49 (1991) 395. References [16] EMU-01 Collaboration, M.I. Adamovich et al., Z. Phys. C 76 (1997) 659. [1] JACEE Collaboration, T.H. Burnett et al., Phys. Rev. Lett. 50 [17] D. Ghosh et al., Phys. Rev. C 59 (1999) 2286. (1983) 2062. [18] D. Ghosh et al., Phys. Rev. C 52 (1995) 2092. [2] UA5 Collaboration, G.J. Alner et al., Phys. 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Azimuthal asymmetry of J/ψ suppression in non-central heavy-ion collisions

Xin-Nian Wang a,FengYuanb

a Nuclear Science Division, MS 70-319, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA b Institut für Theoretische Physik der Universität, Philosophenweg 19, D-69120 Heidelberg, Germany Received 16 February 2002; received in revised form 29 May 2002; accepted 10 June 2002 Editor: W. Haxton

Abstract The azimuthal asymmetry of J/ψ suppression in non-central heavy-ion collisions is studied within a dynamic model of J/ψ suppression in a deconfined partonic medium. Within this model, J/ψ suppression in heavy-ion collisions is caused mainly by the initial state nuclear absorption and dissociation via gluon-J/ψ scattering in deconfined partonic medium. Only the second mechanism gives arise to azimuthal asymmetry of the final J/ψ production. We demonstrate that if there is an onset of suppression by quark–gluon plasma (QGP) in the NA50 data, it must be accompanied by the non-vanishing azimuthal asymmetry. Using the same critical density above which the QGP effect enters, we predict the azimuthal asymmetric coefficient v2 as well as the survival probability for J/ψ at the RHIC energy.  2002 Published by Elsevier Science B.V. PACS: 12.38.Mh; 24.85.+p; 25.75.-q

In the search for quark–gluon plasma (QGP), J/ψ alous suppression unexplained by the normal initial suppression has been proposed as one of the promising nuclear absorption, there are still much debates about signals [1] of the deconfinement in high-energy heavy- the exact nature of the anomalous suppression [5,6], ion collisions. Because of the color screening effect whether it is caused by the formation of QGP or dis- in a quark–gluon plasma, the linear confining poten- sociation by ordinary hadronic matter. tial in vacuum that binds two heavy quarks to form a We propose in this Letter the study of azimuthal quarkonium disappears so that it can be easily broken asymmetry of J/ψ production [7] as additional mea- up causing suppression of the J/ψ production. The surements to distinguish different competing mecha- problem in heavy-ion collisions is however compli- nism of J/ψ suppression. Since the initial state in- cated by other competing mechanisms such as initial teractions such as nuclear absorption or nuclear shad- nuclear absorption [2] and hadronic dissociation [3]. owing of gluon distribution has no preference over While recent precision data from the NA50 [4] experi- the azimuthal direction they will not have any con- ment at the CERN SPS energies clearly show anom- tribution to the azimuthal anisotropy of the J/ψ production. Only suppression by the final state interaction with the produced medium will cause significant E-mail address: [email protected] (X.-N. Wang). azimuthal anisotropy in the final J/ψ distribution in

0370-2693/02/$ – see front matter  2002 Published by Elsevier Science B.V. PII:S0370-2693(02)02121-4 X.-N. Wang, F. Yuan / Physics Letters B 540 (2002) 62–67 63 the transverse direction. If the centrality dependence value of the binding energy, the validity of the pertur- of the J/ψ suppression additional to the initial nu- bative calculations for the gluonic dissociation cross clear absorption is caused by formation of QGP, it section of χc might be questionable. So the above for- must be accompanied by a sudden onset of the az- mula can only be considered more phenomenological. imuthal anisotropy. On the other hand, a hadronic ab- However, keeping this point in mind, we can see that sorption scenario would give a continuous centrality the above expression still qualitatively reflects the fact dependence of the azimuthal anisotropy. In this Letter, that χc states are easier to be broken up than J/ψ, we will study the centrality dependence of both the because of the much lower energy threshold and the averaged J/ψ suppression factor and the azimuthal overall larger factor of 4. Therefore, this perturbative anisotropy with a model in which J/ψ suppression calculation gives us a reasonable estimate and guides is caused by initial nuclear absorption and final state us to include its contribution for a more complete dissociation by QGP above a critical density. Using study of J/ψ suppression in heavy ion collisions. In parameters from fitting the NA50 data, we will also the following we will consider χc contributing to about give predictions for J/ψ suppression and its azimuthal 40% of the initial J/ψ production and use the above anisotropy at the RHIC energies. formula to estimate its suppression in a deconfined We will quantify the azimuthal anisotropy by the partonic system. second Fourier coefficient v2 of the azimuthal angle In the rest frame of a deconfined parton gas, distribution of the final J/ψ distribution, similarly the momentum distribution of thermal gluons will de- to the proposed elliptic flow measurement [8,9]. We pend on the effective temperature T with an approxi- follow the microdynamic approach of J/ψ suppres- mate Bose–Einstein distribution, f(k0; T) ∝ sion [10], in which the J/ψ suppression is caused by [exp(k0/T) − 1]−1. The velocity averaged dissocia- gluonic dissociation. Different from a normal hadron tion cross sections for charmonia is defined as, gas, a deconfined partonic system contains much 3 0 0; d kv relσ(q )f (k T) harder gluons which can easily break up a J/ψ.The vrelσ (T , p) = , (3) perturbative calculations predict the gluon-J/ψ disso- d3kf(k0; T) ciation cross section [11,12], where vrel is the relative velocity between J/ψ and a gluon. In this Letter we are only interested in (q0/ − 1)3/2 σ q0 = N 0 , (1) J/ψ production in the midrapidity region, so these ψ 0 0 5 (q / 0) cross sections will depend on charmonia transverse where momentum pT as well as the effective temperature T . With the velocity averaged dissociation cross sec- 2π 32 2 16π 1 N = . tions, the survival probabilities of charmonia in the de- 0 2 2 3 3 3gs mQ confined quark–gluon plasma will have the following form, Here gs is the strong coupling constant, mQ is charm quark mass, and q0 is the gluon energy in the J/ψ rest Sdeconf.(b, r, v) 0 frame. To break up a J/ψ, q must be larger than the τf binding energy . Using similar approach, we have 0 = exp − v σ ρ(r + vτ, τ)Θ(ρ − ρ )dτ , calculated the dissociation cross section for P -wave rel c states by gluons, τ0 (4) (q0/ − 1)1/2 0 = χ where τ0 is the formation time of the quark–gluon σχ q 4N0 0 7 (q / χ ) plasma, which will be set as τ0 = 1fminthefollowing (q0/ )2 − (q0/ ) + calculations. The upper limit of the time integral τf is × 9 χ 20 χ 12 0 7 , (2) determined by the Θ function. Here we introduce the (q / χ ) critical density ρc above which the QGP dissociation where χ is the binding energy of the P -wave state. effects enters. We assume it is the same for both J/ψ For χc, it is about 0.250 GeV. Because of the small and χc. Because of different binding energies, the 64 X.-N. Wang, F. Yuan / Physics Letters B 540 (2002) 62–67 effective cross sections will have different temperature participant nucleons. At collider energies such as dependences for J/ψ and χc even if ρc is reached. ρ is RHIC the contribution from the hard processes is the local density depending on τ , the initial production more important. We will use fb = 0.34 and fp = 0.88 point r and the velocity v of the charmonium particles. as determined by the√ PHENIX [14] experiment for The velocity v depends on the transverse momentum Au + Au collisions√ at s = 130 GeV. We extrapolate p T and the azimuthal angle φ. The dissociation cross these parameters to s = 200 GeV by just multiplying sections vrelσ depend on the effective temperature, a factor of 1.14 found by PHOBOS [15]. which will also depend on the local density ρτ (b, s). The final expression for the survival probability due In the case of (1+1)D Bjorken longitudinal expansion to QGP suppression is, with the initial plasma density ρ0 = ρ(τ = τ0), deconf. α S (b,r,v) τ0 ρ (b, s) = ρ (b, s) , (5) τf τ 0 τ dτ = exp − vrelσψg (r + vτ,τ) where α = 1. Correspondingly, for the effective tem- τ τ perature, 0 1/3 τ × ρ0(b, r + vτ)Θ(ρ − ρc) . (12) T (b, s) = T (b, s) 0 . (6) τ 0 τ For simplification, we relate the local effective temper- It is interesting to note that with a very large constant ature with the local density in the following way, dissociation cross section the above formula will be equivalent to the model [5] by Blaizot et al., where 1/3 Tτ (b, s) = κ ρτ (b, s) , (7) they assume that all of J/ψ will be dissociated above some critical density. The detailed comparison of this =[ 2 ]1/3 where κ π /16ζ(3) . approach and ours will be presented elsewhere. The initial plasma density is related to the rapidity We can also include the transverse expansion ef- density of gluons, fects on the local parton density following Ref. [16], g = 1 dN ρ0(b, s) (b, s), (8) τ0 dΩvt τ dy d2s ρ(r,τ) = ρ (r − vt τ), (13) 0 τ 2π 0 where d2s is the transverse area of the overlapping re- where v is the average velocity of the transverse gion of two colliding nuclei. We will follow the two- t expansion of the parton system. We will use v = 0.4c component model [13] and include both the soft and t for SPS and v = 0.6c for RHIC in the following hard contribution to the final hadron production. As- t numerical calculations. suming that the initial gluon density is proportional to Apart from the above discussed QGP suppression the final hadron rapidity density, we have phenomeno- for charmonia states, there is also suppression associ- logically, ated with the initial state interaction, i.e., the nuclear g ¯ dN = [ + ] absorption of so-called preresonance of cc pairs, 2 (b, s) c fbnb(b, s) fpnp(b, s) , (9) dy d s − − abs 1 exp( σabsTA(r)) where c is a constant and we will set c = 1inthe S (b,r) = σ 2 T (r)T (|b − r|) following calculations, and abs A B

1 − exp(−σabsTB(|b − r|)) n (b, s) = T (s)[1 − exp(−T (b − s)σ )] × , (14) p A B pp 2 | − | σabsTA(r)TB( b r ) + TB(b − s)[1 − exp(−TA(s)σpp )], (10) where σ is the absorption cross section of the n (b, s) = T (s)T (b − s)σ . (11) abs b A B pp preresonance with nucleons, for which we will set Since mini-jet cross section at the SPS energy is σabs = 5.8mbinthisLetter. very small, we will effectively only have the soft By summing up these two contributions, we get the contribution which is proportional to the number of ﬁnal J/ψ survival probability as X.-N. Wang, F. Yuan / Physics Letters B 540 (2002) 62–67 65

Ssur.(b, p ) T d2rT (r)T (|b − r|)Sabs(b, r)S deconf.(r, v) = A B . 2 d rT A(r)TB(|b − r|) (15) From the above expression, we see that the nuclear absorption has no dependence on the azimuthal angle. This means that there is no contribution to v2 from the initial nuclear absorption. On the other hand, the final state interaction or the dissociation by the deconfined parton gas indeed has azimuthal angular dependence because the parton density is azimuthally asymmetric. Therefore, any finite value of v2 for J/ψ production should come from the final state interaction. It will provide us important information about the early stage of the quark–gluon plasma. The azimuthal asymmetric coefficient v2 then can be calculated as sur. dφS (b,p T ) cos(2φ) v2(b, pT ) = , (16) sur. dφS (b,p T ) where φ is the azimuthal angle between J/ψ transverse momentum p T and the impact parameter b. With this formula we can study both the pT dependence and the centrality (b) dependence of v2. In the following we present the numerical results of the above approach. We first determine the critical density ρc by fitting the SPS data on J/ψ suppression, Fig. 1. The J/ψ survival probability and v at SPS as a function of and then predict the suppression and v2 at RHIC. 2 transverse energy ET : nuclear absorption alone (dot-dashed line); In Fig. 1, we show our results at the SPS energy. nuclear absorption plus QGP dissociation without (dashed line) and The upper plot is the survival probability as a func- with transverse expansion effects. The experimental data are from tion of transverse energy ET compared with the ex- NA50 [4]. perimental data from NA50 [4]. From the fit we deter- −3 mined the critical density ρc = 3.3fm . The corre- of NA50, and taking into account the transverse lation between the impact parameter b and transverse expansion improves the fit especially in the peripheral energy ET [17], region. As expected, v2 vanishes at very peripheral 1 collisions, and becomes sizable when the anomalous P(ET ,b)= suppression makes sense. For more central collisions, 2 2πq aNp(b) because of the symmetric geometry of the collisions,

2 the azimuthal asymmetry v2 vanishes again. Since [ET − qNp(b)] × exp − , we assumed that only final state interactions produce 2q2aN (b) P finite value of v2, its increase with ET is quite has been used in the calculations, where we set the abrupt around the value when the critical density ρc parameters as q = 0.274 GeV and a = 1.27 [5]. In is reached. After taking into account the transverse the calculations, we also include the ET fluctuation expansion, v2 is a little higher than the case without effects as in [5,6,18], which is important for the last transverse expansion. This is quite different from v2 few ET bins. From this figure, we can see that our for jet quenching [16], where transverse expansion is approach can well describe the experimental data found to reduce v2 significantly. 66 X.-N. Wang, F. Yuan / Physics Letters B 540 (2002) 62–67

this part, the dissociation cross section is very large, in the order of a few mb. So χc would be dissociated almost totally when the local density is above the critical value ρc, which is more similar to the Blaizot- like model. In this case, the transverse expansion will increase v2 a little bit because it increases the time duration when χc is being dissociated. This increase of time duration even overcomes the decrease of geometrical asymmetry due to transverse expansion leading to an increased v2. The dominant suppression, however, comes from directly produced J/ψ (about 60%) at the RHIC energies because of the much higher initial density. The transverse expansion accelerates the decrease of the initial density and reduces the initial geometric asymmetry. This leads to the decrease of v2, very similar to the case of jet quenching in dense. Since the final total J/ψ suppression is a mixture of direct suppression and suppression through χc, this reduction is not as large as it is for jet quenching [16]. In conclusion, we have studied the azimuthal asymmetry of J/ψ suppression at both SPS and RHIC energies within a dynamic model of charmonia dissociation in a deconfined partonic system. With a crit- −3 ical density ρc = 3.3fm we can reproduce well the anomalous suppression found by the NA50 experiment at the SPS. We predicted the azimuthal anisotropy v2 of the J/ψ suppression at SPS. The ex- Fig. 2. The J/ψ survival probability and v2 at RHIC as a function istence of a critical density ρc for J/ψ suppression of number of participants NP . leads to a sharp increase of v2 with ET , assuming that v2 only comes from J/ψ dissociation in the decon- The results at RHIC are shown in Fig. 2. The fined matter. At the RHIC energies, we found that the survival probability and v2 of J/ψ are shown as anomalous suppression already plays a role in periph- functions of number of participants. Comparing with eral collisions because of high density. In noncentral the results in Fig. 1, we find that the anomalous collisions there is sizable v2 for J/ψ at pT = 3GeV. suppression enters already at peripheral collisions at The experimental study of v2 for J/ψ suppression RHIC, and the gap between the full suppression and will be complimentary to other studies of J/ψ sup- the nuclear absorption alone is much larger than that pression. Together with measurements, such as high at SPS. And v2 is also much larger. pT hadrons where jet quenching plays an important It is interesting to note the different dependences role [19], these studies will provide valuable informa- of v2 on the transverse expansion at the SPS and tion about the early stage of high-energy heavy-ion RHIC energies. At SPS, it enhances v2 a little bit collisions. while at RHIC it reduces. This is because at these two different energies the dominant suppression sources are different in the region where v2 is sizable. At Acknowledgements SPS, a large part of J/ψ suppression comes from χc suppression, because the density is not so high and it We thank J. Hüfner for interesting discussions and is difficult to dissociate directly produced J/ψ.For critical reading of the manuscript. This work was X.-N. Wang, F. Yuan / Physics Letters B 540 (2002) 62–67 67 supported by the Director, Office of Energy Research, [6] A. Capella, E.G. Ferreiro, A.B. Kaidalov, Phys. Rev. Lett. 85 Office of High Energy and Nuclear Physics, Divisions (2000) 2080, hep-ph/0002300. of Nuclear Physics, of the US Department of Energy [7] H. Heiselberg, R. Mattiello, Phys. Rev. C 60 (1999) 044902. [8] J.Y. Ollitrault, Phys. Rev. D 46 (1992) 229. under Contract No. DE-AC03-76SF00098 and in part [9] A.M. Poskanzer, S.A. Voloshin, Phys. Rev. C 58 (1998) 1671, by NSFC under project No. 19928511. nucl-ex/9805001. [10] X.M. Xu, D. Kharzeev, H. Satz, X.-N. Wang, Phys. Rev. C 53 (1996) 3051. [11] M.E. Peskin, Nucl. Phys. B 156 (1979) 365; References G. Bhanot, M.E. Peskin, Nucl. Phys. B 156 (1979) 391. [12] D. Kharzeev, H. Satz, Phys. Lett. B 334 (1994) 155. [13] X.-N. Wang, M. Gyulassy, Phys. Rev. D 44 (1991) 3501; [1] T. Matsui, H. Satz, Phys. Lett. B 178 (1986) 416. X.-N. Wang, M. Gyulassy, Phys. Rev. Lett. 86 (2001) 3496. [2] D. Kharzeev, H. Satz, Phys. Lett. B 366 (1996) 316, hep- [14] K. Adcox et al., PHENIX Collaboration, Phys. Rev. Lett. 86 ph/9508276. (2001) 3500. [3] C. Gerschel, J. Hufner, Annu. Rev. Nucl. Part. Sci. 49 (1999) [15] B.B. Back et al., PHOBOS Collaboration, nucl-ex/0108009. 255, hep-ph/9802245; [16] M. Gyulassy, I. Vitev, X.-N. Wang, P. Huovinen, Phys. Lett. R. Vogt, Phys. Rep. 310 (1999) 197. B 526 (2002) 301, nucl-th/0109063. [4] M.C. Abreu et al., NA50 Collaboration, Phys. Lett. B 410 [17] D. 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The Curci–Ferrari model with massive quarks at two loops

R.E. Browne, J.A. Gracey

Theoretical Physics Division, Department of Mathematical Sciences, University of Liverpool, Peach Street, Liverpool, L69 7ZF, United Kingdom Received 20 May 2002; accepted 13 June 2002 Editor: P.V. Landshoff

Abstract Massive quarks are included in the Curci–Ferrari model and the theory is renormalized at two loops in the MS scheme in an arbitrary covariant gauge.  2002 Elsevier Science B.V. All rights reserved.

Recently there has been renewed interest in examining covariantly gauge fixed QCD where the gluon and ghost fields are given explicit mass terms. For instance, in [1–3] such models have been used to investigate how a mass gap emerges for the gluon. In particular a dimension two operator obtains a non-zero vacuum expectation value which generates a gluon mass. Indeed in this context the effective potential calculation of [3] demonstrated that the non-perturbative vacuum favoured a non-zero vacuum expectation value for the simplest dimension two operator possible in Yang–Mills theories. In the main these studies did not involve quarks and were effectively based on a version of Yang–Mills theory with massive gluons originally introduced by Curci and Ferrari in [4]. There a renormalizable theory of massive gluons was constructed with the aim of being an alternative to the Higgs mechanism for endowing vector bosons with mass. The main shortcoming of the Curci–Ferrari model, however, was the breaking of unitarity directly as a result of the massive gluon [5–10]. Nevertheless the model has proved useful for a variety of reasons. From a field theoretic point of view it is of interest due to the non-linear nature of the gauge fixing term which introduces a quartic ghost self-interaction as well as modifying the usual ghost gluon interaction present in a linear covariant gauge fixing [4]. The non-linear property has been examined in [10,11]. Moreover, there has been a debate on the BRST symmetry of the Curci–Ferrari model. For instance, Ojima [6] has carried out a comprehensive examination of the BRST algebra with and without a mass term and explicitly constructed a negative norm state which therefore supports the lack of unitarity in the model. Other such states have been determined in [8]. Another feature which emerged in these papers was the non-nilpotency of the BRST charge which appears to follow as a consequence of the exact form of the Lagrangian. A more recent study of this has been given in [12]. A final motivation for considering massive gluons rests in phenomenological considerations. For example, lattice results [13] suggest that at low energies the gluon is massive though the precise form of the propagator at this scale is not known explicitly but is clearly dependent on non-perturbative properties. However, one can use a Curci–Ferrari type theory with the hope that it can provide useful insight into physics which is in

E-mail address: [email protected] (J.A. Gracey).

0370-2693/02/$ – see front matter  2002 Elsevier Science B.V. All rights reserved. PII:S0370-2693(02)02131-7 R.E. Browne, J.A. Gracey / Physics Letters B 540 (2002) 68–74 69 someway dependent on a gluon which is massive. For instance, diffractive scattering has been examined in [14] with such a motivation. Whilst the majority of the papers in this area concentrate on Yang–Mills theory with a massive gluon the real world is based on QCD which involves quarks. Therefore, if complete studies of the property of a massive gluon are to be performed, massive quarks need to be included. In [6] matter fields were considered but were shown not to affect the failings of the Curci–Ferrari model such as lack of unitarity or non-nilpotency of the BRST charge. The multiplicative renormalizability of the model, which was proved in [4,8–10], was unaffected. Given that the Curci–Ferrari model has been renormalized at one loop in [8,12,15] and more recently at two loops in MS in [16], the purpose of this Letter is to extend the renormalization of the Curci–Ferrari model at two loops to the case where massive quarks are included. This is important for various reasons. First, the main motivation of [16] was to provide a calculational tool for renormalizing Green’s functions of Yang–Mills theories where one could not naively nullify the momenta of several external legs without introducing spurious infrared infinities. Ordinarily one can handle such infrared problems if some form of infrared rearrangement [17,18] is performed to eliminate those divergences which cannot be distinguished from ultraviolet ones in a dimensionally regularized calculation. Moreover, given that infrared rearrangement is usually carried out by hand it does not lend itself readily to automatic calculations by computer. One simple resolution of the infrared problem is to introduce an infrared mass regularization. However, in using the Curci–Ferrari model [4] which naturally incorporates a gluon and ghost mass and preserves renormalizability one resolves the infrared problem [10,15,19] and opens the path to automatic calculations. Therefore, extending the Curci–Ferrari model to include quarks is the natural way to proceed in order to provide a tool similar to [14] for QCD. However, there is an additional practical point of view for our work. In conventional QCD perturbative calculations the fields of the Lagrangian are massless. However, in the real world the quarks have a physical mass. Therefore, including an explicit mass term for the quarks which is independent of the gluon mass we will have a Lagrangian which in principle includes as a special case the more realistic situation of massive quarks and massless gluons and ghosts as well as allowing us to interpolate between various different scenarios. For the two loop MS renormalization of the Lagrangian we do not expect significant differences with conventional results for the renormalization group functions. However, the tool we provide here will, for example, be useful in studying the mixing of operators under renormalization to operators of the same and lower dimension since we will have an explicit quark mass at our disposal. We take as our Lagrangian [4,6]

2 ←→ 1 a aµν 1 µ a 2 m a aµ a µ a 2 a a g abc a b µ c L =− G G − ∂ A + A A + ∂µc¯ ∂ c − αm c¯ c − f A c¯ ∂ c 4 µν 2α µ 2 µ 2 µ αg2 + f eabf ecd c¯acbc¯ccd + iψ¯ iI∂ψ/ iI − βmψ¯ iI ψiI − gψ¯ iI γ µT a ψiJ Aa , (1) 8 IJ µ where 1 a NA,1 I NF with NF and NA the dimensions of the colour group fundamental and adjoint abc representations respectively where the structure constants are f ,1 i Nf where Nf is the number of quark flavours,←→g is the coupling constant, m is the gluon mass and hence the basic mass scale of the classical theory ¯a b =¯a b − ¯a b a and c ∂µc c ∂µc (∂µc )c . The field strength, Gµν , follows from the definition of the covariant derivative a = a − a − abc b c as Gµν ∂µAν ∂ν Aµ gf AµAν . We have included the usual covariant gauge fixing term with parameter α which, to ensure that the action is BRST√ invariant [4,6,8,10] dictates the form of the ghost interactions with a gluon of mass m and a ghost of mass αm. To make the two loop calculations easier to perform we have chosen to parameterize the quark mass with the parameter√ β so that there is one basic mass parameter. This means that β will get renormalized but the full quark mass, βm, will be renormalized in such a way that it is independent√ of α. The case of massive quarks but massless gluons is recovered by setting m → 0insuchawaythat βmremains finite. With this Lagrangian the Minkowski space propagators for the gluon, ghost and quark fields are respectively, √ ηµν (1 − α)kµkν iδab iδij (/p − βm) −iδab − , , . (2) (k2 − m2) (k2 − m2)(k2 − αm2) (k2 − αm2) (p2 − βm2) 70 R.E. Browne, J.A. Gracey / Physics Letters B 540 (2002) 68–74

The renormalization of (1) proceeds along usual grounds. First, we introduce the renormalized variables via aµ = aµ a = a ¯a = ¯a = Ao ZA A ,co Zc c , co Zc c ,ψo Zψ ψ, = = = −1 = go Zgg, mo Zmm, αo Zα ZAα, βo Zβ β, (3) where the subscript ‘o’ denotes bare quantities. Since our calculation builds on the Yang–Mills version of the Curci–Ferrari model we have not assumed the usual Slavnov–Taylor identity of Zα = 1. Therefore, we have seven independent renormalization constants to compute. However, we do have various cross-checks on the results we will obtain for the renormalization constants. Given that the parameter m appears in both the gluon and ghost sectors we ought to obtain a consistent renormalization for it from considering independently the gluon and ghost two point functions. Moreover, we can check that our wave function renormalizations are correct by examining the various vertex corrections. The same coupling constant renormalization constant ought to emerge in all cases. Given the explicit values of the renormalization constants the corresponding renormalization group functions follow from the relations − ∂ ln Z ∂ ln Z ∂ ln Z ∂ ln Z 1 γ (a) = β(a) A + αγ (a) A ,γ(a) = β(a) α − γ (a) 1 − α α , A ∂a α ∂α α ∂a A ∂α ∂ ln Z ∂ ln Z ∂ ln Zβ ∂ ln Zβ γ (a) =−β(a) m − αγ (a) m ,γ(a) =−β(a) − αγ (a) , (4) m ∂a α ∂α β ∂a α ∂α where γβ (a) is deﬁned by µ ∂β γ (a) = (5) β β ∂µ and a = g2/(16π2). (The parameter β ought not to be confused with the β-function, β(a).) In ordinary QCD one would have γA(a) =−γα(a) but since Zα = 1 in the Curci–Ferrari model we do not expect this to be restored in the presence of quarks. Moreover, this is the origin of the second terms in the expressions for γm(a) and γβ (a) which will be dependent on α. Given that we are renormalizing a massive version of QCD we need to have an algorithm for computing the ultraviolet divergences of the massive multiscale two loop Feynman integrals which contribute. First, we use dimensional regularization with d = 4 − 2%. Second, we follow the strategy of [20,21] where massive Feynman integrals are expanded in powers of the external momenta based on the identity 1 1 (2kp − p2) = + . (6) ((k − p)2 − m2) (k2 − m2) ((k − p)2 − m2)(k2 − m2) The expansion is terminated by the rule that when the powers of momenta exceed those which can appear in the Green’s function through renormalizability, then they are dropped. So, for example, the gluon propagator is only expanded to O(p2) with the O(p3) terms being dropped where p is the external momentum. Consequently, one is left with massive vacuum bubble graphs where because of the different masses of (1), not all the masses are equal. However, massive vacuum two loop bubbles have been evaluated to the ﬁnite part in [22], for example, though we only require the Laurent expansion to the simple pole in %. For instance, the basic vacuum bubble with three different scales is given by 1 (k2 − m2)(l2 − αm2)[(k − l)2 − βm2] kl 1 3 1 4π 1 m2 = −(1 + α + β) + + ln + (α ln α + β ln β) + O(1) , (7) 2%2 2% % m2eγ % (4π)4

= d d where γ is the Euler–Mascheroni constant and k d k/(2π) . The expression for different powers of the propagators are determined by differentiating with respect to the parameter α, β and m2. It is worth noting that R.E. Browne, J.A. Gracey / Physics Letters B 540 (2002) 68–74 71 the appearance of ln α and ln β terms could in principle lead to a non-analytic renormalization constant. However, these ought to cancel when all contributions from one and two loop graphs are included. In addition, in reducing the integrals to vacuum bubbles, partial fractions have been used which can give rise to other potentially singular terms. For instance, 1 1 1 1 = − . (8) (k2 − αm2)(k2 − βm2) (α − β)m2 (k2 − αm2) (k2 − βm2) This provides another internal check since there are no singular terms in the original Lagrangian and for (1) to be renormalizable they ought not to remain after the two loop calculation. Given the large amount of algebra which arises due to the expansion to vacuum bubbles, to handle their evaluation for different masses we have written an algorithm in a symbolic manipulation language, FORM version 3 [23]. The calculation proceeds automatically, since, for example the Feynman diagrams are generated using the QGRAF package [24]. Moreover, the renormalization constants are extracted by computing the Green’s functions in terms of bare parameters then rescaling by (3) after the pole structure has been determined and following the procedure of [25]. This reproduces the usual method of subtractions automatically. We have renormalized the gluon, ghost and quark two-point functions as well as the three 3-point vertices. We obtain the following renormalization constants 13 α 4 a ZA = 1 + − CA − TF Nf 6 2 3 % 2 3α 17α 13 2 2 1 + − − C + CATF Nf α + 1 16 24 8 A 3 %2 2 α 11α 59 2 5 1 2 3 − + − C + 2CF TF Nf + CATF Nf a + O a , 16 16 16 A 2 % 2 2 α a 2 α 3α 1 α 5α 1 2 3 Zα = 1 − CA + C + − + a + O a , 4 % A 16 16 %2 32 32 % 2 3 α a α 35 2 1 1 Zc = 1 + − CA + − C + CATF Nf 4 4 % 16 32 A 2 %2 2 α α 95 2 5 1 2 3 − − − C + CATF Nf a + O a , 32 32 96 A 12 % 2 2 = − a + α + 3α + α 2 1 Zψ 1 αCF CF CA CF 2 % 8 4 2 % 25 3 2 1 2 3 − CF CA α + − CF TF Nf − C a + O a , 8 4 F % α 35 2 a Z = 1 + − C + T N m 8 24 A 3 F f % 2 α 53α 1435 2 2 2 2 α 19 1 + − − + C + T N + − CATF Nf 128 192 384 A 3 F f 12 6 %2 2 α 11α 449 2 35 1 2 3 + + − C + CF TF Nf + CATF Nf a + O a , 64 64 192 A 24 % 35 1 4 a Zβ = 1 + − CA − TF Nf − 6CF 12 α 3 % 72 R.E. Browne, J.A. Gracey / Physics Letters B 540 (2002) 68–74 2 α 13α 35 2 2 α 1 3α 13 1 + − − C + 18C + + CATF Nf + 4CF TF Nf + − CF CA 16 24 32 A F 3 2 2 2 %2 2 α 11α 449 2 3 2 4 35 97 1 2 3 − + − C + C − CF TF Nf + CATF Nf + CF CA a + O a , 32 32 96 A 2 F 3 12 6 % 2 11 a Zg = 1 + TF Nf − CA 3 6 % 121 2 2 2 2 11 1 5 17 2 1 2 3 + C + T N − CATF Nf + CF TF Nf + CATF Nf − C a + O a , 24 A 3 F f 3 %2 3 6 A % (9) a a acd bcd ab a b ab where T T = CF , f f = CAδ and Tr(T T ) = TF δ . We have recorded these explicitly since the double pole in % follows from the form of the one loop simple pole and therefore provides another check on our computation. Moreover, they are β-independent since we use a mass independent renormalization scheme. From these values we obtain the renormalization group functions at two loops in MS,

2 a 2 2 a 3 γA(a) = (3α − 13)CA + 8TF Nf + α + 11α − 59 C + 40CATF Nf + 32CF TF Nf + O a , 6 A 8 a γα(a) =− (3α − 26)CA + 16TF Nf 12 2 2 2 a 3 − α + 17α − 118 C + 80CATF Nf + 64CF TF Nf + O a , A 16 2 a 2 2 a 3 γc(a) = (α − 3)CA + 3α − 3α − 95 C + 40CATF Nf + O a , 4 A 48 2 a 3 γψ (a) = αCF a + CF (8α + 25)CA − 6CF − 8TF Nf + O a , 4 a γm(a) = (3α − 35)CA + 16TF Nf 24 2 2 2 a 3 + 3α + 33α − 449 C + 280CATF Nf + 192CF TF Nf + O a , A 96 a γβ(a) =− (3α − 35)CA + 72CF + 16TF Nf 12 2 − 2 + − 2 + + + 2 − a + 3 3α 33α 449 CA 1552CF CA 280CATF Nf 144CF 128CF TF Nf O a , 48 11 4 2 34 2 20 3 4 β(a)=− CA − TF Nf a − C − 4CF TF Nf − CATF Nf a + O a . (10) 3 3 3 A 3 The expression for the β-function agrees with the scheme independent results of [26,27]. The renormalization group functions for the wave functions of the fields and α agree with the corresponding results of [25–28] in the Landau gauge, α = 0. Indeed 2 a 2 a 3 γA(a) + γα(a) = α CA + (α + 5)C + O a , (11) 4 A 16 which is independent of Nf and vanishes when α = 0. Moreover, (11) is equivalent to the statement that the ghost gluon vertex does not get renormalized in the Landau gauge [29,30] which was originally verified at one loop for (1) in [29]. With the presence of quarks the gluon mass dimension now depends on Nf at one loop. However, to check we have correctly determined the quark mass anomalous dimension we need to compute the anomalous R.E. Browne, J.A. Gracey / Physics Letters B 540 (2002) 68–74 73 √ dimension of mq = βm.From 1 γm (a) = γm(a) + γβ(a) (12) q 2 we have 2 a 3 γm (a) =−3CF a − CF [97CA + 9CF − 20TF Nf ] + O a (13) q 6 which agrees with the two loop MS result of [31,32] in our conventions and is independent of α. For completeness, the ghost mass dimension is determined from 2 = 2 mc αm (14) which implies 1 γm (a) = γm(a) + γα(a) (15) c 2 giving 2 3 2 a 3 γm (a) =− CAa − (18α + 95)C − 40TF Nf + O a (16) c 8 A 96 which becomes Nf and gauge dependent at two loops. Further checks on the correctness of γm(a) are that the Yang–Mills sector of our expression agrees with the three loop result of [3] and the one loop expression of [33] 1 a aµ− ¯a a where the multiplicative renormalizability of the composite operator 2 AµA αc c was verified by determining the mixing matrix of the renormalization of the constituent dimension two operators. For completeness, we have evaluated γ(a)in the Landau gauge for QCD, Nc = 3andTF = 1/2, and found 2 a a 3 γm(a)|α= = (8Nf − 105) + (548Nf − 4041) + O a . (17) 0 24 96 It is worth noting that both coefficients are negative for Nf < 8 which implies that the gluon mass runs to zero in the ultraviolet limit in this case. In conclusion we have provided the full two loop renormalization of the Curci–Ferrari model with massive quarks in the MS scheme. Whilst the nature of the model may appear unphysical with a gluon mass and lack of unitarity, it is important to recall that one long term aim is to use the model to attack the renormalization of other Green’s functions where quark mass effects will become important and a natural infrared mass regularization is necessary. Further, it is worth noting that models with non-Abelian symmetries and massive gluons may have important phenomenological consequences. For instance, it has been shown in [34] that including O(1/Q2) power corrections in the operator product expansion improves the matching of calculated physical quantities with experiment. Although such corrections derive from a gluon mass, albeit tachyonic in origin, they indicate the potential importance of massive gluons, such as that of (1), to probe non-perturbative physics.

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The polarized and unpolarized photon content of the nucleon

M. Glück, C. Pisano, E. Reya

Universität Dortmund, Institut für Physik, D-44221 Dortmund, Germany Received 8 May 2002; received in revised form 10 June 2002; accepted 11 June 2002 Editor: P.V. Landshoff

Abstract The equivalent photon content of polarized and unpolarized nucleons (protons, neutrons), utilized in Weizsäcker–Williams approximations, are presented. For this purpose a new expression for the elastic photon component of a polarized nucleon is derived. The inelastic photon components are obtained from the corresponding momentum evolution equations subject to the boundary conditions of their vanishing at some low momentum scale. The resulting photon asymmetries, important for estimating cross section asymmetries in photon induced subprocesses are also presented for some typical relevant momentum scales.  2002 Elsevier Science B.V. All rights reserved.

The concept of the photon content of (charged) where t ≡ q2 =−Q2 and fermions is based on the equivalent photon (Weizs- G2 (t) + τG2 (t) äcker–Williams) approximation [1]. Applied to the H (t) ≡ F 2(t) + τF2(t) = E M 1 1 2 + (3) nucleon N = p,n it consists of two parts, an elastic 1 τ one due to N → γN and an inelastic part due to with τ ≡−t/4m2, m being the nucleon mass, and N → γX with X = N. Accordingly the total photon where GE = F1 − τF2 and GM = F1 + F2 are distribution of the nucleon is given by the common elastic (Sachs) form factors which are conveniently parametrized by the well-known dipole 2 2 − 2 −2 γ y,Q = γel(y) + γinel y,Q , (1) form proportional to (1 t/0.71 GeV ) as extracted p = from experiment. For the proton, where F1 (0) 1 p and F p(0) = κ 1.79, we have where the elastic contribution of the proton, γel ,has 2 p been presented in [2] which can be generally written as p = + −2 p p GE(t) (1 aτ) ,GM (t) µpGE(t), + 2 tmax 1 µ τ − α dt 1 2m2y H p(t) = p (1 + aτ) 4 (4) γ (y) =− 2 − 1 + H (t) 1 1 + τ el 2π t y t 1 = + ≡ 2 2 tmin with µp 1 κp 2.79 and a 4m /0.71 GeV 4.96. For the neutron, where F n(0) = 0andF n(0) = + yG2 (t) , (2) 1 2 M κn −1.91, we have n = + −2 GE(t) κnτ(1 aτ) , n = + −2 E-mail address: [email protected] (E. Reya). GM (t) κn(1 aτ) ,

n = 2 + −4 1 H1 (t) κnτ(1 aτ) . (5) α dx y = e2 P 2 q γq In the relevant kinematic region s m the integra- 2π = x x q u,d,s y tion bounds in (2) can be approximated by tmin =−∞ 2 2 × N 2 +¯N 2 and tmax =−m y /(1−y) so as to obtain an universal q x,Q q x,Q (10) process independent γel(x). Eq. (2) can now be analyt- (−) (−) ically integrated which yields for the proton with P (x) =[1 + (1 − x)2]/x and where qp≡ q γq 2 (−) (−) (−) (−) − α 2 y n n n ( ) γ p(y) = 1 − y + 1 + 4a + µ2 I and u = d , d = u, s = s . This equation was inel 2π y 4 p tegrated subject to the ‘minimal’ boundary condition 2 γ N (y, Q2) = 0at[4]Q2 = 0.26 GeV2, which is ob- + 2 − − + y ˜ inel 0 0 µp 1 1 y I viously not compelling and affords further theoreti- 4 1 − y cal and experimental studies. Since for the time be- − (6) ing there are no experimental measurements avail- z3 able, the ‘minimal’ boundary condition provides at and for the neutron present a rough estimate for the inelastic component 2 2 n α 2 y 1 1 at Q Q0. γ (y) = κ I + , (7) N 2 el 2π n 2 3 (z − 1)z3 Clearly, the nucleon’s photon content γ (x, Q ) 2 is not such a fundamental quantity as are its under- ≡ + a y 2 where z 1 4 1−y and lying parton distributions f(x,Q ) = q, q,g¯ or the γ 2 ∞ parton distributions f (x, Q ) of the photon, since 1 γ p(x, Q2) is being derived from these more funda- I = dτ τ(1 + aτ)4 mental quantities. It represents mainly a technical de- y2 vice which allows for a simpler and more efﬁcient cal- 4(1−y) culation of photon-induced subprocesses. For exam- 1 1 1 1 =−ln 1 − − − − , (8) ple, the analysis of the deep inelastic Compton scatter- z z 2z2 3z3 ing process ep → eγ X reduces [3,5] to the calculation ∞ of the 2 → 2 subprocess eγ → eγ instead of having to 1 I˜ = dτ calculate the full 2 → 3 subprocess eq → eγ q. Simi- + + 4 (1 τ)(1 aτ) lar remarks hold for the production of charged heavy y2 4(1−y) particles (e.g., higgses) via γγ fusion at high energy + − 1 a− 1 1 1 pp colliders, pp → γγX→ H H X. The reliabil- =− ln 1 + + − + 4 3 2 2 3 ity of this approximation remains, however, to be stud- a− z a−z 2a−z 3a−z (9) ied. Our main purpose here is to extend these calcula- = − with a− a 1. For arriving at (6) we have also tions to the polarized sector, i.e., to utilized the relation ∞ %γ y,Q2 = %γ (y) + %γ y,Q2 . (11) 1 1 − y el inel dτ =−4aI + 4 τ 2(1 + aτ)4 y2z3 The inelastic contribution derives from a straightfor- y2 ward extension of Eq. (10), 4(1−y) which will be also relevant for the polarized photon N 2 d%γinel(y, Q ) contents to be presented below. Our result in (6) agrees d ln Q2 with the one presented in a somewhat different form 1 in [2]. Finally, the inelastic part in (1) has been given = α 2 dx y in [3] eq %Pγq 2π = x x q u,d,s y dγN (y, Q2) inel × N 2 + ¯ N 2 d ln Q2 %q x,Q %q x,Q , (12) M. Glück et al. / Physics Letters B 540 (2002) 75–80 77

2 where %Pγq(x) =[1 − (1 − x) ]/x = 2 − x.Wein- with q = p − p . It is now straightforward to contract µν tegrate this evolution equation assuming again the not TA with the appropriate antisymmetric part of the µν ; necessarily compelling ‘minimal’ boundary condition tensor WA describing the polarized target a(k s) N 2 = | 2 | %γinel(y, Q0) 0, according to %γinel(y, Q0) in (14) which is expressed in terms of the usual 2 = 2 = 2 polarized structure functions g and g where all terms γinel(y, Q0) 0, at Q0 0.26 GeV using the recent 1 2 LO polarized parton densities of [6]. proportional to g2 drop in TA · WA. This yields The elastic distribution %γ (y) in (11) is deter- el %γ (y) mined via the antisymmetric part of the tensor describ- el tmax ing the photon emitting fermion (nucleon) α dt 2m2y2 =− − + 2 2 y GM (t) µν 1 µ ν 2π t t T = Tr (1 + γ n)(/ p / + m)Γ (/p + m)Γ (13) t 2 5 min m2y2 for the generic process − 2 1 − y + G (t)F (t) t M 2 N(p; n) + a(k; s) → N(p ) + X, (14) tmax α dt 2m2y2 where a being a suitable target (parton, photon, etc.) =− GM (t) 2 − y + F (t) 2π t t 1 with momentum k and n, s are the appropriate tmin polarization vectors [7] satisfying n · p = 0and s · k = 0. In terms of the Dirac and Pauli form factors + yF2(t) (17) µ F1,2(t) of the nucleon the elastic vertices Γ are given by with y = k · q/k · p and the first term proportional 2 to GM in the first line corresponds to the pointlike 1 Γ µ = (F + F )γ µ − F (p + p )µ. (15) result of [7]. Following [2], we again approximate 1 2 2m 2 the integration bounds by tmin =−∞ and tmax = The analysis is now a straightforward extension of the −m2y2/(1 − y) as in (2) in order to obtain an univer- calculation [7] of the polarized Weizsäcker–Williams sal process independent polarized elastic distribution. distribution resulting from a photon emitting fermion Using, in addition to (4) and (5), (electron) where N → e in (14) with Γ µ = γ µ,andall + µ τ relevant definitions and kinematics can be found in [7] p = 1 p + −2 F1 (t) (1 aτ) , as well. The resulting antisymmetric part1 of T µν is 1 + τ p κp −2 µν 2 µνρσ F (t) = (1 + aτ) , (18) T = 2imG ε nρ qσ 2 1 + τ A M + 2iG (F /2m) (p + p )µενρσ σ M 2 n τ −2 F (t) = 2κn (1 + aτ) , − (p + p)ν εµρσ σ 1 1 + τ − × n 1 τ −2 nρ pσ p (16) F (t) = κn (1 + aτ) , (19) σ 2 1 + τ Eq. (17) yields for the proton 1 It should be noted that the symmetric (unpolarized) tensor p α y 2 µν = + µ + ν %γ (y) = µp (2 − y) 1 + κp + 2ay I TS Tr (p/ m)Γ (p/ m)Γ el 2π 2 2 = 2 µ ν + µ ν + q µν y2 − y 4GM p p p p g ˜ 1 2 + 2κp 1 − y + I − 2 3 2 4 z µ ν 1 q 2 − 4(p + p ) (p + p ) GM F2 − 1 − F (20) 2 4m2 2 and for the neutron gives rise to the same Weizsäcker–Williams distribution obtained in a somewhat less transparent way in [2], i.e., to Eq. (2), when α y2 the analysis [8] for a photon emitting pointlike unpolarized fermion n = 2 − + − + ˜ %γel(y) κn y(1 y)I 4 1 y I (electron) is straightforwardly extended to an unpolarized nucleon, 2π 4 N(p)+ a(k) → N(p ) + X, instead to the polarized process (14). (21) 78 M. Glück et al. / Physics Letters B 540 (2002) 75–80

Fig. 2. As in Fig. 1 but for a linear y scale.

Fig. 1. The polarized and unpolarized total photon contents of the proton, %γ p and γ p, according to Eqs. (1) and (11) at some typical fixed values of Q2 (in GeV2). The Q2-independent elastic contributions are given by Eqs. (20) and (6). with I and I˜ being given in (8) and (9). These latter two equations together with (12) for N = p,n yield now the total photon content %γ N (y, Q2) of a polarized nucleon in (11). Our results for %γ p(y, Q2) in (11) are shown in Fig. 1 for some typical values of Q2 up to Q2 = Fig. 3. The asymmetry of the polarized to the unpolarized photon 2 = 2 MW 6467 GeV . For comparison the expectations content of the proton as defined in (22) at various fixed values of Q2 for the unpolarized γ p(y, Q2) in (1) are depicted (in GeV2) according to the results in Fig. 1. The Q2-dependence p 2 as well. The Q2-independent polarized and unpolar- of the elastic contribution to Aγ is caused by the Q -dependent total unpolarized photon content in the denominator of (22). For ized elastic contributions in Eqs. (20) and (6), respec- 2 p p illustration the Q -independent elastic ratio %γel /γel is shown as tively, are also shown separately. Due to the singu- well. lar small-x behavior of the unpolarized parton dis- (−) tributions x q (x, Q2) in (10) as well as of the sin- Fig. 2 where the results of Fig. 1 are plotted versus a p → p 2 p 2 gular yγel (y) in (6) as y 0, the total yγ (y, Q ) linear y scale. The asymmetry Aγ (y, Q ) is shown in in Fig. 1 increases as y → 0, whereas the polarized Fig. 3 where p 2 → → y%γ (y, Q ) 0asy 0 because of the vanish- 2 2 2 (−) Aγ y,Q ≡ %γel(y) + %γinel y,Q γ y,Q ing of the polarized parton distributions x% q (x, Q2) p (22) in (12) at small x and of the vanishing y%γel (y) in (20) at small y. In fact, y%γp(y, Q2) is negligibly with the total unpolarized photon content of the small for y 10−3 as compared to yγp(y, Q2).For nucleon being given by (1). To illustrate the size of −2 p 2 p p larger values of y, y>10 , y%γ (y, Q ) becomes %γel relative to the unpolarized γel , we also show the sizeable and in particular is dominated by the Q2- Q2-independent ratio %γ p(y)/γ p(y) in Fig. 3 which p el el independent elastic contribution y%γel (y) at moder- approaches 1 as y → 1. ate values of Q2, Q2 100 GeV2 (with a similar be- The polarized photon distributions %γ p(y, Q2) havior in the unpolarized sector). This is evident from shown thus far always refer to the so-called ‘valence’ M. Glück et al. / Physics Letters B 540 (2002) 75–80 79

Fig. 5. As in Fig. 4 but for a linear y scale.

Fig. 4. As Fig. 1 but for the neutron, with elastic polarized and unpolarized contributions being given by Eqs. (21) and (7). scenario [6] where the polarized parton distributions in (12) have flavor-broken light sea components %u¯ = %d¯ = %s¯, as is the case (as well as experimentally required) for the unpolarized ones in (10) where u¯ = d¯ = s¯. Using instead the somehow unrealistic ‘standard’ scenario [6] for the polarized parton distributions with a flavor-unbroken sea component %u¯ = %d¯ = %s¯,all results shown in Figs. 1–3 remain practically almost indistinguishable. The same holds true for the photon Fig. 6. As Fig. 3 but for the neutron asymmetry according to the results in Fig. 4. content of a polarized neutron to which we now turn. The results for %γ n(y, Q2) are shown in Fig. 4 n which are sizeably smaller than the ones for the photon the vanishing of y%γel(y) in (21) at small y. Finally, n 2 in Fig. 1 and, furthermore, the elastic contribution is the asymmetry Aγ (y, Q ) defined in (22) is shown dominant while the inelastic ones become marginal at in Fig. 6 which is entirely dominated by the elastic y 0.2. For comparison the unpolarized γ n(y, Q2) contribution for x 0.2. As in Fig. 3 we illustrate the in (1) is shown in Fig. 4 as well. Here, γ n in (7) n el size of the elastic %γel(y) relative to the unpolarized is marginal and yγn(y) is non-singular as y → 0 n n n el γel(y) by showing the ratio %γel/γel in Fig. 6 as well. with a limiting value yγn(y)/α = κ2/(3πa) 0.078. n n → 6 → el n Notice that %γel/γel 7 as y 1incontrasttothe Thus the increase of yγn(y, Q2) at small y is entirely case of the proton. n 2 caused by inelastic component yγinel(y, Q ) in (10), As mentioned at the beginning the knowledge (−) of the unpolarized photon content of the nucleon due to the singular small-x behavior of x q (x, Q2), γ N (y, Q2) allows for a simpler and more efficient whichisincontrasttoyγp(y, Q2) in Fig. 1. These calculation of photon-induced subprocesses in elastic facts are more clearly displayed in Fig. 5 where the and deep inelastic ep and purely hadronic (pp,...) results of Fig. 4 are presented for a linear y scale. n 2 → reactions [2,3,5,9–12]. For example, to consider just Notice that again the polarized y%γ (y, Q ) 0as → → → the simple 2 2 subprocess eγ eγ for the analy- y 0 because of the vanishing of the polarized parton → (−) sis of the deep inelastic Compton process ep distributions x% q (x, Q2) in (12) at small x and of eγ X or eγ → νW for associated νW production in 80 M. Glück et al. / Physics Letters B 540 (2002) 75–80 ep → νWX. Similarly, the γγ fusion process γγ → References 2+2−,cc,H¯ +H −, 2˜+2˜−,... for (heavy) lepton (2), heavy quark (c), charged Higgs (H ±)andslepton [1] C.F. Weizsäcker, Z. Phys. 88 (1934) 612; (2˜) production etc. can be easily analyzed in purely E.J. Williams, Phys. Rev. 45 (1934) 729(L). [2] B.A. Kniehl, Phys. Lett. B 254 (1991) 267. hadronic pp reactions which is also an interesting [3] M. Glück, M. Stratmann, W. Vogelsang, Phys. Lett. B 343 possibility of producing charged particles which do (1995) 399. not have strong interactions. In particular, the γγ → [4] M. Glück, E. Reya, A. Vogt, Eur. Phys. J. C 5 (1998) 461. µ+µ− channel will give access to experimental mea- [5] A. De Rujula, W. Vogelsang, Phys. Lett. B 451 (1999) 437. N 2 2 [6] M. Glück, E. Reya, M. Stratmann, W. Vogelsang, Phys. Rev. surements of γ (y, Q = M + − ) at pp, pd and dd µ µ D 63 (2001) 094005. colliders. [7] D. de Florian, S. Frixione, Phys. Lett. B 457 (1999) 236. Analogous remarks hold for the longitudinally [8] S. Frixione, M.L. Mangano, P. Nason, G. Ridolfi, Phys. Lett. polarized eN and pp, pd and dd reactions where the B 319 (1993) 339. polarized photon content of the nucleon %γ N (y, Q2), [9] M. Drees, D. Zeppenfeld, Phys. Rev. D 39 (1989) 2536. as calculated and studied in this Letter, enters. Very [10] J. Blümlein, G. Levman, H. Spiesberger, J. Phys. G 19 (1993) 1695. interestingly, it remains to be seen whether ongoing [11] M. Drees, R.M. Godbole, M. Nowakowski, S.D. Rindani, experiments at RHIC(BNL) for dimuon production, Phys. Rev. D 50 (1994) 2335; + − pp, dd → µ µ X, can directly delineate and test our J. Ohnemus, T.F. Walsh, P.M. Zerwas, Phys. Lett. B 328 (1994) N 2 369. predictions for %γ (y, M + − ). µ µ [12] C.E. Carlson, K.E. Lassila, Phys. Lett. B 97 (1980) 291. AFORTRAN package (grids) containing our results for %γ N (y, Q2) as well as those for γ N (y, Q2) can be obtained by electronic mail.

Acknowledgements

This work has been supported in part by the ‘Bun- desministerium für Bildung und Forschung’, Berlin/ Bonn. Physics Letters B 540 (2002) 81–88 www.elsevier.com/locate/npe

Relativistic ﬁeld-theoretical representation of the Schrödinger equation and ﬁeld-theoretical generalization of the inverse scattering method

A.I. Machavariani a,b

a Joint Institute for Nuclear Research, LIT Dubna, Moscow region 141980, Russia b High Energy Physics Institute of Tbilisi State University, University str. 9, Tbilisi 380086, Georgia Received 8 May 2002; received in revised form 11 June 2002; accepted 13 June 2002 Editor: P.V. Landshoff

Abstract − | =− 3 | = 2 + 2 The relativistic three-dimensional equation (2Ep 2Ep) p Ψp V(t)d p p Ψp with Ep m p and 2 2 2 2 t = (Ep −Ep ) −(p −p ) is transformed into Schrödinger equation ( r +k )r|φk=−v(r)r|φk with Ep = k /2m+m using the Fourier transformation. The potential v(r) and the wave function r|φk differ from the corresponding Fourier images of the relativistic potential V(t)and the relativistic wave function p |Ψp with the kinematical factors only. In the framework of the standard ﬁeld-theoretical approach the three-dimensional relativistic equations for the NN scattering amplitude with the total NN potential Vtot = V(t)+ Vnonl are derived. The nonlocal part of this NN potential Vnonl is constructed from the πNN vertex functions which are determined from the πN phase shifts. The other part of the NN potential V(t)consists of the π-, σ -, ρ-, ω-, ...meson exchange diagrams and for the NN contact (overlapping) terms. In the case of the renormalizable Lagrangians V(t)reproduces exactly the One Boson Exchange (OBE) Bonn model of the NN potential. The inverse scattering problem for this V(t)potential is reduced to the construction of the potential v(r) of the ordinary Schrödinger equation from the NN phase shifts.  2002 Published by Elsevier Science B.V.

PACS: 11.10.-z; 11.80.-m; 11.80.La; 13.75.Cs; 25.80.Hp

Keywords: Inverse; Scattering; Relativistic; Field-theoretical; Nucleon

The inversion method for the reconstruction of the potential in the Schrödinger equation is needed for the numerous problems in nuclear and particle physics [1–4]. However, the relativistic generalizations of the inverse scattering methods in the quantum scattering theory was carried out for the one particle relativistic equations such as the Dirac or Klein–Gordon equation only [1]. The aim of this Letter is to extend the inversion method for the determination of the potential in the ﬁeld-theoretical equations. In particular, we consider the scheme of the relativistic generalization of the Schrödinger equations for the NN scattering amplitudes [5].

E-mail address: [email protected] (A.I. Machavariani).

0370-2693/02/$ – see front matter  2002 Published by Elsevier Science B.V. PII:S0370-2693(02)02137-8 82 A.I. Machavariani / Physics Letters B 540 (2002) 81–88

The relativistic equation in the time-ordered three-dimensional field-theoretical formulation for the NN amplitude p|t(E(p))|p [6,7] has the form of the Lippmann–Schwinger equation p |U(E(p))|p p |t(E(p))|p=p |U(E(p))|p+ dp p |t(E(p))|p, (1) E(p) + i0 − E(p) = ≡ 2 + 2 where p and p denotes the nucleon relative momentum in the c.m. frame and E(p) 2Ep 2 p mN . The relativistic equation (1) for the NN amplitude was derived in the framework of the standard S-matrix reduction technique by using the completeness condition for the asymptotic “in” states [6,7]. Moreover, in Ref. [7] there were considered the explicit connections of Eq. (1) with the quasipotential reduction of the Bethe–Salpeter equation and with the other three-dimensional time-ordered field-theoretical equations. The brief analysis of the Lippmann–Schwinger type equation (1) is given in Appendix A. The potential in Eq. (1) consist of the on-mass-shell meson exchange potential (A.6c) (Fig. 1) and of the NN potential Y(t) ((A.6a), (A.6b)) which are generated by the equal-time commutators and which dependent on the square of the four-momentum transfer t.TheNN potential Y(t) contains the off-mass-shell meson exchange part (Fig. 2A) and the NN overlapping (contact) potential (Fig. 2B). It must be emphasized, that only the considered in Appendix A version of the field-theoretical Lippmann–Schwinger type equation contains the interesting one- variable potential Y(t). All other three-dimensional Lippmann–Schwinger type equations, derived on the base of the Bethe–Salpeter equation or on the base on the Tomonaga–Schwinger equation [7], contains multivariable potentials only. These multivariable potentials are not the object of construction in the framework of the inverse scattering method. The considered field-theoretical equation (1) is connected analytically with all the other field-

Fig. 1. Diagrammatic representation of the time-ordered NN interaction potential with on-mass-shell intermediate pions. The full circles denote | | ; | ¯ | ; the vertex functions pN JpN (0) pπ1, pπ2 ... in and 0 JpN (0) pN pπ1pπ2 ... in with the one off-mass-shell nucleon.

Fig. 2. The NN interaction with the off-mass-shell π,σ,ρ,ω,... meson exchange-diagram (A) and with the four nucleon overlapping or | | contact term diagram (B). The shaded circle corresponds to the vertex function pN jπ (0) pN with off-mass-shell pi-meson and the other πNN vertex functions in the diagram A (Eqs. (A.6a), (A.6b)) are given in the tree approximation. A.I. Machavariani / Physics Letters B 540 (2002) 81–88 83 theoretical equations [7], because these equations (Bethe–Salpeter equation, Tomonaga–Schwinger equation and the generalized unitarity condition (A.3) or so-called Low type equation) are derived in the framework of the same standard S-matrix reduction technique. Therefore, all results obtained in the framework of the time-ordered three- dimensional equation (1) remain valid in the other field-theoretical approaches as well if we take into account the infinite number of the intermediate states. This formulation is free from the “three-dimensional ambiguities” which emerge during the reduction of the Bethe–Salpeter equation in the three-dimensional form. In addition, the structure of the present field-theoretical equations does not depend on the choice of the form of the effective Lagrangian. | | ; | ¯ | ; → The vertex functions pN JpN (0) pπ1p in and 0 JpN (0) pN pπ1 in of the transition N Nπ1 can be determined from the πN scattering phase shifts by using the dispersion relations [9]. The vertex functions | | ; | ¯ | ; → pN JpN (0) pπ1pπ2 in and 0 JpN (0) pN pπ1pπ2 in of the transition N Nπ1π2 can be obtained by using the results of the calculation of the πN scattering amplitude with one crossed pion, etc. Therefore, we can assume that the nonlocal, on-mass-shell meson exchange potential in Eq. (A.6c)—see Fig. 1—is defined “a priori”. Afterwards one can subtract the contributions of this “a priori” fixed potential from the complete NN phase shifts. Thus the inverse scattering problem for the relativistic Lippmann–Schwinger type equation (1) is reduced to the investigation of the relativistic equation 2 2 2 2 dp p |T |p=Y (Ep − Ep) − (p − p) + Y (Ep − Ep ) − (p − p ) p |T |p, 2Ep + i0 − 2Ep (2) = 2 + 2 Y where Ep mN p and the NN potential ((A.6b), (A.6c)) is depicted on Fig. 2. It is convenient to present Eq. (2) for the wave function p |Ψp − | = Y − 2 − − 2 | (2Ep 2Ep) p Ψp dp (Ep Ep ) (p p ) p Ψp , (3)

where p |T |p=p |Y|Ψp. In order to transform Eq. (3) to the nonrelativistic form we introduce the variables k p k2 = m (2E − 2m ), = , (4) N p N k p and k 2 k 2 = + = + u k 2 1, v k 2 1, (5) 4mN 4mN which satisfy the important condition1

2 2 2 t =−(u − v) = (Ep − Ep ) − (p − p ) . (6) Then Eq. (3) takes the form 2 2 2 k − k k |Ψk= dk Y −(u − v) k |Ψk, (7) where 2 1/2 1/2 2 1/2 p dp J (k )k |Ψ =p |Ψp,J(k )Y −(u − v) J (k ) = Y(t), J (k ) = . k k 2 dk

E +m E +m 1 p N p N For the variables u and v there is a more transparent representation u = p + and v = p + . Ep mN Ep mN 84 A.I. Machavariani / Physics Letters B 540 (2002) 81–88

Now we present Eq. (7) in the coordinate space using the Fourier transform − dk dk + k2 r |Ψ =− e ik r Y −(u − v)2 eik r dr r |Ψ . (8) r k (2π)3 (2π)3 k Inserting − Y −(u − v)2 = dz ei(u v)zY(z) (9) into Eq. (8) we obtain 2 r r r + k r |Ψk=− exp z∇r + 1 − 1 exp −z∇r + 1 − 1 4m2 4m2 N N − − dk dk × e ik r eik r eik ze ik z dr Y(z)dz r |Ψ , (10) (2π)3 (2π)3 k where the operators in the big curly brackets act on the function which are included in this brackets, and =∇2 ≡ 2 r r ∂r . After integration over k and k we ﬁnd 2 r r r + k r |Ψk=− exp z∇r + 1 − 1 exp −z∇r + 1 − 1 4m2 4m2 N N × dr δ(z − r )δ(r − z) Y(z)dz r |Ψk. (11)

∇ 2 + − − ∇ 2 + − The operators exp z r r /4mN 1 1 and exp z r r /4mN 1 1 compensate each other on the surface r = r = z. Therefore, we obtain the standard Schrödinger equation 2 r + k r |Ψk=−Y(r )r |Ψk. (12) Our main result is the explicit reduction of the relativistic Lippmann–Schwinger equation (3) with the potential Y 2 + 2 − 2 = 2 (t) into the standard Schrödinger equation (12) with the auxiliary variable 2mN p mN 2mN k (4). This result allows to determine the relativistic potential Y(t) (or Y(t)) from the nonrelativistic potential Y(r) using the standard Schrödinger equations (12) and the well-known inverse scattering theory [1–3]. Summarizing the above formulation of the NN scattering problem, we see that in order to apply the inverse scattering methods to the present ﬁeld-theoretical equation (1) one must take into account the structure of the NN potential. The complete NN potential consists of the nonlocal (on-mass-shell pions exchange) part (Fig. 1) and of the off-mass-shell meson exchange part Y(t) (Fig. 2). The contribution of the potential shown in Fig. 1 is constructed from the πN vertex functions with one of the nucleons being off mass shell. These vertex functions can be obtained from the πN phase shifts by using the dispersion relations. Therefore, in the present formulation the inverse scattering problem is reduced to the determination of the Y(t) potential in Eq. (2). This is because the other part of the NN potential in Eq. (1) can be constructed from the πN scattering amplitudes. One can build the potential Y from the corresponding NN phase shifts, by using the inverse scattering methods for the nonrelativistic potential v(r) of the Schrödinger equation (12).2

2 Unlike the nonrelativistic case, in the considered relativistic formulation r does not have the physical meaning of the coordinate. This variable should be treated as the auxiliary variable which is conjugate to the momentum p. In Ref. [12] x = i 1 + p2/m2 ∂/∂p was A.I. Machavariani / Physics Letters B 540 (2002) 81–88 85

The potential Y(t) is generated by the equal-time anticommutation relation (A.6a) and it is generally the function of the t-variable. For the simplest Lagrangian (A.5) Y(t) consist of the terms, describing the exchange by the off-mass-shell mesons: π,σ,ρ,ω,... (A.6a) (Fig. 2A) and of the contact terms (A.6b) (Fig. 2B). Therefore, the determination of the potential Y in the framework of the inverse scattering methods can help us to clarify the form of the general meson–nucleon Lagrangians. The formulation considered above can be extended easily to the πN–πN and πN–γN scattering reactions.

Acknowledgements

The author is indebted to H.V. Geramb and A.A. Suzko for discussions and for the interest to this work.

Appendix A

The well-known expression for the NN scattering S-matrix with the local meson and the local nucleon ﬁeld operators Φ(x) and Ψ(x)has the form [14,15] ≡ ; | ; SN N ⇐NN out p1s1p2s2 p1s1p2sN2 in = ; | ; +P P 4 (4) + − − A in p1s1p2s2 p1s1p2s2 in 1 2 12(2π) iδ (p1 p2 p1 p2) N N ⇐NN, (A.1) A ⇐ = P out; p s |J (0)|p1s1p2s1; in N N NN 1 2 1 1 p2s2 = equal-time anticommutators 4 (−ip1x) ¯ ¯ + P12 P12 d xe p s | J (0)θ(−xo)Jp s (x) − Jp s (x)θ(xo)J (0) |pN2s2, 1 1 p2s2 1 1 1 1 p2s2 (A.2) where p s and pi si denote the three momentum and the spin of the nucleon, i = 1, 2 correspondingly in the i i ﬁnal and in the initial states, P12 = (1/2)(1 − P12) stands for the antisymmetrization operator with the nucleon µ transposition operator P12, J (x) =¯u(p s )J (x) ≡¯u(p s )(iγ ∂µ − mN )Ψ (x) is the nucleon source operator p2s2 2 2 2 2 which is determined by the Dirac equation of motion, u(pN s) stands for the Dirac bispinor function and θ(xo) = 1 if xo > 0andθ(xo) = 0ifxo < 0 is the well-known step function. | ; ; |= Substituting the completeness condition n n in in n 1 into Eq. (A.2) between the source operators, we obtain after integration over x

A ⇐ = equal-time anticommutators N N NN δ(p1 + p2 − Pn) + 3P P | | ; (2π) 1 2 12 p s Jp s (0) n in 1 1 2 2 E + E − P o + i0 n=d,NN,πNN,... p1 p2 n ×in; n|J¯ (0)|p s p1s1 2 2 δ(p − p1 − Pm) − | ¯ | ; 1 ; | | p1s1 Jp1s1 (0) m in o in m Jp s (0) p2s2 , (A.3) E − Ep − P 2 2 m=π,ππ,... p1 1 m treated as the generators of translation in the Euclidean p-space with the x2 − L2/m2 Casimir operator of the Lorentz group. In this field- theoretical approach the relativistic analogue of the Fourier transformations (the Shapiro transformation [13] with the complete set of the functions ξ(p, x)) was used. The final equation have the form Ho(2Ep − Ho)r|φp=−(1/2m)V (r, Ep)r|φp with the free Hamiltonian 2 Ho = 2m ch((i/m)(∂/∂r)) + (2i/r)sh((i/m)(∂/∂r)) + θ,φ/(mr ) exp((i/m)(∂/∂r)). 86 A.I. Machavariani / Physics Letters B 540 (2002) 81–88 ≡ = 2 + 2 where pN (EpN , pN ) pN mN , pN denotes the four-momentum of the on-mass-shell nucleon. Comparing Eq. (A.2) with Eq. (A.3), we see, that the time-ordering procedure in Eq. (A.2) is replaced by the linear propagator which consists from the energies of the outside and inside particles. Besides in Eq. (A.2) only the sum of the three-momentums of the all intermediate particles is conserved. However, the main property of the present field-theoretical formulation is that it is the only one, where the one variable vertex functions like | ¯ | | pN Jβ (0) pπ or pN jπ (0)pN (in the equal-time anticommutators) are required as input functions for the construction of the NN potential. The equal-time anticommutators in Eqs. (A.2) and (A.3) have the form † equal-time anticommutators ≡ Y(t) = P P12p s | J (0), b (0) |p2s2, (A.4) 1 2 1 1 p2s2 p1s1 † 3 −ip y where the operator bp s (yo) = d ye 1 Ψ(y)γou(p1s1) tends to the nucleon creation (annihilation) operator 1 1 + + in the asymptotic region limx →±∞ b ps (xo) ⇒ b ps (out or in) and satisfy the anticommutation relations + o { }= − bp s (xo), bps(xo) (Ep/mN )δs sδ(p p). The exact form of the equal-time anticommutators (A.4) can be derived using the usual form of the meson– nucleon Lagrangians fπ µ µ fV µ ν ν µ Lint = gσ ΨΨσ + Ψγ5γµΨ∂ Φπ + gV ΨγµΨV + Ψσµν Ψ ∂ V − ∂ V , (A.5) mπ 4mN where Φσ , Φπ and ΦV denote the field operators of the σ , π and of the V = ρ,ω mesons. Using the equal-time commutation relation between the Heisenberg field operators, we obtain Y ≡ P P | † | (t) 1 2 12 p s Jp s (0), b (0) p2s2 1 1 2 2 p1s1 p s |jσ (0)|p2s2 fπ p s |jπ (0)|p2s2 = P P ¯ 1 1 + ¯ 1 1 1 2 12 gσ u(p2s2)u(p1s1) 2 i u(p2s2)γ5u(p1s1) 2 t − m 2mN mπ t − m σ π | µ | p1s1 jV (0) p2s2 + gV u(¯ p s )γµu(p1s1) + contact terms, (A.6a) 2 2 t − m2 V − Ep Ep1 fπ p s |jπ (0)|p2s2 contact terms = P P 2 u(¯ p s )i γ γ u(p s ) 1 1 1 2 12 2 2 5 o 1 1 − 2 2mN 2mN mπ t mπ + fV ¯ | | u(p2s2)u(p1s1) p1s1 Ψ(0)Ψ (0) pN s2 8mN + fV ¯ | | +··· u(p2s2)γ5γou(p1s1) p1s1 Ψ(0)γ5γoΨ(0) p2s2 , (A.6b) 8mN

2 2 2 where t = (p − pN ) ≡ (E − Ep ) − (p − pN ) and due to the Lorentz-covariance of the scalar (σ ), N pN N N pseudoscalar (π) and vector (V = ρ,ω) vertex functions the following simple expressions | | = ¯ | | = ¯ pN jσ (0) pN gσ Gσ (t)u(pN )u(pN ), pN jπ (0) pN igπ Gπ (t)u(pN )γ5u(pN ), | µ | = ¯ µ pN jV (0) pN gV GV (t)u(pN )γ u(pN ) (A.7) are valid. The diagrammatic representations of the three-dimensional equations (A.6a) and (A.6b) are given in the Fig. 2. The first three terms of Eq. (A.6a) corresponds to the one off-mass-shell meson exchange NN interaction potential and these terms exactly coincides with the OBE NN Bonn potential. However, in the original derivation of the Bonn OBE potential model from quasipotential equation, the dependence of meson–nucleon vertices on the off- 2 = 2 2 = 2 mass-shell variables pN mN and pN mN has been neglected. In the present field-theoretical formulation this approximation is not needed. In this meaning the present derivation can be considered as an additional justification A.I. Machavariani / Physics Letters B 540 (2002) 81–88 87 for the Bonn OBE potential model of the NN interaction. The contact or overlapping terms in Eq. (A.6b) are generated by the nonrenormalizable pseudoscalar and vector parts in the Lagrangian (A.5). If we take the renormalizable pseudoscalar coupling Lps = igπ Ψγ5Ψ instead of the pseudovector coupling in the Lagrangian (A.5), the first term in the right-hand side of Eq. (A.6b) vanishes [6,7]. The other terms of the relation (A.6b) are produced by the nonrenormalizable fourth term of the Lagrangian (A.5). In Ref. [7,8] the structure of the contact (overlapping) terms of the NN interaction potential with quark degrees of freedom was investigated. The contact terms of the NN potential were shown to consist of quark–gluon exchange contributions. In addition it was obtained that due to the structure of the equal-time commutators the quark–gluon exchange terms do not violate the unitarity condition for the NN scattering amplitude. The second term of the Eq. (A.3) is depicted in the Fig. 1A as the on-mass-shell meson exchange diagram. After the cluster decomposition procedure, i.e., after separation of connected and disconnected parts in the amplitudes ¯ in; m|Jps (0)|ps and in; n|Jps(0)|ps, one obtain the 8 skeleton diagrams for the connected parts of the transition amplitudes. We can omit the diagrams with the two and more on-mass-shell meson exchange and we will neglect the contributions of the πd and πNN intermediate states, and of the anti-particle d,¯ NN,... intermediate states in the low energy region. Therefore, after cluster decomposition we obtain only other time-ordered pi-meson exchange diagram which is depicted in the Fig. 1B. Thus we can define the inhomogeneous term of the Eq. (A.3) as | | p1s1p2s2 W p1s1p2s2 = equal-time anticommutators δ(p − p1 − Pm) + 3P P − | ¯ | ; 1 ; | | (2π) 1 2 12 p s Jp s (0) m in c in m J s (0) p2s2 c 1 1 1 1 − − o p2 2 = Ep Ep1 Pm m π,ππ,... 1 δ(−p2 − p1 − Pm) + | ¯ | ; ; | | 0 Jp s (0) p2s2 m in in mp s Jp s (0) 0 , (A.6c) 1 1 −E − E − P o 1 1 2 2 m=π,ππ,... p2 p1 m where the subscript ‘c’ denotes the connected part of the corresponding transition amplitude. Using Eq. (A.6c) we can rewrite Eq. (A.3) in the c.m. frame as | | = | | + | | 1 | †| + | | dp | †| p T p p W p p T Pd Pd T p p T p p T p , E(p) − md E(p) + i0 − E(p ) d (A.8) = 2 + 2 where we have omitted the spin variables for the sake of simplicity, E(p) 2 p mN is the energy of the NN state, Pd = (md , 0) is the four-momentum of deuteron in the c.m. frame p |T |p≡−p s |J (0)|p1s1p s p2s2; in, (A.9a) 1 1 p2s2 1 1 p |T |Pd ≡−p s |J (0)|PdSd ; in. (A.9b) 1 1 p2s2 The potential term V in Eq. (A.8) consist of the on-mass-shell pion exchange diagrams (see the second term of Eq. (A.3) and the diagrams shown in Fig. 1) and of the equal-time commutator (A.4). In the general case the commutator is reduced to the off-mass-shell π-, σ -, ρ-, ω-meson exchange diagrams and to the contact terms shown in Fig. 2. In the framework of the simplest Lagrangian (A.5) the exact form of this commutators is given by Eqs. (A.6a), (A.6b):

W = V(on-mass-shell meson exchange) + Y(t)(off-mass-shell meson exchange). (A.10) The procedure of the exact linearization of the quadratically nonlinear equations (A.8) for πN, NN and γN– πN–γπN–ππN scattering reactions is described in Refs. [6,7,10,11]. 88 A.I. Machavariani / Physics Letters B 540 (2002) 81–88

The on-mass-shell meson exchange part of the NN potential V (Fig. 1) is nonhermitian, but Y†(t) = Y(t). Nevertheless, we can obtain the linear energy-dependent potential p|U(E)|p from p|W|p: p |U E = E(p ) |p=p |W|p, (A.11) so that off-energy shell U(E)is hermitian p|U †(E)|p=p|U(E)|p, but on the half-energy shell U(E)(as well as W) is not hermitian p|U †(E = E(p))|p =p|U(E)|p. This property allows us to derive the Lippmann– Schwinger type equation exactly, by using Eq. (A.8) [6,7], dp p |t(E)|p=p |U(E)|p+ p |U(E)|p p |t(E)|p. (A.12) E + i0 − E(p) On the energy shell the solution of Eqs. (A.8) and (A.12) coincide, i.e., p|t(E(p) = E(p))|p=p|T |p. Eq. (A.12) coincide with Eq. (1) on the half-energy shell E = E(p).

References

[1] K. Chadan, P.C. Sabatier, Inverse Problems in Quantum Scattering Theory, Springer-Verlag, New York, 1977. [2] R.G. Newton, Scattering Theory of Waves and Particles, Springer-Verlag, New York, 1982. [3] B.N. Zakhariev, A.A. Suzko, Direct and Inverse Problems, Springer-Verlag, Berlin, 1982. [4] H.V. Geramb (Ed.), Quantum Inversion Theory and Applications, Lecture Notes in Physics, Vol. 427, Springer-Verlag, Berlin, 1993. [5] See [4], pages 285, 314 and 342. [6] A.I. Machavariani, A.G. Chelidze, J. Phys. G 9 (1993) 128. [7] A.I. Machavariani, Phys. Part. Nucl. (Sov. J. Part. Nucl.) 24 (3) (1993) 317. [8] A.I. Machavariani, A.J. Buchmann, A. Faessler, G.A. Emelyanenko, Ann. Phys. 253 (1997) 149. [9] D.N. Epstein, Phys. Lett. 79B (1978) 195; W.I. Nutt, C.M. Shakin, Phys. Lett. 79B (1978) 290. [10] A.I. Machavariani, A.G. Rusetsky, Nucl. Phys. A 515 (1990) 671. [11] A.I. Machavariani, A. Faessler, nucl-th/0202060, to be published in Nucl. Phys. A. [12] V.G. Kadyshevsky, R.M. Mir-Kasimov, N.B. Skatchkov, Nuovo Cimento A 55 (1968) 233; V.G. Kadyshevsky, R.M. Mir-Kasimov, N.B. Skatchkov, Sov. J. Part. Nucl. 2 (1972) 69. [13] I.S. Shapiro, Dokl. Akad. Nauk SSSR 106 (1956) 647; I.S. Shapiro, Sov. Phys. Doklady 1 (1956) 91. [14] J.D. Bjorken, S.D. Drell, Relativistic Quantum Fields, McGraw–Hill, New York, 1965. [15] C. Itzykson, C. Zuber, Quantum Field theory, McGraw–Hill, New York, 1980. Physics Letters B 540 (2002) 89–96 www.elsevier.com/locate/npe

On masses of unstable particles and their antiparticles in the CPT-invariant system

K. Urbanowski

University of Zielona Gora, Institute of Physics, ul. Podgorna 50, 65-246 Zielona Gora, Poland Received 6 April 2002; received in revised form 6 June 2002; accepted 10 June 2002 Editor: G.F. Giudice

Abstract We show that the diagonal matrix elements of the effective Hamiltonian governing the time evolution in the subspace of states of an unstable particle and its antiparticle need not be equal at t>t0 (t0 is the instant of creation of the pair) when the total system under consideration is CPT invariant but CP noninvariant. To achieve this we use the transition amplitudes for transitions |1→|2, |2→|1 together with the identity expressing the effective Hamiltonian by these amplitudes and their derivatives with respect to time t. This identity must be fulﬁlled by any effective Hamiltonian (both approximate and exact) derived for the two state complex. The unusual consequence of this result is that, contrary to the properties of stable particles, the masses of the unstable particle “1” and its antiparticle “2” need not be equal for t t0 in the case of preserved CPT and violated CP symmetries.  2002 Elsevier Science B.V. All rights reserved.

PACS: 03.65.Ca; 11.30.Er; 11.10.St; 14.40.Aq

Keywords: CPT invariance; Unstable particles; Particle–antiparticle masses; Matter–antimatter asymmetry

1. Introduction CPT symmetry. The CPT symmetry is a fundamental theorem (called the CPT theorem) of axiomatic quan- The knowledge of all the subtleties of the difference tum ﬁeld theory, which follows from locality, Lorentz between a particle and its antiparticle has a fundamen- invariance, and unitarity [1]. One consequence of the tal meaning for understanding our Universe. One can- CPT theorem is that under the product of operations C P T not exclude that the more complete understanding of , and the total Hamiltonian H of the system con- this difference can be helpful in explaining the prob- sidered must be invariant. From this property one usu- lem why the observed Universe contains (according ally infers that stable particles and the related antipar- to the physical and astrophysical data) an excess of ticles have exactly the same mass. Indeed, let H be the H matter over antimatter. General properties of antipar- selfadjoint Hamiltonian acting in the Hilbert space ticles follow from properties of particles through the of states of the system under consideration. Assuming that the normalized eigenstates |ψm of H for eigenvalues m E-mail address: [email protected], [email protected] (K. Urbanowski). H |ψm=m|ψm, (1)

0370-2693/02/$ – see front matter  2002 Elsevier Science B.V. All rights reserved. PII:S0370-2693(02)02101-9 90 K. Urbanowski / Physics Letters B 540 (2002) 89–96 correspond to the particle states, one obtains vectors the mass matrix, Γ is the decay matrix [3–9]. The ¯ |ψm¯ describing the states of the related antiparticles standard method of derivation of a such H bases on def a modification of the Weisskopf–Wigner (WW) ap- from vectors |ψm using the antiunitary operator Θ = CPT [1,2], proximation [10]. Lee, Oehme and Yang have adapted the WW approach to the case of a two particle −iθ ¯ Θ|ψm=e |ψm¯ . (2) subsystem [3,4] to obtain their (approximate) effec- ≡ Now, if the relation tive Hamiltonian H HLOY. Almost all properties of the neutral kaon complex, or another particle– − ΘHΘ 1 = H, (3) antiparticle subsystem can be described by solving the ¯ Schrödinger-like evolution equation [3–8] holds, then one infers from (1) that every |ψm¯ is also an eigenvector of H for the same eigenvalue m and ∂ thus that the mass m¯ of the antiparticle, whose state i |ψ; t = H |ψ; t (t t0 > −∞), (7) ¯ ∂t is described by |ψm¯ , is equal to the mass m of the with the initial condition [8,11] related particle described by vector |ψm, ¯ ¯ m ≡ ψm|H |ψm= ψm¯ |H |ψm¯ ≡m. ¯ (4) |ψ; t = t0 =1, |ψ; t t0 = 0, (8) = = (We use h¯ c 1 units.) for |ψ; t belonging to the subspace H ⊂ H spanned, The properties of singularities of scattering ampli- e.g., by orthonormal neutral kaons states |K0, |K0, tudes appearing in the S-matrix theory provide one and so on (then states corresponding to the decay prod- with reasons for a conclusion similar to (4). Generally, def ucts belong to H H = H⊥). such a conclusion is considered to be universal. Solutions of Eq. (7) can be written in a matrix The property (4) of the particle–antiparticle pairs, form and such a matrix defines the evolution operator obvious for stable particles, is considered to be oblig- (which is usually nonunitary) U (t) acting in H : atory also for unstable particles, though relations of type (4) cannot be extended to the case of unstable par- | ; = | ; = def= | ticles because there are no asymptotic states for these ψ t U (t) ψ t0 0 U (t) ψ , (9) particles and the states of such particles are described where, by vectors belonging to H, which are not eigenvectors for the considered Hamiltonian H . An analogous con- |ψ ≡ a1|1+a2|2, (10) clusion to the relation (4) that masses of unstable par- | | | | ticles and their antiparticles are equal follows from the and 1 stands for vectors of the K0 , B0 , n — | properties of the Lee, Oehme and Yang (LOY) method neutrons, etc., type and 2 denotes antiparticles of the | | |¯ | = of approximate description of time evolution in the particle “1”: K0 , B0 , n , and so on, j k δjk, = subspace of states of a two particle subsystem pre- j,k 1, 2. In many papers it is assumed that the real parts, pared at some initial instant t0 > −∞ andthenevolv- (·), of the diagonal matrix elements of H : ing with time t>t0 [3,4]. Searching for properties of the unstable particle– ≡ = antiparticle pairs one usually uses an effective nonher- (hjj ) Mjj (j 1, 2), (11) mitean Hamiltonian [3–9], say H , which in general where can depend on time t [5], = | | = i hjk j H k (j, k 1, 2), (12) H ≡ M − Γ, (5) 2 correspond to the masses of particle “1” and its where antiparticle “2”, respectively, [3–9], (and such an + + interpretation of (h ) and (h ) will be used in M = M ,Γ= Γ , (6) 11 22 this Letter), whereas the imaginary parts, (·), are (2 × 2) matrices, acting in a two-dimensional subspace H of the total state space H and M is called −2(hjj ) ≡ Γjj (j = 1, 2), (13) K. Urbanowski / Physics Letters B 540 (2002) 89–96 91 are interpreted as the decay widths of these parti- and U(t) is the total unitary evolution operator U(t), cles [3–9]. Such an interpretation seems to be consis- which solves the Schrödinger equation tent with the recent and the early experimental data for ∂ neutral kaon and similar complexes [12]. i U(t)|φ=HU(t)|φ,U(0) = I, (18) ∂t Taking H|| = HLOY and assuming that the property (3) holds in the system considered one easily finds the I is the unit operator in H, |φ≡|φ; t0 = 0∈H is standard result of the LOY approach the initial state of the system. Operator U (t) acts in the subspace of unstable states H ≡ P H. Of course, hLOY = hLOY, 11 22 (14) U (t) has nontrivial form only if which, among others, means that [P,H] = 0, (19) LOY = LOY M11 M22 , (15) and only then transitions of states from H into H⊥ LOY = LOY LOY = | | where Mjj (hjj ) and hjj j HLOY j and vice versa, i.e., decay and regeneration processes, (j = 1, 2). This last relation is interpreted as the are allowed. equality of masses of the unstable particle |1 and its Using the matrix representation one finds antiparticle |2. However, one should remember that in A(t) 0 fact such a conclusion follows from the approximate U (t) ≡ , (20) expressions for the HLOY. The accuracy of the LOY 00 approximation [9] is not sufficient for considering where 0 denotes the suitable zero submatrices and a consequences of the relation (15) as the universal one. submatrix A(t) is the 2 × 2 matrix acting in H | | ¯ Note that vectors ψm and ψm¯ , (1), (2), describe stationary (bound) states of the system considered. A11(t) A12(t) = A(t) = (21) Within this problem the time t can vary from t A21(t) A22(t) −∞ up to t =+∞. On the other hand, the following = | | ≡ | | = supposition seems to be reasonable: the behaviour of and Ajk(t) j U (t) k j U(t) k (j, k 1, 2). a system in which time t varies from t =−∞ to Now assuming (3) and using, e.g., the following t =+∞, and a system in which t can vary only from phase convention [3–6], t = t > −∞ to t =+∞, under CPT transformation 0 | def=−| | def=−| need not be the same. Θ 1 2 ,Θ2 1 , (22) one easily finds that [13–17]

2. Implications of the Khalﬁn theorem A11(t) = A22(t). (23)

The aim of this section is to show that diagonal Note that assumptions (3) and (22) give no relations matrix elements of the exact effective Hamiltonian between A12(t) and A21(t). A (t) H cannot be equal when the total system under The important relation between amplitudes 12 consideration is CPT invariant but CP noninvariant. and A21(t) follows from the famous Khalﬁn theo- Universal properties of the (unstable) particle– rem [13–16]. This theorem states that in the case of antiparticle subsystem of the system described by the unstable states, if amplitudes A12(t) and A21(t) have Hamiltonian H , for which the relation (3) holds, can the same time dependence be extracted from the matrix elements of the exact def A (t) r(t) = 12 = const ≡ r, (24) U (t) appearing in (9). Such U (t) has the following A (t) form 21 then it must be |r|=1. U (t) = P U(t)P, (16) For unstable particles the relation (23) means that where decay laws

def def 2 P =|1 1|+|2 2|, (17) pj (t) =|Ajj (t)| , (25) 92 K. Urbanowski / Physics Letters B 540 (2002) 89–96

(where j = 1, 2), of the particle |1 and its antiparticle More conclusions about the properties of the matrix |2 are equal, elements of H , that is in particular about Mjj , one can infer analyzing the following identity [5,18–22] p1(t) ≡ p2(t). (26) ∂U (t) −1 H ≡ H (t) = i [U (t)] , (33) The consequence of this last property is that the decay ∂t rates of the particle |1 and its antiparticle |2 must be − where [U (t)] 1 is defined as follows equal too. −1 −1 From (23) it does not follow that the masses of the U (t)[U (t)] =[U (t)] U (t) = P. (34) particle “1” and the antiparticle “2” should be equal. (Note that the identity (33) holds, independent of A (t) Indeed, every amplitude jk is a complex number whether [P,H] = 0or[P,H]=0.) The expres- |A (t)| such that jk 1. This means that, e.g., sion (33) can be rewritten using the matrix A(t) − − M = gj (t) i j (t) Ajj (t) e , (27) ∂A(t) −1 H (t) ≡ i [A(t)] . (35) ∂t where gj (t), Mj (t) are real functions of t,and Relations (33), (35) must be fulfilled by the exact as gj (t) 0. These functions can be connected with the def 1 ∂pj (t) well as by every approximate effective Hamiltonian decay rate, γj (t) =− , and mass (energy), pj (t) ∂t governing the time evolution in every two-dimensional Mj (t), of the particle “j”, respectively. There is subspace H of states H [5,18–22]. It is easy to find from (33) the general formulae for ∂gj (t) γj (t) = 2 , (28) the diagonal matrix elements, h ,ofH (t),inwhich ∂t jj we are interested. We have 1 WW which in the case of gj (t) ≡ tγ (within the WW 2 j i ∂A11(t) approximation, or the LOY approximation) leads to h11(t) = A22(t) ≡ WW = detA(t) ∂t γj (t) γj const. In general ∂A12(t) − A21(t) , (36) gj (t) ∼ γj (t), Mj (t) ∼ Mj (t) ∂t (t t0 = 0). (29) i ∂A21(t) h22(t) = − A12(t) From (23) and (26) it only follows that the decay detA(t) ∂t rates must be equal in the case considered, ∂A22(t) + A (t) . (37) ∂t 11 γ1(t) = γ2(t), (30) Now, assuming (3) and using the consequence (23) of which need not be true for the masses. Indeed, one this assumption, one finds finds that i ∂A (t) − = 21 M = M ± h11(t) h22(t) A12(t) 1(t) 2(t) 2nπ detA(t) ∂t (t > ), (n = , , ,...), ∂A12(t) 0 0 1 2 (31) − A (t) . (38) ∂t 21 so, it can occur Next, after some algebra one obtains M (t) = M (t) or M (t) = M (t), (32) 1 2 1 2 − h11(t) h22(t) in a CPT invariant system for t>0. Note that in A12(t)A21(t) ∂ A12(t) the case of the unstable particle–antiparticle pair, the =−i ln . (39) detA(t) ∂t A21(t) property (23) is the only exact relation following from the CPT invariance. So, in the case of unstable This result means that in the considered case for t>0 particles connected by a relation of type (22), when the the following theorem holds: property (3) holds, the only exact relation (23) allows h11(t) − h22(t) = 0 the pair unstable particle–antiparticle to have different ⇐⇒ A12(t) = masses. const (t > 0). (40) A21(t) K. Urbanowski / Physics Letters B 540 (2002) 89–96 93

Thus for t>0 the problem under studies is reduced to 3. Discussion the Khalfin theorem (see the relation (24)). From (36) and (37) it is easy to see that at t = 0 In fact there is nothing strange in the above conclusions about the masses of unstable particles under hjj (0) = j|H |j (j = 1, 2), (41) consideration. From (3) (or from the CPT theorem [1]) it only follows that the masses of particle and antipar- which means that in a CPT invariant system (3) in ticle eigenstates of H (i.e., masses of stationary states the case of pairs of unstable particles, for which for H ) should be the same in the CPT invariant sys- transformations of type (22) hold tem. Such a conclusion cannot be derived from (3) for particle |1 and its antiparticle |2 if they are unstable, = ≡ | | M11(0) M22(0) 1 H 1 , (42) i.e., if states |1, |2 are not eigenstates of H .Simply the unstable particles “1” and “2” are created at the proof of the CPT theorem makes use of the properties of asymptotic states [1]. Such states do not exist t = t0 ≡ 0 as particles with equal masses. The same result can be obtained from the formula (39) by for unstable particles. What is more, one should re- taking t → 0. Note that the properties of the function member that the CPT theorem of axiomatic quantum field theory has been proved for quantum fields cor- Mj (t) following from the definition (27) and from the responding to stable quantum objects and only such properties of the amplitude Ajj (t) and their derivative at t = 0 do not contradict these last conclusions. fields are considered in the axiomatic quantum field Now let us go on to analyze the conclusions theory. There is no axiomatic quantum field theory following from the Khalfin theorem. CP noninvariance of unstable quantum particles. So, all implications of requires that |r| = 1 [13,14,16,17] (see also [3–6, the CPT theorem (including those obtained within the 8,12]). This means that in such a case it must be S-matrix method) need not be valid for decaying par- = r = r(t) = const. So, if in the system considered the ticles prepared at some initial instant t0 0andthen property (3) holds but evolving in time t 0. Simply, the consequences of CPT invariance need not be the same for systems in =−∞ =+∞ [CP,H] = 0, (43) which time t varies from t to t and for systems in which t can vary only from t = t0 > −∞ to and the unstable states “1” and “2” are connected by a t =+∞. Similar doubts about the fundamental nature relation of type (22), then at t>0itmustbe(h11(t) − of the CPT theorem were formulated in [23], where the h22(t)) = 0 in this system. Assuming the LOY inter- applicability of this theorem for QCD was considered. pretation of (hjj (t)) (j = 1, 2), one can conclude One should also remember that conclusions about the from the Khalfin theorem and from the property (40) equality of masses of stable particles and their antipar- that if A12(t), A21(t) = 0fort>0 and if the total sys- ticles following from the properties of the S-matrix tem considered is CPT-invariant, but CP-noninvariant, cannot be extrapolated to the case of unstable states. then M11(t) = M22(t) for t>0, that is, that contrary Simply, there is no S-matrix for unstable states. to the case of stable particles (the bound states), the The following conclusion can be drawn from (42) masses of the simultaneously created unstable parti- and (40): in the case when the particle and antiparticle cle “1” and its antiparticle “2”, which are connected states are connected by the relation (22) and time t by the relation (22), need not be equal for t>t0 = 0. can vary only from t = t0 ∞ to t =+∞,the Of course, such a conclusion contradicts the standard time evolution can cause the masses of the unstable LOY result (14), (15). However, one should remember particle and its simultaneously prepared at t = 0 that the LOY description of neutral K mesons and sim- antiparticle to be different at t t0 in the CPT- ilar complexes is only an approximate one, and that the invariant but CP-noninvariant system. There are other LOY approximation is not perfect. On the other hand possibilities for interpreting the results discussed in the relation (40) and the Khalfin theorem follow from the present Letter. The first is that the interpretation the basic principles of the quantum theory and are rig- of the real and imaginary parts of hjj (t) as the masses orous. Consequently, their implications should also be and decay rates respectively is wrong. This, however considered rigorous. is highly improbable as the experimental data for 94 K. Urbanowski / Physics Letters B 540 (2002) 89–96 neutral K-andB-mesons corroborate the standard This estimation does not contradict the experimen- interpretation of (hjj ) and (hjj ). Moreover, such tal data for neutral K mesons [12] and that the ef- an interpretation follows directly from (33). One can fect following from the relation (40) is very small but think of yet another explanation: from the logical and nonzero. mathematical point of view quantum mechanics is not Note that the relation (40) explains also why the selfconsistent. The possibility of the above statement property (14) of HLOY takes place in the case of being true is very low indeed. preserved CPT symmetry. Simply, there is r = const The consequences of the exact relation (40) and within the LOY approximation. of the Khalfin theorem confirm the results obtained Detailed analysis of assumptions leading to the in [21,25,30] within a different method and in [9, standard form of the LOY effective Hamiltonian gov- 24–28], where a more accurate approximation (based erning the time evolution in a two state (two particle) on the Krolikowski–Rzewuski equation for a distin- subsystem indicates that some assumptions, which guished component of state vector [29]) than the LOY have been used in the LOY treatment of the prob- was used. In particular, this explains why for the lem, and which the WW theory of single line width Fridrichs–Lee model [13,26] assuming CPT-invariance uses, should not be directly applied to the case of two, (i.e., (3)) and (43) it was found in [28] that or more, level subsystems interacting with the rest of − the physical system considered [9]. Namely, when one FL − FL ≡ FL − FL m21Γ12 m12Γ21 h11 h22 h11 h22 i considers the single line width problem in the WW 4(m0 − µ) manner it is quite sufficient to analyze the smallness of (m Γ ) ≡ 12 21 , matrix elements of the interaction Hamiltonian, H (1), − (44) 2(m0 µ) only (we have H = H (0) + H (1) within the LOY ap- FL = →∞ where hjj (j 1, 2), denotes hjj (t ) calculated proach). For the multilevel problem, contrary to the for the Fridrichs–Lee model within using the more single line problem, such a smallness does not ensure accurate approximation mentioned above [26–28], the suitable smallness of components of the evolution mjk ≡ Hjk = j|H |k (j,k = 1, 2), m0 ≡ H11 = equations containing these matrix elements. Moreover, H22 and µ denotes the mass of the decay products. there is no necessity of taking into account the in- Formulae (44) were obtained in [28] assuming that ternal dynamics of the subsystem, which also has an |m12|≡|H12| (m0 − µ).FortheK0, K0-complex effect on the widths of levels (on the masses of the ≡ ∗ 1 ∼ × −12 Γ21 Γ12 (Γ12) 2 Γs 3.7 10 MeV. This particles) in many levels (many particles) subsystems, property, the relation (44) and the assumption that m0 in such a case. The observed masses in a two parti- can be considered as the kaon mass, m0 = mK , enable cle subsystem depend on the interactions of this sub- FL − FL us to find the following estimation of (h11 h22 ) system with the rest, but they also depend on the infor the neutral K-system [28], teractions between the particles forming this subsys- FL − FL ∼ × −15 tem. This means that contrary to the LOY assumption h11 h22 9.25 10 (m12) (1) that H12 = 1|H |2=0, that is 1|H |2=0, one −15 ≡ 9.25 × 10 (H12). (45) should consider the implications of the assumption (1) The assumptions leading to (44) can be used to that 1|H |2 = 0 when one wants to describe the obtain real properties of two particle subsystems. One finds assuming (3) and using the more accurate approxima- FL + FL − − i (H21Γ12) h11 h22 2m0 iΓ0 , (46) tion than LOY approximation that 2 m0 − µ →∞ − →∞ = (where Γ0 ≡ Γ11 = Γ22) which together with (44), h11(t ) h22(t ) 0, (48) (45) give if 1|H (1)|2 = 0 [9,31,32]. The Fridrichs–Lee model (hFL − hFL) calculations performed within the mentioned approx- 11 22 (H12) FL FL Γs imation confirm this conclusion—see (44), (45). On (h + h ) 8m0(m0 − µ) 11 22 the other hand one can show that the assumption ∼ × −18 [ ]−1 9.25 10 (H12) MeV . 1|H (1)|2=0 implies that the more accurate H is (47) equal to HLOY [9,31,32] and thus that for amplitudes K. Urbanowski / Physics Letters B 540 (2002) 89–96 95

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Charmless B decays to ﬁnal states with radially excited vector mesons

Alakabha Datta a, Harry J. Lipkin b,c, Patrick J. O’Donnell d

a Laboratoire René J.-A. Lévesque, Université de Montréal, CP 6128, succ. centre-ville, Montréal, QC, H3C 3J7 Canada b Department of Particle Physics, Weizmann Institute, Rehovot 76100, Israel c School of Physics and Astronomy, Tel-Aviv University, Tel-Aviv 69978, Israel d Department of Physics and Astronomy, University of Toronto, Toronto, ON, Canada Received 24 March 2002; accepted 21 April 2002 Editor: H. Georgi

Abstract We consider the weak decays of a B meson to final states that contain a S-wave radially excited vector meson. We consider vector–pseudoscalar final states and calculate ratios of the type B → ρ π/B → ρπ, B → ω π/B → ωπ and B → φ π/B → φπ where ρ , ω and φ are higher ρ, ω and φ S-wave radial excitations. We find such decays to have larger or similar branching ratios compared to decays where the final state ρ, ω and φ are in the ground state. We also study the effect of radial mixing in the vector system generated from hyperfine interaction and the annihilation term.  2002 Elsevier Science B.V. All rights reserved.

The new data accumulating from B factories and states and probe the high-momentum tails of their other accelerators will include transitions to many new wave functions. final states which have not been previously studied in In this Letter we calculate predictions for the ratios detail; e.g., radially excited meson states. Many decays 0 − + 0 − + R + = BR B → π ρ /BR B → π ρ , (1) involve a transition from a low momentum spectator ρ − − 0 − − 0 quark to a high momentum relativistic meson. The R 0 = BR B → π ρ /BR B → π ρ , (2) ρ form factors for such transitions are expected to be − − − − R = BR B → π ω /BR B → π ω , (3) sensitive to the high momentum components of the ω 0 0 final meson wave function, and therefore to favor Rφ = BR Bs → π φ /BR Bs → π φ , (4) radially excited states. The data on B decays to these where ρ, ω and φ are the radially excited states. states will thus provide important new information, Most studies of two body nonleptonic B decays have particularly for the form factors to the radially excited concentrated on processes of the type B → M1M2 where both M1 and M2 are mesons in the ground E-mail addresses: [email protected] (A. Datta), state configuration. Nonleptonic decays, where one of [email protected] (H.J. Lipkin), the mesons in the final state containing the spectator [email protected] (P.J. O’Donnell). quark is a radially excited state, are expected to have

momentum to make a meson with momentum −p. Radial excitations could be favoured over ground states, if the final momentum p is large, since the radial excitations are expected to have higher kinetic energies. An alternative but equivalent way of understanding this effect is to note that for a heavy b quark one can write [2] − Fig. 1. Factorization for the decay B → Mπ. π (p)M( −p)T |B − = M(−p)J |B π (p) J µ|0 , (8) larger or similar branching ratios compared to decays 1µ 2 where the final state contains the same meson in the where J1,2 are currents that occur in W = J1 × J2. ground state. This is easily seen in a simple model in The transition matrix element for the hadronic decay which B → πM and M is a simple flavor eigenstate can then be written in terms of B → M form factors with no flavor mixing beyond isospin. We follow the and the pion decay constant. The form factors can be inactive spectator approach used [1] to treat B decays expressed as overlap integrals of the B and the M me- to charmonium in which the spectator quark does not son wavefunctions. When M is a light meson, with participate in a flavor-changing interaction and later a mass much smaller than the B meson, the overlap combines with a light antiquark to make the final integrals get contributions mainly from the high mo- light meson as shown in Fig. 1. The decay amplitudes mentum components of the meson wavefunctions. It is are then described as the product of a b-quark decay now clear that for a radially excited meson M,which amplitude and a hadronization function describing the has more higher momentum components, the overlap combination of a quark–antiquark pair to make the integrals will be enhanced compared to the overlap in- final meson. Neglecting the relative Fermi momentum tegral with a ground state meson M. Consequently the of the b quark and the spectator quark in the B meson, B → M form factors are likely to be enhanced rel- the quark transition for the processes in Eqs. (1)–(3) is ative to the B → M form factors which would then translate into higher branching ratio for B → Mπ rel- → − − b π (p)u( p), (5) ative to B → Mπ. where the b quark is at rest and p denotes the final Our discussion above assumed the physical states momentum of the π−. For the process in Eqs. (4), the to be pure radial excitations. However, additional in- quark transition is essentially similar to the one above teractions can mix the various radial excited components. For instance hyperfine interactions can mix ra- b → π0(p)s( −p), (6) dial excitations with the same flavor structure and so in general, in the ρ, ω and φ systems, the various physi- where now p denotes the final momentum of the π0. cal states will be admixtures of radial excitations [3,4]. Concentrating on the processes in Eq. (1)–(3) the Flavor mixing in the vector system is known to be transition matrix for the full decay has the form [1] small but is important in the pseudoscalar sector. We π−(p)M( −p)T |B will consider the pseudoscalar case in a different pub- − lication [5]. To make quantitative predictions we use = M(−p)F u(−p)q(¯ 0) π (p)u( −p)W|b , constituent quark wave functions with several poten- (7) tials to test the dependence of the results on the con- where T denotes the transition matrix for the hadronic fining potential. We shall see that the effects of the po- decay which factors, as shown in Fig. 1, into a weak tential dependence and mixing are small so that the matrix element at the quark level denoted by W and results are reasonably robust and are not seriously de- a recombination matrix element denoted by F .This pendent on the fine details of the model. latter matrix element describes the transition of a Even though our discussion has so far only in- quark with momentum −p and an antiquark with zero cluded vector–pseudoscalar final states we can also A. Datta et al. / Physics Letters B 540 (2002) 97–103 99 consider vector–vector final states such as B → ρρ or Table 1 B → J/ψK∗. However vector–vector final states are Eigenvalues and eigenstates for the ρ system harmonic potential complicated since different partial waves are present. Harmonic N0 N1 N2 Our purpose here is to demonstrate the effects of radial ρ(0.768) 0.990 0.124 −0.066 mixing in the simpler physical system of the vector– ρ(1.545) 0.108 −0.973 0.204 pseudoscalar final state. If the effects of radial en- ρ(2.370) 0.089 −0.195 0.977 hancements are observed in the vector–pseudoscalar case we would expect them to be also present in the Table 2 vector–vector final state. Eigenvalues and eigenstates for the ρ system linear potential We will first review the study the masses and Linear N N N mixing in the vector meson sector following the simple 0 1 2 − nonrelativistic approach in Refs. [3,4]. To obtain ρ(0.775) 0.992 0.112 0.053 ρ(1.515) 0.104 −0.986 0.130 the eigenstates and eigenvalues in the vector meson ρ(2.260) 0.066 −0.122 0.990 system we diagonalize the mass matrix ¯ | | ¯ qaqb,n M qaqb,n Table 3 Eigenvalues and eigenstates for the ρ system quartic potential = δaa δbb δnn (ma + mb + En) B Quartic N0 N1 N2 + · δaa δbb sa sbψn(0)ψn (0), (9) ρ(0.759) 0.988 0.129 −0.077 mamb ρ(1.567) 0.103 −0.955 0.278 where sa,b and ma,b are the quark spin operators and ρ(2.370) 0.11 −0.267 0.957 masses. Here n = 0, 1, 2 and the basis states for the isovector mesons are chosen as |N,I = 1,I3√= 1 = ¯ ¯ is a minimum. This fixes the constant λ in V(r)= λr −|ud , |N,I = 1,I3 = 0 =|uu¯ − dd / 2and and we obtain the eigenvalues and eigenstates in |N,I = 1,I3 =−1 =|du¯ where I,I3 stand for the Table 2. We follow the same procedure for the quartic isospin quantum numbers. In the above equation En is the excitation energy of the nth radially excited state potential and obtain the eigenvalues and eigenstates in and B is the strength of the hyperfine interaction. Table 3. We observe from Tables 1–3 that the mass To begin with, we use the same harmonic confining eigenstates and eigenvalues of the ρ system are not potential as well as the other parameters used in very sensitive to the confining potential and the radial Ref. [4] to obtain the eigenstates and eigenvalues for mixing effects are small. the mass matrix in Eq. (9). The various parameters To obtain the eigenstates and eigenvalues in the used in the calculation are the constituent masses, ω–φ system we diagonalize the mass matrix = = = mu md 0.350 GeV, ms 0.503 GeV, the angular q q¯ ,n|M|q q¯ ,n = = 2 = a b a b frequency, ω 0.365 GeV and b B/mu 0.09. = + + The eigenvalues and eigenstates for the ρ system with δaa δbb δnn (ma mb En) a harmonic potential are shown in Table 1. To see how B + δaa δbb sa ·sbψn(0)ψn (0) this result changes with a different confining potential mamb we use a power law potential V(r) = λrn [6]. We A + δabδab ψn(0)ψn (0). (10) will use a linear and a quartic confining potential mamb and compare the spectrum with that obtained with a This has a similar structure as the ρ system but now harmonic oscillator potential. To fix the coefficient we have the additional annihilation interaction with λ we require that the energy eigenvalues of the strength A that causes flavor mixing [3,7]. Schrödinger equation are similar in the least square Diagonalizing the mass matrix√ in Eq. (10), with sense with the energy eigenvalues used in Ref. [4]. So, the basis states |N =|uu¯ + dd¯ / 2and|S =|ss¯ , for example, for the linear potential, we demand that we obtain the eigenvalues and the eigenstates of 2 the ω–φ system. We use the same value for the F = E (harmonic) − E (linear) n n hyperfine interaction as used for the ρ system. For n 100 A. Datta et al. / Physics Letters B 540 (2002) 97–103

Table 4 Eigenvalues and eigenstates for the ω–φ system linear potential

Linear N0 N1 N2 S0 S1 S2 ω(0.782) 0.991 0.123 −0.058 −0.014 0.004 −0.002 φ(1.05) 0.012 0.011 −0.004 0.997 0.071 −0.034 ω(1.52) −0.113 0.982 0.144 −0.006 −0.034 0.004 φ(1.66) 0.007 −0.030 −0.014 0.068 −0.994 −0.077

Table 5 Eigenvalues and eigenstates for the ω–φ system harmonic potential

Harmonic N0 N1 N2 S0 S1 S2 ω(0.783) 0.984 0.154 −0.081 −0.033 0.011 −0.007 φ(1.05) 0.026 0.029 −0.011 0.994 0.089 −0.048 ω(1.57) −0.126 0.948 0.256 −0.008 −0.139 0.010 φ(1.68) 0.025 −0.12 −0.07 0.082 −0.976 −0.143

Table 6 Eigenvalues and eigenstates for the ω–φ system quartic potential

Quartic N0 N1 N2 S0 S1 S2 ω(0.783) 0.980 0.163 −0.096 −0.049 0.012 −0.009 φ(1.05) 0.041 0.034 −0.015 0.991 0.100 −0.060 ω(1.58) −0.122 0.932 0.322 −0.010 −0.113 0.006 φ(1.7) 0.022 −0.089 −0.067 0.086 −0.968 −0.207

= 2 → 0 the linear potential we obtain with B 0.09 mu Bs π φ decay the QCD penguin is isospin forbid- = 2 and A 0.005 mu the eigenstates and eigenvalues den, the annihilation contribution is OZI forbidden and in Table 4. For the harmonic potential we obtain the electroweak penguin is also described by Eq. (7). = 2 = 2 with B 0.09 mu and A 0.015 mu the eigenstates We obtain, using factorization for the nonleptonic and eigenvalues in Table 5. For the quartic potential amplitude = 2 = 2 we obtain with B 0.09 mu and A 0.023 mu the eigenstates and eigenvalues in Table 6. As in the ρ Rρ+ system we find the mixing to be insensitive to the + − ρ |¯uγ µ(1 − γ )b|B0 π |dγ¯ (1 − γ )u|0 2 confining potential and the effects of radial mixing to = 5 µ 5 +|¯ µ − |0 −| ¯ − | be small. We also find, as expected, a small value for ρ uγ (1 γ5)b B π dγµ(1 γ5)u 0 + ρ 2 3 the annihilation term in the fits to the masses. A P + = 0 ρ We now use these wavefunctions to predict the ra- + , (11) ρ P 3 tios in Eqs. (1)–(4). These decays are dominated by A0 ρ diagrams which satisfy the inactive spectator approach where P is the magnitude of the three momentum [1] and are treated with Eqs. (7) and (8). Some of the of the final states and the form factor A0 is defined diagrams which violate this assumption; e.g., penguin through and annihilation contributions, may not be as negligible here as in the charmonium case treated in Ref. [1] Vf |Aµ|Pi for the decays to the ground state configurations. But *∗.q they are expected to have much smaller form factors = + ∗ − (Mi Mf )A1 *µ 2 qµ for radial excitations. Therefore it is reasonable to ne- q glect them for this preliminary investigation of the *∗.q M2 − M2 − + − i f order of magnitude of these ratios. Note that for the A2 (Pi Pf )µ 2 qµ Mi + Mf q A. Datta et al. / Physics Letters B 540 (2002) 97–103 101

*∗.q + I = d3pφ∗ (p +a)φ (p), 2Mf A0 2 qµ, (12) 1 f i q p. a 1 where Aµ is the axial vector current. Similarly we I = m d3pφ∗ (p +a)φ (p) + , 2 s f i 2 obtain µa mf M± = Mf ± Mi, Rρ0 msq˜ − = = ρ0 |¯uγ µ(1 − γ )b|B0 π |dγ¯ (1 − γ )u|0 2 a 2msβ 2 , ≈ 5 µ 5 M+ 0 µ 0 − ¯ ρ |¯uγ (1 − γ5)b|B π |dγµ(1 − γ5)u|0 2 − 2 2 2 M− q 0 ˜ = ρ 2 3 q M+ 2 , A P 0 M+ − q2 = 0 ρ , (13) 0 3 m m ρ P = i f A0 ρ0 µ (16) mi + mf R ω and φf and φi represent the momentum space wave |¯ µ − |0 −| ¯ − | 2 ω uγ (1 γ5)b B π dγµ(1 γ5)u 0 functions while β is the velocity of the mesons in the ≈ ω|¯uγ µ(1 − γ )b|B0 π−|dγ¯ (1 − γ )u|0 equal velocity frame (also called the Breit frame or 5 µ 5 ω 2 3 the brick wall frame) and mi,f are the nonspectator A P = 0 ω quark masses of the initial and final meson. The equal ω 3 . (14) A0 Pω velocity frame in a convenient frame to calculate the Lorentz invariant form factors where the velocities, Finally, βi and βf of the mesons with masses Mi and Mf R are equal in magnitude but opposite in direction. We φ |¯ µ − | 0|¯ − | 2 use the momentum wavefunction φf obtained from = φ sγ (1 γ5)b Bs π uγµ(1 γ5)u 0 µ 0 0 spectroscopy while for φi we use the wave function φ|¯sγ (1 − γ5)b|B π |¯uγµ(1 − γ5)u|0 − 2 2 φ 3 = = p /pF 2 P φi φB NB e , (17) A0 φ = . (15) φ P 3 where p is the Fermi momentum of the B meson. A0 φ F In our calculations we will take p = 300 MeV. Note To calculate the ratios we need the form factor A . F 0 that in the analysis presented in the introduction we Note that the wavefunctions for the various vector have neglected the Fermi momentum of the b quark, meson states are not enough to calculate nonleptonic since p /m is small. decay amplitudes. In particular, with the factorization F b For transitions to higher resonant states, we use assumption for nonleptonic decays, the calculations the same quark masses as those used in the transition of decay amplitudes require the matrix elements of of the B meson to the lowest resonant state. This is current operators between the physical states. These reasonable, as the spectator quark still comes from matrix elements can be expressed in terms of form the B meson and therefore has the same value for factors and decay constants. In this work we use a its mass irrespective of whether the final state is in constituent quark model (CQM) model for the form the lowest or the first excited state. The values for factors [9] that incorporates some relativistic features the masses of the nonspectator masses are taken to and is relatively simple to adapt to the case of radially be the same as those used for spectroscopy. However excited states. In this model the form factor A is given 0 for the calculation of the velocity β and hence a by defined in Eq. (16) we use the physical mass of M− the higher resonant state. In Table 7 we give our A0 = Z I1 − I2 , M+ predictions for the various ratios defined above. We find that the transitions to higher excited states can where be comparable or enhanced relative to the transitions 4M M M+ = i f to the ground state. From Table 7 we see that the Z 2 , M+ − q2 ratios of branching ratios are slightly sensitive to 102 A. Datta et al. / Physics Letters B 540 (2002) 97–103

+ − −6 Table 7 BR[B → ρ π ]∼28 × 10 [11] we get BR[Bs → Ratios of branching ratios for different confining potentials φπ0]∼6 × 10−7. Hence the large enhancement for 0 Ratio Linear Quadratic Quartic Rφ indicates that BR[Bs → φ π ] can be around ∼ 4 × 10−6. Rρ+ 2.32.01.9 Rρ0 2.32.01.9 Note that in the ρ(ω) system there are two res- Rω 3.52.51.7 onances, ρ(1450)[ω(1420)] and ρ(1700)[ω(1650)], Rφ 6.76.25.2 which can be identified as a S-wave radial excitation (2S) and a D wave orbital excitation in the quark model. However recent studies of the decays of these the confining potential and the ratios of branching resonances show that it is possible that these states ratios increase as we go from the quartic to the linear are mixtures of qq¯ and hybrid states Ref. [10]. Hence potential. This is because the wavefunction for the the state ρ(1450)[ω(1420)] is interpreted as a 2S state linear potential has a longer tail and hence more high with a small mixture of a hybrid state. We do not take momentum components than the wavefunction for the into account such possible mixing with a hybrid state quadratic and the quartic potentials. The wavefunction in our calculation and the meson masses for these ex- for the quadratic potential, has in turn, a longer tail cited states used in our calculation are the ones we and hence more high momentum components than predict here. For the φ system there is only state at the wavefunction for the quartic potential. Hence we φ(1680) which we interpret as a 2S state in the ab- would expect the hierarchy (A ) >(A ) > 0 linear 0 quadratic sence of mixing effects. (A0)quartic and a similar one for the radially excited In conclusion we have considered the weak de- states (A )linear >(A )quadratic >(A )quartic where A0 0 0 0 cays of a B meson to final states that are mixtures and A are the form factors for the transition of 0 of S-wave radially excited components. We calculated B to the ground state and the first radially excited nonleptonic decays of the type B → ρπ/B → ρπ, state of the meson M. We see from Table 7 that this B → ωπ/B → ωπ and B → φπ/B → φπ where hierarchy is maintained for the ratios of form factors ρ , ω and φ are higher ρ, ω and φ resonances. We and so we have (A /A0)linear >(A/A0)quadratic > 0 0 found that the transitions to the excited states can be (A /A ) . Note that the ratio of form factors also 0 0 quartic comparable or enhanced relative to transitions to the depend on the choice of the Fermi momentum of the B ground state. It would, therefore, be extremely inter- meson, as a smaller (larger) Fermi momentum would esting to test these predictions. We also studied the ef- make the form factors more (less) sensitive to the tail fect of radial mixing in the vector system generated of the wavefunction of M, as well as mixing effects from hyperfine interaction and the annihilation term; in the wavefunction of the meson M. The effect of these turn out to be generally small. mixing between the various radially excited states and the ground state is generally small. We observe in Table 7 that there can be a large Acknowledgement enhancement for Rφ . This decay is suppressed in the standard model. One can get a rough estimate of the This work was supported by the US–Israel Bi- branching ratio for B → φπ0 using factorization as s National Science Foundation and by the Natural Sci- 0 BR[Bs → φπ ] ences and Engineering Research Council of Canada. [0 → + −] BR B ρ π 1V V ∗ (c + c /N ) − V V ∗3c /2 ≈ ub us 1 2 c tb ts 9 ≈ . , ∗ + 0 02 References 2 VubVud (c2 c1/Nc) where we have neglected form factor and phase [1] A. Datta, H.J. Lipkin, P.J. O’Donnell, hep-ph/0111336, Phys. 0 0 Lett. B, in press. space differences between Bs → φπ and B → + − [2] M. Beneke, G. Buchalla, M. Neubert, C.T. Sachrajda, hep-ph/ ρ π . The Wilson coefficients ci can be found in 0104110. Ref. [8] while Vub, Vus , Vud , Vtb and Vts are the [3] I. Cohen, H.J. Lipkin, Nucl. Phys. B 151 (1979) 16. various CKM elements [10]. Using the measured [4] M. Frank, P.J. O’Donnell, Phys. Rev. D 29 (1984) 921. A. Datta et al. / Physics Letters B 540 (2002) 97–103 103

[5] A. Datta, H.J. Lipkin, P.J. O’Donnell, hep-ph/0102070; [8] A.J. Buras, hep-ph/9806471. A. Datta, H.J. Lipkin, P.J. O’Donnell, work in progress. [9] R. Aleksan, A. Le Yaouanc, L. Oliver, O. Pene, J.C. Raynal, [6] C. Quigg, J.L. Rosner, Phys. Rep. 56 (1979) 167. Phys. Rev. D 51 (1995) 6235, hep-ph/9408215. [7] A. De Rujula, H. Georgi, S.L. Glashow, Phys. Rev. D 12 (1975) [10] D.E. Groom et al., Eur. Phys. J. 15 (2000) 1. 147; [11] C.P. Jessop et al., CLEO Collaboration, Phys. Rev. Lett. 85 N. Isgur, Phys. Rev. D 12 (1975) 3770; (2000) 2881, hep-ex/0006008. H. Fritzsch, J.D. Jackson, Phys. Lett. B 66 (1977) 365. Physics Letters B 540 (2002) 104–110 www.elsevier.com/locate/npe

More anomalies from fractional branes ✩

M. Bertolini a, P. Di Vecchia a,M.Fraub,A.Lerdac,b,R.Marottad

a NORDITA, Blegdamsvej 17, DK-2100 Copenhagen Ø, Denmark b Dipartimento di Fisica Teorica, Università di Torino and INFN, Sezione di Torino, Via P. Giuria 1, I-10125 Torino, Italy c Dipartimento di Scienze e Tecnologie Avanzate, Università del Piemonte Orientale, I-15100 Alessandria, Italy d Dipartimento di Scienze Fisiche, Università di Napoli, Complesso Universitario Monte S. Angelo, Via Cintia, I-80126 Napoli, Italy Received 19 March 2002; received in revised form 6 June 2002; accepted 10 June 2002 Editor: M. Cveticˇ

Abstract In this Letter we show how the anomalies of both pure and matter coupled N = 1, 2 supersymmetric gauge theories describing the low energy dynamics of fractional branes on orbifolds can be derived from supergravity.  2002 Elsevier Science B.V. All rights reserved.

1. Introduction T 1,1 has been analyzed and the corresponding supergravity solution has been derived. This solution Recently there has been a lot of activity in trying to has been successfully used to describe the dual four- generalize the AdS/CFT correspondence and promote dimensional gauge theory, which has N = 1 super- it to a more general gauge/gravity duality for theories symmetry, a gauge group SU(N + M) × SU(N) and without conformal invariance or with less supersym- a non-trivial matter content [4]. While very interest- metry. Indeed many different approaches have been ing IR properties of this gauge theory have been ob- proposed to this aim, like, for example, the study of tained by considering the non-singular solution on a mass deformations of conformal field theories and of deformed conifold [3], to describe the UV features of the corresponding RG flows, or the study of D-branes the gauge theory it is enough to consider the singular wrapping supersymmetric cycles of K3 and Calabi– solution of Ref. [2] which indeed accounts for the log- Yau spaces, or finally the study of fractional D-branes, arithmic running of the coupling constants and the pre- in conifold and orbifold backgrounds. cise coefficients of the β-functions of the two gauge Fractional branes in conifolds backgrounds were groups [5]. Very recently, in Ref. [6] it has been shown first considered in Refs. [1–3] where a configura- that also the chiral anomaly and the breaking of the tion of N regular and M fractional D3-branes on U(1)R-symmetry to Z2M are correctly incorporated in the classical UV supergravity solution of Ref. [2].

✩ Fractional D-branes in orbifold backgrounds have Work partially supported by the European Community’s been extensively considered in the recent literature. Human Potential Programme under contract HPRN-CT-2000- 00131 Quantum Spacetime and by MIUR under contract PRIN- In Ref. [7] the supergravity solution of M frac- 2 2001025492. tional D3-branes in a C /Z2 orbifold has been ex- E-mail address: [email protected] (M. Bertolini).

0370-2693/02/$ – see front matter  2002 Elsevier Science B.V. All rights reserved. PII:S0370-2693(02)02105-6 M. Bertolini et al. / Physics Letters B 540 (2002) 104–110 105 plicitly obtained (see also Ref. [8]) and used to de- where JR is the R-current and q(x) is the topological rive the exact (perturbative) β-function of the dual charge density. In writing these formulas we have used N = 2 SU(M) Yang–Mills theory. This analysis has standard conventions, namely we have normalized the been subsequently generalized by adding fractional generators of SU(M) so that the quadratic Casimir D7-branes that yield hypermultiplets in the gauge the- invariant is M in the adjoint representation and 1/2 ory [9,10], or by considering D-branes in Zk orb- in the fundamental representation, and have assigned ifolds [11] (see Ref. [12] for a recent review on frac- a R-charge 1 to the gauginos and −1tothechiral tional branes in N = 2 orbifolds and a complete list of fermions in the hypermultiplets.1 references). More recently, similar results have been In supersymmetric theories the scale anomaly, pro- also obtained for fractional D-branes in the N = 1orb- portional to the β-function, and the chiral anomaly are 3 ifold C /(Z2 × Z2) [13]. in the same supersymmetry multiplet. In the theory at In this Letter, prepared while Ref. [6], which hand, this fact can easily be seen by introducing the contains also a brief discussion of the chiral anomaly complex quantity for the N = 2 orbifold, has appeared, we show how θYM 4π the chiral anomalies of both N = 2andN = 1 τ ≡ + i , (2.3) YM 2π g2 supersymmetric gauge theories in four dimensions can be very simply obtained from the explicit supergravity where θYM is the Yang–Mills θ-angle, and observing 2 solutions describing fractional D3-branes in C /Z2 that the term reproducing the anomalies has the 3 and C /(Z2 × Z2) orbifolds presented, respectively, following form [15] in Refs. [7,10,12] and Ref. [13]. In particular, we 2M − N Tr φ2 show how to obtain the correct β-function and the τYM = i log , (2.4) 4π Λ2 chiral anomaly of the pure N = 1 super-Yang–Mills. It is worth to emphasize that our present analysis where Λ is the dynamically generated scale and φ N = does not rely on the probe technique which instead is the (complex) scalar field of the 2 vector N = was used in our previous papers. In fact, here we multiplet. As is well known, due to 2 non- will exploit a simple holographic representation of a renormalization theorems, Eq. (2.4) is the complete gauge theory operator in terms of supergravity bulk perturbative result which is corrected only by instan- quantities and deduce from it how the scale and ton effects. Then the scale and chiral anomalies can chiral transformations are realized in the supergravity be simply obtained by looking at the response of τYM description. to a rescaling of the energy by a factor of µ and to a U(1)R transformation with parameter α ∈[0, 2π), respectively. Under these transformations the scalar 2. N = 2 gauge theories and fractional branes in field φ, which has the scale dimension of a mass and a R-charge 2 (remember that the gauginos have Z2 orbifolds R-charge 1), transforms as follows N = Let us consider a 2 super-Yang–Mills theory φ → µe2iαφ, (2.5) in four dimensions with gauge group SU(M) and N fundamental hypermultiplets. As is well known (see, and thus from Eq. (2.4) we easily find that for example, Ref. [14]), this theory has a β-function 2M − N τ → τ + i (log µ + 2iα). (2.6) given by YM YM 2π (2M − N) This equation implies that β(g) =− g3, (2.1) 16π2 1 1 2M − N → + log µ and where g is the running coupling constant, and a U(1)R g2 g2 8π2 anomaly given by α = − ∂αJR 2(2M N)q(x) , 1 We note that in Ref. [6] a different R-charge assignment has been given to the various fields. In particular, their charges differ 1 a αβ q(x)= F F (2.2) from ours by a factor of 1/3. 32π2 αβ a 106 M. Bertolini et al. / Physics Letters B 540 (2002) 104–110 − 2Mgs ρ θYM → θYM − 2(2M − N)α, (2.7) N 1 π log * c =− 2πα gs 2M − θ, (2.14) 2 − Ngs ρ which are equivalent to Eqs. (2.1) and (2.2), respec- 1 2π log * tively. where gs is the string coupling constant and * is We now show how these results can be obtained in a a regulator. Notice that the explicit appearance of the very simple way by using the supergravity solutions of angle θ in the above expressions of C0 and c implies fractional D-branes that we presented in Refs. [7,10]. that this supergravity solution is not invariant under Let us recall that in string theory the simplest way to rotations in the transverse plane, a fact that is similar realize a four-dimensional N = 2 Yang–Mills theory to what has been recently emphasized in Ref. [6] for with gauge group SU(M) and N hypermultiplets is the fractional D-branes of the conifold. to consider a stack of M fractional D3-branes and N Let us now consider the world-volume theory of 2 D7-branes of type IIB in the orbifold C /Z2 [9,10]. the fractional D3/D7-brane system which, in the limit

For definiteness we assume that the Z2 parity acts as α → 0, reduces to N = 2 super-Yang–Mills theory a reflection in the directions 6789 (labeled by indices with gauge group SU(M) and N fundamental hyper- ,m,...), and that the D3-branes extend along the multiplets in four dimensions. In particular, the action directions 0123 (labeled by indices α,β,...) while the SYM for the bosonic fields in the vector multiplet can D7-branes wrap the directions 01236789. With this be simply obtained by taking the Dirac–Born–Infeld arrangement, the directions 4 and 5 are transverse to action plus the Wess–Zumino term of a fractional both types of branes and define a plane where they can D3-brane and expanding it in the supergravity back- move. In this plane it is convenient to use the complex ground previously considered. Taking the limit α → 0 coordinate and keeping fixed the combination − z ≡ x4 + ix5 = ρeiθ . (2.8) φ = 2πα 1z, (2.15) The supergravity background created by this D3/D7- which plays the role of the scalar field of the N = 2 brane system comprises the dilaton ϕ,the(Einstein vector multiplet, we easily find [10] frame) metric of the form 1 1 1 S =− d4x F a F αβ + D φ¯ a Dαφ 2 = −1/2 α β + −ϕ 1/2 2 + 2 2 YM 2 αβ a α a ds H ηαβ dx dx e H dρ ρ dθ g 4 2 1/2 m θYM 4 a αβ + H δm dx dx , (2.9) + d xF F , (2.16) 32π2 αβ a a R–R 0-form C , a R–R 4-form C ,andtwo (0) (4) where 2 2-forms, namely, B(2) from the NS–NS sector and C(2) − from the R–R sector, which are given by 1 e ϕb 1 2M − N ρ = = + log , (2.17) g2 16π3α g 8πg 8π2 * = = s s C(2) cω(2),B(2) bω(2), (2.10) + = c C(0)b =− − θYM (2M N)θ. (2.18) where ω(2) is the antiself dual 2-form associated to 2πα gs the vanishing 2-cycle of the orbifold ALE space. The Inserting these expressions in Eq. (2.3), we can see explicit expressions for the various fields have been that τYM is a simple holomorphic function of z, derived in Ref. [10] (see also Ref. [9]), but for our namely, present purposes it is enough to recall that M − N z = 2 1 τYM i log , (2.19) eϕ = , (2.11) 2π ρe − Ngs ρ 1 2π log * −π/(2M−N)g where ρe = *e s is the distance in the Ng C = s θ, (2.12) z-plane where the enhançon phenomenon takes (0) 2π − (2M N)gs ρ 1 + log 2 b = 2π2α π * , (2.13) We take this opportunity to correct a sign misprint in Eq. (5.5) − Ngs ρ of Ref. [10]. 1 2π log * M. Bertolini et al. / Physics Letters B 540 (2002) 104–110 107 place [16]. Finally, using Eqs. (2.5) and (2.15) we de- 3. N = 1 gauge theories and fractional branes in duce that the scale and chiral transformations are real- Z2 × Z2 orbifolds ized on the supergravity coordinate z as follows Let us now consider type IIB string theory in the orbifold C3/(Z × Z ) which provides the simplest z → µe2iαz, (2.20) 2 2 set up to realize supersymmetric gauge theories with N = 1 supersymmetry in four dimensions by means and hence the field theory results (2.6) and (2.7) are of fractional D3-branes, whose supergravity solution precisely reproduced from the supergravity classical was obtained in Ref. [13]. For definiteness we take the solution. orbifold directions to be x4,...,x9, introduce three The above analysis can be easily generalized by complex coordinates defined by discussing a more general bound state in which both 4 5 iθ types of fractional D3-branes present in the Z2 orb- z1 ≡ x + ix = ρ1e 1 , ifold are considered [12]. The most general configura- 6 7 iθ2 z2 ≡ x + ix = ρ2e , tion one can have is made of N1 branes of type 1 and 8 9 iθ3 z3 ≡ x + ix = ρ3e , (3.1) N2 branes of type 2. In this case the world-volume theory is a N = 2 Yang–Mills theory with gauge group and consider fractional D3-branes that are completely 3 SU(N1) × SU(N2) and two hypermultiplets in the transverse to the orbifold, i.e., that are extended along α bifundamental representations (N1, N2) and (N1, N2), x with α = 0, 1, 2, 3. As explained in Refs. [13, respectively. In this case our previous analysis can be 17], there are four types of such fractional D3-branes generalized in a straightforward way and the correct corresponding to the four irreducible representations results are simply obtained by replacing in all formulas of Z2 × Z2, none of which is free to move in the the quantity (2M − N) with 2(N1 − N2) if we refer to transverse space. The most general configuration we the first factor of the gauge group, or with 2(N2 − N1) can consider is therefore a stack of N1 branes of type 1, if instead we refer to the second factor of the gauge N2 of type 2, N3 of type 3 and N4 of type 4, all located group. at the orbifold fixed point. On the world volume of this Finally, we would like to point out that the method bound state there is a four-dimensional N = 1 Yang– of obtaining the β-function and the chiral anomaly Mills theory with gauge group SU(N1) × SU(N2) × 4 from the fractional brane solutions as we have de- SU(N3) × SU(N4) and bifundamental matter. In scribed it now, does not rely on the use of the probe particular, for each factor SU(NI ) of the gauge group technique which instead was employed in Refs. [7,10, one finds 6 chiral multiplets, 3 of them transforming 11], and, in particular, does not require that the analy- in the bifundamental representation (NI , NJ )and3in sis be made in the Coulomb branch of the N = 2 the conjugate representation (NI , NJ ) with J = I,for theory. The only necessary ingredients are the holo- a total of 12 chiral multiplets. In the explicit string graphic identification of a world-volume field in terms realization these fields are equipped with a suitable of bulk supergravity quantities, as we did, for exam- 4 × 4 Chan–Paton matrix that specifies on which of ple, in Eq. (2.15), and its behavior under scale and chi- the four types of branes the two end-points of the open ral transformations. Everything else then follows from string are attached. Taking this fact into account and the explicit expressions Eqs. (2.17) and (2.18) of the picking the same complex structure as in Eq. (3.1), gauge coupling constant and the θ-angle in terms of under the same conventions as those of Ref. [13], one the supergravity fields, expressions that are dictated by can rearrange the 12 chiral multiplets into three 4 × 4 the low-energy limit of the world-volume action of the matrices given by fractional branes. Thus this method can be in principle   0 A1 00 applied also to N = 1 models as we will discuss in the   = B1 000 next section. Φ1   , 000C1 00D1 0

3 We neglect a diagonal U(1) factor which is decoupled, and the relative U(1) which are subleading in the large N-limit. 4 Again we do not consider irrelevant U(1) factors. 108 M. Bertolini et al. / Physics Letters B 540 (2002) 104–110   00A 0 2 where q(x) is the topological charge density for  000B  Φ =  2  , SU(N ) (see Eq. (2.2)). Just like in the N = 2 theories, 2 C 000 1 2 also in this case we can combine the effect of the scale 0 D 00  2  and chiral anomalies together by writing 000A3 (3N − N − N − N )  00B3 0  → + 1 2 3 4 Φ3 =   , (3.2) τYM τYM i 0 C3 00 2π D 000 2 3 × log µ + iα , (3.6) 3 where Ai,...,Di are each a chiral multiplet. The position of these multiplets inside the matrices indicates where τYM is defined as in Eq. (2.3) in terms of the from which types of open strings they originate, for coupling constant and θ-angle of the first factor of the example, A1 arises from strings stretched between gauge group. Of course similar expressions hold for branes of type 1 and branes of type 2, whereas C3 from the other factors and can be obtained from the previous strings stretched between branes of type 3 and branes formulas in a straightforward way. Notice that while of type 2 and so on. Each field matrix Φi encodes those the chiral anomaly is a one-loop effect, in N = 1 chiral superfields having dynamics in the zi plane. gauge theories the β-function receives corrections at There are several important points that we would all loops. The reason why it is possible to construct like to emphasize. First of all, the Lagrangian of this the complex combination (3.6) is because here we N = 1 theory contains a cubic superpotential of the are discussing the Wilsonian β-function which is form W = Tr(Φ1[Φ2,Φ3]), which is renormalizable perturbatively exact at one loop [18]. in the UV. This is to be contrasted to what happens Let us now consider the supergravity background in the conifold theory where the matter fields have corresponding to our bound state of fractional a quartic unrenormalizable superpotential [3,4]. Sec- D3-branes [13]. This is characterized by a metric of ondly, the Lagrangian of the orbifold theory is classi- the form cally invariant under scale transformations of the en- 2 = −1/2 α β + 1/2 m ergy and U(1)R transformations. Due to the presence ds H ηαβ dx dx H δm dx dx , (3.7) of the cubic superpotential, the superfields Φi have a R–R 4-form C(4) and three pairs of scalars bi and R-charge 2/3, and hence their scalar components φ i ci (i = 1, 2, 3) which correspond to the components of have charge 2/3 while the chiral fermions have charge the 2-forms B(2) and C(2) along the antiself dual forms −1/3. Therefore, under a scale transformation with i ω(2) associated to the three exceptional vanishing parameter µ and a U(1)R transformation with para- cycles of the orbifold, namely, meter α we have i i C(2) = ci ω ,B(2) = biω . (3.8) 2 iα (2) (2) φi → µe 3 φi , (3.3) The explicit form of the solution can be found in for i = 1, 2, 3. As is well known, these transformations Ref. [13]; here we simply recall that become anomalous in the quantum theory. Indeed, 2g ρ focusing for simplicity on the first factor of the gauge b = 2π2α 1 + s f (N ) log i , (3.9) i π i I * group (similar considerations hold for any gauge groups), we find a scale anomaly proportional to the ci =− 4πα gs fi (NI )θi, (3.10) (Wilsonian) β-function where − − − (3N1 N2 N3 N4) 3 β(g) =− g , (3.4) f (N ) = N + N − N − N , 16π2 1 I 1 2 3 4 f2(NI ) = N1 − N2 + N3 − N4, where g is the running coupling constant of SU(N1), and a U(1)R anomaly given by f3(NI ) = N1 − N2 − N3 + N4. (3.11) As we mentioned before, the world-volume action of α 1 ∂αJ = 2 N − (N + N + N ) q(x), (3.5) R 1 3 2 3 4 our bound state of fractional D3-branes in the limit M. Bertolini et al. / Physics Letters B 540 (2002) 104–110 109

α → 0 reduces to a N = 1 super-Yang–Mills the- We conclude by observing that a pure N = 1 super- ory in four dimensions with gauge group SU(N1) × Yang–Mills theory in four dimensions can be realized SU(N2) × SU(N3) × SU(N4) and bifundamental mat- on a stack of M fractional D3-branes of just one ter. The bosonic part of this action can be obtained type, for example, by putting N1 = M and N2 = by expanding the Dirac–Born–Infeld action plus the N3 = N4 = 0 in our previous analysis. Therefore, the Wess–Zumino term for fractional D3-branes in the method we have described allows to obtain the correct corresponding supergravity background, and then ob- β-function and chiral anomaly of this theory from the serving that the complex coordinates zi of Eq. (3.1) supergravity solution, namely, can be traded for the scalar components φi of the chiral 1 1 3M superfields Φ of Eq. (3.2), similarly to what we have → + logµ and i g2 g2 8π2 done in Eq. (2.15) with the scalar component of the → − N = 2 vector multiplet. This correspondence can also θYM θYM 2Mα. (3.16) be understood by looking at the explicit open-string Notice that the equation above correctly accounts for realization of the scalars φi , each of which is indeed the breaking of the U(1)R-symmetry to Z2M . To our related to a position in the zi plane [13]. Using this knowledge this is the first quantitative derivation of identification between φi and zi , and Eq. (3.3), we can these results for pure N = 1 super-Yang–Mills theory find the realization of the scale and chiral transforma- using a supergravity dual background. Since these are tions on the supergravity coordinates, namely, UV results, the naked singularity of the supergravity 2 solution does not play any role. It would be very → 3 iα zi µe zi. (3.12) interesting to explore the possibility of resolving this We now focus for simplicity on those terms of world- singularity, for example, by suitably deforming the 3 volume action that, in the low-energy limit, depend orbifold C /(Z2 × Z2), and see whether in this way only on the gauge fields of the first factor SU(N1) of one can get some information on the IR behavior of the gauge group. These terms are simply [13] the dual gauge theory. 1 4 a αβ SYM =− d xF F g2 αβ a 4 Acknowledgements θ + YM 4 a αβ d xFαβ Fa , (3.13) 32π2 M.F. and A.L. thank M. Billó and I. Pesando where and M.B. thanks G. Ferretti, E. Imeroni, E. Lozano- Tellechea and J.L. Petersen for useful discussions 1 1 1 = − and exchange of ideas. M.B. is supported by a EC 2 2 bi 1 g 8πgs 4π α i Marie Curie Postdoc Fellowship under contract num- 1 1 ρi ber HPMF-CT-2000-00847. = + fi (NI ) log , 16πg 8π2 * s i = 1 =− References θYM ci fi(NI )θi. (3.14) 4πα gs i i [1] I.R. Klebanov, N.A. Nekrasov, Nucl. Phys. B 574 (2000) 263, Using these formulas, we see that the supergravity hep-th/9911096. realization of the complex coupling τYM is [2] I.R. Klebanov, A.A. Tseytlin, Nucl. Phys. B 578 (2000) 123, hep-th/0002159. = 1 + 1 zi [3] I.R. Klebanov, M.J. Strassler, JHEP 0008 (2000) 052, hep- τYM i fi (NI ) log . (3.15) th/0007191. 4gs 2π * i [4] I.R. Klebanov, E. Witten, Nucl. Phys. B 536 (1998) 199, hep- From this equation it is now immediate to see that th/9807080. [5] C.P. Herzog, I.R. Klebanov, P. Ouyang, hep-th/0108101. the field theory result (3.6) is correctly reproduced [6] I.R. Klebanov, P. Ouyang, E. Witten, hep-th/0202056. if we use the transformation (3.12) and the explicit [7] M. Bertolini, P. Di Vecchia, M. Frau, A. Lerda, R. Marotta, definitions of the functions fi (NI ) given in Eq. (3.11). I. Pesando, JHEP 02 (2001) 014, hep-th/0011077. 110 M. Bertolini et al. / Physics Letters B 540 (2002) 104–110

[8] J. Polchinski, Int. J. Mod. Phys. A 16 (2001) 707, hep- [14] P. Di Vecchia, hep-th/9803026. th/0011193. [15] P. Di Vecchia, R. Musto, F. Nicodemi, R. Pettorino, Nucl. Phys. [9] M. Graña, J. Polchinski, hep-th/0106014. B 252 (1985) 635. [10] M. Bertolini, P. Di Vecchia, M. Frau, A. Lerda, R. Marotta, [16] C.V. Johnson, A.W. Peet, J. Polchinski, Phys. Rev. D 61 (2000) Nucl. Phys. B 621 (2002) 157, hep-th/0107057. 086001, hep-th/9911161. [11] M. Billò, L. Gallot, A. Liccardo, Nucl. Phys. B 614 (2001) 254, [17] D.R. Morrison, M. Ronen Plesser, Adv. Theor. Math. Phys. 3 hep-th/0105258. (1999) 1, hep-th/9810201. [12] M. Bertolini, P. Di Vecchia, R. Marotta, hep-th/0112195. [18] M.A. Shifman, A.I. Vainshtein, Nucl. Phys. B 277 (1986) 456, [13] M. Bertolini, P. Di Vecchia, G. Ferretti, R. Marotta, hep- Sov. Phys. JETP 64 (1986) 428. th/0112187. Physics Letters B 540 (2002) 111–118 www.elsevier.com/locate/npe

1,1,1 Comments on Penrose limit of AdS4 × M Changhyun Ahn

Department of Physics, Kyungpook National University, Taegu 702-701, South Korea Received 24 May 2002; accepted 13 June 2002 Editor: M. Cveticˇ

Abstract × 1,1,1 1,1,1 = SU(3)×SU(2)×U(1) We construct a Penrose limit of AdS4 M where M SU(2)×U(1)×U(1) that provides the pp-wave geometry 7 equal to the one in the Penrose limit of AdS4 × S . There exists a subsector of three-dimensional N = 2 dual gauge theory which has enhanced N = 8 maximal supersymmetry. We identify operators in the N = 2 gauge theory with supergravity KK excitations in the pp-wave geometry and describe how the gauge theory operators made out of two kinds of chiral ﬁelds of conformal dimension 4/9, 1/3fallintoN = 8 supermultiplets.  2002 Published by Elsevier Science B.V.

1. Introduction supersymmetry. The corresponding operators in the gauge theory side were identified with the stringy Recently [1], it was found that the large-N limit excitations in the pp-wave geometry and some of N = of a subsector of four-dimensional N = 4 SU(N) the gauge theory operators are combined into 4 supersymmetric gauge theory is dual to type IIB supersymmetry multiplets [8]. Moreover, it was found = N = string theory in the pp-wave background [2,3]. In in [52] that a subsector of d 3, 2 gauge theory × 1,1,1 N = the N →∞ with the finiteness of string coupling dual to AdS4 Q has enhanced 8 maximal constant, this subspace of the gauge theory and the supersymmetry and the gauge theory operators are N = operator algebra are described by string theory in the combined into 8 chiral multiplets. pp-wave geometry. By considering a scale limit of the In this Letter, we consider a similar duality that is 5 present between a certain three-dimensional N = 2 geometry near a null geodesic in AdS5 × S , it leads to the appropriate subspace of the gauge theory. The gauge theory and 11-dimensional supergravity the- operators in the subsector of N = 4 gauge theory can ory in a pp-wave background with the same spirit as be identified with the excited states in the pp-wave in [7,8,10,52]. We describe this duality by taking a background. There are many papers [4–51] related to scaling limit of the duality between 11-dimensional × 1,1,1 1,1,1 the work of [1]. There exist some works [7,8,10] on supergravity on AdS4 M where M was con- 1,1 sidered first in [53] and three-dimensional supercon- the Penrose limit of AdS5 × T that gives the pp- 5 formal field theory that consists of an N = 2 SU(N)× wave geometry of AdS5 × S . There is a subsector of SU(N) gauge theory with two kinds of chiral fields N = 1 gauge theory that contains an enhanced N = 4 ∗ Ui , i = 1, 2, 3, transforming in the ( , ) color representation and VA, A = 1, 2, transforming in the ∗ E-mail address: [email protected] (C. Ahn). ( , ) color representation [54]. The com-

0370-2693/02/$ – see front matter  2002 Published by Elsevier Science B.V. PII:S0370-2693(02)02134-2 112 C. Ahn / Physics Letters B 540 (2002) 111–118

1,1,1 2 2 plete analysis on the spectrum of AdS4 × M was bundle over CP × S . The spherical coordinates found by [55]. This gives the theory of [54] that lives (θ, φ) parametrize two sphere, as usual, and the angle on N M2-branes at the conical singularity of a Calabi– µ and three real one forms σ1,σ2 and σ3 parametrize Yau four-fold. The scaling limit is obtained by con- CP2 satisfying the SU(2) algebra and the angle τ sidering the geometry near a null geodesic carrying parametrizes the U(1) Hopf fiber. The angles vary large angular momentum in the U(1)R isometry of the over the ranges, 0 θ π,0 φ 2π,0 τ 1,1,1 M space which is dual to the U(1)R R-symmetry 4π and 0 µ π/2. The SU(3) × SU(2) × U(1) in the N = 2 superconformal field theory. In Section 2, isometry group of M1,1,1 consists of SU(3) × SU(2) we construct the scaling limit around a null geodesic global symmetry and U(1)R symmetry of the dual 1,1,1 in AdS4 ×M and obtain a pp-wave background. In superconformal field theory of [54]. Section 3, we identify supergravity KK excitations ob- Let us make a scaling limit around a null geodesic 1,1,1 tained by [55] in the Penrose limit with gauge theory in AdS4 × M that rotates along the τ coordinate operators. In Section 4, we summarize our results. of M1,1,1 whose shift symmetry corresponds to the U(1)R symmetry in the dual superconformal field theory. Let us introduce coordinates which label the 1,1,1 2. Penrose limit of AdS4 × M geodesic + 1 1 Let us start with the supergravity solution dual x = t + (τ + 2φ) , 2 8 to the N = 2 superconformal field theory [54]. By 2 putting a large number of N coincident M2 branes at − L 1 x = t − (τ + 2φ) , the conifold singularity and taking the near horizon 2 8 limit, the metric becomes that [56–58] of AdS × 4 and make a scaling limit around ρ = 0 = µ = θ in the M1,1,1 (see also [59,60]) above geometry (1). By taking the limit L →∞while ds2 = ds2 + ds2 , (1) rescaling the coordinates 11 AdS4 M1,1,1 √ r ζ 2 ξ where ρ = ,µ= √ ,θ= , L 3 L L 2 = 2 − 2 2 + 2 + 2 2 dsAdS L cosh ρdt dρ sinh ρdΩ2 , 4 the Penrose limit of the AdS × M1,1,1 becomes 2 4 2 = L + 2 + 2 dsM1,1,1 dτ 3sin µσ3 2cosθdφ 3 64 + − + + ds2 =−4 dx dx + dri dri − ri ri dx dx 3L2 1 11 + dµ2 + sin2 µ σ 2 + σ 2 i=1 4 4 1 2 1 + + dξ2 + ξ 2 dφ2 − 2ξ 2 dφdx + cos2 µσ 2 4 3 + 1 2 + 2 ˜ 2 +˜2 +˜2 2 dζ ζ σ1 σ2 σ3 + L 2 + 2 2 4 dθ sin θdφ , 2 + 8 + 2ζ σ˜3 dx 2 3 where dΩ2 is the volume form of a unit S and + − + + 6 =− + i i − i i the curvature radius L of AdS4 is given by (2L) = 4 dx dx dr dr r r dx dx 2 6 1,1,11 i=1 32π pN. Topologically M is a nontrivial U(1) 1 + dwdw¯ + i(wdw¯ − wdw)dx¯ + 4 1 Sometimes instead of using the notation Mpqr , M(m,n) space with two parameters m and n (that are two winding numbers of the U(1) gauge field over the CP2 and S2 of the base manifold) SU(3) × SU(2) × U(1).ForM(3, 2) = M1,1,1,thereisanN = 2 rather than three parameters p,q, and r in Mpqr is used where supersymmetry while for all other M(m,n) no supersymmetry the parameters are related by m/n = 3p/(2q). The integers p,q survives. In particular, M(0, 1) = M0,1,0 = CP2 × S3, M(1, 0) = 5 2 1,0,1 5 2 and r characterize the embedding of SU(2) × U(1) × U(1) in S × S and M = (S /Z3) × S . C. Ahn / Physics Letters B 540 (2002) 111–118 113

2 1 + SU(3) × SU(2) global symmetry. We identify states in + dz dz¯ − i(z¯ dz − z dz¯ )dx , i i i i i i the supergravity containing both short and long multi- 4 = i 1 (2) plets with operators in the gauge theory and focus only where in the last line we introduce the complex coor- on the bosonic excitations of the theory. In each multi- dinate w = ξeiφ for R2 and a pair of complex coor- plet, we specify a SU(3)×SU(2) representation,3 con- 4 dinates z1 and z2 for R . Since the metric has a co- formal weight and R-charge. variantly constant null Killing vector ∂/∂x− ,itisalso pp-wave metric. The pp-wave has a decomposition of Massless (or ultrashort) multiplets [55] the R9 transverse space into R3 ×R2 ×R4 where R3 is i 2 4 parametrized by r , R by w and R by z1 and z2.The (1) Massless graviton multiplet: symmetries of this background are the SO(3) rotations (1, 1), ∆ = 2,R= 0. in R3. In the gauge theory side, the SO(3) symmetry corresponds to the subgroup of the SO(2, 3) confor- There exists a stress-energy tensor superfield Tαβ (x, θ) ± × mal group. Note that the pp-wave geometry (2) in the that satisfies the equation for conserved current Dα αβ scaling limit reduces to the maximally N = 8 super- T (x, θ) = 0. This Tαβ (x, θ) is a singlet with respect 7 symmetric pp-wave solution of AdS4 × S [17,18,61, to the flavor group SU(3) × SU(2) and its conformal ix+ −ix+ 62] through w = e w and zi = e z˜i dimension is 2. Moreover, R charge is 0. So this corresponds to the massless graviton multiplet that prop- 9 + − agates in the AdS bulk. ds2 =−4 dx dx + dri dri 4 11 (2) Massless vector multiplet: = i 1 3 9 = = 1 (8, 1) or (1, 3), ∆ 1,R0. − riri + ri ri dx+ dx+. 4 There exists a conserved vector current, a scalar su- i=1 i=4 perfield JSU(3)(x, θ), to the generator of the flavor The supersymmetry enhancement in the Penrose limit symmetry group SU(3) through Noether theorem sat- N = implies that a hidden 8 supersymmetry is present isfying the conservation equations D±αD± × in the corresponding subsector of the dual N = 2 α JSU(3)(x, θ) = 0. This JSU(3)(x, θ) transforms in the superconformal field theory. In the next section, we adjoint representation 8 of the first factor SU(3) of the provide precise description of how to understand flavor group and its conformal dimension is 1 with the excited states in the supergravity theory that vanishing R-charge. This corresponds to the mass- corresponds in the dual superconformal field theory to less vector multiplet (8, 1) propagating in the AdS4 operators with a given conformal dimension. bulk. There is also massless vector multiplet denoted by (1, 3). There exists a corresponding scalar superfield JSU(2)(x, θ), to the generator of the flavor 3. Gauge theory spectrum symmetry group SU(2) satisfying the conservation D±αD±J (x, θ) = J (x, θ) × equations α SU(2) 0. This SU(2) The 11-dimensional supergravity theory in AdS4 transforms in the adjoint representation 3 of the SU(2) M1,1,1 is dual to the N = 2 gauge theory encoded in a quiver diagram with gauge group SU(N) × SU(N) with two kinds of chiral fields U ,i = 1, 2, 3, trans- the dimensions of the baryons by computing the ratio of volume of i 1,1,1 forming in the ( , ∗) color representation and 5-cycle to the one of M . It turns out that N product of U is = ∗ equal to 4N/9andN product of V is equal to N/3. Therefore this VA,A 1, 2, transforming in the ( , ) color tells us that the conformal dimensions of U and V are 4/9and1/3, representation [54]. At the fixed point, these chiral su- respectively, [54]. perfields U,V have conformal weights 4/9, 1/3, re- 3 A representation of SU(3) can be identified by a Young dia- 2 spectively and transform as (3, 1) and (1, 2) under the gram and when we denote the Dynkin label (M1,M2) so that totally we have M1 + 2M2 boxes, the dimensionality of an irreducible rep- + + = + + 2 M1 M2 resentation is N(M1,M2) (1 M1)(1 M2) 2 .Also 2 There exist two supersymmetric 5-cycles that are the restric- an irreducible representation of SU(2) can be described by a Young ∗ tions of the U(1) fibration over P2 and P1 ×P1. One can determine diagram with 2J boxes. Its dimensionality is 2J + 1. 114 C. Ahn / Physics Letters B 540 (2002) 111–118

= = + 3Y of the ﬂavor group and its conformal dimension is 1 hypercharge Y : M1 0, 1, 2,..., M2 M1 2 , with vanishing R-charge. J =|Y/2|, |Y/2|+1,...and Y = 0, ±2, ±4,.... For = = + 3|Y | (3) Two massless vector multiplets: Y<0, M2 0, 1, 2,...and M1 M2 2 . The corresponding eigenmodes occur in (M1,M1 + (1, 1), ∆ = 1,R= 0. 3|Y | + 3|Y | 2 ) for Y>0or,(M2 2 ,M2) for Y<0

It is known that the Betti current Jbetti = 2UU − 3V V SU(3) representation, the angular momentum J SU(2) of M1,1,1 is obtained from the toric description [54], representation and U(1) charge Y/4. The eigenvalues it is conserved and corresponds to additional massless (4) as a linear combination of the quadratic Casimirs vector multiplets. for the symmetry group SU(3)×SU(2)×U(1) are the form for a coset manifold [58] sometime ago. Therefore massless multiplets (1), (2) and (3) sat- The U(1) part of the isometry group of M1,1,1 urate the unitary bound and have a conformal weight acts by shifting U(1) weak hypercharge Y . The half- related to the maximal spin. integer R-charge, R is related to U(1) charge Y by R = Y .LetustakeR 0. One can do similar case for Short multiplets [55] R<0. One can find the lowest value of ∆ is equal to R corresponding to a mode scalar with M1 = 0,M2 = It is known that the dimension of the scalar opera- 3R/2andJ = R/2 because E becomes 16R(R + 3) tor in terms of energy labels, in the dual SCFT corre- and plugging back to (3) then one obtains ∆ = R. × 1,1,1 Thus we find a set of operators filling out a sponding AdS4 M is (1+ 3R )(2+ 3R ) 2 2 , R + 1 multiplet of SU(3) × SU(2) × 2 2 3 1 m (1+ 3R )(2+ 3R ) ∆ = + 1 + U(1) where the number 2 2 is the dimension 2 2 4 2 of SU(3) representation while R + 1 is the dimension √ 3 1 E ( ) ∆ = R = + 45 + − 6 36 + E. (3) of SU 2 representation. The condition satu- 2 2 4 rates the bound on ∆ from superconformal algebra. The energy spectrum on M1,1,1 exhibits an interesting The fact that the R-charge of a chiral operator is equal feature which is relevant to superconformal algebra to the dimension was observed in [6] in the context and it is given by [55,60]4 of R symmetry gauge field. 1 E = 64 (M + M + M M ) (1) One hypermultiplet: 3 1 2 1 2 1 + 3R 2 + 3R 1 1 2 2 , R + 1 ,∆= R. + J(J + 1) + Y 2 , (4) 2 2 8 According to [55], the information on the Laplacian where the eigenvalue E is classified by SU(3) quan- eigenvalues allows us to get the spectrum of hyper- tum numbers (M1,M2), SU(2) isospin J and weak multiplets of the theory corresponding to the chiral operators of the SCFT. This part of spectrum was given 4 Of course, for general Mpqr space, it is known that the energy in [54] and the form of operators is spectrum on this space is given by 3 2 R/2 Tr Φc ≡ Tr U V , (5) 2 2 2 q E = γ α (M1 + M2 + M1M2) 3 p2 where the flavor SU(3) and SU(2) indices are totally 1 1 symmetrized and the chiral superfield Φc(x, θ) satis- + 2β J + J 2 − q2Y 2 + q2Y 2 , + = 4 4 fies Dα Φc(x, θ) 0. The hypermultiplet spectrum in the KK harmonic expansions on M1,1,1 agrees with = = + 3 =|1 | | 1 |+ where M1 0, 1, 2,..., M2 M1 2 pY, J 2 qY , 2 qY 1,..., and Y = 0, ±2, ±4,... For Y<0, M = 0, 1,..., and the chiral superfield predicted by the conformal gauge 2 3 2 = + 3 | | theory. From this, the dimension of U V should be 2 M1 M2 2 p Y .Hereα, β and γ are related to the scale parameters. In particular, on the supersymmetric space M1,1,1,we to match the spectrum. In fact, the conformal weight of = = = 1 = 1 = have p q 1, α 2 , β 4 and γ 8. a product of chiral fields equals the sum of the weights C. Ahn / Physics Letters B 540 (2002) 111–118 115 of the single components. This is due to the relation of Therefore the short OSp(2|4) multiplets (1), (2) and ∆ = R satisfied by chiral superfields and to the addi- (3) saturate the unitary bound and have a conformal tivity of the R-charge. dimension related to the R-charge and maximal spin. (2) One short graviton multiplet: Long multiplets [55]6 1 + 3R 2 + 3R 2 2 , R + 1 ,∆= R + 2. Although the dimensions of nonchiral operators are 2 in general irrational, there exist special integer values of n such that for M = n ,M = n + 3R/2andJ = The gauge theory interpretation of this multiplet is i 1 1 2 1 R/2 + n , one can see the Diophantine like condition obtained by adding a dimension 2 singlet operator with 2 (see also [64]), respect to flavor group into the above chiral superfield Φ (x, θ). We consider Tr Φ ≡ Tr(T Φ ), where 2 − + 2 − − = c αβ αβ c n1 n1 3 n2 n2 6n1n2 0 (6) Tαβ (x, θ) is a stress energy tensor and Φc(x, θ) is a √ chiral superfield (5). All color indices are symmetrized make 36 + E be equal to 4R + 2(2n1 + 2n2 + before taking the contraction. This composite operator 3). Furthermore in order to make the dimension be + αβ = rational (their conformal dimensions are protected), satisfies Dα Φ (x, θ) 0. √ (3) One short vector multiplet: 45 + E/4 − 6 36 + E in (3) should be square of something. It turns out this is the case without any further restrictions on ni ’s. Therefore we have ∆ = 1 + 3R 2 + 3R + + 2 2 , R + 3 R n1 n2 which is ∆+ for ∆ 3/2and∆− 2 for ∆ 3/2. This is true if we are describing states with finite ∆ and R. Since we are studying the 3R 3R or 2 + 4 + , R + 1 , scaling limit ∆,R →∞,wehavetomodifytheabove 2 2 analysis. This constraint (6) comes from the fact that ∆ = R + 1. the energy eigenvalue of the Laplacian on M1,1,1 for the supergravity mode (4) takes the form One can construct the following gauge theory object, = 64 2 + 2 + 64 3 + corresponding to the short vector multiplet E n1 32n2 R 2 n1 + 3R + 3R 3 3 2 (1 2 )(2 2 ) , R + 3 ,TrΦ ≡ Tr(J ( )Φc), where 2 SU 2 + 32(R + 1)n2 + 16R(R + 3). (7) JSU(2)(x, θ) is a conserved vector current transforming in the adjoint representation of SU(2) flavor group and One can show that the conformal weight of the long Φc(x, θ) is a chiral superfield (5). There exists other vector multiplet A below becomes rational if the + 3R + 3R + condition (6) is satisfied. short multiplet by (2 2 )(4 2 ), R 1 . Simi- larly one can consider Tr Φ ≡ Tr(JSU(3)Φc), where JSU(3)(x, θ) is a conserved vector current transform- 6 The complete KK spectrum of the round S7 compactification ing in the adjoint representation of SU(3) flavor group. consists of short OSp(8|4) multiplets characterized by a +α + = 5 In this case, we have D Dα Φ(x,θ) 0. quantization of masses in terms of the SO(8)R-symmetry representation [63]. All the KK states are BPS states and their spectrum can be obtained by analyzing the short unitary irreducible representation of OSp(8|4). In the dual theory, all the composite primary operators have conformal weight equal to their naive 5 Of course there exist two short gravitino multiplets specified dimensions. No anomalous dimensions are generated. However, by the KK states with lower supersymmetry, for example, in 1,1,1 AdS4 × M do not fall into short multiplets of OSp(2|4) and do not necessarily have quantized masses. Their masses depend 3R 3R 1 + 3R 2 + 3R 2 + 4 + , R + 2 , 2 2 , R not only on the R-symmetry representation but also on the gauge 2 2 2 group SU(3) × SU(2) in the supergravity side (or flavor group in the dual field theory). This implies that according to AdS/CFT with conformal weight, ∆ = R − 1/2andR + 3/2, respectively. correspondence, anomalous dimensions are generated. 116 C. Ahn / Physics Letters B 540 (2002) 111–118

(1) One long vector multiplet A: enhanced N = 8 superconformal symmetry in the 3R 3R Penrose limit. This implies that the spectrum of the (1 + n1) 1 + n1 + 2 + 2n1 + 2 2 , gauge theory operators in this subsector should fall N = 2 into 8 multiplets. We expect that both the chiral 3 1√ primary fields of the form Tr(U 3V 2)R/2 (5) and the 2n + R + 1 ,∆=− + E + 36. 2 2 4 semi-conserved multiplets of the form (9) combine into make N = 8 multiplets in the limit. Note that for →∞ However, as we take the limit of R ,this finite R, the semi-conserved multiplets should obey − constraint (6) is relaxed. The combination of ∆ R the Diophantine constraint (6) in order for them to is given by possess rational conformal weights. 1 In the remaining multiplets we consider the follow- ∆ − R = n + n + O , (8) 1 2 R ing particular representations in the global symmetry group: where the right-hand side is definitely rational and 3R 3R they are integers. So the constraint (6) is not relevant 1 + n1)(1 + n1 + 2 + 2n1 + 2 2 , in the subsector of the Hilbert space we are interested 2 in. Candidates for such states in the gauge theory side are given in terms of semi-conserved superfields [54]. 2n2 + R + 1 . Although they are not chiral primaries, their conformal dimensions are protected. The ones we are interested (2) One long graviton multiplet h: in take the following form, 1 1√ ∆ = + E + 36. n1 n2 3 2 R/2 2 4 Tr Φs.c. ≡ Tr (JSU(3)) (JSU(2)) U V , (9) For finite R with rational dimension, after inserting where the scalar superfields JSU(3)(x, θ) transform the E into the above, we will arrive at the relation with in the adjoint representation of flavor group SU(3) same constraint (6) which is greater than (8) by 2: ±α ± and satisfy D D JSU(3)(x, θ) = 0 with conformal α 1 dimension 1 and zero U(1)R charge. Similarly, the ∆ − R = 2 + n1 + n2 + O . (10) scalar superfields JSU(2)(x, θ) transform in the adjoint R representation of the flavor group SU(2).Alsowe The gauge theory interpretation of this multiplet is +α + = have D Dα Φs.c.(x, θ) 0. Since the singleton su- quite simple. If we take a semi-conserved current ac perfields U carry indices a,c in the of SU(N) Φ (x, θ) defined in (9) and multiply it by a stress- i,bd ∗ s.c. and indices b,d in the of the SU(N),thefields energy tensor superfield Tαβ (x, θ) that is a singlet bdf ∗ VA,ace carry indices a,c,e in the of SU(N) with respect to the flavor group, namely Tr(Tαβ Φs.c.), and indices b,d,f in the of the SU(N), one we reproduce the right OSp(2|4) × SU(3) × SU(2) can construct the following conserved flavor currents representations of the long graviton multiplet. Also transforming (8, 1) and (1, 3) under SU(3) × SU(2), one can expect that other candidate for this multiplet respectively, with different representation by multiplying a semi- conserved current into a quadratic conserved scalar su- δj1 j1 = j1 − i1 perfield: Tr(JSU(3)JSU(3)Φs.c.),Tr(JSU(3)JSU(2)Φs.c.) (JSU(3))i U Ui1 UU, 1 3 or Tr(JSU(2)JSU(2)Φs.c.). In this case, the constraint j → + → δ 2 for finite ∆ and R is shifted as n1 n1 2, n1 j2 j2 i2 (JSU(2)) = V Vi − V V, n1 + 1,n2 → n2 + 1andn2 → n2 + 2, respectively. i2 2 2 (3) One long vector multiplet Z: where the color indices are contracted in the right- 1 1√ hand side. Note that the conformal dimension of these ∆ = + E + 32R + 36. currents is not the one of naive sum of U and U 2 4 and V and V . As we discussed in the last section, Although there exists no rational dimension for this 1,1,1 supergravity theory in AdS4 × M acquires an case with any choice of ni ’s when ∆ and R are finite, C. Ahn / Physics Letters B 540 (2002) 111–118 117 the combination of ∆ − R with Penrose limit R →∞ In addition to the above (1)–(5) multiplets, there are in the gauge theory side becomes also six long gravitino multiplets.7 1 ∆ − R = 3 + n1 + n2 + O . R 4. Conclusion Since we do not have any singlet of conformal dimension 3 with respect to the flavor group, one cannot in- We described an explicit example of an N = 2 crease a conformal dimension by simply tensoring any superconformal field theory that has a subsector of extra superfields into a semi-conserved current in or- the Hilbert space with enhanced N = 8 superconfor- der to match the spectrum. So the only way to do this mal symmetry, in the large-N limit from the study of 1,1,1 is to increase the number of conserved scalar super- AdS4 × M . The pp-wave geometry in the scal- field. In order to produce the following gauge theory ing limit produced to the maximally N = 8 super- × 7 operator Tr(Tαβ JSU(3)Φs.c.) or Tr(Tαβ JSU(2)Φs.c.) cor- symmetric pp-wave solution of AdS4 S .Theresult responding to this vector multiplet, one can think of a of this Letter shares common characteristic feature of 1,1,1 higher-dimensional representation in the global sym- previous case of AdS4 × Q [52]. This subsector metry SU(3) or SU(2). Then the constraint coming of gauge theory is achieved by Penrose limit which from the requirement of rationality of conformal di- constrains strictly the states of the gauge theory to mension is also changed for finite ∆ and R. One can those whose conformal dimension and R charge di- describe also the product of cubic J ’s and Φs.c. simi- verge in the large-N limit but possesses finite value larly. ∆ − R. We predicted for the spectrum of ∆ − R of (4) One long vector multiplet W: the N = 2 superconformal field theory and proposed how the excited states in the supergravity correspond 5 1√ ∆ = + E + 36. to gauge theory operators. In particular, both the chi- 2 4 ral multiplets (5) and semi-conserved multiplets (9) of In this case, we get ∆ − R by adding 2 to the one in N = 2 supersymmetry should combine into N = 8 (10) chiral multiplets. 1 ∆ − R = 4 + n1 + n2 + O . R Acknowledgements One can describe corresponding gauge theory operator αβ by taking quadratic stress-energy tensor Tαβ T (x, θ) This research was supported by grant 2000-1- and multiplying it into a semi-conserved current 11200-001-3 from the Basic Research Program of the Φs.c.(x, θ) in order to match with the conformal di- Korea Science and Engineering Foundation. αβ mension. That is, one obtains Tr(Tαβ T Φs.c.).Sim- ilarly one can construct the following gauge theory operators related to this vector multiplet References Tr(Tαβ JSU(3)JSU(3)Φs.c.),Tr(Tαβ JSU(3)JSU(2)Φs.c.) or Tr(T J J Φ ). For four J ’s with Φ , one [1] D. Berenstein, J. Maldacena, H. Nastase, JHEP 0204 (2002) αβ SU(2) SU(2) s.c. s.c. 013, hep-th/0202021. can analyze similarly. [2] M. Blau, J. Figueroa-O’Farrill, C.M. Hull, G. Papadopoulos, (5) One long vector multiplet Z: hep-th/0201081. 1 1√ ∆ = + E + 4. + 7 There exist three of them [55] χ characterized by ∆ =−1 + 2 4 √ √ 2 − 1 E + 16R + 32, or − 1 + 1 E + 16R and three of them χ Although there exists no rational dimension for this 4 2 √ 4 √ characterized by ∆ = 3 + 1 E + 16R + 32, or 3 + 1 E + 16R. case with any choice of ni ’s when ∆ and R are finite, 2 4 2 4 − →∞ Similar analysis can be done in this case. Although for finite ∆ the combination of ∆ R with Penrose limit R and R, both do not provide rational conformal dimensions, in in the gauge theory side becomes ∆ − R = 2 + n1 + the Penrose limit there is no constraint on the integer values and n2 + O(1/R). R →∞limit will give us a rational conformal dimension. 118 C. Ahn / Physics Letters B 540 (2002) 111–118

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Born–Infeld kinematics and correction to the Thomas precession

Frederic P. Schuller

DAMTP, University of Cambridge, Cambridge CB3 0WA, UK Received 4 April 2002; received in revised form 4 June 2002; accepted 14 June 2002 Editor: P.V. Landshoff

Abstract Dynamical symmetries of Born–Infeld theory associated with its maximal field strength are encoded in a geometry on the tangent bundle of spacetime manifolds. The resulting extension of general relativity respecting a finite upper bound on accelerations is put to use in the discussion of particle dynamics, first quantization, and the derivation of a correction to the Thomas precession.  2002 Elsevier Science B.V. All rights reserved.

Keywords: Born–Infeld; Pseudo-complex manifolds; Maximal acceleration; Relativistic phase space; Thomas precession; Non-commutative geometry

1. Introduction Over the last two decades, there has been some interest in and speculation on a finite upper bound on accelerations. This is mainly due to the fact In classical mechanics, the study of phase space that although special relativity allows arbitrarily high geometry yields deep insights which would be hidden accelerations, upon quantization, a finite upper bound in a mere configuration space formulation. In particu- enters through the back door [2]. This raises the lar, there is no well-defined distinction between coor- question of whether to use kinematics respecting a dinates and momenta, as these mix under symplectic distinguished acceleration from start. Indeed, careful transformations. quantization of a particle with dynamically enforced Geometric quantization aims at exploiting the sym- submaximal acceleration [3] nourishes the hope that plectic structure of phase space in order to understand a finite upper bound on accelerations might positively the transition to quantum systems. influence the convergence behaviour of field theoretic In contrast [1], special and general relativity are amplitudes. This approach, however, makes an ad-hoc formulated merely on spacetime. This is of course assumption of a maximal acceleration and admittedly likewise true for all theories built on this framework, lacks a proper kinematical framework. most notably quantum field theory and string theory. The aim of this Letter is to devise such kinematics, by kinematizing dynamical symmetries of the Born– Infeld action on the velocity phase space of the spacetime manifold, and to derive a correction to the E-mail address: [email protected] (F.P. Schuller). Thomas precession within this framework.

2. Kinematization is equivalent to restricting the acceleration of the projection π(x∗) = x to values less than a.Thiswas A particle of mass m and electric charge e mini- first observed by Caianiello [5] for flat spacetime. mally coupled to Born–Infeld theory [4] L = 1/2 + BI det (gµν bFµν) (1) 4. Complex tangent bundles? can at most experience an acceleration a = eb−1m−1, as the maximal field strength is given by b−1. There were attempts [7,8] to devise a maximal ac- Denoting a point of the tangent bundle TM of celeration modification of special and general relativ- the n-dimensional spacetime manifold M by Xm ≡ ity by equipping the tangent bundle with the metric gD (axµ,uµ),whereuµ is the n-velocity of the particle, and an additional complex structure F , analogous to the Born–Infeld Lagrangian (1) can be written the phase space structure of non-relativistic systems. However, invoking a strong principle of equivalence, L = 1/4 [ m n] BI det X ,X , (2) one must require both structures to be covariantly con- if one assumes a b2-suppressed coordinate non-com- stant, mutativity of spacetime in the presence of an electro- ∇gD = 0and∇F = 0, (8) magnetic field F µν , where ∇ is the Levi-Cività connection with respect [ µ ν]=− −3 2 µν x ,x ie b F , (3) to gD . Then according to the Tachibana–Okumura [xµ,pν ]=−igµν , (4) theorem [9], conditions (8) are satisfied if, and only [pµ,pν ]=−ieFµν. (5) if, the base manifold M is flat. Hence complex tangent bundles can never provide Associated with the existence of a distinguished ac- a theory of gravity with finite upper bound on acceler- celeration a there must be dynamical symmetries of ations. Born–Infeld theory. The form (2) suggests encod- ing these in the geometry of the spacetime tangent bundle. We show that this indeed results in a con- 5. Bimetric tangent bundles sistent kinematical framework that extends relativity such as to respect a finite upper bound on accelera- The Tachibana–Okumura no-go theorem can be tions. Note that shifting the upper bound a to infin- circumvented by using the horizontal lift (here given ity restores coordinate commutativity. Hence space- in induced tangent bundle coordinates (xµ,uµ)) time non-commutativity is a signature of a finite upper bound on accelerations. ua∂ g g gH = a ij ij (9) gji 0 of the spacetime metric instead of a complex structure. 3. Maximal acceleration geometry It can be shown [6] that the horizontal lift ∇H of the Levi-Cività connection on spacetime is then the Consider the diagonal lift [6] (here given in induced unique linear connection on TM satisfying tangent bundle coordinates (xµ,uµ)) H D H H + t s t ∇ g = 0and∇ g = 0 (10) D = gij gtsΓ iΓ j Γ j gti g t (6) Γ igtj gij for arbitrary curvatures of the base manifold M.This, of the spacetime metric g to the tangent bundle, where however, comes at the cost of introducing torsion to t a t the tangent bundle. A tangent bundle curve X : R → Γ i ≡ u Γa i ,andΓ are the Christoffel symbols of ∗ = g. Requiring positivity of the natural lift [6] x ≡ TMis called an orbit if there exists a frame where X µ ∗ (axµ, dx ) of a timelike spacetime curve x, π(X) . An orbit is called an orbidesic if it is a geodesic dτ with respect to some metric, or an orbiparallel if it is ∗ ∗ gD(dx ,dx )>0 (7) an autoparallel with respect to some connection. F.P. Schuller / Physics Letters B 540 (2002) 119–124 121

The lifting and projection properties of spacetime For non-gH -geodesic motion, the metric gD as- geodesics and tangent bundle orbidesics and orbipar- sumes a non-trivial role, as it restricts the deviation of allels are well known [6] and summarized in the dia- orbits from gH -orbidesics to the gD-positive ones. gram Condition (11) enforces the orthogonality of the n-velocity and n-acceleration of a particle in any frame. Thus we now recognize that in standard general relativity already, the horizontal lift gH is a naturally induced structure on the associated tangent bundle. Postulate III predicts a deviation from the relativistic physical time for accelerated particles. From decay experiments [10] we get a lower bound for the maximal acceleration a of about 1019 ms−2. From high- precision measurements of the Thomas precession, we get a better lower bound in Section 10. In particular, note that gH -orbidesics coincide with ∇H -orbiparallels, despite ∇H having non-vanishing H torsion [6]. Further, one can show that all g -orbi- 7. Field equations desics are gH -null and gD-positive. It is conceptually inevitable to formulate the extended theory entirely on the tangent bundle without 6. Extended general relativity recurring to spacetime objects. The latter are viewed as derived concepts via the canonical bundle projec- The mathematical structure outlined above allows tion. In this spirit we find [11] the lifted Einstein field one to formulate physical postulates for the kinematics equations of an extension of general relativity respecting a finite 1 upper bound on accelerations. Gambn − Gabmn RV = 8πGTD , (14) 2 mn mn I. Submaximally accelerated particles are described where RV denotes the vertical lift [6] of the spacetime by orbits X satisfying Ricci tensor R,and

gH (dX, dX) = 0, (11) Gabcd ≡ gDab gDcd + gHab gHcd . (15) gD(dX, dX) > 0. (12) The lifted equations (14) are equivalent to the Einstein ﬁeld equations on spacetime. II. In the absence of non-gravitational interaction, orbits of particles are gH -(null-)geodesics. III. The physical time experienced by an observer 8. Extended special relativity with orbit X is measured by The case of ﬂat Minkowski spacetime can be ≡ D 1/2 dω g (dX, dX) . (13) studied in a very concise and illuminating way. The diagonal and horizontal lifts of the Minkowski metric H Note that g -orbidesics on TM coincide with g-geo- η can be written as desics on M, and are automatically gD -positive. Hence, for unaccelerated motion, the kinematics of standard ηD = η ⊗ 1andηH = η ⊗ I, general relativity are recovered. This explains why for with accelerations that are small compared to the upper limit a, we only need one single metric g on space- 10 01 1 ≡ ,I≡ . (16) time. 01 10 122 F.P. Schuller / Physics Letters B 540 (2002) 119–124

This motivates the study of the pseudo-complex num- can always arrange for this motion to be in 1-direction bers in a rest frame at coordinate time zero. As dω is P ≡ + ∈ R 2 =+ an SOP(1, 3)-invariant, we may study the action of a Ib a,b ,I 1 . ≡ dX ≡ a pseudo-boost on the covariant velocity U dω As these build a commutative ring, pseudo-complex u˜ + Ia˜. For hyperbolic motion of spacetime curva- Lie algebras can be defined [13]. The pseudo-com- ture g ≡ a tanh(α), the projections π01 and π10 of plexification V of a real vectorspace V is a free U into the u˜0–a˜ 1 and u˜1–a˜ 0 planes, respectively, are module of pseudo-complex dimension dimR V ,and straight lines through the origin of hyperbolic angle ∼ − we have the isomorphism of VP = TV as real vec- 1 g tanh ( a ): torspaces. If V is a representation space for a real Lie g algebra L,thenLP acts naturally on VP. a˜ 0 = u˜1, (20) We are particularly interested in the pseudo-com- a g plexification of Minkowski spacetime (Rn,η) and the a˜ 1 = u˜0. (21) real Lorentz group SOR(1,n−1). On the algebra level a one gets A pseudo-boost with boost parameter Iβ can be easily seen to hyperbolically rotate both of these − = µν = µν µν soP(1,n 1) M P M ,IM R (17) lines by an angle β within their respective planes. and obtains the connection component of the identity Hence, the π01 and π10 projections of the transformed of the pseudo-complex Lorentz group by exponentia- curve coincide with the projections for a U-curve µν ≡ 1 µν + tion. Changing the basis in (17) to G 2 (M corresponding to a hyperbolic motion with spacetime µν µν ≡ 1 µν − µν curvature a tanh(α + β). Carefully counting degrees IM ) and G 2 (M IM ) we get the decomposition of freedom, one checks that the transformation of the two projections already determines the transformation = ⊕ soP(1,m) soR(1,m) soR(1,m). of the whole U-curve. Hence, the pseudo-boosts are Thus, the representation theory in the pseudo-complex transformations to relatively uniformly accelerated case can be easily obtained from the real case. frames, respecting the maximal acceleration a. Clearly, for a pseudo-complex n-vector U µ ≡ uµ + A pseudo-complex Lorentz transformation of an or- Iaµ, the expression bit X clearly induces a transformation of the spacetime projection π(X). For not purely real transforma- η(U,U) ≡ U µU ν η µν tion parameters, however, these transformations can- µ µ µ = u uµ + a aµ + I 2u aµ (18) not be understood as maps M → M, simply because : → is SOP(1,n− 1)-invariant, separately in the real and the components projected out by π TM M con- pseudo-imaginary parts. This shows the isomorphism tribute to the transformation. of In other words, spacetime events fail to be well- defined under non-real Lorentz transformations. Hence, n D H ∼ n T R ,η ,η = P ,η (19) we observe the breakdown of the classical space- as inner product spaces. Hence, in the flat case we time particle concept when changing to an accelerated can trade a bimetric real vectorspace against a metric frame. This anticipates the Unruh effect [12] on a clas- module. Essentially, special relativity with pseudo- sical level already. complex coordinates is extended special relativity. In particular, according to (17) all pseudo-complex Lorentz transformations can be composed of the stan- 9. Dynamics dard boost and rotation transformations with pure real and pure pseudo-imaginary arguments. For the inter- Prior approaches [14] to maximal acceleration dy- pretation of the pseudo-boosts, it is instructive to con- namics enforce the finite upper bound on accelerations sider the orbit X induced by a spacetime curve de- dynamically, i.e., by modified Lagrangians, but still in scribing a submaximally accelerated hyperbolic mo- the kinematical framework of special or general relation. Using the Lorentz invariance of the theory, we tivity. The prototypical example is the massive particle F.P. Schuller / Physics Letters B 540 (2002) 119–124 123 studied by Nesterenko et al. [15], of the spatial coordinate system of an orbiting ob- 2 2 1/2 server, where γSR = (1 + R ω ) . Exactly along the µ 2 µ 2 L = x¨ x¨µ − a x˙ x˙µ dt . (22) same lines, the non-commutativity of the pseudo-boost generators This dynamical enforcement inevitably results in La- grangians containing second order derivatives, being IM0i,IM0j = Mij (27) inconvenient for technical and conceptual reasons, especially in the transition to quantum theory. leads to an additional precession around the same axis, In contrast, pseudo-complexification of merely rel- effecting a total precession rate ativistic Lagrangians, e.g., for the massive relativistic dθESR = − particle to (γESR 1)ω, (28) dt 2 2 2 4 −2 1/2 ˙ µ ˙ where γ = (1 + R ω + R ω a ) .Fromex- LP = X Xµ dt, (23) ESR perimental data [17], obtained at Rω2 = 1018 ms−2, where γSR = 1.2, the ratio γESR/γSR deviates from unity by × −9 X ≡ ax + Iu, (24) less than 5 10 . This yields a lower bound 22 −2 results in first order Lagrangians. From relation (24) a 10 ms (29) this might appear to be a mere notational trick. How- and hence an upper bound on the Born–Infeld parame- ever, this is not the case as u and x are independent ter degrees of freedom in extended relativity, only linked − C by the pseudo-complex kinematics, in particular, the b 10 11 . (30) orthogonality condition (11). N Studying the equations of motion for LP and mak- Thus high-precision measurements of the Thomas ing full use of the pseudo-complex Lorentz symmetry, precession might be able to discriminate between we indeed get a free particle [11]. A fully SOP(1, 3)- Maxwell and Born–Infeld electrodynamics in the fu- invariant coupling term to Born–Infeld theory can be ture. constructed from the diagonal lift of the Kaluza–Klein metric from spacetime to the tangent bundle [11]. It turns out that mere pseudo-complexification of the Acknowledgements standard minimal coupling term achieves the same with much less labour. I would like to thank Gary Gibbons for very helpful Pseudo-complexification applies to relativistic discussions and remarks on the material of this paper. spaces, symmetry algebras and Lagrangians alike I have also benefited from remarks by Paul Townsend to give their extended relativistic counterparts. This and discussions with Sven Kerstan. This work is makes the pseudo-complex formalism so worthwhile. funded by EPSRC and Studienstiftung des deutschen Vo l ke s.

10. Correction to Thomas precession References

Due to the non-commutativity of the boost genera- [1] M. Born, Proc. R. Soc. A 165 (1938) 291. tors [2] H.E. Brandt, Found. Phys. Lett. 2 (1) (1989). [3] V.V. Nesterenko, A. Feoli, G. Lambiase, G. Scarpetta, Phys. M0i,M0j = Mij (25) Rev. D 60 (1999) 065001, hep-th/9812130. [4] M. Born, L. Infeld, Proc. R. Soc. London A 144 (1934) 425. in special relativity, an observer resting at the center [5] E.R. Caianiello, Lett. Nuovo Cimento 32 (1981) 65. of a uniform circular motion of angular velocity ω and [6] K. Yano, S. Ishihara, Tangent and Cotangent Bundles, Marcel radius R observes the Thomas precession [16] Dekker, New York, 1973. [7] S.G. Low, J. Math. Phys. 38 (1997) 2197. dθSR [8] H.E. Brandt, Found. Phys. Lett. 4 (6) (1991). = (γ − 1)ω (26) [9] S. Tachibana, M. Okumura, Tohoku Math. J. 14 (1962) 156. dt SR 124 F.P. Schuller / Physics Letters B 540 (2002) 119–124

[10] Farley et al., Nuovo Cimento 45 (1966) 281. [15] V.V. Nesterenko, A. Feoli, G. Scarpetta, J. Math. Phys. 36 (10) [11] F.P. Schuller, Ann. Phys. 299 (2002) 174, hep-th/0203079. (1995) 5552. [12] W.G. Unruh, Phys. Rev. D 14 (1968) 870. [16] L.H. Thomas, Nature 117 (1926) 514; [13] S. Lang, Algebra, Springer Graduate Texts in Mathematics, L.H. Thomas, Philos. Mag. 3 (1927) 1. Springer, Berlin, 2002. [17] D. Newman et al., Phys. Rev. Lett. 40 (21) (1978) 1355. [14] V.V. Nesterenko, J. Phys. A: Math. Gen. 22 (1989) 1673. Physics Letters B 540 (2002) 125–136 www.elsevier.com/locate/npe

Axisymmetric gravitational solutions as possible classical backgrounds around closed string mass distributions

Hitoshi Nishino, Subhash Rajpoot

Department of Physics & Astronomy, California State University, Long Beach, CA 90840, USA Received 26 April 2002; received in revised form 9 June 2002; accepted 12 June 2002 Editor: H. Georgi

Abstract By studying singularities in stationary axisymmetric Kerr and Tomimatsu–Sato solutions with distortion parameter δ = 2, 3,... in general relativity, we conclude that these singularities can be regarded as nothing other than closed string- like circular mass distributions. We use two different regularizations to identify δ-function type singularities in the energy– momentum tensor for these solutions, realizing a regulator independent result. This result gives supporting evidence that these axisymmetric exact solutions may well be the classical solutions around closed string-like mass distributions, just like Schwarzschild solution corresponding to a point mass distribution. In other words, these axisymmetric exact solutions may well provide the classical backgrounds around closed strings.  2002 Elsevier Science B.V. All rights reserved.

PACS: 04.20.F; 11.27; 04.70

Keywords: Exact solutions; General relativity; Singularities; Closed strings

1. Introduction

Since its first discovery in general relativity in 1916 [1] it has been well understood that the Schwarzschild solution describes the classical gravitational background around a point mass. If there is a mathematically rigorous ‘point mass’ in reality, it is natural to expect a basic and simple classical solution in general relativity that describes the spacetime around such a ‘point mass’. On the other hand, developments in string theory [2] tell us that such a ‘point mass’ description of a particle is unrealistic due to divergent self energy, but instead, it must be replaced by a more ‘smoothed’ one, like a circular mass distribution in closed string theory [2]. However, if that is really the case, and such a circular mass distribution is a fundamental physical entity, it seems unnatural that no basic exact classical solution has ever been discovered as the fundamental classical background around such a circular mass distribution, as an improved version of Schwarzschild solution, or as more physically realistic mass distributions. For example, Chazy–Curzon solution [3] gives a simplest metric around a static axisymmetric mass distribution. However, its actual mass distribution is not at a finite radius, but is confined

E-mail addresses: [email protected] (H. Nishino), [email protected] (S. Rajpoot).

0370-2693/02/$ – see front matter  2002 Elsevier Science B.V. All rights reserved. PII:S0370-2693(02)02128-7 126 H. Nishino, S. Rajpoot / Physics Letters B 540 (2002) 125–136 to the origin as a certain limit. What is missing is the classical background solution around a circular closed string like mass distribution with a finite radius. The idea of studying classical solutions in general relativity around singular mass distributions is also motivated by the recent developments in brane theory known as braneworld scenario [4,5], or Randall–Sundrum scenario [6] in which the matter distributions are limited to the four-dimensional (4D) brane hypersurface embedded into 5D with δ-function type singularities. Such singularities on submanifolds or boundaries play an important role in brane theories, inducing a special effect such as desirable mass hierarchies crucial to phenomenology in 4D [6], or embedding 10D supergravity into 11D [4]. As a matter of fact, some ideas relating the singularities in the Kerr solution in general relativity [7] to strings [2] have been proposed in [8], or a new concept of Bekenstein–Hawking black hole entropy interpreted microscopically in terms of elementary string excitations has been presented [9]. From these developments, it seems natural to consider the physical significance of the singularities in stationary axisymmetric exact solutions in vacuum in general relativity, in particular, their possible link with singular mass distributions that resemble closed strings. Some of these exact solutions may well be analogous to Schwarzschild solution [1] that corresponds to a point mass distribution. In our present Letter, we try to show that the series of axisymmetric solutions in vacuum, starting with the Kerr solution [7] and the Tomimatsu–Sato (TS) solutions [10,11] with the distortion parameter δ 2 seem to describe nothing other than the classical backgrounds around circular mass distributions with coaxial radii with naked singularities for q>1 [10], resembling closed strings themselves. To put it differently, we try to show that the naked singularities in TS solutions with δ 2 [10,11] can be interpreted as closed string-like mass distributions. As a methodology, we adopt regularization schemes similar to that in [12], i.e., we use Yukawa-potential type regularizations with an exponential damping factor, that give the right coefficients for the δ-function like mass distributions. In Ref. [12], this regularization was applied successfully to the cases of Schwarzschild and Kerr solutions. In our present Letter, we generalize this result to the more general case of TS solutions with the distortion parameter δ = 2, 3,... [10,11]. We use two types of regulators: the first one is equivalent to that in [12], while the second one is its slight modification. By comparing the results of these two regulators, we will demonstrate no regulator dependence in our results, which may be taken as the correctness of our physical interpretation of the ring singularities as closed string-like circular mass distributions.

2. A preliminary with circular charge distribution for Coulomb potential

We first analyze the case of circular charge distribution for the case of an electric field with a Coulomb potential, as a preliminary for the later case of TS solutions in general relativity. Consider a circular charge distribution at radius ρ = a on the equatorial xy-plane with constant line charge density σ in cylindrical coordinates (ρ,ϕ,z).We need to get the Coulomb potential φ(ρ,ϕ,z) around such a distribution, satisfying the Laplace equation ∆3φ = 0. Next, given such a result for φ(ρ,ϕ,z), we want to reconstruct the original charge distribution, which should contain δ(ρ − a)δ(z). The latter scenario is highly non-trivial, because if we blindly substitute the result for φ(ρ,ϕ,z) back into the Laplace equation, we get zero everywhere, by definition. What is required here is the careful treatment of the singularity at ρ = a and z = 0. In other words, we need a good regularization to handle the singularity. There are several regularizations to treat such singularities. In this Letter, we adopt finite mass regularizations, based on the 3D ‘massive’ Laplace equation 2 ∆3φ − κ φ = 0, (2.1) that replaces the original ‘massless’ Laplace equation. The Coulomb potential φ(ρ,ϕ,z) around the circular charge distribution can be obtained by the use of a massive Green’s function of the Yukawa-potential type e−κr/r,wherer =[(ρ cosϕ − a cosϑ)2 + (ρ sin ϕ − a sin ϑ)2 + z2]1/2 is the distance from the charge between the azimuthal angles ϑ ∼ ϑ + δϑ on the circle ρ = a on the xy-plane and the point (ρ cosϕ,ρ sin ϕ,z). Let us call such a small contribution δφ to the total potential φ.The H. Nishino, S. Rajpoot / Physics Letters B 540 (2002) 125–136 127 total potential φ is obtained by superimposing δφ over 0 ϑ<2π at ρ = a:

2π σaexp(−κ (ρ cos ϕ − a cosϑ)2 + (ρ sin ϕ − a sin ϑ)2 + z2 ) φ(ρ,ϕ,z)= δφ = dφ = dϑ (ρ cosϕ − a cosϑ)2 + (ρ sin ϕ − a sin ϑ)2 + z2 0 2π exp(−κ ρ2 + z2 + a2 − 2aρ cosϑ) = σa dϑ , (2.2) ρ2 + z2 + a2 − 2aρ cosϑ 0 where the ϕ-dependence has disappeared in the final expression, because of the periodicity of the original integrand under ϑ − ϕ → ϑ − ϕ + 2π, also consistent with the axial symmetry. Due to its complicated nature, the final result in a closed form is more involved than elliptic integrals, and thus the ϑ-integral here is not analytically performed. Given such a Coulomb potential φ(ρ,z) in (2.2), our next question is whether we can ‘reconstruct’ the original charge distribution singular at ρ = a, z = 0. As has been mentioned, putting simply κ = 0 from the outset will 3 lead to a vanishing result by definition. One method to avoid this is to compute d x∆ 3φ by equating it with 3 2 2 d xκ φ, under the massive Laplace equation ∆3φ − κ φ = 0:

2π exp(−κ ρ2 + z2 + a2 − 2aρ cosϑ) I(κ)≡ d3xκ 2φ = σa d3x dϑ κ2 ρ2 + z2 + a2 − 2aρ cosϑ 0 ∞ ∞ 2π 2π exp(−κ ρ2 + z2 + a2 − 2aρ cosϑ) = σaκ2 dρ dz dϕ dϑ ρ . (2.3) ρ2 + z2 + a2 − 2aρ cosϑ 0 −∞ 0 0 The question now is whether or not the integral I(κ)in (2.3) gives a non-vanishing ﬁnite result that corresponds to the right total charge q ≡ 2πaσ, after the limit κ → 0+. To this end, we ﬁrst change the coordinate variables from (ρ,ϕ,z)to (r,θ,ϕ), such that its Jacobian equals the familiar value r:

2π ∞ π √ exp(−κ r2 + a2 sin ϑ) I(κ)= 2πσaκ2 dϑ dr dθ r (r cosθ + a cos ϑ) r2 + a2 sin2 ϑ 0 0 0 2π ∞ √ r exp(−κ r2 + a2 sin ϑ) = 2πσaκ2 dϑ dr (2r + 2πacosϑ). (2.4) r2 + a2 sin2 ϑ 0 0 Here we have performed the θ-integral. We have developed in [12] the following lemma about the r-integration such as in (2.4) for m +1: ∞ − 0 (m = 0, −1, −2,...), + κ2 dη(η + 1)me κη → as κ → 0 . (2.5) 1 (m =+1), 0 √ To use this lemma in our case, we use a new variable η ≡ r2 + b2 − b(b≡ a sin ϑ), and expand the integrand around η =−1. Thus the ﬁrst term in (2.4) is

2π ∞ √ 2r2 exp(−κ r2 + b2 ) 2πσaκ2 dϑ dr √ r2 + b2 0 0 128 H. Nishino, S. Rajpoot / Physics Letters B 540 (2002) 125–136

2π ∞ 2 η + 2bη − + = 4πσaκ2 dϑ dη(η + b) e κ(η b) η + b 0 0 2π ∞ − b − 1 − − = 4πσaκ2 dϑ e κb dη(η + 1) 1 + + O (η + 1) 2 e κη η + 1 0 0 2π 2π − = 4πσa dϑ e κasinϑ + O(κ) = 4πσa dϑ (1 − κasin ϑ)+ O(κ)

0 0 + → 8π2σa as κ → 0 . (2.6) Similarly, the second term in (2.4) can be shown to be zero after κ → 0+, and, therefore, we get the desirable result

I(κ)→ 8π2σa = 4π(2πaσ)= 4πq, (2.7) for the total charge q ≡ 2πaσ on the circle. Since we have obtained a non-vanishing finite result for I(κ), it is natural to identify the integrand with a combination of δ-function such as δ(ρ − a)δ(z). However, there are a few caveat for such an identification. For example, we have to make sure that the integrand before the d3x-integration vanishes almost everywhere except for (ρ, z) = (a, 0). We can itemize such criterions for the identification with the δ-functions, as follows, by letting f(ρ,z; κ) be the integrand in I(κ)≡ d3xf(ρ,z ; κ):

(i) lim f(ρ,z; κ) = 0, (2.8) + ρ =a,z =0 κ→0 (ii) lim lim f(ρ,z; κ) = 0, (2.9) + → z =0 κ→0 ρ a ; = (iii) lim lim f(ρ,z κ) = 0, (2.10) κ→0+ z→0 ρ 0 (iv) lim lim f(ρ,z; κ)=∞, (2.11) → + → → κ 0 ρ a,z 0 (v) lim d3xf(ρ,z ; κ)= 8π2σa. (2.12) κ→0+ Eq. (2.8) is to guarantee that the integrand is almost everywhere zero, while (2.9), (2.10) and (2.11) imply that the integrand has a singularity only at points satisfying both ρ = a and z = 0, i.e., the circle with the radius a from the origin on the equatorial xy-plane. Eq. (2.12) has been already satisfied by (2.6). Now, (2.8) is manifestly satisfied, because there is no other singularities in f(ρ,z; κ) other than the points on (ρ, z) = (a, 0). For the same reason, Eqs. (2.9) and (2.10) are also satisfied. The remaining one (2.11) can be also confirmed as

π/4 exp(−κasin x) lim f(ρ,z; κ)= 4κ2 dx =∞, (2.13) ρ→a,z→0 2a sin x 0 for x ≡ ϑ/2. This is due to the logarithmic divergence at x → 0. Since the criterion (i) through (v) are satisﬁed, we can identify

2π exp(−κ ρ2 + z2 + a2 − 2aρ cosϑ) lim κ2ρ dϑ = 4πδ(ρ − a)δ(z). (2.14) κ→0+ ρ2 + z2 + a2 − 2aρ cosϑ 0 H. Nishino, S. Rajpoot / Physics Letters B 540 (2002) 125–136 129

3. Ring singularities and circular mass distributions for TS-solutions

Once we have understood the basic case of circular charge distribution for Coulomb potential, it is much easier to handle similar singularities in TS solutions [10,11]. As has been mentioned, we will adopt the previous method in [12], which is also consistent with the criterions above. The Kerr metric [7] is classiﬁed as a generalization of the Schwarzschild metric [1] due to rotation, while Weyl metric [13] is another generalization due to deformation. The generalizations due to both rotation and deformation yield TS metrics [10,11] with the distortion parameter δ = 2, 3,.... In particular, the case δ = 1 of the TS metric [10] is equivalent to the Kerr metric [7], while in the cases of δ 2 there are naked ring singularities lying outside non-singular event horizons. These ‘naked’ ring singularities outside event horizons serve as actual circular mass distributions in our work. We start with the metric of axisymmetric TS solutions with the general distortion parameter δ = 1, 2,...[10]:

ds2 =−e2ν(dt)2 + m2e2ψ (dϕ − Ωdt)2 + m2e2µ2 (dx)2 + m2e2µ3 (dy)2, (3.1)

p2(x2 − 1)B (1 − y2)D 2δ2qC e2ν ≡ ,e2ψ ≡ ,Ω≡− , D δ2B mD B B e2µ2 ≡ ,e2µ3 ≡ , (3.2) δ2p2δ−2(x2 − 1)(x2 − y2)δ2−1 δ2p2δ−2(1 − y2)(x2 − y2)δ2−1 where A, B, C and D are, respectively, the polynomials of x and/or y with the respective degrees 2δ2,2δ2,2δ2 − 1 and 2δ2 + 2. The ring singularities exist as the zeros of the algebraic equation B = 0 [10,11], lying on the y = 0 equatorial plane at ﬁnite coaxial radii x = x1,x2,...,xδ which are the zeros of B|y=0 = 0. Our task is to show that the 00 corresponding energy–momentum tensor T have δ-function type ring singularities at radii x = x1,...,xδ,onthe equatorial plane at y = 0. To this end, we look into the corresponding Einstein tensor G00 ≡ R00 − (1/2)g00R, with an appropriate regularization like that in [12]. The ‘regularized’ total energy or mass for such an exact solution is given in a closed form by [14] √ √ 1 1 P 0 = −gT00 dS = −ge−νe 0 R(0)(0) + R dS 0 (0) 8π 2 0 1 ∞ m − −˜ ˜ − −˜ ˜ − −˜ −˜ = dy dx −eψ ν e µ2 eµ3 − eψ ν e µ3 eµ2 − e ν e µ3 µ2 eψ 4 x x y y x x − 1 x0 − −˜ −˜ 1 −˜ +˜ − 1 +˜ −˜ − − e ν e µ2 µ3 eψ − e3ψ µ2 µ3 3ν Ω2 − e3ψ µ2 µ3 3ν Ω2 . (3.3) y y 4 x 4 y

The x0 is the lower limit of the x-integration [10]. All subscripts of coordinates denote differentiations with respect ψ ψ ψ to the coordinates, e.g., (e )y ≡ ∂y e ≡ ∂e /∂y [14]. Following Ref. [12], we have regularized the functions µ2 and µ3 by a Yukawa-potential type damping factor, denoted by µ˜ 2 and µ˜ 3: ˜ ˜ − eµ2 ≡ eµ2 S, eµ3 ≡ eµ3 S, S ≡ 1 + bαe αξ, (3.4) where b is a non-zero real constant independent of α>0, to be fixed shortly. Since we are interested in the case a>m(equivalently q>1) to see naked singularities in TS solutions, we use also x =+ix,ˆ p =−ipˆ appropriately [10,11,15]. The ξ in (3.4) is defined by ξ ≡ˆx −ˆx0 ≡−i(x − x0) with the range 0 ξ<∞. We use in this Letter the exponent −αξ in the regulator mimicking the previous Coulomb case (2.2) or as in [12]. However, in the next section, we will use a slightly modified regulator, in order to see possible regulator dependence. If we take the limit 130 H. Nishino, S. Rajpoot / Physics Letters B 540 (2002) 125–136

α → 0+ before the x and y-integrations, we simply get zero, satisfying the basic criterion like (2.8) in the Coulomb case. Since the regulator S in (3.4) is inserted only into eµ2 and eµ3 , the effect of the regulator in the integrand will arise only when the derivatives ∂x and/or ∂y hit the regulator S at least once: 1 ∞ m − −˜ ˜ − −˜ ˜ P 0 =+ dy dξ +eψ ν e µ2 eµ3 − eψ ν e µ3 eµ2 4 ξ ξ y y −1 ξ0 1 ∞ m − − − − − − =+ dy dξ − eψ ν e µ2 S 1 eµ3 S + eψ ν e µ3 S 1 eµ2 S 4 ξ ξ y y −1 0 1 ∞ m − − − ξ=∞ m − − − y=1 + dy eψ µ2 ν S 1 eµ3 S − dξ eψ µ3 ν S 1 eµ2 S . (3.5) 4 ξ ξ=0 4 y y=−1 −1 0

In the first term we have a sign flip caused by the imaginary unit in ξ ≡ˆx −ˆx0 ≡−i(x − x0).Wehavealso performed partial integrations under the x and y-integrations. The last two terms on the boundaries have no contributions when α → 0+, since they are at O(α2). The second term in (3.5) is seen to be zero because at least one derivative ∂y should hit S which in turn is independent of y, yielding a zero contribution. This is because unless there is a y-derivative on S, there will be a cancellation between S−1 and S, which is exactly the same as the non-regularized case. The only term remaining in (3.5) is 1 ∞ 2 2 m 1 1 − y D x − 1 − P 0 =− dy dξ, S 1S , (3.6) 2 − − 2 ξ 4 δp x 1 B ξ 1 y −1 0 which can be shown to be non-zero and finite. To this end, we need a special lemma for the factor eψ−ν involving B and D for a general δ: 2 2− + 2+ 2+ B = p2δx2δ + O x2δ 1 ,D= p2δ 2x2δ 2 + O x2δ 1 , (3.7) justified by the solution for a general δ for the polynomial A [10]: δ D 2 2 r+1 A ≡|α| −|β| = 2 (−1) d + f c F = , (3.8a) r 1 r δ,r r ∆ r=1 D ≡ r D ≡ fr+r −1 ≡ 1 Fr , det ,∆det , (3.8b) ∆r r + r − 1 r + r − 1 + − ! − !! 2r−1 δ(δ r 1) (2r 3) 2 2 r 2 2 r cδ,r ≡ 2 ,dr ≡ ,fr ≡ p (x − 1) + q (y − 1) , (3.9) (δ − r)!(2r)! (2r − 2)!! where Dr (or ∆r ) is the co-factor of the determinant D (or ∆)forthe(1,r)component. We can prove that only the term with D1, among δ terms in the sum in (3.8), has the highest power in x: 2δ−2 2δ2−2 2δ2−3 D1 ≈ p x + O(x ). Note that Dr is proportional to 1 i ···ir ···iδ Dr = 7 1 D i D i ···Dri ···Dδi (δ − 1)! 1 1 2 2 r δ ······ (1+2+···+ˆr+···+δ)+(i +i +···+i+···+i )−(δ−1) ≈ 7i1 ir iδ O x2 1 2 r δ , (3.10) H. Nishino, S. Rajpoot / Physics Letters B 540 (2002) 125–136 131

2 2 2+i −1 where Dij is the ij -component of the determinant matrix in (3.8b), e.g., D2i ≈ O(p (x − 1) 2 + + − + − 2 q2(y2 − 1)2 i2 1) ≈ O(x2 i2 1), etc., while a ‘hatted’ factor or term is to be skipped in the product or sum. Now it is clear that the highest power of x arises only in the case of r = 1 with i1 = 1, so that 2 δ−1 2 2(δ−1)(δ−2)/2−δ+1 2δ−2 2δ2−2 D1 ≈ p x ≈ p x . (3.11)

Therefore, only the term with F1 in (3.8a) has the highest power: 2 2− A ≈ p2δx2δ + O x2δ 1 . (3.12) Recalling other relevant relationships [10,11], such as A ≡|α|2 −|β|2 = u2 + v2 − m2 − n2,B≡ + m)2 + (v + n)2, α ξ ≡ ,α≡ u + iv, β ≡ m + in (u,v,m,n∈ R), β δ 2 2 2 2 2 2 2 2 G ≡|β| = cδ,rFr ,DA= p x − 1 B − 4δ q 1 − y C , (3.13) r=1 we see that 2 2δ−2 2δ2−2 2δ2−3 2 δ−1 δ2−1 G ≈ cδ,1F1 ≈ δ p x + O x ≈|β| ⇒ β ≈ δp x ≈ m, 2 2 A ≈ u2 − m2 ≈ p2δx2δ ⇒ u ≈ pδxδ , (3.14)

2 2 2 2 p x B + 2+ B ≈ + m)2 ≈ p2δx2δ ,D≈ ≈ p2δ 2x2δ 2, (3.15) A yielding (3.7). The symbol ≈ denotes the omission of lower power of x. We now use these leading terms for the evaluation of (3.6): 1 ∞ 2 bmpαˆ − P 0 = dy dξ (ξ + ξ )2 + 1 e αξ 1 + O(α) 4δ 0 −1 0 1 ∞ 2 bmpαˆ − bmpˆ = dy dξ (ξ + 1) + O (ξ + 1)0 e αξ + O(α) → , (3.16) 4∆ 2δ −1 0 as α → 0+, where we have relied on the lemma (2.5). This gives the total mass in (3.5) lim P 0 = m, (3.17) α→0+ agreeing with the asymptotic mass [16] after the identification 2δ b = . (3.18) pˆ Our remaining task is to confirm the criterions for identification of the integrand with a desirable combination of δ-functions, like (2.8)–(2.12) in the Coulomb case. Our present ones for the integrand g(x,y; α) under dy dx in (3.3) can be dictated as

(i) lim g(x,y; α) = 0, (3.19) + x =x ,y =0 α→0 i (ii) lim lim g(x,y; α) = 0, (3.20) + y =0 α→0 x→xi 132 H. Nishino, S. Rajpoot / Physics Letters B 540 (2002) 125–136

; = (iii) lim lim g(x,y α) = 0, (3.21) α→0+ y→0 x xi (iv) lim lim g(x,y; α) =∞, (3.22) + → → α→0 x xi ,y 0 (v) lim d3xg(x,y ; α) = (finite and non-zero). (3.23) α→0+ As before, (3.19), (3.20) and (3.21) are easily seen to be satisfied, while (3.23) has been already confirmed in (3.16), leaving (3.22). The only non-trivial singularity at (x, y) → (xi, 0)(i= 1, 2,...,δ) in (3.22) must arise from the first term in (3.5) out of its four terms, because only this term had non-vanishing contribution at α → 0+: √ 2 2 −αξ m − − − m D x − 1 bα e − eψ ν e µ2 S 1 eµ3 S = √ , (3.24) ξ ξ ˆ 2 + −αξ 4 4δp x − 1 B ξ 1 bαe y=0 after the limit y → 0 is taken. Now the question is about the behaviour of the combination D/B at y = 0. To answer this question, we have to postulate the general polynomial structures of A, B and C at y = 0as | = − − ··· − A y=0 (x x1)(x x2) (x xδ)A2δ2−δ(x), | = − 2 − 2 ··· − 2 B y=0 (x x1) (x x2) (x xδ) B2δ2−2δ(x), | = − − ··· − C y=0 (x x1)(x x2) (x xδ)C2δ2−δ−1(x). (3.25)

The A2δ2−δ(x), B2δ2−2δ(x) and C2δ2−δ−1(x) are, respectively, polynomials of x whose degrees are denoted by their subscripts, and they are supposed to have no zeros at x = xi (i = 1, 2,...,δ). These postulates are based on the following facts [10]. First, the spacetime singularities causing the Riemann tensor to blow up, occurs when two conditions u + m = 0andy = 0 are met [10]. In particular, there are only n zeros at x = xi (i = 1, 2,...,δ) for the equation u + m = 0. Second, in terms of α = u + iv, β = m + in as in [10], v = n = 0aty = 0. Third, the 2 polynomial + m) at y = 0 is supposed to have simple zeros at x = xi , and, therefore, B ≈ + m) has double zeros at x = xi . Fourth, A ≡ + m) − m) + (v + n)(v − n) has simple zeros at x = xi , due to the structure + m) − m). Fifth, we assume the form of C above, based on the explicit cases of δ = 1andδ = 2 [10]. Once we specify the structures (3.25), it follows from (3.13) that 2 2 2 2 3 3 (B2δ2−2δ) 2 2 (C2δ2−δ−1) D|y=0 =+p x − 1 (x − x1) ···(x − xδ) − 4δ q (x − x1) ···(x − xδ) . (3.26) A2δ2−δ A2δ2−δ

Obviously there are simple poles in the combination D/B|y=0 at x = xi , because the second term here will generate simple poles after being divided by B in (3.25). Therefore, at least one of such poles remains after the differentiation ∂ξ = ∂x in (D/B)ξ in (3.24). This establishes the singularity (3.22), and, therefore, we have the satisfaction of all the criterions for the identiﬁcation of the integrand g(x,y; α) with the combination of the δ-functions: δ 00 m T = √ δ(xˆ −ˆxi)δ(y). (3.27) 2πδ −g i=1 The previous case for Kerr solution in [12] can be now re-obtained as the special case of δ = 1. Recall that the coordinates (x,y)ˆ are related to the familiar Weyl’s canonical coordinates (ρ, z) [10,15] as mpˆ mpˆ ρ = xˆ2 + 1 1 − y2 ,z= xy, (3.28) δ δ = ≡ ˆ ˆ2 + = so that (3.27) implies that we have the ring singularities at the radii ρ ρi mp xi 1/δ (i 1,...,δ) corresponding to xˆ =ˆxi on the equatorial plane at z = 0 or equivalently y = 0, as desired. To turn the table around, our result can be re-interpreted in the following way: suppose we are provided with the closed string-like multiple circular mass distributions (3.27). In total, there are δ closed strings with different radii H. Nishino, S. Rajpoot / Physics Letters B 540 (2002) 125–136 133 at ρ = ρi . We can then obtain the exact classical gravitational background solution around such mass distributions, which are nothing other than the series of TS solutions with the distortion parameter δ [10,11]. In other words, our result has the signiﬁcance that the classical gravitational background around multi closed strings can be given by the series of TS solutions [10,11].

4. Regularization independence

We have chosen so far the particular regulator (3.4), mimicking the case of Coulomb potential (2.3). However, this does not have to be the only regulator for gravitational singularities. As a matter of fact, the exponent of a regulator does not have to be a linear function like −αξ, but it may be a quadratic function, such as −βξ2, if its purpose is just to get the ﬁnite ξ-integral at ξ →∞. Namely, we now consider replacing S in (3.4) by

− 2 2 S ≡ 1 + cβe β ξ , (4.1) with a non-zero constant c, and we take the limit 0 <β→ 0+ at the end of computations. The particular power of β in the exponent or that in the outside of the exponential function have been chosen, such that the final result is neither trivially zero nor divergent. In what follows, we briefly study the potential difference between this and the previous result (3.16), and show that there is eventually no regulator dependence. Among the four terms in (3.5), the last two terms on the boundaries do not contribute, while the second term is also zero for reasons similar to the previous regulator. The remaining first term in (3.5) is now 1 ∞ 2 2 3 −β2ξ 2 m pˆ 1 − y − xˆ + 1 −2cβ ξe dy dξ + O ξ 1 1 + O(β) 4 δ 1 − y2 1 + cβe−β2ξ 2 −1 0 ∞ 3 mpcβˆ a− a− − − 2 2 = dξ a (ξ + 1)2 + a (ξ + 1) + a + 1 + 2 + O (ξ + 1) 3 e β ξ 2δ 2 1 0 ξ + 1 (ξ + 1)2 0 mpcˆ = a I + a I + a I + a− I− + a− I− +··· . (4.2) 2δ 2 2 1 1 0 0 1 1 2 2 2 Here we have expanded the integrand other then the factor e−βξ in terms of ξ + 1 instead of ξ itself in order to avoid the singularity at ξ = 0, with the coefficients a2,a1,....The In’s are defined by ∞ 2−n n −ζ 2 In ≡ dζ β (ζ + β) e , (4.3) 0 + with ζ ≡ βξ. Among these In’s, we can show that it is only I2 whose limit is non-zero when β → 0 , similarly to the lemma (2.5): 0 (n = 1, 0, −1, −2,...), √ + I → π as β → 0 . (4.4) n (n =+2), 4

In fact, I1 → 0, I0 → 0 are easily proven, while we need the inequality 0

∞ −ζ 2 2+(m−1) e β dζ = I− − . (4.5) (ζ + β)m−1 (m 1) 0

Therefore, all the I−m’s are bounded from above by I−1 which in turn goes to zero: ∞ 1 ∞ 3 3 3 β −ζ 2 β −ζ 2 β −ζ 2 0

lim P 0 = m, (4.7) β→0+ √ if the constant c is normalized as c = 8δ/( π p)ˆ ,sincea2 = 1 from (4.2). √ 2 As for the criterions (3.19)–(3.23), the most important pole structure in (3.24) out of ( x − 1 D/B)ξ remains −1 the same, even though the last factor is now replaced by S Sξ . Therefore, all criterions are satisfied as in the previous regularization, leading us to the same identification with the δ-functions as in (3.27). This independence of regularizations is the reflection of the fact that the contribution to the total integral comes solely from the leading terms in the energy–momentum tensor T 00 as the integrand. This also strongly indicates the ‘physical’ correctness of our interpretation of the singularity as the closed string-like circular mass distributions.

5. Concluding remarks

In this Letter, we have shown by explicit computation, with the appropriate regulator (3.4) that there exist ring singularities in the energy–momentum tensor for general TS solutions at the coaxial radii ρ = ρi (i = 1,...,δ) on the equatorial plane at z = 0, that can be identified with the δ-function like singularity (3.27). As the guiding principle, we followed the case of circular charge distribution for Coulomb potential with a massive Yukawa- type damping factor, which also provides the criterion to be satisfied for the identification of the integrand with a combination of δ-functions. Thanks to the generalization in Ref. [11] of the TS solutions in [10] to 1 δ<∞,we have been able to generalize our result to an arbitrary number of the distortion parameter δ which corresponds to the number of ring singularities. In particular, there are naked singularities outside the event horizons in the cases δ 2 which may well be actual closed string-like mass distributions. It has been well-known that the asymptotic mass for a given exact axisymmetric gravitational solution can be obtained by the asymptotic behaviour of such a solution at r ≈∞[16]. However, such a method is not explicit enough for us to identify the singularities in the solution with the closed string type mass distribution. One of the reasons is that the asymptotic behaviors are associated with global boundary effects, which are not detailed enough to describe ‘microscopic’ mass distributions. In other words, our methodology provides more direct and explicit links between exact solutions and mass distributions. We stress that the mass distribution in energy–momentum tensor is a ‘physical entity’, and it is not just an ‘effective mass’ observed only at infinity which is indirect or implicit. We have also used in this Letter two distinct regularizations (3.4) and (4.1), and have demonstrated that our result P 0 = m is independent of a particular choice of regulators. This is because the contributions to the total integral came from the ‘leading’ terms in the expansions, such as the highest power in (3.16), or the structure of zeros in H. Nishino, S. Rajpoot / Physics Letters B 540 (2002) 125–136 135

(3.25) and (3.26). The analogous result in the Coulomb case also provides a justification of such a regularization based on a ‘physical’ consideration. In other words, our method is not a reflection of a mathematical artifact, but is a ‘physical entity’ with the total mass m. We also mention that, as is usually the case in physics, once physical interpretation is correct, a regularized finite result does not depend on the choice of regularizations. If superstring theory [2] really describes the ultimate microscopic scales around the Planck mass, it implies the existence of singular mass distributions as ‘physical objects’. These mass distributions are required to be rigorously singular, i.e., there must be no smearing effect on such singularities. From this viewpoint, it is natural that the gravitational background around such singular mass distributions exist in a relatively simple and fundamental form, such as generalized TS solutions for δ 2 [10,11]. Our result in this Letter has provided supporting evidence for such philosophy. In the Randall–Sundrum scenario [6], the 4D non-gravitational fields are supposed to exist only on branes embedded in 5D. Accordingly, all the non-gravitational physical fields are represented as a δ-function singularity in the energy–momentum tensor [6]. Similarly in the braneworld scenario [4,5], 10D supergravity can be embedded into 11D supergravity, again within certain δ-function type singularities. The common feature between the braneworld scenarios and our result is that the δ-function singularities represent important physical entities. Therefore, it is natural to identify the singularities in the energy–momentum tensor in certain axisymmetric exact gravitational solutions in vacuum with δ-function type closed string mass distributions. In the cases of δ 2 of TS solutions, there are in total δ ring singularities, some of which may be completely hidden by event horizons, in addition to ‘naked’ ones lying outside of non-singular event horizons [10,11]. Our original motivation for studying the TS solutions with δ 2 was that these naked singularities are more ‘physical’ than the hidden ones, because they are more likely to be actual closed string-like mass distribution. However, in our actual computation (3.16) for the identification P 0 = m, we did not distinguish them. This is due to the philosophy that it is the total mass m that should agree with the asymptotic mass m [16], which of course includes those mass distributions inside event horizons. Or more intuitively, any mass distribution, even if it is hidden inside of event horizon, is ‘visible’ from outside the horizon. To put it differently, we know as a simple fact that even a point mass inside the event horizon in Schwarzschild metric definitely attracts a test particle outside of the event horizon. From these viewpoints, there is nothing contradictory in our prescription of identifying all the ring singularities. It is also true that the cases of δ 2 [10,11] have more ‘naked’ closed string-like mass distributions which are much more ‘real’ than those hidden by event horizons. Despite the fact that we may need more supporting evidence based on alternative regularizations, our first results provide strong evidence of closed string like mass distribution for axially symmetric exact solutions. In other words, the general TS solutions can be regarded as classical backgrounds around closed strings. Since the distortion parameter δ corresponds also to the number of the ring singularities, i.e., the number of rings, it is legitimate to regard δ as nothing other than the number of real closed strings with δ-function type mass singularities. We believe that the encouraging results in this Letter provide an entirely new directions to be explored both in the context of superstring theory and general relativity.

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Massive IIA supergravity as a non-linear realisation

Igor Schnakenburg 1, Peter West

Department of Mathematics, King’s College, London, UK Received 22 May 2002; accepted 11 June 2002 Editor: P.V. Landshoff

Abstract A description of the bosonic sector of massive IIA supergravity as a non-linear realisation is given. An essential feature of this construction is that the momentum generators have non-trivial commutation relations with the generators associated with the gauge ﬁelds.  2002 Elsevier Science B.V. All rights reserved.

1. Introduction

Long ago Nahm [1] pointed out that supergravity theories could only exist in eleven and less space–time dimensions and that the maximum number of supersymmetries they could possess were contained in spinors that had in total no more than 32 real components. The supergravity theories with 32 components are called maximal supergravity theories. There is a unique such theory in eleven dimensions which was constructed [2] as an application of the Noether method that was used to construct the first supergravity theory, the N = 1, D = 4 supergravity theory. However, in ten dimensions there are two such maximal supergravity theories called IIA and IIB. These two theories possess different supersymmetries which when decomposed in terms of Majorana–Weyl spinors are of opposite and the same chirality, respectively. While the construction of the IIA theory [3] was found by dimensional reduction of the eleven-dimensional supergravity theory, the construction of the IIB theory [4–6] required new techniques. There is a modification of the IIA theory that preserves the number of supersymmetries, but introduces a dimensionfull parameter [7]. This theory is called massive IIA supergravity and it possesses a cosmological constant. One of the most remarkable features of supergravity theories is that the scalars in the supergravity multiplets always occur in a coset structure [8]. While this can be viewed as a consequence of supersymmetry, the groups that occur in these cosets are rather mysterious [9–11]. It has been conjectured [12] that the symmetries found in these cosets are symmetries of the associated non-perturbative string theory. The coset construction for the description of the scalars was extended to include the gauge fields for the maximal supergravity theories [13]. In this construction all gauge fields and scalars were introduced along with their duals.

E-mail addresses: [email protected] (I. Schnakenburg), [email protected] (P. West). 1 Financially supported by DAAD (D/00/09914).

The advantage of this approach is that the equations of motion can be reduced to first order equations in form of a generalised self-duality condition. This method was subsequently applied to massive IIA [14]. However, in these papers the vector indices on the gauge fields did not arise from the underlying group, but were introduced by hand. As such, it is difficult to see how this construction could be extended to include the other degrees of freedom of the theory, namely the graviton and the fermions. Recently, it was shown that the entire bosonic sectors, including gravity, of the eleven-dimensional supergravity theory and the ten-dimensional IIA [15] and IIB supergravity theories [16], could be formulated as a non-linear realisation. It was also conjectured, that when suitably formulated eleven-dimensional supergravity would be invariant under a Kac–Moody algebra [17]. Although, this conjecture was not proved in [17] some evidence was given and the Kac–Moody algebra was identified. It had rank eleven and was denoted E11 [17]. A similar analysis found that E11 was also the Kac–Moody algebra that would underlie the ten-dimensional IIA and IIB theories [16,17]. As has been pointed out by one of the authors (P.C.W.) in a number of seminars it could be that E11 is part of an even larger underlying algebra that is a Borcherds algebra. This is consistent with the general belief that all these theories are part of a larger theory which has been called M-theory [18]. It was conjectured [17] that E11 over an appropriate field was a symmetry of M-theory. However, as seen from this perspective M-theory does not necessarily live in eleven dimensions, but rather has its dimension undetermined. A particular theory results from a choice of “vacuum” in M-theory and the dimension of the resulting theory is a consequence of the Lorentz group that is contained in the subgroup of E11 that is preserved by the “vacuum” under consideration. In this Letter we extend some of these considerations to the massive IIA theory and as a first step show that its entire bosonic sector can be described as a non-linear realisation.

2. Massive IIA supergravity

The bosonic sector of massive IIA supergravity was originally [7] described in terms of the same ﬁelds as in the b ten-dimensional chiral theory without the cosmological constant. These ﬁelds were the graviton, ha , the scalar A = (dilaton) and the p-forms Aa1...ap for p 1, 2, 3. However, the massive theory also contains a constant which is related to the cosmological constant. The bosonic part of the original Lagrangian given by Romans in [7] was

−1 1 µ 1 −1/2A µνρσ 1 A µνρ 2 −3/2A µν 1 2 −5/2A e L = R − ∂µA∂ A − e F Fµνρσ − e G Gµνρ − m e B Bµν − m e 2 12 3 2 −1 e µνρσ µ ...µ + 1 6 16∂µA ∂µ A Bµ µ + 16m∂µAνρσ Bµ µ Bµ µ Bµ µ · νρσ 1 µ2µ3µ4 5 6 1 2 3 4 5 6 6 48 36 2 + m Bµν Bρσ Bµ µ Bµ µ Bµ µ , (2.1) 5 1 2 3 4 5 6 where = = + Gµνρ 3∂[µBνρ], and Fµνρσ 4 ∂[µAνρσ ] 6mB[µν Bρσ] . (2.2) By redeﬁning the ﬁelds according to

2 6 B = A + ∂[ A ],A= A − 6A[ A ] − A[ ∂ A ], (2.3) µν µν m µ ν µνρ µνρ µ νρ m µ µ ρ as explained in [19], it can be rewritten in the form

−1 1 µ 1 1/2A µνρσ 1 −A µνρ 3/2A µν 1 2 5/2A e L = R − ∂µA∂ A − e Fµνρσ F − e GµνρG − e Fµν F − m e 2 12 3 2 I. Schnakenburg, P. West / Physics Letters B 540 (2002) 137–145 139 e−1 + µ1...µ10 + ∂µ1 Aµ2µ3µ4 ∂µ5 Aµ6µ7µ8 Aµ9µ10 m∂µ1 Aµ2µ3µ4 Aµ5µ6 Aµ7µ8 Aµ9µ10 18 9 2 + m Aµ µ Aµ µ Aµ µ Aµ µ Aµ µ + 18m∂µ Aµ ...µ , (2.4) 20 1 2 3 4 5 6 7 8 9 10 1 2 10 where the gauge invariant field strengths are now given by 1 Fµν = 2 ∂[µAν] + mAµν , and Gµνρ = 3∂[µAνρ], (2.5) 2 3 Fµνρσ = 4 ∂[µAνρσ ] + 6A[µ∂ν Aρσ] + mA[µνAρσ] . (2.6) 2 The terms containing negative powers of m form a total divergence and can be dropped. In this second formulation the one form gauge field, which had been absorbed in the two form gauge field to make it massive in the former formulation, appears explicitly and one can take m → 0 to find the Lagrangian of massless IIA supergravity in a straightforward way. Following [19], in this formulation we may treat m as a dynamical field. The field equation for the new nine form gauge fields states that ∂µm = 0, i.e., m is a constant. While the field equation for m sets the field strength of the nine form gauge plus a combination of other forms in the theory equal to the epsilon symbol. We note that unlike for the other gauge fields these equations are first order. In [19] it was argued that one can derive the field equations for the purely bosonic sector of massive IIA supergravity from a Lagrangian which does not contain the mass parameter m at all. To obtain this formulation the field equation for m was plugged back into the Lagrangian (2.4). The advantage of this approach, as was explained in [19] is that this theory then naturally couples to an eight brane. Refs. [13,14] introduced duals of all the original gauge fields. As a result, the field equations for the gauge potentials could be reduced to first order. In the case of massive IIA supergravity [14] the dual of the nine form gauge field was taken to be a “minus one form” whose field strength was then dual to the ten form field strength of the nine form gauge field. We have noted in the last paragraph that the field equation for the nine form gauge potential is necessarily first order. We therefore do not introduce a dual of the nine form potential. In this Letter = we will introduce dual gauge fields for all the original gauge field, namely Aa1...aq for q 5, 6, 7, 8, but not for the nine form gauge field. We will find that the momentum operator plays an important role in place of the generator associated with the minus one form. Although to see the full symmetry one will have to introduce “duals of gravity” we will not do this here. The complete bosonic field content we require is thus given by: b ha ,A,Ac,Ac1c2 ,Ac1c2c3 ,Ac1...c5 ,Ac1...c6 ,Ac1...c7 ,Ac1...c8 ,Ac1...c9 . (2.7) In a non-linear realisation these fields are considered as the Goldstone bosons. We therefore introduce the corresponding generators

a c c1c2 c1c2c3 c1...c5 c1...c6 c1...c7 c1...c8 c1...c9 K b,R,R,R ,R ,R ,R ,R ,R ,R . (2.8) a The generators K b satisfy the commutation relations of GL(10, R) and the non-zero commutation relations between all generators mentioned above are given by Ka ,Kc = δcKa − δaKc , Ka ,P =−δaP , Ka ,Rc1...cp = δc1 Rac2...cp +···, b d b d d b b c c b b b c ...cp c ...cp c ...cp c ...cq c ...cp+q R,R 1 = cpR 1 , R 1 ,R 1 = cp,qR 1 , (2.9) where ‘+···’ means the appropriate antisymmetrisations. We also include the momentum generator Pa in the symmetry algebra and its non-trivial relations with the other generators are given by c1...cp =− c1 c2...cp +··· [ ]=− Pa,R mbp δa R , Pa,R mb0Pa. (2.10)

In what follows it will often be useful to denote c−1 = mb0, since in this way one can view the last commutator of Eq. (2.10) as an extension of those of Eq. (2.9). 140 I. Schnakenburg, P. West / Physics Letters B 540 (2002) 137–145

If the coefficients are taken to be 1 1 3 5 c =−c =− ,c=−c = ,c=−c =− ,c=−c− = , 3 5 4 2 6 2 1 7 4 9 1 4 c1,2 =−c2,3 =−c3,3 = c1,5 = c2,5 = 2,c3,5 = 1,c2,6 = 2,c1,7 = 3, 1 1 5 c , =−4,b=− ,b=− ,b= (2.11) 2 7 2 2 7 2 9 8 (all not mentioned coefficients are equal to zero) then we can verify that all Jacobi identities are satisfied. For example, the generators corresponding to the gauge fields (all c’s) fulfill the condition

cq,rcp,q+r = cp,qcp+q,r + cp,rcq,p+r , (2.12) where q,p and r indicate the rank of the generators. The Jacobi identities which involve b0 obey the above relation provided we take mb0 = c−1 where appropriate. The Jacobi identities that involve the bp,p = 0 structure constants in the new commutators of Eq. (2.10) obey the relation

bp+qcp,q = bpcp−1,q + bq cp,q−1. (2.13) ab c ...c One such example is given by the Jacobi identity involving R , R 1 7 and Pd which is satisfied provided c2,7b9 = c2,6b7 + b2c1,7. The relations of Eqs. (2.9) are the same as the algebra denoted GIIA [16] relevant to the IIA supergravity theory except that they also include the rank nine generator. For the IIA theory the commutators of Eq. (2.10) vanish. Since the above mentioned commutation relations include those of IIA supergravity [16], we call the modified group which is generated by the above generators GmIIA. We can write a general group element of the corresponding group as µ b a g = exp x Pµ exp ha K b gA ≡ gxghgA, where a ...a a ...a a ...a (1/9!)Aa ...a R 1 9 (1/8!)Aa ...a R 1 8 (1/7!)Aa ...a R 1 7 gA = e 1 9 e 1 8 e 1 7 (1/6!)A Ra1...a6 (1/5!)A Ra1...a5 (1/3!)A Ra1a2a3 (1/2!)A Ra1a2 A Ra1 AR × e a1...a6 e a1...a5 e a1a2a3 e a1a2 e a1 e . (2.14) Of course, we could have chosen any other representation, but the calculations turn out to be simplest in this particular exponential representation. We now construct a non-linear realisation of the GmIIA algebra taking the local subalgebra to be the Lorentz group. As such, we demand that the theory is invariant under −1 g → g0gh , (2.15) where g0 is a rigid element from the whole group GmIIA and h is a local Lorentz transformation. We calculate the Maurer–Cartan form − V = g 1dg − ω (2.16) = 1 µ a b in the presence of the Lorentz connection ω 2 dx ωµb J a , which transforms as − − ω → hωh 1 + hdh 1. (2.17) As a result, V transforms as − V → hVh 1. (2.18) We split the Cartan form in gravity and gauge field parts according to V = −1 − + −1 + −1 −1 − −1 gh dgh ω gA dgA gA gh dgh gA gh dgh , (2.19) I. Schnakenburg, P. West / Physics Letters B 540 (2002) 137–145 141 and set the result to be equal to 9 1 − V ≡ dxµ eˆ aP + Ω bKa + dxµ e cp−1AD A Ra1...ap , (2.20) µ a µa b p! µ a1...ap p=1,p =4 where c ≡ ˆ−1 µ −1 c − c Ωab e a e ∂µe b ωµb , (2.21) and ˆ a = −5/4A h a a = h a eµ e e µ ,eµ e µ . (2.22) a We see that the object in front of the momentum generator gets altered with respect to IIA (where it was eµ ) c due to the non-vanishing commutator in Eq. (2.10). We also see that the vielbeins in the Ωµb Eq. (2.21) are the −5/4A c unhatted vielbeins. The additional factor of e just multiplies the usual expression of Ωab from the massless case [16]. As we will see in Section 3, the physical vielbein therefore remains unchanged. The objects DµAa1...ap defined in (2.20) will be explicitly stated below. Massive IIA supergravity is the non-linear realisation of the group that is the closure of the GmIIA algebra given above and the conformal group in ten dimensions. We therefore take only those combinations of the Cartan forms of GmIIA that can be rewritten as Cartan forms of the conformal group (see Section 3). Lorentz covariant objects which are also covariant under the full non-linear realisation of the closure of the conformal and the GmIIA algebra are then, for example, the completely anti-symmetrised derivatives − − = 5/4A cp−1A Fa1...ap pe e D[a1 Aa2...ap], (2.23) µ wherewehavetouseDa =êa Dµ to convert the curved index to a flat index group-covariantly. As the physical vielbein is the unhatted one, we gain an additional factor e−5/4A in front of every field strength in comparison with the IIA case. A discussion of the closure with the conformal group is postponed to Section 3. We now give the explicit form of the field strengths. They are given for the scalar by −5/4A Fa = e DaA, (2.24) for the 1-form: −5/4A (3/4)A −5/4A (3/4)A 1 Fa a = 2e e D[a Aa ] = 2e e D[a Aa ] + mAa a , (2.25) 1 2 1 2 1 2 2 1 2 for the 2-form: = −5/4A −(1/2)A Fa1a2a3 3e e D[a1 Aa2a3], (2.26) for the 3-form: −5/4A (1/4)A 3 Fa ...a = 4e e D[a Aa ...a ] + 6A[a Da Aa a ] + mA[a a Aa a ] , (2.27) 1 4 1 2 4 1 2 3 4 2 1 2 3 4 for the 5-form: −5/4A −(1/4)A 1 Fa ...a = 6e e D[a Aa ...a ] + 20 D[a Aa ...a + mA[a a Aa a Aa a ] , (2.28) 1 6 1 2 6 1 2 4 2 1 2 3 4 5 6 for the 6-form: = −5/4A (1/2)A + 1 − Fa1...a7 7e e D[a1 Aa2...a7] mAa1a2...a7 20A[a1a2a3 Da4 Aa5a6a7] 7 1 + 12A[a D[a Aa ...a ] + 20Aa a D[a Aa a a ] + mAa a Aa a ] , (2.29) 1 2 3 7 2 3 4 5 6 7 2 4 5 6 7 142 I. Schnakenburg, P. West / Physics Letters B 540 (2002) 137–145 for the 7-form: = −5/4A −(3/4)A − + Fa1...a8 8e e D[a1 Aa2...a8] 42A[a1a2 Da3 Aa4...a8] 10Da3 Aa4a5a6 Aa7a8] 5 + mAa a Aa a Aa a ] , (2.30) 2 3 4 5 6 7 8 for the 8-form: = −5/4A − + Fa1...a9 9e D[a1 Aa2...a9] 24A[a1 Da2 Aa3...a9] 56A[a1a2 Da3 Aa4...a9] − + + 56A[a1a2a3 Da4 Aa5...a9] 1008A[a1Aa2a3 Da4 Aa5...a9] 8mA[a1a2 Aa3...a9] ! ! + 7 + 8 mA[a1 Aa2a3 Aa4a5 Aa6a7 Aa8a9] A[a1 Aa2a3 Aa4a5 Da6 Aa7a8a9] 2 4 5 − 1120A[a a Aa a a Da Aa a a ] − mAa a ...a , (2.31) 1 2 3 4 5 6 7 8 9 36 1 2 9 and finally for the 9-form: = −5/4A −(5/4)A − + Fa1...a10 10e e D[a1 Aa2...a10] 144A[a1a2 Da3 Aa4...a10] 3024A[a1a2 Aa3a4 Da5 Aa6...a10] 9! + A[a a Aa a Aa a Da Aa a a ] + 3024mA[a a Aa a Aa a Aa a Aa a ] , 18 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 (2.32) where Da is the covariant derivative µ −1 c DaAa ...a = ea ∂µAa ...a + e ∂µe Aca ...a +··· (2.33) 1 p 1 p a1 2 p −1 and ··· indicates the terms where (e ∂µe) acts on the other indices of the gauge field. Also, we have written the exponential e−5/4A separately in front of every field strengths to indicate that it is common to all of them. Using the Cartan forms which transform only under the local Lorentz group in a manner that their indices suggest we must write down a set of invariant equations. If we ask that they be first order in derivatives they can only be given by 1 a1...ap a1...a10 F = Fa + ...a ,p= 1, 2, 3, 4. (2.34) (10 − p)! p 1 10 Here we see that the common exponential factor e−5/4A indeed peels off each equation. The nine form gauge field does not possess a dual field, however, its ten form field strength can be taken to be a constant m times the epsilon symbol

−5/4A 1 a ...a m = e 1 10 Fa ...a . (2.35) 10! 1 10 All the above equations of motion and the Einstein equation are equivalent to those one can derive from the Lagrangian formulation given at the beginning of this section. We can see that the simple ﬁeld strengths of Eqs. (2.5), (2.6) indeed match with those given in our group approach of Eqs. (2.25)–(2.27) and one can indeed verify that the above equations of motion for the gauge sector (2.34) and (2.35) are the same as those one can derive from the Lagrangian (2.4) once we take m to be a dynamical ﬁeld. To recover the massless case, we simply switch off the commutators with the momentum generator by setting m = 0. However, because c−1 = mb0 we also have c−1 = 0. Then using the Jacobi relation

c−1 =−c9 (2.36) I. Schnakenburg, P. West / Physics Letters B 540 (2002) 137–145 143 we deduce c9 = 0,andthenas

c2,7c9 = c2,7(c2 + c7), (2.37) we also need c2,7 = 0. Thereby the nine form is made redundant (F(10) = dA(9)) and we are indeed left with the massless case.

3. Closure with the conformal group

The massive IIA supergravity theory is the non-linear realisation of the group that is the closure of the GmIIA group given above with the ten-dimensional conformal group. The closure of these two groups is an infinite- dimensional group, but rather than working with this group we can perform a simultaneous realisation of the GmIIA and the conformal group. What this actually means is that we construct the equations of motion only from combinations of the Cartan forms of the GmIIA group, given above, that can be rewritten in terms of the Cartan or other covariant forms of the conformal group. In doing this one gains invariance under both conformal group and GmIIA, and so necessarily we find invariance under the group which is the closure of GmIIA and the conformal group. This is discussed at length in reference [15], but here we briefly discuss the novel features that arise in this procedure when applied to the massive IIA theory. b The two groups only have one Goldstone boson in common namely the trace of ha which is related to the a h a h¯+δσ a conformal field σ . In fact we have to identify these two fields via eµ ≡ (e )µ = (e )µ (as in [15]). All the other fields that occur as Goldstone bosons in the GmIIA algebra are viewed as matter fields from the conformal group viewpoint. The conformal covariant derivative of a field B transforming under a representation Σ of the Lorentz group is: −σ ν ∆µB = e ∂µ + ∂ σΣµν B. (3.1)

In contrast, the GmIIA covariant derivative of a matter field is given by −1 µ 1 c b DaB = e ∂µ + ωµb Σ c B (3.2) a 2 multiplied by a suitable exponential of A. As the latter is a Lorentz scalar it plays no part for the discussion of the closure with the conformal group given in this section. Solving (3.1) for ∂µB and plugging the result into (3.2), we get −1 µ σ ν 1 c b DaB = e e ∆µB − ∂ σΣµν B + ωµb Σ cB . (3.3) a 2 If we demand that the whole σ dependence be through the conformal covariant derivative, then we learn from this c equation that ωµb must be solved for by a GmIIA invariant condition on the Cartan forms in such a way as to cancel the derivatives of σ on the right-hand side. This is carried out below and we find that the usual expression of the spin connection in terms of the vielbein (stated below in (3.8)) has precisely the right form to do the job. We now illustrate the procedure of taking the closure of the conformal and the GmIIA group by considering the vector field Aa instead of a general matter field B. The other gauge fields are treated in a very similar way. The conformal covariant derivative of a vector (3.1) is given by (see also [15]) = −σ + c − c ∆µAa e ∂µAa ηµa∂ σAc ∂aσδµAc . (3.4)

Using this equation we may rewrite the GmIIA covariant derivative of Aa given in Eq. (2.25), respecting (2.33) as =¯ µ − −σ c − c − − ¯−1 ¯ c − 1 Da1 Aa2 ea1 ∆µAa2 e ηµa2 ∂ σAc ∂a2 σδµAc ∂µσAa2 e ∂µe Ac mAµa2 , (3.5) a2 2 144 I. Schnakenburg, P. West / Physics Letters B 540 (2002) 137–145

¯ a a a hµ +δ σ σ a where the vielbein with the overbar stands for the traceless part eµ = e µ = (ee¯ )µ . We realise that only if we take the combination D[aAb] does the σ dependence only appear through the conformal covariant derivative alone, as then the 3σ -dependent terms in the second bracket vanish since they are symmetric in µ and a2.Ifwe want to use expressions covariant under both the conformal group and GmIIA, then we have to demand that all σ dependence be implicitly through the conformal derivative alone. As such we conclude that only the totally 3/4A µ antisymmetrised object 2e eˆ[a| DµA|b] ≡ Fab is covariant under both groups. We know that the closure of the GmIIA group and the conformal group generates gauge transformations and general coordinate transformations and so the above object should be covariant under these transformations. We observe that −1 c µ ν [ ] = [ ] + [ + = [ ] + D aAb 2 ∂ aAb e ∂ ae b] Ac mb2Aab 2ea eb (∂ µAν mb2Aµν ), (3.6) making it clear that it is covariant under gauge and general coordinate transformations. The GmIIA covariant derivatives of IIA supergravity only differ from those of massive IIA supergravity by terms containing m,and the nine form potential. However, as these new terms do not contain derivatives the closure with the conformal group is not spoilt by the presence of these terms. We now turn to the gravity sector of the theory. Clearly, the constraint

Ωa,[bc] − Ωb,(ac) + Ωc,(ab) = 0 (3.7) is GmIIA covariant, but one can show in much the same way as for the IIA and the eleven-dimensional supergravity cases [15] that it is also conformally covariant. Examining the definition of Ωa,bc in Eq. (2.21), we see that it µ −5/4A µ −5/4A involves an undifferentiated factor of eâ = e ea . The factor of e can then be removed and we find that it is exactly the same constraint as in the other cases (eleven-dimensional supergravity, IIA and IIB supergravity) and therefore results in the usual expression for the spin connection in terms of the vielbein, namely, 1 ρ ρ 1 ρ ρ 1 λ ρ λ ρ a ωµbc = eb ∂µeρc − ec ∂µeρb − eb ∂ρeµc − ec ∂ρeµb − eb ec ∂λeρa − ec eb ∂λeρa eµ . (3.8) 2 2 2 a We conclude that the physical vielbein of general relativity is just eµ . The upshot of this discussion is that the simultaneously covariant objects that transform covariantly under the = group GmIIA and the conformal group are the Fa1...ap for p 1,...,10 (except 5) defined in (2.24)–(2.32) and the Riemann tensor

c c d c Rµνb ≡ ∂µωνb + ωµb ωνd − (µ ↔ ν). (3.9) The invariant ﬁeld equations for all the ﬁelds except that of gravity are given in Eqs. (2.34) and (2.35), while that for gravity must be of the form c ν µ 1 2 5/2A 1 −A (3)cd (3) 1 (3)cdf (3) Rµνb ec ea = m e ηab + ∂aA∂bA + e F F − ηabF F 16 2 a bcd 12 cdf 3/2A (2)c (2) 1 (2)cd (2) + 2me F F − ηabF F a bc 16 cd 1 1/2A (4)cdf (4) 3 (4)cdfg (4) + e F F − ηabF F . (3.10) 3 a bcdf 32 cdfg

This equation does not look GmIIA covariant as we seem to have used the unhatted vielbeins only. However, the same factor of e−5/2A turns up in every single term and can therefore again be dropped. We note that, one cannot know the factor in front of each term on the right-hand side. These factors can only be determined if we additionally use information from supersymmetry or the Kac–Moody algebra. We have just put in the correct values for those constants. I. Schnakenburg, P. West / Physics Letters B 540 (2002) 137–145 145

4. Discussion

We have shown that like all the other maximal supergravity theories the entire bosonic sector of massive IIA supergravity can also be described as a non-linear realisation. Apart from introducing dual fields for all the gauge fields of the original formulation [7] of the massive IIA theory we also have included, following [19], a nine form gauge which is associated with the introduction of the cosmological constant. The correct theory requires that the momentum generator has non-trivial commutation relations with the generators associated with the gauge fields as given in Eq. (2.10). This is natural as the nine form is associated with the cosmological constant and so with gravity. This is in contrast to the work of Ref. [14] which takes a different approach and does not include gravity, but does introduce a dual form for the nine form gauge field which was called a “minus one form”. The properties of this minus one form are not very explicitly spelt out. In effect we find in this Letter that the momentum generator plays the role of the generator associated with the “minus one form” of Ref. [14]. It would be interesting to examine if the non-linear realisation could be extended, in ways explained in Ref. [17], to be invariant under a Kac–Moody, or Borcherds algebra, and to conjecture what this algebra is. In the previous non-linear realisation of the maximal supergravities the momentum generator has not played a central part in the Kac–Moody algebra that has been identified. However, the non-trivial relations of Eqs. (2.10) imply that this generator must occur in a non-trivial way in the corresponding algebra. Progress in this direction may also shed light on the place that the massive IIA theory has in M-theory.

Acknowledgements

I.S. would like to thank André Miemiec, who has given support when calculating the equations of motion for various SUGRAs. I.S. is also supported by DAAD (D/00/09914).

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Skyrmed monopoles

D.Yu. Grigoriev a,1,P.M.Sutcliffeb, D.H. Tchrakian a

a Mathematical Physics, National University of Ireland Maynooth (NUIM), Maynooth, Co. Kildare, Ireland b Institute of Mathematics, University of Kent at Canterbury, Canterbury, CT2 7NF, UK Received 15 May 2002; received in revised form 11 June 2002; accepted 17 June 2002 Editor: P.V. Landshoff

Abstract We investigate multi-monopole solutions of a modified version of the BPS Yang–Mills–Higgs model in which a term quartic in the covariant derivatives of the Higgs field (a Skyrme term) is included in the Lagrangian. Using numerical methods we find that this modification leads to multi-monopole bound states. We compute axially symmetric monopoles up to charge five and also monopoles with Platonic symmetry for charges three, four and five. The numerical evidence suggests that, in contrast to Skyrmions, the minimal energy Skyrmed monopoles are axially symmetric.  2002 Elsevier Science B.V. All rights reserved.

1. Introduction is a 4N-dimensional moduli space of static solutions which are degenerate in energy, so in this sense there Two of the most interesting kinds of topological are no stable bound states since any static charge N so- solitons in three space dimensions are BPS monopoles lution has the same energy as N well-separated charge and Skyrmions. Although there are some similarities one monopoles. Contrast these features with those of between monopoles and Skyrmions, which we shall Skyrmions, which are soliton solutions of a nonlin- discuss shortly, there are a number of important differ- ear sigma model with target space SU(2). The Skyrme ences which we first recall. SU(2) BPS monopoles are field is constant on the two-sphere at spatial infinity soliton solutions of a Yang–Mills–Higgs gauge the- and this yields a compactification of Euclidean three- ory (with a massless Higgs) in which the topologi- space to a three-sphere. The integer-valued topological cal charge N is an element of the second homotopy charge (baryon number) is an element of the third ho- group of the two-sphere, identified as the Higgs field motopy group of the target space and counts the num- vacuum manifold. The topological charge is therefore ber of times that the target space is covered by the associated with a winding of the Higgs field on the Skyrme field throughout space. There are static forces two-sphere at spatial infinity. In the BPS limit there between Skyrmions, which for a suitable relative internal orientation are attractive, and this leads to multi- Skyrmion bound states. E-mail addresses: [email protected] (D.Yu. Grigoriev), To summarize, three main differences between [email protected] (P.M. Sutcliffe), [email protected] (D.H. Tchrakian). Skyrmions and monopoles are the basic fields of the 1 On leave of absence from Institute for Nuclear Research of model, the way the topological charge arises, and the Russian Academy of Sciences. existence (or not) of bound states. Given these facts it

0370-2693/02/$ – see front matter  2002 Elsevier Science B.V. All rights reserved. PII:S0370-2693(02)02141-X D.Yu. Grigoriev et al. / Physics Letters B 540 (2002) 146–152 147 1 1 is rather surprising that there appears to be some sim- E = − Tr Fij Fij + Di ΦDi Φ ilarity between various monopole and Skyrmion so- 8π 2 lutions. There are axially symmetric monopoles and 2 + µ [ ] Skyrmions for all charges greater than one (although Di Φ,Dj Φ 2 above charge two these are not the minimal energy ×[ ] 3 Skyrmions) and both have solutions with Platonic DiΦ,Dj Φ d x. (1) symmetries for the same certain charges. For example, there is a tetrahedral monopole for N = 3, a cubic Here Latin indices run over the spatial values 1, 2, 3, ∈ monopole for N = 4 and a dodecahedral monopole for the Higgs field and gauge potential are Φ,Ai su(2), = +[ ] N = 7 [4,6]. All BPS monopoles of a given charge the covariant derivative is Di Φ ∂i Φ Ai,Φ have the same energy but these particular monopole and Fij is the field strength. | |2 =−1 2 solutions are selected out by being mathematically The boundary condition is that Φ 2 Tr Φ more tractable than an arbitrary solution. For these equals one at spatial infinity. The Higgs field at three values of the charge N = 3, 4, 7 the minimal en- infinity then defines a map between two-spheres and ergy Skyrmion has precisely the same symmetry as the the winding number of this map is the monopole above monopoles and energy density isosurfaces are number N. qualitatively similar [1,2]. These and other similarities If µ = 0 then the energy (1) is the usual BPS Yang– can be partially understood by relating both types of Mills–Higgs energy and monopole solutions satisfy soliton to rational maps between Riemann spheres [5]. the first order Bogomolny equations. All members of The obvious differences and yet remarkable simi- the 4N-dimensional moduli space of solutions have larities between monopoles and Skyrmions is the mo- energy E = N, and include solutions describing N tivation for the present work, where we aim to mod- well-separated monopoles as well as axially symmet- ify the BPS monopole Lagrangian by the addition ric N-monopoles. For µ = 0 the additional term is of a Skyrme-like term with the goal of breaking the the gauge analogue of the Skyrme term for the sigma energy degeneracy and producing monopole bound model. In the sigma model context the presence of the states. That such a modification might yield monopole Skyrme term is necessary to have stable soliton solu- bound states is suggested by the fact that more compli- tions but in the monopole context it is optional. Clearly cated models, involving Skyrme-like terms, have been the energy degeneracy of the BPS model will be bro- shown to have this property [7]. ken for µ = 0 and as we shall describe below this pro- We refer to the soliton solutions of our modified duces monopole bound states, rather than the familiar model as Skyrmed monopoles, though in the follow- monopole–monopole repulsion induced by the addi- ing for brevity we mainly use the term monopoles, tion of a Higgs potential, which is an alternative way and refer to monopole solutions of the unmodified to lift the energy degeneracy of the BPS model. model as BPS monopoles. Our numerical computations, for monopoles up to charge five, show that the modified model does indeed have multi-monopole 2. Numerical methods and results bound states, though perhaps surprisingly our numerical results suggest that the minimal energy multi- In order to construct static solutions of the field monopoles are all axially symmetric and do not share equations which follow from the variation of the the Platonic symmetries of the corresponding minimal energy (1) we apply a simulated annealing algo- energy Skyrmions. Platonic monopole solutions are rithm [11] to minimize the energy using a finite dif- computed, and although they have low energies they ference discretization on a grid containing 813 points are very slightly above those of the axially symmetric with a lattice spacing dx = 0.25. Note that this grid is solutions. a little smaller than those currently in use to study sim- Explicitly, the model we consider is defined by ilar problems for Skyrmions [1] (although simulated the following energy function (we deal only with annealing computations of Skyrmions on grids con- static solutions in this Letter but the extension to the taining 803 points do provide accurate results [3]), but relativistic Lagrangian is obvious) the large number of fields which need to be dealt with 148 D.Yu. Grigoriev et al. / Physics Letters B 540 (2002) 146–152 in studying a Yang–Mills–Higgs gauge theory make it E = 1.007 whose deviation from unity is an indica- difficult to handle grids much larger than this with our tion of the error associated with the energy values we current resources. However, by testing our codes on quote. Another important test is to compare the ener- the BPS limit (µ = 0) where exact results are known, gies of different BPS multi-monopole solutions which we are able to estimate the numerical errors involved have the same charge. Of course a perfect calculation and have confidence in our results being accurate to would produce energies equal to the charge for any so- the level that we discuss later. lution. As an example, using the rational map R = z3 In order to apply our annealing code we need of the axially symmetric 3-monopole in the initial con- to provide initial conditions which have the correct dition produces the energy E = 3.021, whereas the ra- topological winding of the Higgs field at infinity. tional map To provide these initial conditions, and be able to √ √ R = iz2 − z3 − iz prescribe any particular symmetry that we may want 3 1 3 to impose, we make use of a formula relating the anneals to produce a tetrahedrally symmetric 3-mon- asymptotic Higgs field to a rational map between opole with energy E = 3.018. This illustrates the Riemann spheres [8]. Explicitly, the initial Higgs field fact that our energies are accurate to around 1% but is given by that comparisons between different configurations are likely to be more accurate, in this case the error is if (r ) 1 −|R|2 2R Φ = , (2) around 0.1%. Similar results were obtained for other +| |2 2R |R|2 − 1 1 R BPS examples. where f(r) is a real profile function, which depends We now turn to the modified model with µ = 0, on the radius r, and satisfies the boundary conditions and the first issue to address is a suitable choice for that f(0) = 0andf = 1 on the boundary of the nu- the value of µ. To facilitate numerical comparisons merical grid. Here R(z) is a rational map of degree N it is useful to choose a value of µ large enough so in the complex variable z, i.e., a ratio of two poly- that the additional term raises the energy of the N = 1 nomials of degree no greater than N, which have no monopole by something of the order of 50% from common factors and at least one of the polynomi- the BPS value, since it then has an effect significant als has degree precisely N.Thevariablez is a Rie- enough to be calculated numerically but does not mann sphere coordinate on the unit sphere around the dominate over the usual terms. In Fig. 1 we plot the origin in space, i.e., it is given by z = eiφ tan(θ/2) energy of the N = 1 monopole as a function of µ2. where θ and φ are the usual polar coordinates. In the This calculation is performed by using a hedgehog BPS case there is a one-to-one correspondence be- ansatz and computing the energy minimizing profile tween charge N monopole solutions and (an equiva- functions. From Fig. 1 we see that a reasonable choice lence class of) degree N rational maps [9] and the ex- is µ = 5, which we use from now on, and this gives istence of certain symmetric monopole solutions can E = 1.591. Using the full three-dimensional annealing be proved by the construction of the associated sym- code we compute the 1-monopole energy to be E1 = metric maps [5]. Although there is clearly no such cor- 1.602 which is in reasonable agreement with the more respondence in our modified model we shall make use accurate one-dimensional calculation. of some of the relevant symmetric maps in our initial The crucial calculation is now to compute the en- conditions. We take all gauge potentials to be zero ini- ergy of the axially symmetric 2-monopole. Using the tially and this preserves any symmetry that the Higgs rational map R = z2 we compute the axially sym- field may initially have. On the boundary of the grid metric 2-monopole, whose energy density isosurface the Higgs field is fixed to the initial form (2), which, is displayed in Fig. 2(A), and find the energy E2 = in particular, ensures that the winding number remains 2.777. The important point is that E2/2 = 1.388 < equal to N, but the gauge potential is annealed to min- 1.602 = E1 so a 2-monopole bound state exists. It imize the energy given the fixed boundary Higgs field. seems reasonable to conclude that the minimal energy As a test of the accuracy of our code we first com- 2-monopole is axially symmetric, though clearly we pute several BPS monopoles. For the N = 1 mono- have not proved this. Note that 2E1 − E2 = 0.427 and, pole (with rational map R = z) we find an energy as we mentioned above, this is expected to be signif- D.Yu. Grigoriev et al. / Physics Letters B 540 (2002) 146–152 149

Fig. 1. The 1-monopole energy as a function of µ2.

Fig. 2. Energy density isosurfaces (to scale) of various Skyrmed monopoles. (A) N = 2axial,(B)N = 3axial,(C)N = 4axial,(D)N = 5 axial, (E) N = 3 tetrahedral, (F) N = 4 octahedral, (G) N = 5 dihedral, (H) N = 5 octahedral.

icantly larger than the numerical errors present in our Table 1 energy comparisons. If required a more accurate cal- The monopole charge N, the symmetry group G of the energy culation of the 2-monopole energy could be performed density, the energy E and energy per monopole E/N for several examples of Skyrmed monopoles by making use of the axial symmetry to reduce to an effective two-dimensional computation. NG EE/N For higher charges we ﬁrst look at axially symmet- 1 O(3) 1.602 1.602 × ric monopoles by using the rational maps R = zN . For 2 O(2) Z2 2.777 1.388 3 O(2) × Z2 3.807 1.269 N = 2, 3, 4, 5 the energies EN and energies per mono- 3 Td 3.869 1.290 pole EN /N are presented in Table 1 and we display 4 O(2) × Z2 4.847 1.212 energy density isosurfaces in Figs. 2(A)–(D). We also 4 Oh 4.974 1.244 × plot the energy per monopole for these axially sym- 5 O(2) Z2 5.924 1.185 metric solutions as a function of monopole number in 5 D2d 5.982 1.196 5 O 5.987 1.197 Fig. 3. This plot demonstrates that all these solutions h 150 D.Yu. Grigoriev et al. / Physics Letters B 540 (2002) 146–152

Fig. 3. The energy per monopole E/N for the axially symmetric monopoles (crosses) and Platonic monopoles (stars).

are stable against the break-up into N well separated one mentioned earlier, monopoles, and also into any well-separated clusters √ √ = 2 − 3 − containing single or axially symmetric monopoles. R 3 iz 1 z 3 iz . Note that for these axially symmetric solitons the en- Annealing produces the tetrahedral 3-monopole dis- ergy per monopole decreases as the monopole number T = played in Fig. 2(F) which has an energy E3 3.869. increases and this contrasts sharply with Skyrmions. This is very slightly higher than the energy of the axial For axially symmetric Skyrmions with N 2the = T − = 3-monopole E3 3.807, and since E3 E3 0.062 energy per Skyrmion increases with the number of we expect that even though this difference is almost Skyrmions [10], and only the N = 2 minimal en- as large as the likely overall error in the computa- ergy Skyrmion has an axial symmetry. Furthermore, tion of each individual energy, it is an order of mag- for N>4 the axially symmetric charge N Skyrmion nitude greater than the errors we estimate in the com- is not even bound against the break-up into N well- parison between two energies. This calculation sug- separated single Skyrmions. gests that the axial 3-monopole has less energy than The fact that for the axially symmetric solutions the tetrahedral 3-monopole, in contrast to Skyrmions, the energy per monopole decreases as a function of and hence that it is likely to be the minimal energy increasing monopole number (we have also checked 3-monopole. Of course, since the energy differences that this trend continues up to N = 10, using larger are small it is desirable to have a more accurate calcu- grids) makes it possible that the minimal energy lation of both these energies using larger grids, but this monopole is axially symmetric for all N 2. In is beyond our current resources. We have verified that order to test this we have computed some non-axially the axial 3-monopole has less energy than the tetrahe- symmetric monopoles with N>2 which have the dral 3-monopole for a number of other values of the symmetries of the known minimal energy Skyrmions, parameter µ and also performed another consistency since these are the obvious non-axial contenders for check by computing the energy of the additional term minimal energy monopoles. given the two different BPS 3-monopoles. This will The minimal energy N = 3 Skyrmion has tetrahe- be a good approximation to the excess above the BPS dral symmetry Td and the relevant rational map is the bound in the limit where µ is small, so that the fields D.Yu. Grigoriev et al. / Physics Letters B 540 (2002) 146–152 151 vary little from the BPS configurations. This result is not share the symmetries of the minimal energy in agreement with the full nonlinear computation since Skyrmions. it yields an excess energy which is slightly less for the The minimal energy N = 5 Skyrmion has only the axial 3-monopole than for the tetrahedral 3-monopole, dihedral symmetry D2d and corresponds to a rational though we must point out that this calculation does map of the form appear to be very sensitive to obtaining the BPS so- = 5 + 3 + 4 − 2 + lution to a very high accuracy. In principle, given the R z bz az az bz 1 correspondence between BPS monopoles and rational where a and b are particular real constants. Using the maps, the additional energy contribution should pro- values associated with the minimal energy 5-Skyrmion vide an interesting energy function on the space of ra- produces the monopole displayed in Fig. 2(G) with D = tional maps, though it does not seem possible to obtain an energy E5 5.982 which is larger than the any explicit information about this energy function axial energy E5 = 5.924. For charge 5 there is also without first computing the monopole fields, which another obvious minimal energy candidate, which is can only be done numerically and is computationally an octahedrally symmetric Oh monopole associated expensive. with the above rational map in which the parameter b An interesting question, given that our results is zero and a =−5. The annealed monopole has O = suggest that the tetrahedral 3-monopole is not the energy E5 5.987 and is presented in Fig. 2(H). minimal energy solution, is whether this is a stable Deforming the dihedral monopole to the octahedral local minimum or a saddle point solution. We are monopole produces a tiny change in energy, and unable to answer this question at this stage, since the difference is even within the numerical errors the algorithm requires the Higgs field to be fixed expected when comparing two energies, so we can on the boundary of the grid with a prescribed form, only conclude that the numerical results suggest that and hence symmetry. In principle, since we have both have higher energy than the axial 5-monopole, not explicitly fixed a gauge, any Higgs field which but which of these two has the lower energy is not has a winding number equal to N is equivalent to clear. any other, so it should be possible to move between For all the charges and examples discussed above different configurations if the symmetry is initially we have performed several other computations using broken by the gauge potentials, but in practice this both larger and smaller values for the parameter µ does not happen since the energy differences between and found qualitatively similar results. In all cases the various configurations are too small and the gauge axially symmetric monopoles are always those with potentials quickly anneal to match the symmetry of the lowest energy, suggesting that this is the case for all the Higgs field. It is this technical difficulty which µ>0. We have also examined the replacement of the prevents us from simply finding the minimal energy fourth-order Skyrme term by a sixth-order term and N-monopole by starting from an asymmetric initial found similar results. condition, which is the method used for Skyrmions but in that case the Skyrme field is fixed on the boundary of the grid to be a constant and contains no information 3. Conclusion about the structure and symmetry of the Skyrmion. The minimal energy 4-Skyrmion has octahedral Motivated by the similarities and differences be- symmetry Oh and is described by the rational map tween BPS monopoles and Skyrmions we have inves- √ √ tigated a modification of the usual BPS Yang–Mills– R = z4 + 2 3 iz2 + 1 z4 − 2 3 iz2 + 1 . Higgs model by including a Skyrme-term formed from the covariant derivatives of the Higgs field. We found Using this map we compute the cubic 4-monopole that this modification indeed produces monopoles O = displayed in Fig. 2(F) with energy E4 4.974. which are more like Skyrmions, in the sense that This is again slightly larger than the energy of the bound states now exist, but that the numerical re- axial 4-monopole E4 = 4.847 and further supports sults suggest that the minimal energy monopoles for our findings that the minimal energy monopoles do charges greater than two do not share the symmetries 152 D.Yu. Grigoriev et al. / Physics Letters B 540 (2002) 146–152 of the minimal energy Skyrmions, but instead appear nience, but the qualitative features of our results, such to be axially symmetric. The energy differences we as monopole bound states, remain valid. have found are not substantial, so further more accurate computations would be desirable, but we have Acknowledgements demonstrated a significant difference (with values well beyond our expected numerical errors) between the Many thanks to R. Flume, C.J. Houghton, B. Klei- behaviour of axially symmetric monopoles in our haus, J. Kunz and N.S. Manton for useful discussions. modified model and axially symmetric Skyrmions; We thank Enterprise–Ireland and the British Council in the axial monopole case the energy per soliton is for financial support under project BC/2001/021. PMS a decreasing function of soliton number and in the acknowledges the EPSRC for an Advanced Fellow- Skyrmion case it is an increasing function. This prop- ship. The research of D.G. is supported by Enterprise– erty alone demonstrates that our modified monopoles Ireland grant SC/2000/020. have qualitative differences with Skyrmions. There are a number of interesting properties of monopoles in the modified model which require fur- References ther investigation. These include a study of the energy of a 2-monopole configuration as a function of the [1] R.A. Battye, P.M. Sutcliffe, Phys. Rev. Lett. 79 (1997) 363; monopoles separation and the related issue of how the R.A. Battye, P.M. Sutcliffe, Phys. Rev. Lett. 86 (2001) 3989; interaction between two well-separated monopoles de- R.A. Battye, P.M. Sutcliffe, Rev. Math. Phys. 14 (2002) 29. [2] E. Braaten, S. Townsend, L. Carson, Phys. Lett. B 235 (1990) pends on their relative phase. The dynamics and scat- 147. tering of monopoles in this model would also seem [3] M. Hale, O. Schwindt, T. Weidig, Phys. Rev. E 62 (2000) 4333. worth investigating, both using full field simulations [4] N.J. Hitchin, N.S. Manton, M.K. Murray, Nonlinearity 8 and approximate techniques. In principle the mod- (1995) 661. uli space approximation could be applied to Skyrmed [5] C.J. Houghton, N.S. Manton, P.M. Sutcliffe, Nucl. Phys. B 510 (1998) 507. monopoles by treating the modification as a perturba- [6] C.J. Houghton, P.M. Sutcliffe, Commun. Math. Phys. 180 tion to the BPS monopole metric together with an in- (1996) 343; duced potential function on the BPS monopole moduli C.J. Houghton, P.M. Sutcliffe, Nonlinearity 9 (1996) 385. space. [7] B. Kleihaus, D. O’Keeffe, D.H. Tchrakian, Phys. Lett. B 427 Although the main motivation for this work is to (1998) 327; B. Kleihaus, D. O’Keeffe, D.H. Tchrakian, Nucl. Phys. B 536 explore connections between various types of three- (1999) 381. dimensional topological solitons, the additional Sky- [8] T. Ioannidou, P.M. Sutcliffe, J. Math. Phys. 40 (1999) 5440. rme term that we have included is a natural modifi- [9] S. Jarvis, J. Reine Angew. Math. 524 (2000) 17. cation that might arise in an effective theory. In this [10] V.B. Kopeliovich, B.E. Stern, JETP Lett. 45 (1987) 203. context the value of µ is expected to be much smaller [11] P.J.M. van Laarhoven, E.H.L. Aarts, Simulated Annealing: Theory and Applications, Kluwer Academic, Dordrecht, 1987. than the value we have studied for numerical conve- Physics Letters B 540 (2002) 153–158 www.elsevier.com/locate/npe

Constituents of doubly periodic instantons

C. Ford a,J.M.Pawlowskib

a Instituut-Lorentz for Theoretical Physics, Niels Bohrweg 2, 2300 RA Leiden, The Netherlands b Institut für Theoretische Physik III, Universität Erlangen, Staudtstraße 7, D-91058 Erlangen, Germany Received 17 May 2002; accepted 13 June 2002 Editor: P.V. Landshoff

Abstract Using the Nahm transform we investigate doubly periodic charge one SU(2) instantons with radial symmetry. Two special points where the Nahm zero modes have softer singularities are identiﬁed as constituent locations. To support this picture, the action density is computed analytically and numerically within a two-dimensional slice containing the two constituents. For 1 particular values of the parameters the torus can be cut in half yielding two copies of a twisted charge 2 instanton. Such objects comprise a single constituent.  2002 Elsevier Science B.V. All rights reserved.

Topologically non-trivial objects play a pivotal rôle There is an almost complete lack of explicit results in most conﬁnement scenarios. A prominent example for multiply periodic instantons in the literature. How- being the dual Meissner effect via the condensation of ever, the existence of higher charge instantons on T4 magnetic monopoles. To date, there is no gauge in- was established by Taubes. More recently, doubly pe- dependent way of identifying the monopole content riodic charge one instantons have been discussed [4,5]. of a given gauge potential. However, some progress In [5] doubly periodic charge one SU(2) instantons has been made with regard to instantons. In particular, with radial symmetry in the non-compact R2 direc- charge one SU(N) calorons, or periodic instantons, tions were considered. Under the Nahm transforma- can be viewed as bound states of N monopole con- tion these instantons are mapped to abelian potentials stituents [1,2]. This identiﬁcation does not hinge on a on the dual torus T 2. Our basic approach is to start particular gauge choice as the constituents are clearly with these rather simple abelian potentials and then visible as peaks in the action density. Presumably, sim- Nahm transform to recover the original SU(2) instan- ilar results hold for ‘higher’ tori, that is, instantons on ton. This involves solving certain Weyl–Dirac equa- T 2 × R2 (doubly periodic instantons), T 3 × Randthe tions; for each x in T 2 × R2 one has a different Weyl four torus T4. In the latter case twists must play a rôle equation on T 2. The Weyl zero modes were deter- since untwisted charge one instantons do not exist [3]. mined explicitly for a two-dimensional subspace [5] (see also [6]) of T 2 × R2 (this subspace corresponds to the origin of R2). Although this falls short of a complete solution, E-mail addresses: [email protected] (C. Ford), this subspace is, by virtue of the radial symmetry, [email protected] (J.M. Pawlowski). exactly where any constituents are expected to lie. In

0370-2693/02/$ – see front matter  2002 Elsevier Science B.V. All rights reserved. PII:S0370-2693(02)02130-5 154 C. Ford, J.M. Pawlowski / Physics Letters B 540 (2002) 153–158 this Letter we use the explicit zero modes to develop respect to x1 → x1 + L1, x2 → x2 + L2 its Nahm a constituent picture of the associated doubly periodic transform is instantons. There are two points in the subspace where ∂ the zero modes have different singularity profiles. Aîj (z) = d4xψi †(x; z) ψj (x; z), µ ∂zµ These are obvious candidates for constituent locations. T 2×R2 The behaviour of the field strengths in the vicinity µ = 1, 2, of the proposed constituent locations is investigated îj = 4 i † ; j ; analytically and numerically. It is possible to arrange Aµ (z) d xψ (x z)ixµψ (x z), that the two constituents are identical lumps. If we T 2×R2 further fix their separation to be a half period, the torus = can be cut in two. In each half of the torus we have a µ 0, 3, (3) 1 i twisted instanton of topological charge 2 comprising where the ψ (x; z) with i = 1,...,k are orthonormal † a single constituent. zero modes of Dz (A). Note that we may gauge z0 and 2 × 2 Before we specialise to T R let us briefly re- z3 to zero. The Nahm potential is a self-dual U(k) 4 2 call how the Nahm transform is formulated on T gauge field with N singularities in T (periods 2π/L1 [3]. Start with a self-dual anti-hermitian SU(N) poten- and 2π/L2). ˆ ˆ tial, Aµ(x), on a euclidean four-torus with topological One can regard A1(z) and A2(z) as the compo- 4 4 charge k. A gauge field on T is understood to be an R nents of a two-dimensional gauge potential, and com- ˆ ˆ potential which is periodic (modulo gauge transforma- bine A0(z) and A3(z) into a ‘Higgs’ field Φ(z) = tions) with respect to xµ → xµ + Lµ (µ = 0, 1, 2, 3), 1 ˆ − ˆ 2 (A0(z) iA3(z)). The next step is to seek solutions the Lµ being the four periods of the torus. The next, of the dimensionally reduced self-duality (or Hitchin) apparently trivial, step is to turn the SU(N) instanton equations. In the one-instanton sector this is rather into a U(N) instanton by adding a constant U(1) po- straightforward since the corresponding Nahm poten- tential, Aµ(x) → Aµ(x) − izµ,wherethezµ are real tial is abelian. What is more tricky, however, is to ex- numbers. We can regard the zµ as coordinates of the ecute the second Nahm transformation to recover the 4 dual torus, T , since the shifts zµ → zµ + 2π/Lµ can instanton itself. We shall restrict ourselves to the spe- be effected via periodic U(1) gauge transformations. cial case of a zero Higgs field. This means that the cor- Now consider the U(N) Weyl operator responding instanton will be radially symmetric: local = µ gauge invariants such as the action density depend on Dz(A) σµDz (A), x , x and r = x2 + x2 only. With this restriction the Dµ(A) = ∂µ + Aµ(x) − izµ, (1) 1 2 0 3 z self-duality equations are just σµ = (1,iτ1,iτ2,iτ3) where the τi are Pauli matrices. ˆ ˆ Provided certain mathematical technicalities are met Fyy¯ = ∂yAy¯ − ∂y¯ Ay = 0. (4) † =− † µ Dz (A) σµDz (A) has k square-integrable zero = + i Here we have used complex coordinates; y z1 modes ψ (x; z) with i = 1, 2,...,k. The Nahm po- 1 1 iz2, y¯ = z1 − iz2, ∂y = (∂z − i∂z ), ∂y¯ = (∂z + tential is defined as 2 1 2 2 1 ˆ = 1 ˆ − ˆ ˆ ¯ = 1 ˆ + i∂z2 ), Ay 2 (A1(z) iA2(z)) and Ay 2 (A1(z) îj 4 i † ∂ j ˆ A (z) = d xψ (x; z) ψ (x; z). (2) iA2(z)). Consider the ansatz µ ∂zµ T4 ˆ ˆ Ay = ∂y φ, Ay¯ =−∂y¯φ, (5) Here the zero modes are taken to be orthonormal. Re- =− markably, A(z)ˆ is a U(k) instanton on the dual torus which gives Fyy¯ 2∂y∂y¯ φ. Then, in order to satisfy with topological charge N. If we execute a second (4), φ must be harmonic except at two singularities, Nahm transformation on Aˆ, the original SU(N) instan- sinceweareaimingforanSU(2) instanton. A suitable ton will be recovered. φ satisfies 2 × 2 Formally, one can obtain the T R Nahm 2 + 2 ∂z ∂z φ(z) transform by taking two of the periods, say L0 and L3, 1 2 to be infinite. Given an SU(N) instanton periodic with =−2πκ δ2(z − ω) − δ2(z + ω) , (6) C. Ford, J.M. Pawlowski / Physics Letters B 540 (2002) 153–158 155 where κ is a constant and ±ω are the positions The SU(2) instanton we seek, Aµ(x), is the Nahm of the two singularities (we have used translational transform of the abelian potential A(z)ˆ invariance to shift the ‘centre of gravity’ of the ∂ singularities to the origin). The delta functions should Apq(x) = d2zψp †(z; x) ψq (z; x), (10) µ ∂xµ be read as periodic (with respect to z1 → z1 + 2π/L1 T 2 and z2 → z2 +2π/L2). Physically, the Nahm potential p describes two Aharonov–Bohm fluxes of strength κ where the ψ (z; x),p = 1, 2 are orthonormal zero − modes of and κ threading the dual torus. They must have equal ˆ 1 i and opposite strength to ensure a periodic A.Wemay i x¯⊥ ∂y + ∂yφ − x¯ − D†(A)ˆ = 2 2 . assume that κ lies between 0 and 1 since it is possible x i 1 2 ∂y¯ − ∂y¯ φ − x x⊥ via a (singular) gauge transformation to shift κ by an 2 2 (11) integer amount (under such a transformation the total 2 flux through T 2 remains zero). In addition to the complex coordinates y, y¯ on T we One can write φ explicitly in terms of Jacobi theta have introduced two sets of complex coordinates for 2 × 2 = + functions T R ; in the ‘parallel’ directions x x1 ix2, x¯ = x1 − ix2, and in the ‘transverse’ non-compact directions x⊥ = x + ix , x¯⊥ = x − ix . φ(z) 0 3 0 3 When x⊥ = 0 the Weyl equation decouples and the L1 1 iL1 iL1 2 κ θ (y + ω1 + iω2) + + , two zero modes have a simple form [5] = log 2π 2 2L2 L2 L1 1 iL1 iL1 2 2 θ (y − ω1 − iω2) + + , 1 0 2π 2 2L2 L2 ψ (z; x)= − , e φ(z)G+(z − ω) iL L ω + 1 2 2 −¯ − eφ(z)G−(z + ω) (y y) 2ω2L1 . (7) ψ2(z; x)= , (12) π 0 The theta function is defined as where G±(z) are periodic Green’s functions satisfying − − 1 ¯ = 1 2 ∞ i∂y 2 x G+(z) 2 δ (z), 2 θ(w,τ)= eiπn τ+2πinw, − − 1 = 1 2 i∂y¯ 2 x G−(z) 2 δ (z). (13) n=−∞ G−(z) has a theta function representation Im τ>0, (8) G−(z) and has the periodicity properties θ(w + 1,τ) = − − iL1 1 ix (y¯−y) + = iπτ 2πiw = e 2 θ(w,τ) and θ(w τ,τ) e θ(w,τ).In 2 4π each cell θ(w,τ) has a single zero located at the 1 iL iL 1 iL i iL 1 1 θ + 1 , 1 θ y + + 1 − x , 1 w = + τ 2 2L2 L2 2 2L2 L2 L2 centre of the torus ( 2 2 ). We have chosen the × , 1 iL i 1 iL iL constant term in (7) so that the integral of φ over the θ + 1 − x ,τ θ y + + 1 , 1 2 2L2 L2 2 2L2 L2 dual torus is zero. This renders φ(z) an odd function, (14) φ(−z) =−φ(z). The Nahm potential derived from (7) with θ (w, τ) = ∂wθ(w,τ). The corresponding result was obtained in [5] via the ADHM construction. Here ∗ G+(z) G+(z) = G (−z) the flux strength κ is related to the ADHM ‘size’, λ,of for can be obtained via − . The zero modes have square-integrable singularities at an instanton centred at xµ = 0, both fluxes (see Table 1). There are two values of x 2 = πλ Table 1 κ . (9) = L1L2 Singularities of the x⊥ 0 zero modes z ∼ ωz∼−ω Although we do not directly use the ADHM formalism | 1|2 ∝| − − |2(κ−1) ∝| + + |−2κ in the present Letter the relation (9) proves useful in ψ y ω1 iω2 y ω1 iω2 | 2|2 ∝| − − |−2κ ∝| + + |2(κ−1) interpreting our results. ψ y ω1 iω2 y ω1 iω2 156 C. Ford, J.M. Pawlowski / Physics Letters B 540 (2002) 153–158

(in each copy of T 2) for which Table 1 does not hold. We have established that any lumps centred at the When x = 0 the Green’s function’s G± do not exist. constituent points have a simple exchange property. This does not mean there are no zero modes, indeed To actually see the lumps we must compute the field one can see that strengths. We can do this explicitly in the x⊥ = 0 slice, since here we have the exact zero modes (12). 0 ψ1(z; x = 0) = , Before we insert these modes into (10), they must be e−φ(z) normalised.√ This can be done by dividing both modes eφ(z) ρ ψ2(z; x = 0) = , (15) by ,where 0 2 2φ(z) 2 ρ = d ze |G−(z + ω)| . (18) are solutions of the x = 0 Weyl equation. ψ1 has the expected square-integrable singularity at z =−ω,but T 2 for z = ω, ψ1(z; x = 0) is zero. On the other hand In fact, one can write two components of the gauge ψ2(z, x = 0) diverges at z = ω but not at z =−ω.The potential solely in terms of this normalisation factor finiteness of G−(2ω) was used to derive Table 1. But if =−1 it is zero, ψ1 will lose its singularity at z =−ω.From Ax 2 τ3∂x logρ, the theta function representation of G−(z) one can = 1 Ax¯ 2 τ3∂x¯ log ρ, (19) see that there is exactly one value of x in T 2 where so that this occurs, πx =−iL1L2(ω1 + iω2).Wewishto investigate whether these ‘soft’ points are constituent Fx x¯ = τ3∂x ∂x¯ log ρ. (20) locations. Notice that the singularity profiles of ψ1 and ψ2 The computation of the other components is more are exchanged under the replacement κ → 1 − κ.This involved since the derivatives of the zero modes with suggests that the constituents are exchanged under this respect to x⊥ are required. However, the field strengths mapping. That is, if there are indeed lumps at the two can be more easily accessed via Green’s function points, then κ → 1 − κ swaps the two lumps. The techniques. Here we just quote the results; more details following result formalises this idea will be given elsewhere 2 ν F ¯ = 2πi(τ + iτ )κρ∂ , (21) F (x ,x⊥,κ) x x⊥ 1 2 x µν ρ −1 L1L2 where = V (x)Fµν −x + (ω1 + iω2), iπ 2 2φ(z) ν = d zG+(ω − z)e G−(z + ω). (22) −x⊥, 1 − κ V(x), (16) T 2 where V(x)is some U(2) gauge transformation. The The other components are fixed by self-duality, i.e., + = = proof of Eq. (16) goes as follows: make the change of Fx⊥x¯⊥ Fx x¯ 0andFx x⊥ 0. Note that ρ is → variables z →−z in (10). The zero mode ψp(−z; x) dimensionless, real and periodic (with respect to x1 + → + satisfies the same Weyl equation as ψp(z; x) ex- x1 L1, x2 x2 L2), while ν is dimensionless, complex and periodic up to constant phases. Both cept that the signs of κ and the xµ are flipped. ρ and ν diverge at x = 0, but the field strengths Under periodic gauge transformations A(z,ˆ −κ) and ˆ − should not. A careful analysis of ρ and ν in the A(z, 1 κ) are equivalent up to a constant potential = neighbourhood of x 0 shows that the field strengths ˆ −1 ˆ Aµ(z, −κ)= U (z) ∂µ + Aµ(z, 1 − κ)+ Bµ U(z), are well defined at this point. Alternatively, one can just note that ρ and ν are well behaved at the second (17) soft point and invoke (16). =− = where πB1 iL1L2ω2 and πB2 iL1L2ω1.Inthe Since we have all field strengths within the x⊥ = 0 Weyl equation Bµ can be absorbed into x1 and x2. slice we can compute the action density −1 p Thus U (z)ψ (z, −x − iL1L2(ω1 + iω2)/π, −x⊥) p − ; − 1 µν = 2 satisfies the same Weyl equation as ψ ( z x ,x⊥). 2 Tr Fµν F 16 ∂x ∂x¯ logρ C. Ford, J.M. Pawlowski / Physics Letters B 540 (2002) 153–158 157

2 2 2 2 2 ν + 128π κ ρ ∂ . (23) x ρ

We have provided integral representations of ρ and ν where the integrands are expressible in terms of standard functions. Using these results, we have made numerical plots of the action density for various values of κ and ω, and for the periods we have taken L1 = L2 = 2π, i.e., a square torus. The most symmetric = 1 = = 1 case, κ 2 and ω1 ω2 4 is plotted in Fig. 1. Two equal sized lumps are observed, one at x = 0, the other at x = (1 + i)π. Decreasing κ at fixed ω reduces the size of the x = 0 constituent but = = 1 increases its contribution to the action. Conversely, the Fig. 1. Plot of action density for x⊥ 0andκ 2 and = = 1 second constituent becomes larger but contributes less ω1 ω2 4 . to the action density. These two effects tend to (rather quickly) flatten the second peak as κ is reduced. Here = 7 = 3 we provide plots with κ 16 (Fig. 2) and κ 8 = = 1 = 3 (Fig. 3), again with ω1 ω2 4 . Already at κ 8 we see that the x = (1 + i)π constituent is more of a plateau than a peak. If κ is decreased much further the first peak will dominate completely; this peak will be essentially that of a BPST instanton on R4 with a scale parameter given by the ADHM formula (9). Indeed, if we take (9) at face value, and use (16), our two constituents√ are BPST instanton cores, one with scale λ = √κL1L2/π at x = x⊥ = 0, the other with scale λ = (1 − κ)L1L2/π at the second soft point. When L1 = L2 at least one λ is of the order of the size of the torus in line with the strong overlap seen in the = = 7 plots. Fig. 2. Plot of action density for x⊥ 0andκ 16 and 1 = 1 ω1 = ω2 = . The peaks are best resolved at κ 2 , when the 4 constituents are identical. The overlap can be avoided by taking one period, say L2, smaller than the other, and then taking a sufficiently large constituent separation in the x1 direction. However, then the size of one or both cores will exceed L2, tending to suppress or smooth out the dependence on x2. In this case the constituents become periodic monopoles. Actually, it is possible to see some hint of this monopole limit = 1 = = 1 even for a square torus; taking ω1 4 , ω2 0, κ 2 with L1 = L2 = 2π the two constituents (at x = 0 and x = iπ) overlap so strongly that they merge into a monopole worldline, in that the dependence of the field strengths on x2 is very weak. = 1 The κ 2 case has another interesting feature. If we choose the constituent locations so that they are = = 3 = = 1 separated by half periods the charge one instanton Fig. 3. Plot of action density for x⊥ 0andκ 8 , ω1 ω2 4 . 158 C. Ford, J.M. Pawlowski / Physics Letters B 540 (2002) 153–158

∗ τ3 ν Ax⊥ =− ∂x⊥ log ρ + 2πi(τ − iτ )κρ∂x¯ . (25) 2 1 2 ρ Here ρ and ν depend on r.Theρ and ν respectively given in (18) and (22) should be understood as the zeroth order terms of power series in r2 for the ‘full’ ρ and ν. Assuming that ν falls off exponentially for large r, a decay ρ ∼ r−1, would account for the twist. 1 Doubly periodic charge 2 instantons have also been found in simulations [8,9]. Many qualitative features reported are in accord with our ﬁndings. In particular, we agree that these objects are single lumps with radial symmetry, and that the action density is never zero in the x⊥ = 0 plane. Another striking observation in [8,9] is that the action density decays exponentially for = 1 = 1 = Fig. 4. When κ 2 , the choices ω1 2 π/L1, ω2 0and large r. To make a comparison here, more information = 1 = 1 ω1 2 π/L1, ω2 2 π/L2 yield constituent separations allowing about the large r properties of ρ and ν is required. 1 the torus to be cut (dashed line) to yield charge 2 instantons.

1 Acknowledgements can be ‘cut’ to yield a charge 2 instanton (see also 1 [7]). This happens when (ω1,ω2) is ( π/L1, 0), 2 We thank F. Bruckmann, M. Engelhardt, O. Jahn (0, 1 π/L ) or ( 1 π/L , 1 π/L ); the former and the 2 2 2 1 2 2 and P. van Baal for helpful discussions. C.F. was latter are shown in Fig. 4. After cutting we have a supported through a European Community Marie twist Z =−1 in the half torus. But to get a half- 12 Curie Fellowship (contract HPMF-CT-2000-00841). integer topological charge we also require a twist in the non-compact directions, Z03 =−1, since non- orthogonal twists are a prerequisite for fractional References instanton number. Far away from x⊥ = 0, the potential must be a pure gauge [1] K.M. Lee, C.H. Lu, Phys. Rev. D 58 (1998) 025011, hep-th/ −1 9802108. Aµ(x) → V (x ,x ,θ)∂µV(x ,x ,θ), 1 2 1 2 [2] T.C. Kraan, P. van Baal, Nucl. Phys. B 533 (1998) 627, hep-th/ r →∞, (24) 9805168. [3] P.J. Braam, P. van Baal, Commun. Math. Phys. 122 (1989) 267. iθ where x⊥ = re . The non-compact twist, Z03 =−1, [4] M. Jardim, Commun. Math. Phys. 216 (2001) 1, math.dg/ translates into a double-valued gauge function, 9909069. [5] C. Ford, J.M. Pawlowski, T. Tok, A. Wipf, Nucl. Phys. B 596 V(x1,x2,θ + 2π)=−V(x1,x2,θ).However,theex- plicit results we have given are restricted to the r = 0 (2001) 387, hep-th/0005221. = [6] H. Reinhard, O. Schröder, T. Tok, V.C. Zhukovski, hep-th/ slice. What can we say about the full r 0 potential? 0203022. It can be argued that as for r = 0 (as well as SU(2) [7] A. Gonzalez-Arroyo, Nucl. Phys. B 548 (1999) 626, hep-th/ calorons) two functions, one real the other complex, 9811041. are sufﬁcient [8] A. Gonzalez-Arroyo, A. Montero, Phys. Lett. B 442 (1998) 273, hep-th/9809037. ∗ τ3 ν [9] A. Montero, Phys. Lett. B 517 (2001) 142, hep-lat/0104008. Ax =− ∂x logρ − 2πi(τ − iτ )κρ∂x¯⊥ , 2 1 2 ρ Physics Letters B 540 (2002) 159–165 www.elsevier.com/locate/npe

Higher order hybrid Monte Carlo at ﬁnite temperature

Tetsuya Takaishi

Hiroshima University of Economics, Hiroshima 731-0192, Japan Received 2 April 2002; received in revised form 28 May 2002; accepted 7 June 2002 Editor: T. Yanagida

Abstract The standard hybrid Monte Carlo algorithm uses the second order integrator at the molecular dynamics step. This choice of the integrator is not always the best. Using the Wilson fermion action, we study the performance of the hybrid Monte Carlo algorithm for lattice QCD with higher order integrators in both zero and finite temperature phases and find that in the finite temperature phase the performance of the algorithm can be raised by use of the 4th order integrator.  2002 Elsevier Science B.V. All rights reserved.

PACS: 12.38.Gc; 11.15.Ha

Keywords: Lattice QCD; Hybrid Monte Carlo algorithm

1. Introduction molecular dynamics (MD) step. The integrator causes O(t3) integration errors, where t denotes the step- In lattice QCD the hybrid Monte Carlo (HMC) al- size. Due to the integration errors the Hamiltonian gorithm [1] is widely used for simulations of even- is not conserved. The errors introduced by this in- flavorofquarks.1 These simulations are usually dif- tegrator have to be removed by the Metropolis test, ficult tasks, especially at small quark masses where i.e., accept the new configuration with a probability the computational cost of the matrix solver which is ∼ min(1, exp(−H )) where H = Hnew − Hold is the most time consuming part of the HMC algorithm an energy difference between the starting Hamiltonian grows. In order to obtain reliable results within limited Hold and the new Hamiltonian Hnew at the end of the computer resources it is important to find an efficient trajectory. way to implement the HMC algorithm so that the total The acceptance at the Metropolis step depends on computational cost of the algorithm is minimized [3]. the magnitude of the energy difference induced by The basic idea of the HMC is a combination of the integration errors. If a higher order integrator is (1) molecular dynamics and (2) Metropolis test. Usu- used at the MD step, the integration errors can be ally the second order leapfrog integrator is used at the reduced. Therefore one may easily imagine that the performance of the HMC increases with the higher order integrator. However this is not always true E-mail address: [email protected] (T. Takaishi). 1 Odd-flavor simulations of the HMC algorithm are also possible since the higher order integrator has more arithmetic by modifying the Hamiltonian [2]. operations than the lower one and this might decrease

0370-2693/02/$ – see front matter  2002 Elsevier Science B.V. All rights reserved. PII:S0370-2693(02)02139-1 160 T. Takaishi / Physics Letters B 540 (2002) 159–165 the performance. The total performance should be the equations. This scheme is written as  measured by taking account of two effects: acceptance + t = + t  q t 2 q(t) 2 p(t), and the number of arithmetic operations. In Ref. [4]  ∂S q t+ t the performance of the higher order integrator at zero + = − 2  p(t t) p(t) t , (3) temperature (β = 0) was studied systematically and  ∂q + = + t + t + it turned out that for the simulation parameters used q(t t) q t 2 2 p(t t). for the current large-scale simulations with the Wilson Eq. (3) forms an elementary MD step. This elemen- fermion action the 2nd order integrator is the best one. tary MD step is performed repeatedly N times. The The main reason why the higher order integrators are trajectory length τ is given by τ = N × t. not so effective is that the energy difference caused Any integrator which satisfies two conditions: (1) by the errors of the higher order integrator increases time reversible and (2) area preserving can be used more rapidly than that of the lower one as the quark for the MD step of the HMC. These conditions are mass decreases. In Ref. [5] it is shown that at finite needed to satisfy the detailed balance. When we use temperature the quark mass dependence of the energy the Lie algebraic formalism [6–9] we can easily con- difference is small. If so, the conclusion of Ref. [4] struct higher order integrators which satisfy the above might change at finite temperature. In this Letter we conditions. From the Lie algebraic formalism we find test higher order integrators at finite temperature and that higher order integrators can be constructed by demonstrate that they can actually perform better than combining lower order integrators. Let G2nd(t) be the lower order. an elementary MD step of the 2nd order integrator with a step-size t. The 4th order integrator G4th(t) is constructed from a product of three 2nd order inte- 2. Higher order integrator grators as [6–10]

G4th(t) = G2nd(a1t)G2nd(a2t)G2nd(a1t), (4) In this section we define higher order integrators where the coefficients a are given by studied here. Our definition is parallel to that of i Ref. [4]. Let H be a Hamiltonian given by 1 a = , (5) 1 2 − 21/3 1 21/3 H = p2 + S(q), (1) a =− . (6) 2 2 2 − 21/3 Eq. (4) means that there are three elementary MD where q = (q1,q2,...)and p = (p1,p2,...)are coordinate variables and conjugate momenta, respectively. steps: (i) first we integrate the equations by Eq. (3) S(q) is a potential term of the system considered. For with a step-size of a1t, (ii) then proceed the 2nd the lattice QCD, q correspond to link variables and order integration with a step-size of a2t, (iii) finally S(q) consists of gauge and fermion actions. integrate the equations with a step-size of a1t. In the MD step we solve Hamilton’s equations of After performing these three elementary MD steps motion, sequentially we obtain the 4th order integrator with the step-size t. This construction scheme can be  generalized to the higher even-order integrators. The  dqi = ∂H dt ∂p , (2k + 2)th order integrator is given recursively by i (2)  dp i =−∂H . = dt ∂qi G2k+2(t) G2k(b1t)G2k(b2t)G2k(b1t), (7) where the coefficient b are In general these equations are not solvable analyti- i cally. Introducing a step-size t, the discretized ver- 1 b1 = , (8) sion of the equations are solved. In the conventional 2 − 21/(2k+1) HMC simulations the 2nd order leapfrog scheme, 21/(2k+1) which causes O(t3) step-size error, is used to solve b2 =− . (9) 2 − 21/(2k+1) T. Takaishi / Physics Letters B 540 (2002) 159–165 161

Table 1 Parameter sets (Y1–Y3) of the 6th order integrators by Yoshida [8]. w0 is given by w0 = 1 − w1 − w2 − w3 Y1 Y2 Y3 w1 −0.117767998417887e−1 −0.213228522200144e+10.152886228424922e−2 w2 0.235573213359357e+00.426068187079180e−2 −0.214403531630539e+1 w3 0.784513610477560e+00.143984816797678e+10.144778256239930e+1

Compared with the 2nd order integrator, the compu- Although mathematically speaking Eq. (13) is valid tational cost of the nth order one constructed from only for H 21/2 1, the numerical study [4] shows Eq. (7) grows as 3n/2−1. For instance the 6th order in- that Eq. (13) approximates the acceptance quit well for 2 1/2 tegrator has 9 times more arithmetic operations than H 3 which corresponds to Pacc 20%. those of the 2nd order one. Yoshida [8] found three This is enough for our purpose since typically the parameter sets of 6th order integrators with less arith- acceptance of the HMC is taken to be Pacc 50%. metic operations (Table 1). Yoshida’s 6th order inte- In the lowest order of t, H 21/2 of the nth grators consist of seven G2nd’s as order integrator is expressed as [4,5] 2 1/2 = 1/2 n + O n+1 G6th(t) = G2nd(w3t)G2nd(w2t)G2nd(w1t) H CnV t t , (14)

× G2nd(w0t)G2nd(w1t) where V is volume of the system and Cn is a Hamiltonian dependent coefﬁcient. × G (w t)G (w t). (10) 2nd 2 2nd 3 Substituting Eq. (14) into Eq. (13) one obtains In Ref. [4] these 6th order integrators were examined 1/2 n Pacc=exp −CnV t , (15) and one of three parameter sets, denoted by Y1, was √ found to give smaller integration errors than those of where Cn ≡ Cn/ 2π. If one uses Eq. (15) for Eq. (11) others. In this study the parameter set Y1 (see Table 1) one can easily obtain the optimal step-size which gives is used for the 6th order integrator. the maximum efﬁciency:

n 1 topt = . (16) 1/2 3. Efficiency of the HMC algorithm nCnV Furthermore substituting topt to Eq. (15) one obtains In order to compare among various integrators one the optimal acceptance2 as [4] needs a criterion which ranks integrators. Following 1 Ref. [4] we utilize the efficiency function Eff con- P = exp − (17) acc opt n structed from a product of acceptance Pacc and step- size t: 0.61 2nd, = 0.78 4th, (18) Eff = Pacct. (11) 0.85 6th. Note that the above result does not depend on the This function has one maximum at a certain step-size specific Hamiltonian and can be applied for any which we denote t . Using the energy difference opt model. H the acceptance is expressed by [5] Using Eq. (16) and (17) we obtain the optimal 1 efficiency of the nth order integrator P =erfc H 1/2 , (12) acc 2 nth 1 n 1 (Eff) = exp − . (19) where erfc is the complementary error function. In opt 1/2 n nCnV stead of using Eq. (12), when H is small, we may use 2 1/2 A recent study [11] shows that if one considers higher order 2 1 2 effects of t the optimal acceptance might be slightly changed. In Pacc=exp −√ H . (13) π 8 the current study we omit this small effect. 162 T. Takaishi / Physics Letters B 540 (2002) 159–165

We use Eq. (19) to compare among integrators. Eq. (19) is easy to handle because Eq. (19) has one unknown parameter Cn and the value of Cn can be estimated easily from a single simulation on a rather small lattice. Now let us compare nth and mth order integrators (n > m). If one obtains a gain G for the nth order integrator over the mth order one, the following condition must be satisﬁed

nth = mth (Eff)opt Gknm(Eff)opt , (20) where knm is a relative cost factor needed to implement the nth order integrator against the mth one and G Fig. 1. H 21/2 on a 163 × 4 lattice at β = 5.75 and κ = 0.155 is defined so that both nth and mth order integrators as a function of t. The lines proportional to tn are also drawn. are equally effective with G = 1. In our case, k42 = 3 The simulations were done with the Wilson fermion action. and k64 = 7/3. Substituting Eq. (19) to Eq. (20) and rewriting the equation one obtains the lattice volume corresponds to H 21/2 1.3 It seems that for needed to have the gain G H 21/2 1 the effect of the higher order terms is nm n small. n − m 1 1 1 2 2 = − + V Gknm exp m n mCm m 4. Simulation results × (nCn) . (21) Our formulas are based on the assumption that We use the plaquette gauge action and standard H 21/2 satisfies Eq. (14). The validity of Eq. (14) Wilson fermion action with two flavors of degenerate depends on the action we take and simulation parame- quarks (nf = 2). We first determine the coefficients ters. In general it is expected that the contribution of Cn at zero and finite temperature. This can be done the higher order terms to Eq. (14) becomes small as by measuring H 21/2 at a small step-size and the lattice size increases. Let us rewrite Eq. (14) as substituting the results into Eq. (14). The trajectory = length τ is set to 1. We choose β 5.75 and make 2 1/2 1/2 n 1/2 n+1 4 3 × H = CnV t + Cn+1V t , (22) simulations on both 12 and 12 4 lattices. The 4 critical kappa κc of nf = 2atthisβ is around 0.157. 1/2 n+1 where Cn+1V t stands for the first relevant We use several κ’s in a range of 0.1 κ 0.155. higher order term. We measure the contribution of the In this range we maintain the confinement phase on higher order term by the ratio the 124 lattice and the deconfinement phase on the 123 × 4 lattice. In this study we refer to the results 1/2 n+1 Cn+1V t Cn+1 4 3 × = t. (23) on the 12 (12 4) lattice as those at zero (finite) 1/2 n CnV t Cn temperature. In order to keep the constant acceptance, t must be Figs. 2–4 show Cn as a function of mqa.Heremq a = − taken to be small as the lattice size increases. Thus is defined by mq a (1/κ 1/κc)/2. At large quark Eq. (23), i.e., the contribution of the higher order term, masses where the fermionic effects are negligible, becomes small for larger lattices. values of Cn at zero and finite temperature coincide 3 each other. This may indicate that the contribution Fig. 1 shows a typical example of Cn on a 16 × 4 lattice as a function of t. H 21/2 are well- to Cn from the gauge sector is almost same at zero expressed by the functions proportional to tn except for large t. We are only interested in the acceptance 3 See Fig. 1 in [4]. 4 region indicated by Eq. (18): Pacc 60% which This value is estimated from Fig. 5 in Ref. [12]. T. Takaishi / Physics Letters B 540 (2002) 159–165 163

Fig. 2. C2 as a function of the quark mass mq a. In the ﬁgure, C2 is Fig. 5. Le as a function of the quark mass mq a. Le stands for the 1/2 normalized as C2/(2π) ≡ C2. lattice size for which the higher order integrator and the lower order one are equally effective. Circles are from comparison between the 2nd order integrator and the 4th order one, and squares are comparison between the 4th order integrator and the 6th order one.

and finite temperature. At small quark masses Cn at zero temperature increases more rapidly than those at finite temperature as mq a decreases. The quark mass dependence of Cn at finite temperature, compared to that at zero temperature, is small for all the integrators studied here. This behavior is consistent with the result of Ref. [5]. Substituting values of Cn at finite temperature into Eq. (21) with G = 1, we calculate the = 3 × lattice size Le (here V Le Nt ). This lattice size Le is the one with which the higher order integrator and the lower order one perform equally. For a lattice size Fig. 3. Same as in Fig. 2 but for C . 4 L>Le the higher order integrator is more effective than the lower one. Fig. 5 shows Le from comparison between the 2nd and 4th order integrators (2nd vs. 4th) and between the 4th and 6th order ones (4th vs. 6th). For the case of 4th vs. 6th, Le increases as mq a decreases, which we do not appreciate. On the other hand for the 2nd vs. 4th, Le remainslessthan20even at small mq a. This result is contrast to that obtained at zero temperature where Le increases as mq a decreases [4]. The above result encourages us to use the 4th order integrator at finite temperature. The results in Fig. 5, however, just show the lattice on which the both integrators are equally effective. To use the higher order integrator in simulations one must obtain some gain over the lower order one. In Fig. 6, using Eq. (21), we show the expected gain (the region Fig. 4. Same as in Fig. 2 but for C6. between the solid lines) at κ = 0.1525 (mq a ≈ 0.094) 164 T. Takaishi / Physics Letters B 540 (2002) 159–165

Table 2 Step-size and acceptance at κ = 0.1525 κ = 0.1525 183 × 4283 × 4 2nd 4th 2nd 4th t 1/24 1/10 1/36 1/12 Acceptance 0.57(2) 0.81(2) 0.60(3) 0.81(3)

5. Conclusions

We have studied higher order integrators for the HMC algorithm with the Wilson fermion action at finite temperature. Contrast to the zero temperature = 3 × Fig. 6. The gain G as a function of L (here V L 4) at case, the 4th order integrator at finite temperature can κ = 0.1525. The solid lines are calculated from Eq. (21). Circles are from Monte Carlo simulations. be more effective than the 2nd order one. This was demonstrated by the simulations at β = 5.75 on L3 ×4 lattices. The gain is dependent of the lattice size. It was shown that on the 283 × 4 lattice at β = 5.75 and as a function of lattice size L.TohaveG = 2gain κ = 0.1525 the 4th order integrator is about 35% faster (which means 2 times faster) a lattice size L ≈ 100 is than the 2nd order one. required. This huge lattice size is still not accessible When large-scale simulations at finite temperature in the current large-scale simulations. Probably the are planned, it is recommended to check which inte- maximum lattice size accessible at the moment is L grator is effective for the lattice considered. This check 50. Therefore we cannot expect a large gain from the can be done easily. First we measure C and C .This 4th order integrator even if it is used now. If we use 2 4 first step does not take much computational time since a lattice with L ≈ 50, G ≈ 1.5 can be achieved. Thus they can be measured on a small lattice. Then substi- at the level of the current large-scale simulations, we tuting values of C and C to Eq. (21), we obtain a expect to obtain G 1.5. 2 4 relation between G and L. If we obtain G>1onthe We also make simulations at κ = 0.1525 to confirm lattice planned for the simulations, we should consider that we can actually obtain some gain for the 4th order to use the 4th order integrator. integrator over the 2nd one. At κ = 0.1525, L is e Eq. (21) is obtained by using the approximation estimated to be 17.5(9). We choose 183 ×4 and 283 ×4 at small t. There might exist the systematic errors lattices. On the 183 ×4 lattice we expect G ≈ 1 and on caused by the approximation. Although for the present the 283 × 4 lattice, G>1. The step-size is adjusted so study we considered the errors to be small, for other that the acceptance gives a similar value with Eq. (18). actions and simulation parameters they might con- The gain G is calculated by tribute largely. We have to keep in mind that there might exist such contributions depending on the simu- (P × t) G = acc 4th , (24) lation details. 3(Pacc × t)2nd where a factor of 3 in the denominator comes from the relative cost factor k42 = 3. Table 2 shows the Acknowledgements simulation results. Using these results, we obtain G = 1.08(3) on the 183 × 4 lattice and G = 1.35(8) on the The simulations were done on the NEC SX-5 at IN- 283 × 4 lattice (see also Fig. 6). As expected the gain SAM Hiroshima University and at Yukawa Institute. increases with L. The result on the 283 × 4 lattice is an The author would like to thank Atsushi Nakamura for example showing that the 4th order integrator is more useful discussion and comments. This work was sup- effective than the 2nd order one. ported by the Grant in Aid for Scientific Research by T. Takaishi / Physics Letters B 540 (2002) 159–165 165 the Ministry of Education, Culture, Sports, Science [4] T. Takaishi, Comput. Phys. Commun. 133 (2000) 6. and Technology (No. 13740164). [5] S. Gupta, A. Irbäck, F. Karch, B. Petersson, Phys. Lett. B 242 (1990) 437. [6] J.C. Sexton, D.H. Weingarten, Nucl. Phys. B 380 (1992) 665. [7] M. Creutz, A. Gocksch, Phys. Rev. Lett. 63 (1989) 9. References [8] H. Yoshida, Phys. Lett. A 150 (1990) 262. [9] M. Suzuki, Phys. Lett. A 146 (1990) 319. [1] S. Duane, A.D. Kennedy, B.J. Pendleton, D. Roweth, Phys. [10] M. Campostrini, P. Rossi, Nucl. Phys. B 329 (1990) 753. Lett. B 195 (1987) 216; [11] JLQCD Collaboration, S. Aoki et al., Phys. Rev. D 65 (2002) S. Gottlieb, W. Liu, D. Toussaint, R.L. Renken, R.L. Sugar, 094507. Phys. Rev. D 35 (1987) 2531. [12] Y. Iwasaki, K. Kanaya, S. Kaya, S. Sakai, T. Yoshie, Phys. Rev. [2] T. Takaishi, Ph. de Forcrand, Int. J. Mod. Phys. C 13 (2002) D 54 (1996) 7010. 343, hep-lat/0108012. [3] For recent studies on this subject see, e.g., M. Peardon, Nucl. Phys. B (Proc. Suppl.) 106 (2002) 3. Physics Letters B 540 (2002) 167–172 www.elsevier.com/locate/npe

Entropy bounds and Cardy–Verlinde formula in Yang–Mills theory

Shin’ichi Nojiri a, Sergei D. Odintsov b

a Department of Applied Physics, National Defence Academy, Hashirimizu Yokosuka 239-8686, Japan b Lab. for Fundamental Study, Tomsk State Pedagogical University, 634041 Tomsk, Russia Received 28 May 2002; accepted 17 June 2002 Editor: M. Cveticˇ

Abstract Using gauge formulation of gravity the three-dimensional SU(2) YM theory equations of motion are presented in equivalent form as FRW cosmological equations. With the radiation, the particular (periodic, big bang – big crunch) three-dimensional universe is constructed. Cosmological entropy bounds (so-called Cardy–Verlinde formula) have the standard form in such universe. Mapping such universe back to YM formulation we got the thermal solution of YM theory. The corresponding holographic entropy bounds (Cardy–Verlinde formula) in YM theory are constructed. This indicates to universal character of holographic relations.  2002 Elsevier Science B.V. All rights reserved.

It becomes popular in theoretical physics to apply tion of Hubble, Bekenstein and Bekenstein–Hawking different formulations (and even theories) in the de- entropies maybe done. As a result FRW equation is scription of the same phenomenon. For example, it is represented in the form similar to two-dimensional expected that Einstein theory presented in Yang–Mills CFT entropy (so-called Cardy–Verlinde formula [2]). (YM) form is easier to quantize. Moreover, its renor- Introducing the radiation (matter) the explicit solu- malizability properties seem to be better in YM-like tion of such three-dimensional periodic universe is form [1]. From another point, there exists the YM the- obtained. The cosmological entropy bounds (holo- ory presentation in Einstein-like form. In this relation graphic Cardy–Verlinde formula) are quite simple for the natural question is: can one achieve some new re- such universe. In particular, the Bekenstein entropy sults in YM theory using recent studies of cosmology is constant. Such three-dimensional universe is then based on holographic principle? mapped back to YM theory form. (Such procedure In this Letter we start the investigation in this di- maybe considered as an interesting way of generat- rection. Using three-dimensional YM theory as an ex- ing of YM solutions from FRW cosmological solu- ample we first rewrite it in the gravitational form. tions). The correspondent thermal YM solution on The YM equations of motion are then rewritten as which YM action diverges is constructed. The di- FRW cosmological equations where natural defini- vergence of action is absorbed into the renormalization of the gauge coupling constant. After the identification of YM entropy with Bekenstein entropy the E-mail addresses: [email protected] (S. Nojiri), Cardy–Verlinde formula defines the entropy bounds in [email protected] (S.D. Odintsov). YM theory.

The action of Yang–Mills theory with the gauge the expression of the spin connection by the dreibein group SU(2) is given by fields. In the following, as a most simple and non-trivial 3 1 d a aij 2 a ai case, we consider the case that the dimension of the SYM = d x F F + 4g A J , 4g2 ij YM i spacetime is 3 (d = 3) and the spacetime is Euclid- YM a=1 (1) ean, where the Lorentz group is SO(3) or SU(2).In 3 case that the spacetime is 3-dimensional Euclidean a = a − a − abc b c Fij ∂i Aj ∂iAj Ai Aj . (2) space, the Poincare algebra is generated by the gen- b,c=1 erators of the translation T a (a = 1, 2, 3) and the ro- a = Here d is the dimension of the spacetime and J ai is tation R (a 1, 2, 3) around a-axis. The algebra is the external source. For d = 3, one can rewrite the explicitly given by action (1) in the following form: Ra,Rb = abcRc, Ra ,Tb = abcT c, 3 a b = 1 d a ai 2 a ai T ,T 0. (7) SYM = d x B B + 2g A J . (3) 2g2 i YM i YM a=1 Denote the gauge fields corresponding to the transla- a a = a tion and the rotation by ui and Ai (a 1, 2, 3), re- Here Bi is defined by Ta spectively. Then the curvatures Rij corresponding to ak = kij a − 1 abc b c the translation and RRa corresponding to the rotation B ∂i Aj Ai Aj . (4) ij 2 are given by The equations of motion are Ta = a − a − abc b c − b c Rij ∂i uj ∂i uj Ai uj Aj ui , = kij a − 2 ai 0 DkBj gYMJ Ra = a − a − abc b c Rij ∂iAj ∂iAj Ai Aj . (8) kij a 1 abc b c 2 ai = ∂kB − A B − g J . (5) The relation j 2 k j YM RTa = 0 (9) Here Di expresses the covariant derivative. On the ij other hand, Bianchi identities are: corresponds to the torsionless condition in gravity. The a ai condition (9) can be solved with respect to A : 0 = Di B . (6) i a =− abc b − b −1 j There exists a formulation of gravity starting from Ai ∂iuj ∂j ui u c the gauge theory, whose gauge group is the Poincare 1 bcd b b −1 j −1 k a group composed of translations and Lorentz transfor- + ∂ku − ∂j u u u u 4 j k c d i mations. The gauge field of the translation corresponds nmk b a b 1 b a ∂mu (u u − u u ) to the vierbein (or vielbein) and that of the Lorentz = k n i 2 n i . (10) transformation to the spin-connection. The gauge cur- detu vature of the translation corresponds to the torsion. Then the field strength Bak defined by (4) is related Then if we impose a condition that the curvature van- ij = ij − 1 ij with the Einstein tensor G R 2 g R by ishes, we can solve the spin connection in terms of the √ ai = a ij vier(viel)bein field if the dimension of the spacetime is B guj G . (11) larger than 2. As the Lorentz group is S(d − 1, 1) for Here the metric tensor g is defined by d-dimensional spacetime, one can rewrite the gauge ij field of SO(d − 1, 1) by the vier(viel)bein field. Such a = a b gij ui uj (12) formulation for the supergravity is developed in [4]. In the similar way in Ref. [5], 4-dimensional Yang–Mills and Rij and R are the Ricci curvature and the scalar theory is presented in the Hamiltonian formulation in curvature constructed on gij . Let denote the covariant terms of 3-dimensional gravity. The parametrization derivative with respect to the SO(3) gauge group by Di of the gauge field used in the Letter corresponds to and the covariant derivative with respect to the gravity S. Nojiri, S.D. Odintsov / Physics Letters B 540 (2002) 167–172 169

2t 3t 1θ 3θ 1φ 2φ by ∇i . Then Eq. (11) tells that B = B = B = B = B = B = 0, √ 1t =− 2 2A ˙2 − ai = a ∇ ij − i kl + l kl B l sin θe A sin θ, Dj B guj j G ΓjlG Γlj G , (13) B2θ = l sin θ A¨ + A˙2 eA, which leads 3φ = ¨ + ˙2 A √ B l A A e . (22) D Bai = gua∇ Gij . (14) i j i If we define Bab by The Bianchi identity ab = b ai B ui B , (23) ∇ ij = i G 0 (15) Eq. (11) tells Bab is symmetric: can be consistent with the gauge Bianchi identity (6). Bab = Bba. (24) By using (11), the Einstein equation Any N × N symmetric matrix M, in general, can be Gij = κ2T ij , (16) diagonalized by a matrix O from SO(N),   with the energy–momentum tensor Tµν can be rewrit- m1 0 ··· 0  ···  ten in terms of the gauge fields:  0 m2 0 0   .  → −1 =  .. .  κ2 √ M O MO  00 . .  . Ba = Bext a,Bext a ≡ guaT ij . (17)  . . ..  i i i 2 j . . . 0 ext a 00··· 0 mN Here one can regard Bi as an external filed, which is related with J ai (5) by Then using SO(3) gauge transformation, one can diagonalize Bab. The gauge transformation removes kij ext a = 2 ai ab DkBj gYMJ . (18) 3 components in B , which corresponds to the degree of the freedom. Eq. (19) and also Eq. (22) manifest Since the conservation of the energy–momentum ten- such a special gauge choice. ∇ ij = sor i T 0 corresponds to the “conservation law” The stress-energy tensor may be presented in terms of the external fields, if the energy–momentum con- of the energy density ρ and the pressure p as servation holds one can construct the solution of the − − Bianchi identity (6) from the solution of the Einstein T tt =−ρ, T θθ = l 2e 2Ap, equation (16) using (11). 1 − T φφ = e 2Ap, Now we consider more concrete solution. Assume l2 sin θ the dreibein fields have a form of other components = 0. (25) 1 = 2 = A(t) 3 = A(t) ut 1,uθ e ,uφ sin θe , Thus other components = 0, (19) Bext 2t = Bext 3t = Bext 1θ = Bext 3θ = Bext 1φ which give the Euclidean FRW metric = Bext 2φ = 0, κ2 ds2 = dt2 + l2e2A(t) dθ2 + sin2 θdφ2 . (20) Bext 1t =− l2 sin θe2Aρ, 2 The Minkowski signature metric can be obtained κ2 Bext 2θ = l sin θeAp, changing the compact gauge group SU(2) or SO(3) to 2 the non-compact ones SU(1, 1), SO(2, 1) or SL(2,C). κ2 Bext 3φ = leAp. (26) The assumptions (19) give 2 1 = 2 = 3 = 1 = 2 = 3 = Then by combining (17), (21) and (26), we obtain the At At At Aθ Aθ Aφ 0, FRW equations: A3 = leAA,˙ A2 =−l sin θeAA,˙ θ φ 2 ˙2 1 κ A1 = cosθ, (21) A + = ρ, φ a2 2 170 S. Nojiri, S.D. Odintsov / Physics Letters B 540 (2002) 167–172

κ2 A¨ + A˙2 = p. (27) corresponds to the total energy 2 Φext =−E. (34) If we define J ai by (18), any solution of (27) satisfies 2 − − 2 the equations of motion (5) by the identification (22). Since Eq. (31) tells SB (SBH SB) 0, Eqs. (28) Moreover, if the energy momentum tensor (25) satis- and (34) give some bound between the external mag- ij fies the conservation law ∇i T = 0, the solution of netic flux and the spacial volume V : Eq. (27) also satisfies the Bianchi identity (6). = A 4V 2 Here a le . Combining (21) and (26) with (18), Φext 2 + Φext . (35) we may obtain the external source J ai. κ2a2 In gravity one can define the Hubble, Bekenstein We should note, however, the spacial volume V is and Bekenstein–Hawking entropies SH, SB and SBH given from the gravity side. by [2] In order to proceed further, we consider the radi- 2 ation as a matter. Then the energy density ρ and the 2 (4π) V 1t V S =− dθ dφB + pressure p have the following forms: H κ4 a2 − (4π)2V 2A˙2 ρ = 2p = ρ e 3A. (36) = , 0 4 κ The FRW equations (27) are −2A 2 = ext 1t = e κ − SB πa dθ dφB πaE, A˙2 + = ρ e 3A, l2 2 0 4πV 2 S = , (28) ¨ ˙2 κ −3A BH 2 A + A = ρ e . (37) κ a 4 0 with space volume V and the total energy E defined The solution of (38) is given by using an intermediate by variable η as follows, √ = = 2 A 2 V dθ dφ g 2πa , a = le = a0 sin η, √ 1 = t = a0 η − sin 2η . (38) E dθ dφ gρ. (29) 2

Then from FRW equations it follows the relation [2] The maximum radius of the universe a0 is given by reminding about two-dimensional CFT entropy [3] κ2l3 (so-called Cardy–Verlinde formula) a = ρ . (39) 0 2 0 2 + 2 = SH SBH 2SBHSB. (30) The universe starts with the big bang at η = 0(t = 0) = Eq. (30) follows from the first equation in (27). and the universe ends with the big crunch at η π = This equation can be rewritten as (t πa0). Then the universe has the period of πa0. As this is Euclidean spacetime, one can regard period 2 = 2 − − 2 SH SB (SBH SB) . (31) as the inverse of the temperature T The well-known square root like in two-dimensional 1 T = . (40) CFT entropy appears when one evaluates Hubble πa0 entropy from above relation. We should note that Since the energy E (29) is given by = 1t 2 −3A 3 −1 Φ dθ dφB (32) E = 4πa ρ0e = 4πl ρ0a , (41) is total magnetic flux. Then the external flux we find that the Bekenstein entropy SB is constant: 2 ext = ext 1t 2 3 8π 2 Φ dθ dφB (33) SB = 4π l ρ0 = a0 = 8πκ T. (42) κ2 S. Nojiri, S.D. Odintsov / Physics Letters B 540 (2002) 167–172 171

Here Eqs. (39) and (40) are used. When the radius of 1 1 3 =− 3 √ ext a ext ai = d x Bi B . (49) the universe is maximal (a a0), the Hubble entropy 2g2 g YM a=1 SH and the Bekenstein–Hawking entropy SBH are Then with the help of (19) and (26) one gets 2 SH = 0,SBH = 16πκ T. (43) 4 κ t 2A 2 2 In this case CV formula becomes the identity. SYM =− d e ρ + 2p . (50) 8g2 The interesting question is: what happens in terms YM of Yang–Mills theory? We now evaluate the Yang– By further substituting the solution (38) and using Mills action by substituting the classical solution. (36), (39) and (40), we obtain the following expression Being in the Euclidean spacetime, the action can for the action be regarded as the free energy F divided by the π 6π2 dη temperature T : S =− T . (51) YM 2 2 gYM sin η F 0 SYM = . (44) T Th integration in (51) diverges. The divergence may be Using the obtained free energy, the Yang–Mills theory absorbed into the renormalization of the Yang–Mills entropy may be evaluated. coupling g: In curved spacetime, the action (1) has the follow- π ing form: 1 1 dη ≡ . (52) g2 g2 sin2 η √ 3 R YM 1 3 a aij 2 a ai 0 SYM = d x g F F + 4g A J . 4g2 ij YM i YM a=1 Using the renormalized coupling constant gR,the (45) action has the following form: Then (3) looks as: 6π2 S =− T (53) 1 YM 2 = 3 gR SYM 2 d x 2gYM given by 3 √ 1 a ai 2 a ai =− 2 2 2 × √ B B + 2g gA J . (46) F 6π gRT . (54) g i YM i a=1 Then the entropy is Even in curved spacetime, the Bianchi identity (6) dF 12π2 is always satisfied since the Bianchi identity is a S =− = T. (55) 2 topological equation. On the other hand, the equation dT gR of motion (5) is modified as If we identify a 1 Bj 2 = √ kij √ − 2 ai 12π 2 0 Dk gYMJ . (47) = 8πκ , (56) g g 2 gR If we define, however, J ai, instead of (18), the entropy (55) is equal to the Bekenstein entropy Bext a SB (42). As we consider curved spacetime, the spacial 1 kij j = 2 ai √ Dk √ gYMJ , (48) volume V is given by (29). Therefore the identifica- g g tion (56) gives a bound for the entropy of the Yang– Eq. (47) is always satisfied. Mills theory with the external source J ai (48): By using the equations of motion (47) and (17), the S2 + S2 = 2S S. (57) action (46) can be rewritten in the following form: H BH BH As we are now considering the curved spacetime, S 3 BH 1 3 1 a ai and SB (57) is given, as in (28), by the structures SYM =− d x √ B B 2g2 g i of spacetime where the Yang–Mills field lives. Using YM a=1 172 S. Nojiri, S.D. Odintsov / Physics Letters B 540 (2002) 167–172

(28), (32) and the identiﬁcation S = SB, the entropy dimensional CFT entropy (see [6] for a recent review) in the Yang–Mills theory maybe expressed in terms of or cosmological entropy bounds. The main lesson of the magnetic ﬂux. We can also rewrite Eq. (57) in the our investigation is that similar results are in some form of (31): sense universal. They should be expected in other theories, like in the case of YM theory under considera- 2 = S2 − − S 2 SH (SBH ) , (58) tion. which gives the bound for the entropy S: S2 2 S2 − S 2 References SH, (SBH ) . (59)

If necessary the above YM entropy bound found [1] D. Diakonov, V. Petrov, hep-th/0108097, Grav. Cosm. 8 (2002), for the particular thermal solution mapped to three- to appear. dimensional FRW cosmology maybe completely ex- [2] E. Verlinde, hep-th/0008140. [3] J.L. Cardy, Nucl. Phys. B 270 (1986) 186; pressed in terms of gauge ﬁelds and sources. More- H.W.J. Bloete, J.L. Cardy, M.P. Nightingale, Phys. Rev. Lett. 56 over, it is clear that similar procedure maybe applied (1986) 742. to study new solutions and entropy bounds for other [4] T. Kugo, S. Uehara, Nucl. Phys. B 226 (1983) 49; cases, including YM theory in four and higher dimen- T. Kugo, S. Uehara, Prog. Theor. Phys. 73 (1985) 235. sions. [5] P.E. Haagensen, K. Johnson, Nucl. Phys. B 439 (1995) 597, hep- th/9408164. There are recent gravitational theory results which [6] S. Nojiri, S.D. Odintsov, S. Ogushi, hep-th/0205187. have the holographic origin: like FRW cosmology presentation in the form reminding about two- Physics Letters B 540 (2002) 173–178 www.elsevier.com/locate/npe

Symmetry breaking and time variation of gauge couplings

Xavier Calmet a,1, Harald Fritzsch a,b,2

a Ludwig-Maximilians-University Munich, Sektion Physik, Theresienstraße 37, D-80333 Munich, Germany b Stanford Linear Accelerator Center, Stanford University, Stanford, CA 94309, USA Received 25 April 2002; received in revised form 12 June 2002; accepted 18 June 2002 Editor: G.F. Giudice

Abstract Astrophysical indications that the fine structure constant has undergone a small time variation during the cosmological evolution are discussed within the framework of the standard model of the electroweak and strong interactions and of grand unification. A variation of the electromagnetic coupling constant could either be generated by a corresponding time variation of the unified coupling constant or by a time variation of the unification scale, of by both. The various possibilities, differing substantially in their implications for the variation of low energy physics parameters like the nuclear mass scale, are discussed. The case in which the variation is caused by a time variation of the unification scale is of special interest. It is supported in addition by recent hints towards a time change of the proton–electron mass ratio. Implications for the analysis of the Oklo remains and for quantum optics tests are discussed.  2002 Elsevier Science B.V. All rights reserved.

The Standard Model of the electroweak and strong six constants are of importance: the mass of the elec- interactions has at least 18 parameters, which have tron, setting the scale of the Rydberg constant, the to be adjusted in accordance with experimental ob- masses of the u-andd-quarks setting the scale of the servations. These include the three electroweak cou- breaking of isotopic spin, and the strong interaction pling strengths g1, g2 g3, the scale of the electroweak coupling constant αs . The latter, often parametrized by symmetry breaking, given by the universal Fermi con- the QCD scale parameter Λ, sets the scale for the nu- stant, the 9 Yukawa couplings of the six quarks and cleon mass. The mass of the strange quark can also the three charged leptons, and the four electroweak be included since the mass term of the s-quarks is ex- mixing parameters. One parameter, the mass of the pected to contribute to the nucleon mass, although the hypothetical scalar boson, is still undetermined. For exact amount of strangeness contribution to the nu- the physics of stable matter, i.e., atomic physics, solid cleon mass is still being discussed—it can range from state physics and a large part of nuclear physics, only several tenth of MeV till more than 100 MeV. As far as macro-physical aspects are concerned, Newton’s constant must be added, which sets the scale for the Planck E-mail address: [email protected] units of energy, space and time. (X. Calmet). 1 Partially supported by the Deutsche Forschungsgemeinschaft, Since within the Standard Model the number of free DFG No. FR 412/27-2. parameters cannot be reduced, and thus far theoreti- 2 Partially supported by the VW-Stiftung Hannover (I-77495). cal speculations about theories beyond the model have

0370-2693/02/$ – see front matter  2002 Elsevier Science B.V. All rights reserved. PII:S0370-2693(02)02147-0 174 X. Calmet, H. Fritzsch / Physics Letters B 540 (2002) 173–178 not led to a well-defined framework, in view of lack of electromagnetic coupling constant and the QCD scale guidance by experiment, one may consider the possi- parameter Λ, implying a physical time variation of bility that these parameters are time and possibly also the nucleon mass, when measured in units given by space variant on a cosmological scale. Speculations an energy scale independent of QCD, like the electron about a time-change of coupling constants have a long mass or the Planck mass. The main assumption is that history, starting with early speculations about a cos- the physics responsible for a cosmic time evolution of mological time change of Newton’s constant G [1–4]. the coupling constants takes place at energies above Since in particular the masses of the fermions as well the unification scale. This allows to use the usual as the electroweak mass scale are related to the vac- relations from grand unified theories to evolve the uum expectation values of a scalar field, time changes unified coupling constant down to low energy. Of of these parameters are conceivable. In some theories particular interest is the relatively large time change beyond the Standard Model also the gauge coupling of the proton mass in comparison to the time change constants are related to expectation values of scalar of α: fields which could be time dependent [5]. We should like to emphasize that a time or space M˙ Λ˙ α˙ ≈ ≈ 38 . (2) variation of these coupling parameters would imply a M Λ α violation of Poincaré invariance. The observed region of our universe seems to be isotropic and homoge- Considering the six basic parameters mentioned neous to a high degree, suggesting that the subgroup of above plus Newton’s constant G, one can in gen- the Poincaré group concerning space translations re- eral consider seven relative time changes: G/G˙ , α/α˙ , ˙ mains a valid symmetry to a high degree. However the Λ/Λ, m˙ e/me, m˙ u/mu, m˙ d /md and m˙ s/ms . Thus in symmetry under time translations is certainly violated principle seven different functions of time do enter the by the Big Bang about 14 billion years ago, and such discussion. However not all of them could be mea- a violation might show up in a time dependence of the sured, even not in principle. Only dimensionless ra- fundamental constants. For this reason a time varia- tios, e.g., the ratio Λ/me or the fine-structure constant tion might be considered as more likely and larger in could be considered as candidates for a time variation. magnitude than a space variation, and the relevant time The time derivative of the ratio Λ/me describes scale for a time variation should be the observed age a possible time change of the atomic scale in com- of the universe of the order of 14 billion years. parison to the nuclear scale. In the absence of quark We note that a time variation of a fundamental masses there is only one mass scale in QCD, unlike in parameter like a gauge coupling constant is meant to atomic physics, where the two parameters α and me be a variation with respect to the cosmological time, enter. The parameter α is directly measurable by com- defined to be the time coordinate of a system, in which paring the energy differences describing the atomic 2 4 the cosmic background radiation is isotropic. fine structure (of order mec α ) to the Rydberg energy 2 2 Recent observations in astrophysics concerning the hcR∞ = mec α /2 ≈ 13.606 eV. atomic fine-structure of elements in distant objects The mass of the strange quark enters in QCD as suggest a time change of the fine structure constant a shift in the nuclear mass scale. Its effects and a [6]. The data suggest that α was lower in the past, at a possible time shift of ms can be absorbed in a time redshift of z ≈ 0.5,...,3.5: shift of the nucleon mass. The masses of the u and − d quarks, however, influence the proton and neutron α/α = (−0.72 ± 0.18) × 10 5. (1) mass, as well as many effects in nuclear physics. The ˙ If α is indeed time dependent, the other two ratio ((m˙ d −˙mu)/(md − mu))/(Λ/Λ) is in principle gauge coupling constants of the Standard Model are measurable and can be considered to be the QCD also expected to depend on time, as pointed out analog of the fine-structure effects in atomic physics. recently [7] (see also [8,9]), if the Standard Model is Note that a determination of this ratio, which, for embedded into a grand unified theory. Moreover the example could be seen by monitoring the ratio (Mn − idea of a grand unification of the coupling constants Mp)/Mp in time, would only give information of the leads to a relation between the time variation of the relative change of the quark masses in comparison X. Calmet, H. Fritzsch / Physics Letters B 540 (2002) 173–178 175 to Λ. It might imply that Λ is changing, or the quark symmetry Λw, the scale of the onset of supersymme- masses in comparison to Λ, or both. try Λs and the scale ΛG where the grand unification The ratio (md + mu)/Λ is given by the magnitude sets in. of the σ -term, which leads to a small shift of the If a change of α is observed, it would imply nucleon mass of about 40 MeV [11]. If Λ stays necessarily, that at least one mass scale is changing invariant, but the quark masses change, the effect as well. The magnitudes of the three gauge coupling could be seen by a time variation of the ratio Mp/me. constants are in particular given by the value of the Thus a discovery of a time variation of this ratio would unified coupling constant at the scale ΛG. Variations not necessarily imply that Λ would change in time. of the coupling constants at low energies will result Both astrophysics experiments as well as high pre- if this coupling constant changes in time, or if the cision experiments in atomic physics in the laboratory unification scale ΛG changes with respect to the other could in the future give indications about a time varia- scales, e.g., the electron mass, or both. tion of three dimensionless quantities: α, Mp/me and According to the renormalization group equations, (Mn −Mp)/me. The time variation of α reported in [6] considered here in lowest order, the behaviour of the implies, assuming a simple linear extrapolation, a rel- coupling constants changes according to ative rate of change per year of about 1.0 × 10−15/yr. α (µ)−1 This poses a problem with respect to the limit given by i an analysis of the remains of the naturally occurring 1 1 ΛG = + bS ln θ(µ− Λ ) nuclear reactor at Oklo in Gabon (Africa), which was 0 i S α (ΛG) 2π µ active close to 2 billion years ago. One finds a limit i − Λ of α/α˙ = (−0.2 ± 0.8) × 10 17/yr. This limit was + 1 + 1 SM S − 0 bi ln θ(ΛS µ), derived in [12] under the assumption that other para- αi (ΛS) 2π µ meters, especially those related to the nuclear physics, (3) did not change during the last 2 billion years. It was SM = SM where the parameters bi are given by bi (b1 , recently pointed out [7,10], that this limit must be re- SM SM = − − b2 ,b3 ) (41/10, 19/6, 7) below the super- considered if a time change of nuclear physics para- S = S S S = symmetric scale and by bi (b1 ,b2 ,b3 ) (33/5, meters is taken into account. In particular it could be 1, −3) when N = 1 supersymmetry is restored, and that the effects of a time change of α are compensated where by a time change of the nuclear scale parameter. For this reason we study in this Letter several scenarios for 1 1 1 MZ = + bSM ln (4) time changes of the QCD scale, depending on differ- 0 0 i αi (ΛS) αi (MZ) 2π ΛS ent assumptions about the primary origin of the time 0 variation. where MZ is the Z-boson mass and αi (MZ) is the Without a specific theoretical framework for the value of the coupling constant under consideration physics beyond the Standard Model the relative time measured at MZ . We use the following definitions for changes of the three dimensionless numbers men- the coupling constants: tioned above are unrelated. We shall incorporate the 5 5α idea of grand unification and assume for simplicity the α = = , 1 2 3cos2(θ) simplest model of this kind, consistent with present 3g14π MS observations, the minimal extension of the supersym- 2 g2 α α2 = = , metric version of the Standard Model (MSSM), based 4π sin2(θ) on the gauge group SU(5). In this model the three MS 2 coupling constants of the Standard Model converge at g3 αs = . (5) high energies at the scale ΛG. In particular the QCD 4π scale Λ and the fine structure constant α are related Assuming αu = αu(t) and ΛG = ΛG(t), one finds to each other. In the model there are besides the electron mass and the quark masses three further scales ˙ ˙ bS ˙ 1 αi = 1 αu − i ΛG entering, the scale for the breaking of the electroweak (6) αi αi αu αu 2π ΛG 176 X. Calmet, H. Fritzsch / Physics Letters B 540 (2002) 173–178 which leads to other via the Planck scale, e.g., ˙ ˙ ˙ 1 α = 8 1 αs − 1 S + 5 S − 8 S ΛG 1 = 1 + bG ΛPl b2 b1 b3 . (7) ln , (14) α α 3 αs αs 2π 3 3 ΛG αu αPl 2π ΛG

One may consider the following scenarios. where ΛPl is the Planck scale, αPl the value of the GUT group coupling constant at the Planck scale and (1) ΛG invariant, αu = αu(t).Thisisthecase bG depends on the GUT group under consideration. considered in [7] (see also [8]), and one ﬁnds This leads to ˙ 1 α˙ 8 1 α˙ ΛG 2π 1 α˙u = s (8) = (15) α α 3 αs αs ΛG bG αu αu and and thus to ˙ bS Λ 3 2π 1 α˙ ˙ 1 − 3 ˙ =− . Λ 2π bG 1 α SM (9) = (16) Λ 8 b3 α α − SM bS+ 5 bS Λ b3 8 − 2 3 1 α α 3 bG (2) αu invariant, ΛG = ΛG(t). One ﬁnds or ˙ ˙ 1 α 1 S 5 S ΛG − SM S + 5 S ˙ =− b + b , (10) α˙ b 8 b b bG Λ 2 1 = 3 − 2 3 1 α . (17) α α 2π 3 ΛG − S α 2π 3 bG bG b3 Λ with

SM S Finally, it should be mentioned that the scale b3 /b3 = Λ − 2π 1 of supersymmetry could also vary with time. One ΛG ΛS exp SM (11) ΛS b3 αu obtains: which follows from the extraction of the Landau pole ˙ ˙ bS ˙ 1 αi = 1 αu − i ΛG using (3). One obtains αi αi αu αu 2π ΛG ˙ ˙ S − ˙ ˙ 1 ΛS Λ b3 2π 1 α α + S − SM − = ≈−30.8 . (12) bi bi θ(ΛS µ). (18) Λ SM S + 5 S α α α 2π ΛS b3 b2 3 b1 However without a specific model for supersymmetry (3) α = α (t) and Λ = Λ (t). One has u u G G breaking relating the supersymmetry breaking scale ˙ ˙ bS ˙ to, e.g., the GUT scale, this expression is not very Λ =− 2π 1 αu + 3 ΛG SM SM useful. Λ b αu αu b ΛG 3 3 One should also mention that in principle all the ˙ 3 2π 1 α˙ 3 1 5 8 ΛG other parameters, i.e., vacuum expectation values of =− − bS + bS − bS SM SM 2 1 3 Higgs fields, Yukawa couplings, Higgs bosons masses 8 b3 α α 8 b3 3 3 ΛG ˙ may have a time dependence. α˙ ΛG = 46 + 1.07 , (13) The case in which the time variation of α is not re- α ΛG lated to a time variation of the unified coupling con- where theoretical uncertainties in the factor R = stant, but rather to a time variation of the unifica- (Λ/Λ)/(˙ α/α)˙ = 46 have been discussed in [7]. The tion scale, is of particular interest. Unified theories, in actual value of this factor is sensitive to the inclusion which the Standard Model arises as a low energy ap- of the quark masses and the associated thresholds, just proximation, might well provide a numerical value for like in the determination of Λ. Furthermore higher the unified coupling constant, but allow for a smooth order terms in the QCD evolution of αs will play a time variation of the unification scale, related in spe- role. In Ref. [7] it was estimated: R = 38 ± 6. cific models to vacuum expectation values of scalar (4) In a grand unified theory, the GUT scale and fields. Since the universe expands, one might expect a the unified coupling constant may be related to each decrease of the unification scale due to a dilution of X. Calmet, H. Fritzsch / Physics Letters B 540 (2002) 173–178 177 the scalar field. A lowering of ΛG implies according one finds: to (10) Λ˙ −2π 1 α˙ α˙ ˙ ˙ = =−234.8 . (22) α˙ 1 5 ΛG ΛG Λ bSM + 5 bSM α α α =− α bS + bS =−0.014 . (19) 2 3 1 α 2π 2 3 1 Λ Λ G G This shows that the case with and without supersym- ˙ If ΛG/ΛG is negative, α/α˙ increases in time, consis- metry differ from each other by about a factor 7.6, as tent with the experimental observation. Taking far as the relative time changes are concerned. α/α =−0.72 × 10−5, we would conclude The fact that we find opposite signs for the time −4 ΛG/ΛG = 5.1 × 10 , i.e., the scale of grand unifi- changes of α and Λ is interesting with respect to cation about 8 billion years ago was about the limit on the time change of α deduced from the 8.3×1012 GeV higher than today. If the rate of change Oklo reactors remains. The Oklo constraint comes is extrapolated linearly, ΛG is decreasing at a rate from the fact that the neutron capture cross section for ˙ ΛG =−7 × 10−14/yr. thermal neutrons off samarium 149 is dominated by a ΛG According to (12) the relative changes of Λ and nuclear resonance just above threshold. According to α are opposite in sign. While α is increasing with a the analysis given in [12] the position of the resonance rate of 1.0 × 10−15/yr, Λ and the nucleon mass is could not have changed much during the last 2 billion decreasing, Λ and the nucleon mass are decreasing, years. This gives a constraint for a time change of e.g., with a rate of 1.9 × 10−14/yr. The magnetic alpha, given above. Due to the Coulomb repulsion in α moments of the proton µp as well of nuclei would the nucleus an increase of would increase the energy increase according to of the resonance. However a corresponding decrease of αs would have the opposite effect. Thus in any µ˙ p α˙ − = 30.8 ≈ 3.1 × 10 14/yr. (20) case the Oklo constraint will be less restrictive, and µp α there might even be a nearly complete cancellation of the effect. A more detailed analysis would be The effect can be seen by monitoring the ratio beyond the scope of this Letter and will be discussed µ = M /m . Measuring the vibrational lines of H , p e 2 elsewhere. We emphasize, however, that a partial or a small effect was seen [13] recently. The data allow complete cancellation would not be in sight, if both two different interpretations: time changes have the same sign, as, e.g., in Eq. (8). The time variation of the ratio M /m and α (a) µ/µ = (5.7 ± 3.8) × 10−5; p e discussed here are such that they could by discovered (b) µ/µ = (12.5 ± 4.5) × 10−5. by precise measurements in quantum optics. The wave length of the light emitted in hyperfine transitions, e.g., The interpretation (b) agrees with the expectation the ones used in the cesium clocks being proportional based on (12): 4 to α me/Λ will vary in time like µ = × −5 ˙ ˙ 22 10 . (21) λ α˙ Λ − µ hf = 4 − ≈ 3.5 × 10 14/yr (23) λhf α Λ It is interesting that the data suggest that µ is indeed − decreasing, while α seems to increase. If confirmed, taking α/α˙ ≈ 1.0 × 10 15/yr [6]. The wavelength of − this would be a strong indication that the time variation the light emitted in atomic transitions varies like α 2: of α at low energies is caused by a time variation of the λ˙ α˙ unification scale. at =−2 . (24) Finally we should like to mention the case of uni- λat α fication based on SU(5) or SO(10) without supersym- ˙ −15 One has λat/λat ≈−2.0 × 10 /yr. A comparison metry. Although in this case no unification is achieved, gives: based on the experimental data, one may discuss a uni- ˙ ˙ fication of the electromagnetic and the strong interac- λhf/λhf 4α/α˙ − Λ/Λ 13 =− ≈− . . tions at a scale of 1.0 × 10 GeV. Varying this scale, ˙ ˙ 17 4 (25) λat/λat 2α/α 178 X. Calmet, H. Fritzsch / Physics Letters B 540 (2002) 173–178

At present the time unit second is defined as the References duration of 6 192 631 770 cycles of microwave light emitted or absorbed by the hyperfine transmission of [1] P.M. Dirac, Nature 192 (1937) 235. cesium-133 atoms. If Λ indeed changes, as described [2] E.A. Milne, Relativity, Gravitation and World Structure, in (12), it would imply that the time flow measured by Clarendon, Oxford, 1935; E.A. Milne, Proc. R. Soc. A 3 (1937) 242. the cesium clocks does not fully correspond with the [3] P. Jordan, Naturwiss 25 (1937) 513; time flow defined by atomic transitions. P. Jordan, Z. Physik 113 (1939) 660. It remains to be seen whether the effects discussed [4] L.D. Landau, in: W. Pauli (Ed.), Niels Bohr and the Develop- in this Letter can soon be observed in astrophysics or ment of Physics, McGraw–Hill, New York, 1955, p. 52. in quantum optics. A determination of the double ratio [5] M.B. Green, J.H. Schwarz, E. Witten, Superstring Theory, ˙ ˙ = Vols. 1 and 2, Cambridge Univ. Press, Cambridge, 1987; (Λ/Λ)/(α/α) R would be of crucial importance, T. Damour, A.M. Polyakov, Nucl. Phys. B 423 (1994) 532, both in sign and in magnitude. If one finds the hep-th/9401069. ratio to be about −20, it would be considered as a [6] J.K. Webb et al., Phys. Rev. Lett. 87 (2001) 091301, astro- strong indication of a unification of the strong and ph/0012539. electroweak interactions based on a supersymmetric [7] X. Calmet, H. Fritzsch, hep-ph/0112110, to appear in Eur. Phys. J. C. extension of the Standard Model. In any case the [8] P. Langacker, G. Segre, M.J. Strassler, Phys. Lett. B 528 (2002) numerical value of R would be of high interest towards 121, hep-ph/0112233; a better theoretical understanding of time variation and T. Dent, M. Fairbairn, hep-ph/0112279. unification. [9] H. Fritzsch, hep-ph/0201198, Invited talk at Symposium on 100 Years Werner Heisenberg: Works and Impact, Bamberg, Germany, 26–30 September, 2001; S.J. Landau, H. Vucetich, astro-ph/0005316; V.V. Flambaum, E.V. Shuryak, hep-ph/0201303; Acknowledgements C. Wetterich, hep-ph/0203266; Z. Chacko, C. Grojean, M. Perelstein, hep-ph/0204142. [10] V.V. Flambaum, E.V. Shuryak, in [9]. We should like to thank A. Albrecht, J.D. Bjorken, [11] J. Gasser, H. Leutwyler, Phys. Rep. 87 (1982) 77. E. Bloom, G. Boerner, S. Brodsky, P. Chen, S. Drell, [12] T. Damour, F. Dyson, Nucl. Phys. B 480 (1996) 37, hep- ph/9606486. G. Goldhaber, T. Haensch, M. Jacob, P. Minkowski, [13] A.V. Ivanchik, E. Rodriguez, P. Petitjean, D.A. Varshalovich, A. Odian, H. Walther and A. Wolfe for useful discus- astro-ph/0112323. sions. Physics Letters B 540 (2002) 179–184 www.elsevier.com/locate/npe

Effect of cosmic rays on the resonant gravitational wave detector NAUTILUS at temperature T = 1.5K

P. Astone a,D.Babuscib,M.Bassanc,P.Bonifazid,P.Carellie, E. Coccia c, S. D’Antonio c,V.Fafoneb, G. Giordano b,A.Marinib,G.Mazzitellib,Y.Minenkovc, I. Modena c,G.Modestinob,A.Moletic, G.V. Pallottino f,G.Pizzellag,L.Quintierib, A. Rocchi g, F. Ronga b, R. Terenzi d,M.Viscod

a Istituto Nazionale di Fisica Nucleare INFN, Rome 1, Italy b Istituto Nazionale di Fisica Nucleare INFN, LNF, Frascati, Italy c University of Rome “Tor Vergata” and INFN, Rome 2, Italy d IFSI-CNR and INFN, Roma, Italy e University of L’Aquila and INFN, Rome 2, Italy f University of Rome “La Sapienza” and INFN, Rome 1, Italy g University of Rome “Tor Vergata” and INFN, LNF, Frascati, Italy Received 28 May 2002; received in revised form 28 May 2002; accepted 17 June 2002 Editor: L. Rolandi

Abstract

The interaction between cosmic rays and the gravitational wave bar detector NAUTILUS is experimentally studied with the aluminum bar at temperature of T = 1.5 K. The results are compared with those obtained in the previous runs when the bar was at T = 0.14 K. The results of the run at T = 1.5 K are in agreement with the thermo-acoustic model; no large signals at unexpected rate are noticed, unlike the data taken in the run at T = 0.14 K. The observations suggest a larger efﬁciency in the mechanism of conversion of the particle energy into vibrational mode energy when the aluminum bar is in the superconductive status.  2002 Elsevier Science B.V. All rights reserved.

PACS: 04.80; 04.30; 96.40.Jj

The gravitational wave (GW) detector NAUTILUS that the mechanical vibrations originate from local recently recorded signals due to the passage of cosmic thermal expansion caused by warming up due to the rays (CRs) [1–3]. Several authors [4–11] estimated the energy lost by the particles crossing the material. It possible acoustic effects due to the passage of particles was predicted that for the vibrational energy in the in a metallic bar. The mechanism adopted assumes longitudinal fundamental mode of a metallic bar the following formula would hold: 2 E-mail address: [email protected] 4 γ 2 dW E = (G. Modestino). 9π ρLv2 dx 0370-2693/02/$ – see front matter  2002 Elsevier Science B.V. All rights reserved. PII:S0370-2693(02)02143-3 180 P. Astone et al. / Physics Letters B 540 (2002) 179–184 πz sin(πl cos(θ )/2L) 2 × sin 0 0 0 , (1) sured [1]. In a further investigation, we found very L πRcos(θ0)/L large NAUTILUS signals at a rate much greater than expected [2,3]. (We notice that a GW bar detector, used where L is the bar length, R the bar radius, l0 the as particle detector, has characteristics very different length of the particle’s track inside the bar, z0 the from the usual particle detectors which are sensitive distance of the track midpoint from one end of the bar, only to ionization losses.) Since the bar temperature θ0 the angle between the particle track and the axis of was about 0.14 K, i.e., the aluminum alloy was super- the bar, E the energy of the excited vibration mode, conductor, one could consider some unexpected be- dW/dx the energy loss of the particle in the bar, haviour due to the transition to the normal state along ρ the density, v the sound velocity in the material the particle trajectories. These effects were estimated and γ is the Grüneisen coefficient (depending on the [7,8] for type I superconductor (as aluminum). They ratio of the material thermal expansion coefficient to are very small and cannot account for our observa- the specific heat) which is considered constant with tions, if the showers include only electromagnetic and temperature. hadronic particles. The resonant-mass GW detector NAUTILUS [12], In the present Letter, the results of the effect of the operating at the INFN Frascati Laboratory, consists of CR passage on NAUTILUS during the years 2000 and an aluminum alloy 2300 kg bar which can be cooled 2001 are reported together with comparison with the to very low temperatures, of the order of 0.1 K, previous observations. During this period NAUTILUS below the superconducting transition temperature of operated at different thermodynamic temperatures. In this alloy, TC = 0.92 K [13]. The bar is equipped 2000, until July, the NAUTILUS bar cryogenic temper- with a capacitive resonant transducer, providing the ature was 0.14 K; then, between August and Decem- read-out. Bar and transducer form a coupled oscillator ber, it was brought at 1.1 K. In the period 1 March system with two resonant modes, whose frequencies 2001 through 30 September 2001 NAUTILUS operated are 906.4 and 922.0 Hz. The transducer converts the at a temperature of 1.5 K. We proceeded to apply to mechanical vibrations into an electrical signal and these data the same data analysis algorithms used for is followed by a dcSQUID electronic amplifier. The the previous runs: coincidence search [2,3] and zero NAUTILUS data, recorded with a sampling time of threshold search [1], latter being more efficient for de- 4.54 ms, are processed with a filter [14] optimized to tecting small amplitude signals. detect pulse signals applied to the bar, such as those due to a short burst of GW. NAUTILUS is equipped with a CR detector system consisting of seven layers of streamer tubes for a Coincidence search total of 116 counters [16]. Three superimposed layers, each with an area of 36 m2, are located over the The event list employed in the analysis was gener- cryostat (top detector). Four superimposed layers are ated by considering only the time periods with noise set under the cryostat, each with an area of 16.5m2 temperature (expressing the minimum detectable in- (bottom detector). Each counter measures the charge, novation) less than 5 mK, and imposing the amplitude which is proportional to the number of particles. The threshold at SNR = 4.4 on the data filtered with an al- CR detector is able to measure particle density up gorithm matched to detect short bursts. The threshold to 5000 par m−2 without large saturation effects and value was established trough IGEC Collaboration [15] gives a rate of showers in good agreement with the for data exchange among the GW groups to search expected number [16,17], as verified by measuring for coincident events. For each threshold crossing we the particle density in the top detector, which is not take the maximum value and its time of occurrence. affected by the interaction in the NAUTILUS bar. These two quantities define the event of the GW detec- In a previous paper we reported the results of a tor. The CR shower list was generated by considering − search for correlation between the NAUTILUS data and those events giving a particle density 300 par m 2 the data of the CR detector, when for the first time in the bottom detector. Comparing the two lists, we acoustic signals generated by CR showers were mea- searched for coincidence in a window of ±0.1 s cen- P. Astone et al. / Physics Letters B 540 (2002) 179–184 181

Table 1 Coincidences during the years 1998, 2000 and 2001, using a coincidence window of ±0.1s.NAUTILUS temperature, duration of the analysis period, expected number of accidental coincidences n¯ and number of coincidences nc

Time period NAUTILUS Duration nc n¯ Rate temperature (K) hours (eV/day) September–December 1998 0.14 2002 12 0.47 February–July 2000 0.14 707 9 0.42 Total 2709 21 0.89 0.178 ± 0.041 August–December 2000 1.1 118 0 0.03 March–September 2001 1.5 2003 1 0.42 Total 2121 1 0.45 0.006 ± 0.011 tered at CR arrival time. The expected number of accidental coincidences was experimentally estimated, by means of the time shifting algorithm [18,19]. By shifting the events time in one of the two data sets by an amount δt the number of coincidence n(δt) is deter- minated. Repeating for N different values of the time delay, the expected number of coincidence is

1 n¯ = n(δt). N − 1 With these criteria, i.e., the temperature noise less than 5 mK, the coincidence time window of ±0.1s,andthe particle density showers larger than 300 par m−2,we found, with NAUTILUS temperature at 0.14 K, 12 coincident events on 1998 and 9 coincident events during February–Jully 2000. For both periods, the energy values of the events are concentrated in the 0.1 K range. For the remaining part of 2000, with NAUTILUS temperature at 1.1 K, we found no coincident event. In Fig. 1. The NAUTILUS response to the CR shower with particle − 2001, with NAUTILUS bar temperature at 1.5 K, we density 2812 par m 2, filtered energy (K) versus time (s), centered found just one coincidence. We report the result of the at the CR shower arrival time. The lower figure is a zoom of the analysis and comparison in Table 1. This table affords upper one. We note the oscillation related to the beating of the two resonant modes and the decay due to the detector bandwidth, evidence at about 4σ level that the observed coinci- δf ∼ 0.4Hz. dence rate is related to the bar temperature. In 2001, the single coincidence event had high NAUTILUS energy, E ∼ 0.5 K, and very large particle density M = −2 2812 par m in the bottom detector. The response to where E is expressed in Kelvin units, W in GeV units this CR shower is shown in Fig. 1, filtered energy ver- is the energy delivered by the particle to the bar and f sus time centered at the CR shower arrival time. is a geometrical factor of the order of unity. This is the typical response expected for a delta- We get for this event W ∼ 8 TeV. From the data like excitation acting on the bar. To estimate the shown in Ref. [2] (our calculations and experimental energy absorbed by the incoming CR shower, we data from the CASCADE Collaboration) we expect in apply Eq. (1) to the case of NAUTILUS: 83.5 days of NAUTILUS data taking about one event with energy greater than 0.1 K due to the hadrons, able − E = 7.64 × 10 9W 2f, (2) to deliver to the GW detector an energy of a few TeV. 182 P. Astone et al. / Physics Letters B 540 (2002) 179–184

Zero threshold search data recorded during 1998 [1]. On the same figure, on graph (b), we report the result of the same analysis ap- AgainweusedthesedatawhenNAUTILUS noise plied to the 968 stretches data of the year 2001, with temperature was less than 5 mK and the shower multi- the aluminum bar cooled to 1.5 K. The major contribu- plicity was larger than 300 par m−2 in the bottom CR tion to the signal at zero time is due to the single event detector. In correspondence with each CR shower we of Fig. 1. Removing from the data set that event, we considered the NAUTILUS filtered data in a time pe- obtain graph (c). Comparison shows clearly the differ- riod of ±19 s centered at the CR arrival time. With ent response of NAUTILUS in the two time periods. this selection, in 2000, there were 308 data stretches The question arises whether NAUTILUS, operating corresponding to as many CR showers during a total at temperature T = 1.5 K (in a normal non supercon- period of observation of 707 non-continuous hours. ductive status) is sensitive to the CR showers as pre- The selected stretches were superimposed and aver- dicted by the thermo-acoustic models. For a quantita- aged at the same relative time with respect to the ar- tive estimation of a possible effect due to CR we pro- rival time of the CR showers. The result of this proceeded as follows. We consider NAUTILUS stretches cedure is shown in Fig. 2, where we plot the aver- for the year 2001 corresponding to CR in various con- ages for each data sample (136.3 ms) versus time, for tiguous multiplicity intervals. For the stretches of each the 308 CR events with particle density greater than of the selected multiplicity intervals we calculate the 300 par m−2, graph (a). Several events with energy of energy averages over thirty contiguous sampling times the order of 0.1 K contribute to the large response at corresponding to 136.3 ms. At zero delay we take the zero time, which confirms the results obtained by the average at time 0 ± 68.2 ms. We recall that the beat period in the filtered signal, due to the two resonance modes, is 64 ms, as we can see from Fig. 1. With this averaging procedure we avoid the problem of taking either a maximum value or a minimum value, which are not exactly in phase among the various stretches. By doing so we get an average value smaller than the maximum by a factor 3.6, as we find by numerically averaging the data of Fig. 1. For each multiplicity range, the measured signal (average at time 0 ± 68.2 ms) is compared with the signal we expect due to the electromagnetic component of the shower. The theoretical value is given by [2] 2 −10 Eth = Λ · 4.7 × 10 K, (3) where Λ is the number of secondaries through the bar. The measured multiplicity might be affected by a systematic error of the order of ±25% [2]. As an estimate of the background we take the average energy during the periods from −4000 to −3000 sampling times (from −18.18 to −13.63 s) and from 3000 to 4000 samplings times (from 13.63 to 18.18 s), for a total Fig. 2. The energy response of NAUTILUS to the CRs passage at zero time period of 2000 sampling times, 9.088 seconds. time. In (a), we show the average energy (K) vs. time for 308 data In Fig. 3 we show the difference in mK units stretches detected during 2000, with NAUTILUS bar temperature at between the average energy at zero time delay and 0.14 K. In (b), the result of the same analysis is shown for 968 data stretches detected during 2001, with bar temperature at 1.5 K. The the background versus the expected signal due to − CR showers particle density is larger than 300 par m 2 for the both the electromagnetic component of the CR showers. periods. Excluding from the last data set the event of Fig. 1, the The straight line is a least square fit through the average energy for 967 data stretches is shown in (c). origin and the vertical bars indicate statistical errors P. Astone et al. / Physics Letters B 540 (2002) 179–184 183

Table 2 The average NAUTILUS signal Eobs, and its standard deviation, vs the multiplicity of CR events. multiplicity selections. Also indicated are the number of stretches for each selection and the difference between the signal at zero delay and the background, with its standard deviation. The theoretical values are calculated with Eq. (3) (valid for the electromagnetic component of the shower) divided by 3.6, using the measured particle density in the lower part of CR detector and taking the average. The big event of Fig. 1 has been excluded from the last row 2 − particles/m Number of Eobs σEobs Eobs bkg σEobs−bkg Eth stretches (mK) (mK) (mK) (mK) (mK) 300–600 688 3.690 0.085 0.310 0.085 0.069 600–900 138 3.67 0.21 0.20 0.21 0.228 900–1200 63 3.91 0.37 0.40 0.37 0.453 1200–1500 34 4.10 0.26 0.98 0.26 0.783 1500–1800 16 3.35 0.77 −0.24 0.79 1.119 1800–2100 9 4.76 0.80 0.91 0.84 1.517 2100–2400 11 4.75 0.58 1.82 0.60 2.200 2400–2700 3 4.41.21.31.42.744 2700–3000 5 6.21.62.51.73.478

Conclusions

Comparing the previous and the present measurements, two different behaviours of the aluminum bar detector are noticed, with evidence at 4σ level. In the run with the bar temperature above the superconductive transition we find a result in good agreement with the theoretical predictions of the thermo- acoustic model. These measurements are a record for the GW detectors, as signals of the order of 10−4 K, corresponding to 10−8 eV, were extracted from noise. Fig. 3. Experimental signal versus the expected signal due to the The unexpected behaviour of NAUTILUS noticed in electromagnetic component of the CR shower (see text and Table 2). the previous runs [2,3], i.e., very large signals at a rate The straight line is a least square fit and the vertical bars indicate statistical errors (± one standard deviation). higher than expected, occurs only at ultracryogenic temperatures. The observed phenomenology suggests a larger efficiency in the mechanism of conversion of the particles energy into the vibrational mode energy, at least for some type of particles, when the aluminum bar is in the superconductive status. (± one standard deviation). The χ2 calculated for a null hypothesis (signal = background) gives χ2 = 42.4 with 9 degrees of freedom for a probability of 2.8 × 10−6. The slope of the straight line has value ± 0.85 0.13. If we take into account the systematic Acknowledgements error on the experimental value of Λ (∼±25%) and the error on the calibration of the NAUTILUS event energy, of the order of 10%, we get for the slope 0.85 ± 0.16 ± 0.42, showing a good agreement with We thank F. Campolungo, G. Federici, R. Lenci, the thermo-acoustic model. The χ2 calculated for the G. Martinelli, E. Serrani, R. Simonetti and F. Tabac- hypothesis that the individual data be along the straight chioni for precious technical assistance. We also thank line is χ2 = 13.3 for a probability of 0.10. Dr. R. Elia for her contribution. 184 P. Astone et al. / Physics Letters B 540 (2002) 179–184

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Measurement of genuine three-particle Bose–Einstein correlations in hadronic Z decay

L3 Collaboration P. Achard u,O.Adrianir, M. Aguilar-Benitez y, J. Alcaraz y,s, G. Alemanni w,J.Allabys, A. Aloisio ac,M.G.Alviggiac, H. Anderhub au,V.P.Andreevf,ah,F.Anselmoi, A. Areﬁev ab, T. Azemoon c, T. Aziz j,s,P.Bagnaiaam,A.Bajoy,G.Baksayz, L. Baksay z,S.V.Baldewb,S.Banerjeej,Sw.Banerjeed, A. Barczyk au,as,R.Barillères, P. Bartalini w,M.Basilei, N. Batalova ar, R. Battiston ag,A.Bayw, F. Becattini r, U. Becker n, F. Behner au,L.Belluccir, R. Berbeco c, J. Berdugo y,P.Bergesn, B. Bertucci ag,B.L.Betevau,M.Biasiniag, M. Biglietti ac,A.Bilandau, J.J. Blaising d, S.C. Blyth ai, G.J. Bobbink b,A.Böhma,L.Boldizsarm,B.Borgiaam,S.Bottair, D. Bourilkov au, M. Bourquin u, S. Braccini u,J.G.Bransonao,F.Brochud, J.D. Burger n, W.J. Burger ag,X.D.Cain,M.Capelln,G.CaraRomeoi,G.Carlinoac, A. Cartacci r, J. Casaus y,F.Cavallariam, N. Cavallo aj, C. Cecchi ag,M.Cerraday,M.Chamizou, Y.H. Chang aw,M.Chemarinx,A.Chenaw,G.Cheng,G.M.Cheng,H.F.Chenv, H.S. Chen g,G.Chiefariac, L. Cifarelli an, F. Cindolo i,I.Claren,R.Clareal, G. Coignet d, N. Colino y, S. Costantini am,B.delaCruzy, S. Cucciarelli ag, J.A. van Dalen ae, R. de Asmundis ac,P.Déglonu, J. Debreczeni m,A.Degréd,K.Dehmeltz,K.Deitersas, D. della Volpe ac,E.Delmeireu, P. Denes ak, F. DeNotaristefani am,A.DeSalvoau, M. Diemoz am,M.Dierckxsensb,C.Dionisiam, M. Dittmar au,s,A.Doriaac, M.T. Dova k,5, D. Duchesneau d, B. Echenard u, A. Eline s, H. El Mamouni x, A. Engler ai, F.J. Eppling n, A. Ewers a, P. Extermann u, M.A. Falagan y, S. Falciano am,A.Favaraaf, J. Fay x,O.Fedinah,M.Felciniau, T. Ferguson ai, H. Fesefeldt a, E. Fiandrini ag, J.H. Field u, F. Filthaut ae,P.H.Fishern,W.Fisherak,I.Fiskao, G. Forconi n, K. Freudenreich au,C.Furettaaa, Yu. Galaktionov ab,n, S.N. Ganguli j, P. Garcia-Abia e,s, M. Gataullin af, S. Gentile am,S.Giaguam,Z.F.Gongv,G.Grenierx,O.Grimmau, M.W. Gruenewald q,M.Guidaan,R.vanGulikb, V.K. Gupta ak,A.Gurtuj,L.J.Gutayar, D. Haas e, R.Sh. Hakobyan ae, D. Hatzifotiadou i,T.Hebbekera,A.Hervés, J. Hirschfelder ai,H.Hoferau, M. Hohlmann z,G.Holznerau,S.R.Houaw,Y.Huae, B.N. Jin g,L.W.Jonesc,P.deJongb, I. Josa-Mutuberría y, D. Käfer a,M.Kauro, M.N. Kienzle-Focacci u,J.K.Kimaq,J.Kirkbys, W. Kittel ae,A.Klimentovn,ab,

A.C. König ae, M. Kopal ar, V. Koutsenko n,ab,M.Kräberau, R.W. Kraemer ai,W.Krenza, A. Krüger at,A.Kuninn,P.LadrondeGuevaray,I.Laktinehx, G. Landi r, M. Lebeau s, A. Lebedev n,P.Lebrunx,P.Lecomteau,P.Lecoqs,P.LeCoultreau,J.M.LeGoffs, R. Leiste at,M.Levtchenkoaa,P.Levtchenkoah,C.Liv, S. Likhoded at,C.H.Linaw, W.T. Lin aw,F.L.Lindeb,L.Listaac,Z.A.Liug, W. Lohmann at, E. Longo am,Y.S.Lug, K. Lübelsmeyer a,C.Luciam, L. Luminari am,W.Lustermannau,W.G.Mav, L. Malgeri u, A. Malinin ab,C.Mañay, D. Mangeol ae,J.Mansak,J.P.Martinx, F. Marzano am, K. Mazumdar j, R.R. McNeil f,S.Meles,ac,L.Merolaac,M.Meschinir, W.J. Metzger ae,A.Mihull,H.Milcents,G.Mirabelliam,J.Mnicha, G.B. Mohanty j, G.S. Muanza x, A.J.M. Muijs b, B. Musicar ao,M.Musyam,S.Nagyp, S. Natale u, M. Napolitano ac, F. Nessi-Tedaldi au,H.Newmanaf,T.Niessena, A. Nisati am, H. Nowak at, R. Ofierzynski au,G.Organtiniam,C.Palomaress, D. Pandoulas a, P. Paolucci ac, R. Paramatti am,G.Passalevar,S.Patricelliac,T.Paulk, M. Pauluzzi ag, C. Paus n,F.Paussau, M. Pedace am,S.Pensottiaa, D. Perret-Gallix d,B.Petersenae, D. Piccolo ac, F. Pierella i, M. Pioppi ag,P.A.Pirouéak, E. Pistolesi aa,V.Plyaskinab, M. Pohl u,V.Pojidaevr,J.Pothiers,D.O.Prokofievar,D.Prokofievah,J.Quartierian, G. Rahal-Callot au, M.A. Rahaman j,P.Raicsp,N.Rajaj,R.Ramelliau,P.G.Rancoitaaa, R. Ranieri r, A. Raspereza at,P.Razisad,D.Renau,M.Rescignoam,S.Reucroftk, S. Riemann at,K.Rilesc,B.P.Roec,L.Romeroy,A.Roscah,S.Rosier-Leesd,S.Rotha, C. Rosenbleck a,B.Rouxae,J.A.Rubios, G. Ruggiero r, H. Rykaczewski au, A. Sakharov au,S.Saremif,S.Sarkaram, J. Salicio s, E. Sanchez y,M.P.Sandersae, C. Schäfer s,V.Schegelskyah, S. Schmidt-Kaerst a, D. Schmitz a, H. Schopper av, D.J. Schotanus ae, G. Schwering a, C. Sciacca ac,L.Servoliag, S. Shevchenko af, N. Shivarov ap, V. Shoutko n,E.Shumilovab, A. Shvorob af, T. Siedenburg a,D.Sonaq, C. Souga x, P. Spillantini r,M.Steuern,D.P.Sticklandak,B.Stoyanovap, A. Straessner s, K. Sudhakar j, G. Sultanov ap,L.Z.Sunv,S.Sushkovh,H.Suterau,J.D.Swaink, Z. Szillasi z,3,X.W.Tangg,P.Tarjanp,L.Tauschere,L.Taylork, B. Tellili x, D. Teyssier x,C.Timmermansae,SamuelC.C.Tingn,S.M.Tingn,S.C.Tonwarj,s, J. Tóth m,C.Tullyak,K.L.Tungg,J.Ulbrichtau, E. Valente am,R.T.VandeWalleae, R. Vasquez ar,V.Veszpremiz, G. Vesztergombi m, I. Vetlitsky ab,D.Vicinanzaan, G. Viertel au, S. Villa al,M.Vivargentd,S.Vlachose, I. Vodopianov ah,H.Vogelai, H. Vogt at, I. Vorobiev ai,ab, A.A. Vorobyov ah,M.Wadhwae,W.Wallraffa,X.L.Wangv, Z.M. Wang v, M. Weber a, P. Wienemann a,H.Wilkensae, S. Wynhoff ak,L.Xiaaf, Z.Z. Xu v,J.Yamamotoc,B.Z.Yangv,C.G.Yangg,H.J.Yangc,M.Yangg,S.C.Yehax, An. Zalite ah, Yu. Zalite ah,Z.P.Zhangv,J.Zhaov,G.Y.Zhug,R.Y.Zhuaf, H.L. Zhuang g,A.Zichichii,s,t,B.Zimmermannau,M.Zöllera

a I. Physikalisches Institut, RWTH, D-52056 Aachen, FRG, 1 III. Physikalisches Institut, RWTH, D-52056 Aachen, FRG 1 L3 Collaboration / Physics Letters B 540 (2002) 185–198 187

b National Institute for High Energy Physics, NIKHEF, and University of Amsterdam, NL-1009 DB Amsterdam, The Netherlands c University of Michigan, Ann Arbor, MI 48109, USA d Laboratoire d’Annecy-le-Vieux de Physique des Particules, LAPP, IN2P3-CNRS, BP 110, F-74941 Annecy-le-Vieux cedex, France e Institute of Physics, University of Basel, CH-4056 Basel, Switzerland f Louisiana State University, Baton Rouge, LA 70803, USA g Institute of High Energy Physics, IHEP, 100039 Beijing, PR China 6 h Humboldt University, D-10099 Berlin, Germany 1 i University of Bologna and INFN, Sezione di Bologna, I-40126 Bologna, Italy j Tata Institute of Fundamental Research, Mumbai (Bombay) 400 005, India k Northeastern University, Boston, MA 02115, USA l Institute of Atomic Physics and University of Bucharest, R-76900 Bucharest, Romania m Central Research Institute for Physics of the Hungarian Academy of Sciences, H-1525 Budapest 114, Hungary 2 n Massachusetts Institute of Technology, Cambridge, MA 02139, USA o Panjab University, Chandigarh 160 014, India p KLTE-ATOMKI, H-4010 Debrecen, Hungary 3 q Department of Experimental Physics, University College Dublin, Belﬁeld, Dublin 4, Ireland r INFN, Sezione di Firenze and University of Florence, I-50125 Florence, Italy s European Laboratory for Particle Physics, CERN, CH-1211 Geneva 23, Switzerland t World Laboratory, FBLJA Project, CH-1211 Geneva 23, Switzerland u University of Geneva, CH-1211 Geneva 4, Switzerland v Chinese University of Science and Technology, USTC, Hefei, Anhui 230 029, PR China 6 w University of Lausanne, CH-1015 Lausanne, Switzerland x Institut de Physique Nucléaire de Lyon, IN2P3-CNRS, Université Claude Bernard, F-69622 Villeurbanne, France y Centro de Investigaciones Energéticas, Medioambientales y Tecnológicas, CIEMAT, E-28040 Madrid, Spain 4 z Florida Institute of Technology, Melbourne, FL 32901, USA aa INFN, Sezione di Milano, I-20133 Milan, Italy ab Institute of Theoretical and Experimental Physics, ITEP, Moscow, Russia ac INFN, Sezione di Napoli and University of Naples, I-80125 Naples, Italy ad Department of Physics, University of Cyprus, Nicosia, Cyprus ae University of Nijmegen and NIKHEF, NL-6525 ED Nijmegen, The Netherlands af California Institute of Technology, Pasadena, CA 91125, USA ag INFN, Sezione di Perugia and Università Degli Studi di Perugia, I-06100 Perugia, Italy ah Nuclear Physics Institute, St. Petersburg, Russia ai Carnegie Mellon University, Pittsburgh, PA 15213, USA aj INFN, Sezione di Napoli and University of Potenza, I-85100 Potenza, Italy ak Princeton University, Princeton, NJ 08544, USA al University of California, Riverside, CA 92521, USA am INFN, Sezione di Roma and University of Rome “La Sapienza”, I-00185 Rome, Italy an University and INFN, Salerno, I-84100 Salerno, Italy ao University of California, San Diego, CA 92093, USA ap Bulgarian Academy of Sciences, Central Laboratory of Mechatronics and Instrumentation, BU-1113 Soﬁa, Bulgaria aq The Center for High Energy Physics, Kyungpook National University, 702-701 Taegu, South Korea ar Purdue University, West Lafayette, IN 47907, USA as Paul Scherrer Institut, PSI, CH-5232 Villigen, Switzerland at DESY, D-15738 Zeuthen, Germany au Eidgenössische Technische Hochschule, ETH Zürich, CH-8093 Zürich, Switzerland av University of Hamburg, D-22761 Hamburg, Germany aw National Central University, Chung-Li, Taiwan, ROC ax Department of Physics, National Tsing Hua University, Taiwan, ROC Received 16 May 2002; received in revised form 12 June 2002; accepted 17 June 2002 Editor: L. Rolandi

1 Supported by the German Bundesministerium für Bildung, Wissenschaft, Forschung und Technologie. 2 Supported by the Hungarian OTKA fund under contract numbers T019181, F023259 and T037350. 3 Also supported by the Hungarian OTKA fund under contract number T026178. 188 L3 Collaboration / Physics Letters B 540 (2002) 185–198

Abstract We measure three-particle Bose–Einstein correlations in hadronic Z decay with the L3 detector at LEP. Genuine three-particle Bose–Einstein correlations are observed. By comparing two- and three-particle correlations we ﬁnd that the data are consistent with fully incoherent pion production.  2002 Elsevier Science B.V. All rights reserved.

1. Introduction

So far, no theory exists which can describe the non-perturbative process of hadron production in general and Bose–Einstein (BE) effects in particular. The latter are expected from general spin statistics considerations. To help understand these phenomena, studies of identical-boson correlations in e+e− collisions at LEP have been performed in terms of the absolute four-momentum difference Q [1], as well as in two- and three-dimensional distributions in components of Q [2,3]. It has long been realized that the shape and size in spacetime of a source of pions can be determined, as a consequence of the interference of identical bosons, from the shape and size of the correlation function of two identical pions in energy–momentum space [4]. Additional information can be derived from higher-order correlations. Furthermore, such correlations constitute an important theoretical issue for the understanding of Bose– Einstein correlations (BEC) [5]. In this Letter three-particle correlations are analysed. These correlations are sensitive to asymmetries in the particle production mechanism [6,7] which cannot be studied by two-particle correlations. In addition, the combination of two- and three-particle correlation analyses gives access to the degree of coherence of pion production [8,9], which is very difﬁcult to investigate from two-particle correlations alone due to the effect of long-lived resonances on the correlation function. The DELPHI [10] and OPAL [11] Collaborations have both studied three-particle correlations but did not investigate the degree of coherence.

2. The data and Monte Carlo

The data used in this analysis were collected by the L3 detector [12] in 1994 at a centre-of-mass energy of 91.2 GeV and correspond to a total integrated luminosity of 48.1pb−1. The Monte Carlo (MC) event generators JETSET [13] and HERWIG [14] are used to simulate the signal process. Within JETSET, BEC are simulated using 7 the BE0 algorithm [15,16]. The generated events are passed through the L3 detector simulation program, which is based on the GEANT [17] and GHEISHA [18] programs, reconstructed and subjected to the same selection criteria as the data. The event selection is identical to that presented in Ref. [2], resulting in about one million hadronic Z decay events, with an average track multiplicity of about 12. Two additional cuts are performed in order to reduce the dependence of the detector correction on the MC model used: tracks with measured momentum greater than 1 GeV are rejected, as are pairs of like-sign tracks with opening angle below 3◦. This results in an average track multiplicity of about 7. For the computation of three-particle correlations, each possible triplet of like-sign tracks is used ≡ 2 + 2 + 2 ≡ − − 2 to compute the variable Q3 Q12 Q23 Q31,whereQij (pi pj ) is the absolute four-momentum difference between particles i and j.SinceQij , and thus Q3, depends both on the energy of the particles and on the

4 Supported also by the Comisión Interministerial de Ciencia y Tecnología. 5 Also supported by CONICET and Universidad Nacional de La Plata, CC 67, 1900 La Plata, Argentina. 6 Supported by the National Natural Science Foundation of China. 7 The Bose–Einstein simulation is done by the subroutine LUBOEI, with the values PARJ(92) = 1.5andPARJ(93) = 0.33 GeV. L3 Collaboration / Physics Letters B 540 (2002) 185–198 189

Fig. 1. Normalized distributions of the sum of the difference in (a) azimuthal and (b) polar angle of pairs of tracks in a triplet, δφ and δθ, and of (c) Q3. Data, JETSET, with and without BE effects, and HERWIG are displayed. The ratios between the data and MC distributions are shown in (d), (e) and (f).

angle between them, small Qij can be due to small angles or low energies. In a MC generator with BE effects, the fraction of pairs at small Qij with small angle is larger than in one without. Consequently, the estimated detection efﬁciency depends on the MC model used. The momentum and opening angle cuts reduce this model dependence. After selection, the average triplet multiplicity is about 6. In the region of interest, Q3 < 1 GeV, the loss of triplets by the momentum and opening angle cut is about 40%. The momentum cut improves the resolution of Q3 by a factor three with respect to that for the full momentum spectrum. Using MC events, its average is estimated to be 26 MeV for triplets of tracks with Q3 < 0.8GeV.We choose a bin size of 40 MeV, somewhat larger than this resolution. In Fig. 1, the data are compared to JETSET (with and without BE effects) and HERWIG (not having a BE option) at the detector level, after performing all the cuts mentioned above, in the three-particle distributions δφ, δθ,andQ3. The sums run over the three pairs of like-sign tracks in the triplet and δφ and δθ are the absolute differences in azimuthal and polar angle between two tracks, respectively. Within 10%, the angular distributions of the MC models agree with those of the data. None of the models describes the Q3 distribution: JETSET with BE 190 L3 Collaboration / Physics Letters B 540 (2002) 185–198 effects overestimates the data by approximately 20% at low Q3, even though we found good agreement for δφ, δθ, and Q [2]. JETSET without BE effects and HERWIG grossly underestimate the data at low Q3. The statistics for Q3 < 160 MeV are so poor, that this region is rejected from the analysis.

3. The analysis

The three-particle number density ρ3(p1,p2,p3) of particles with four-momenta p1,p2 and p3 can be described in terms of single-particle, two-particle and genuine three-particle densities as ρ3(p1,p2,p3) = ρ1(p1)ρ1(p2)ρ1(p3) + ρ1(p1)[ρ2(p2,p3) − ρ1(p2)ρ1(p3)] + C3(p1,p2,p3), (1) (3) where the sum is over the three possible permutations and C3 is the third-order cumulant, which measures the genuine three-particle correlations. The ρ1ρ2 terms contain all the two-particle correlations. In order to focus on the correlation due to BE interference, we replace products of single-particle densities by the corresponding two- or three-particle density, ρ0, which would occur in the absence of BEC, and deﬁne the correlation functions

ρ2(p1,p2) ρ3(p1,p2,p3) R2(p1,p2) ≡ ,R3(p1,p2,p3) ≡ . (2) ρ0(p1,p2) ρ0(p1,p2,p3)

Assuming the absence of two-particle correlations, i.e., ρ2(p1,p2) = ρ1(p1)ρ1(p2), results in

genuine ≡ + C3(p1,p2,p3) R3 (p1,p2,p3) 1 . (3) ρ0(p1,p2,p3) The kinematical variable Q3 is used to study three-particle correlations. For a three-pion system, Q3 = 2 − 2 M123 9mπ , with M123 the invariant mass of the pion triplet and mπ the mass of the pion. In this Letter, ρ3 is deﬁned as

1 dntriplets ρ3(Q3) ≡ , (4) Nev dQ3 with Nev the number of selected events and ntriplets the number of triplets of like-sign tracks, and ρ2 is defined analoguously. Assuming totally incoherent production of particles and a source density f(x)in spacetime with no dependence on the four-momentum of the emitted particle, the BE correlation functions is related to the source density by [8, 19] 2 R2(Qij ) = 1 +|F(Qij )| , (5) 2 2 2 R3(Q12,Q23,Q31) = 1 +|F(Q12)| +|F(Q23)| +|F(Q31)| + 2Re{F(Q12)F (Q23)F (Q31)}, (6) genuine = + { } R3 (Q12,Q23,Q31) 1 2Re F(Q12)F (Q23)F (Q31) , (7) where F(Qij ) is the Fourier transform of f(x). R2 does not depend on the phase φij contained in F(Qij ) ≡|F(Qij )| exp(ιφij ). However, this phase survives in the three-particle BE correlation functions, Eqs. (6) and (7). Assuming fully incoherent particle production, the phase φij can be non-zero only if the spacetime distribution of the source is asymmetric and Qij > 0. Defining genuine − R3 (Q12,Q23,Q31) 1 ω(Q12,Q23,Q31) = √ , (8) 2 (R2(Q12) − 1)(R2(Q23) − 1)(R2(Q31) − 1) then for an incoherent source Eqs. (5) and (7) imply that ω = cosφ,whereφ ≡ φ12 + φ23 + φ31.Furthermore,as Qij → 0, then φij → 0, and hence ω → 1. For Qij > 0, a deviation from unity can be caused by an asymmetry L3 Collaboration / Physics Letters B 540 (2002) 185–198 191 in the production. However, this will only result in a small (a few percent) reduction of ω [6,7], and this only in the case where the asymmetry occurs around the point of highest emissivity. It is important to emphasize that for (partially) coherent sources, ω can still be defined by Eq. (8), but Eqs. (5)–(7) are no longer valid, in which case more complicated expressions are needed [7], and one can no longer deduce that ω = cosφ or that ω → 1as Qij → 0. In at least one type of model, one can make the stronger statement that the limit ω = 1atQij → 0 can only be reached if the source is fully incoherent [20].

genuine 4. Determination of R3 and R3

The reference sample, from which ρ0 is determined, is formed by mixing particles from different data events in the following way. Firstly, 1000 events are rotated to a system with the z-axis along the thrust axis and are stored in a “pool”. Then, tracks of each new event outside the pool are exchanged with tracks of the same charge from events in the pool having about the same (within about 20%) multiplicity, under the condition that all tracks originate from different events. Thus, after this procedure the new event consists of tracks originating from different events in the pool, and its original tracks have entered the pool. This updating process prevents any regularities in the reference sample. Finally, Q3 is calculated for each triplet of like-sign tracks, resulting in the density ρmix. This mixing procedure removes more correlations than just those of BE, e.g., those from energy–momentum conservation and from resonances. This effect is estimated using a MC model with no BE effects (JETSET or HERWIG) at generator level and using pions only. Thus, in the absence of BEC, the corrected three-particle density is given by ρ3(Q3) ρ0(Q3) = ρmix(Q3)Cmix(Q3), where Cmix(Q3) = . (9) ρmix(Q3) MC,noBE

The density ρ3, measured in data, must be corrected for detector resolution, acceptance, efficiency and for particle misidentification. For this we use a multiplicative factor, Cdet, derived from MC studies. Since no hadrons are identified in the analysis, Cdet is given by the ratio of the three-pion correlation function found from MC events at generator level to the three-particle correlation function found using all particles after full detector simulation, reconstruction and selection. Combining this correction factor with Eqs. (2) and (9) results in

ρ3(Q3)Cdet(Q3) R3(Q3) = . (10) ρmix(Q3) Cmix(Q3) genuine The genuine three-particle BE correlation function, R3 , is obtained via genuine = − + R3 R3 R1,2 1, (11) where R1,2 ≡ ( ρ1ρ2)/ρ0 − 2 is the contribution due to two-particle correlations, as may be seen from Eqs. (1) and (2). The product of densities ρ1(p1)ρ2(p2,p3) is determined by a similar mixing procedure, as deﬁned earlier, where two like-sign tracks from the same event are combined with one track having the same charge from another event with the same multiplicity. Finally, the variable Q3 is calculated from these three tracks. This procedure is similar to that given in Ref. [21]. The ratio ( ρ1ρ2)/ρ0 is also corrected for detector effects as ρ3/ρmix. In our analysis, we use JETSET without BEC and HERWIG to determine Cmix and JETSET with and without BEC as well as HERWIG to determine Cdet. These six MC combinations serve to estimate systematic uncertainties. The corrections are largest at small Q3.AtQ3 = 0.16 GeV, these corrections to R3 are Cmix ≈ 5–30% and Cdet ≈ 20–30%, depending on which MC is used. These corrections for R3 and R1,2 are correlated and largely genuine cancel in calculating R3 by Eq. (11). 192 L3 Collaboration / Physics Letters B 540 (2002) 185–198

To correct the data for two-pion Coulomb repulsion in calculating ρ2, each pair of pions is weighted by the inverse Gamow factor [22] ( πη ) − −1 = exp 2 ij 1 = mπ α G2 (ηij ) , where ηij (12) 2πηij Qij and α is the ﬁne-structure constant. It has been shown [23] that this Gamow factor is an approximation suitable for our purposes. For ρ3, the weight of each triplet is taken as the product of the weights of the three pairs within it. For ρ2ρ1 we use the same weight but with G2(Qij ) ≡ 1 when particles i and j come from different events. At the lowest Q3 values under consideration, the Coulomb correction is approximately 10%, 3% and 2%, for ρ3, ρ1ρ2 and ρ2, respectively.

5. Results

The measurements of R3, R1,2 and R2 are shown in Fig. 2. The full circles correspond to the averages of the data points obtained from the six possible MC combinations used to determine Cmix and Cdet. The error bars, σ1, include both the statistical uncertainty and the systematic uncertainty of the MC modeling, which is taken as the r.m.s. of the values obtained using the different MC combinations. This dominant systematic uncertainty is, for Q3 < 0.8 GeV, about a factor 5 to 7 larger than the statistical uncertainty and is correlated between the R3, R1,2 and R2 distributions of Fig. 2 and between bins. Fig. 2(a) shows the existence of three-particle correlations and from Fig. 2(b) it is clear that about half is due to two-particle correlations. Fig. 2(c) shows the two-particle correlations. As a check, R3, R1,2 and R2 are also computed for MC models without BEC, both HERWIG and JETSET, after detector simulation, reconstruction and selection. For the mixing and detector corrections all possible MC combinations, giving non-trivial results, are studied. The results of this check are shown in Fig. 2 as open circles and, as expected, ﬂat distributions around unity are observed. genuine Fig. 3(a) shows the genuine three-particle BE correlation function R3 . The data points show the existence of genuine three-particle BE correlations. The MC systematic uncertainty is highly correlated from bin to bin. At Q3 < 0.8 GeV, it is about a factor 1.5 to 3.5 larger than the statistical uncertainty, the higher value corresponding to the lowest Q3 value used. The open circles correspond to MC without BEC and form a ﬂat distribution around unity, as expected.

5.1. Gaussian parametrizations

A fit from Q3 = 0.16 to 1.40 GeV using the covariance matrix including both the statistical uncertainty and the systematic uncertainty due to the MC modeling, σ1, is performed on the data points with the commonly used [8, 10,11,21] parametrization genuine =˜ + ˜ 1.5 − 2 2 +˜ R3 (Q3) γ 1 2λ exp R Q3/2 (1 εQ3), (13) where γ˜ is an overall normalization factor, λ˜ measures the strength of the correlation, R is a measure for the effective source size in spacetime and the term (1 +˜εQ3) takes into account possible long-range momentum = | |= correlations.√ The form of this parametrization is a consequence of the assumptions that ω 1andthat F(Qij ) − 2 2 λ exp( R Qij /2), as would be expected for a Gaussian source density. The fit results are given in the first column of Table 1 and shown as the full line in Fig. 3(a). In addition to the MC modeling, we investigate four other sources of systematic uncertainties on the fit parameters. Firstly, the influence of a different mixing sample is studied by removing the conditions that tracks are replaced by tracks with the same charge and coming from events with approximately the same multiplicity. For each of the six MC combinations, the difference in the fit results between the two mixing methods is taken as an L3 Collaboration / Physics Letters B 540 (2002) 185–198 193

Fig. 2. (a) The three-particle BE correlation function, R3, from Eq. (10), (b) the contribution of two-particle correlations, R1,2 ≡ ( ρ2ρ1)/ρ0 − 2, and (c) R2 from Eq. (5). The full circles correspond to the data and the error bars to σ1 (see text). The open circles correspond to the results from MC models without BEC. In (c) the dashed and full lines show the ﬁts of Eqs. (14) and (15), respectively.

estimate of the systematic uncertainty. The square root of the mean of the squares of these differences is assigned as the systematic uncertainty from this source. In the same way, systematic uncertainties related to track and event selection and to the choice of the fit range are evaluated. The analysis is repeated with stronger and weaker selection criteria, changing the number of events by about ±11% and the number of tracks by about ±12%. The fit range is varied by removing the first point of the fit and varying the end point by ±200 MeV. Finally, we study the influence of removing like-sign track pairs with small polar and azimuthal opening angles. The maximum deviation that is found by varying the cuts on these angles up to 6◦, is taken as the systematic uncertainty from this source. The total systematic uncertainty due to these four sources is obtained by adding the four uncertainties in quadrature. We refer to this systematic uncertainty as σ2. For all fit parameters, the largest part of the total uncertainty is due to the six possible combinations of mixing and detector MC corrections and amounts to 50–90%. Table 2 shows the uncertainties for each of the sources for the fit parameters of Eq. (13). As a cross-check, the analysis is repeated without the momentum cut of 1 GeV and without the cut of 3◦ on the opening angle of like-sign track pairs. The results agree with those given in Table 1 well within quoted uncertainties, but the systematic uncertainties are approximately twice as large. 194 L3 Collaboration / Physics Letters B 540 (2002) 185–198

genuine Fig. 3. The genuine three-particle BE correlation function R3 , Eq. (11). The full circles correspond to the data and the error bars to σ1. The open circles correspond to results from MC models without BEC. In (a) the full line shows the ﬁt of Eq. (13), the dashed line the prediction of completely incoherent pion production and a Gaussian source density in spacetime, derived from parametrizing R2 with Eq. (14). In (b) Eqs. (16) and (15) are used, respectively.

Table 1 genuine Values of the fit parameters for the genuine three-particle BE correlation function R3 , using the parametrizations of Eqs. (13) and (16). The first uncertainty corresponds to σ1, the second to σ2, defined in the text Parameter Eq. (13) Eq. (16) γ˜ 0.96 ± 0.03 ± 0.02 0.95 ± 0.03 ± 0.02 λ˜ 0.47 ± 0.07 ± 0.03 0.75 ± 0.10 ± 0.03 R ,fm 0.65 ± 0.06 ± 0.03 0.72 ± 0.08 ± 0.03 − ε˜,GeV 1 0.02 ± 0.02 ± 0.02 0.02 ± 0.02 ± 0.02 κ˜ –0.79 ± 0.26 ± 0.15 χ2/NDF 29.9/27 17.7/26

To measure the ratio ω, we also need to determine the two-particle BE correlation function R2(Q). This is done in the same way as the three-particle BE correlation function. The correlation function R2 is parametrized as a Gaussian: 2 2 R2(Q) = γ 1 + λ exp −R Q (1 + εQ). (14) L3 Collaboration / Physics Letters B 540 (2002) 185–198 195

Table 2 Contribution to the uncertainty on the ﬁt parameters of the parametrizations of Eqs. (13) and (16), respectively. The ﬁrst uncertainty corresponds to σ1, the others added in quadrature give σ2 Parametrization Eq. (13) Eq. (16) − − Fit parameter γ˜ λ˜ R ,fm ε˜,GeV 1 γ˜ λ˜ R ,fm ε˜,GeV 1 κ˜

σ1 (stat. + modeling) 0.029 0.071 0.056 0.022 0.031 0.103 0.078 0.024 0.26 Mixing 0.004 0.006 0.009 0.007 0.010 0.009 0.011 0.011 0.04 Fit range 0.008 0.019 0.020 0.013 0.010 0.022 0.017 0.019 0.14 Track/event sel. 0.010 0.013 0.012 0.008 0.011 0.016 0.012 0.007 0.10 δφ + δθ cut 0.013 0.014 0.012 0.009 0.014 0.017 0.010 0.008 0.11

σ2 0.019 0.028 0.028 0.020 0.023 0.033 0.026 0.024 0.15

Fig. 4. The ratio ω as a function of Q3 assuming R2 is described (a) by the Gaussian, Eq. (14), and (b) by the ﬁrst-order Edgeworth expansion of the Gaussian, Eq. (15). The error bars correspond to σ1. For completely incoherent production, ω = 1.

8 The parametrization starts at Q = 0.08 GeV, consistent with the study of R3 from Q3 = 0.16 GeV. The fit results are given in the first column of Table 3 and in Fig. 2(c). If the spacetime structure of the pion source is Gaussian and the pion production mechanism is completely incoherent, λ˜ and R as derived from the fit to Eq. (13) measure the same correlation strength and effective source size as λ and R of Eq. (14). The values of λ and R are consistent with λ˜ and R , as expected for fully incoherent production of pions (ω = 1). Using the values of λ and R instead of λ˜ and R in Eq. (13), which is justified if ω = 1, results in the dashed line in Fig. 3(a). It is only slightly different from the result of the fit to Eq. (13), indicating that ω is indeed near unity. genuine Another way to see how well R3 corresponds to a completely incoherent pion production interpretation and a Gaussian source density in spacetime, is to compute ω with Eq. (8), for each bin in Q3 (from 0.16 to 0.80 GeV), genuine using the measured R3 and R2 derived from the parametrization of Eq. (14). The result is shown in Fig. 4(a). At low Q3, ω appears to be higher than unity.

8 Due to the use of a different fit range, these fit results differ from those found in Ref. [24]. The same fit range gives similar results. 196 L3 Collaboration / Physics Letters B 540 (2002) 185–198

Table 3 Values of the ﬁt parameters for the two-particle BE correlation function, R2, using the parametrizations of Eqs. (14) and (15). The ﬁrst uncertainty corresponds to σ1, the second to σ2 Parameter Eq. (14) Eq. (15) γ 0.98 ± 0.03 ± 0.02 0.96 ± 0.03 ± 0.02 λ 0.45 ± 0.06 ± 0.03 0.72 ± 0.08 ± 0.03 R,fm 0.65 ± 0.03 ± 0.03 0.74 ± 0.06 ± 0.02 − ε,GeV 1 0.01 ± 0.01 ± 0.02 0.01 ± 0.02 ± 0.02 κ –0.74 ± 0.21 ± 0.15 χ2/NDF 60.2/29 26.0/28

5.2. Extended Gaussian parametrizations

However, the assumption of a Gaussian source density is only an approximation, as observed in Ref. [2] and confirmed by the χ2 of the fit to Eq. (14). Deviations from a Gaussian can be studied by expanding in terms of derivatives of the Gaussian, which are related to Hermite polynomials. Taking only the lowest-order non-Gaussian term into account, this so-called Edgeworth expansion [25] replaces the parametrization of Eq. (14) by √ 2 2 R2(Q) = γ 1 + λ exp −R Q 1 + κH3( 2 RQ)/6 (1 + εQ), (15) 3 where κ measures the deviation from the Gaussian and H3(x) ≡ x − 3x is the third-order Hermite polynomial. The fit results for the two-particle BE correlation function with this parametrization are given in the second column of Table 3. Using the first-order Edgeworth expansion of the Gaussian, Eq. (15), and using Eq. (8), assuming ω = 1, the parametrization of Eq. (13) becomes √ 3 H ( 2 RQij ) Rgenuine(Q ) =˜γ 1 + 2λ˜ 1.5 exp −R 2Q2/2 1 + 3 κ˜ (1 +˜εQ ) 3 3 3 6 3 i,j=1,j>i √ H ( 2 RQ /2) 1.5 ˜γ 1 + 2λ˜ 1.5 exp −R 2Q2/2 1 + 3 3 κ˜ (1 +˜εQ ). (16) 3 6 3

= genuine In the second line the approximation is made that Qij Q3/2. The effect of this approximation on R3 is small compared to the statistical uncertainty. The results of a fit to Eq. (16) are given in the second column of Table 1. The uncertainties are summarized in Table 2. genuine 2 ˜ For both R3 and R2, a better χ /NDF is found using the Edgeworth expansion, and the values of λ and λ are significantly higher, as shown in Tables 1 and 3 and in Figs. 3(b) and 2(c). The values for λ˜ and R are still consistent with the corresponding λ and R, as would be expected for a fully incoherent production mechanism of pions. genuine In Fig. 3(b), as in Fig. 3(a), we observe good agreement between the fit of R3 using the parametrization of Eq. (16) and the prediction of a completely incoherent pion production mechanism, derived from parametrizing R2 with Eq. (15), over the full range of Q3. In Fig. 4(b), no deviation from unity is observed for the ratio ω.This indicates that the data agree with the assumption of fully incoherent pion production. 9 Fits to samples generated with JETSET with BE effects modelled by BE0 or BE32 [16] result in values of R in agreement with the data but in significantly higher values of λ˜ . This confirms the observation in Fig. 1(f) that the standard BE implementations of JETSET overestimate the genuine three-particle BEC.

9 The BE32 algorithm uses the values PARJ(92) = 1.68 and PARJ(93) = 0.38 GeV. L3 Collaboration / Physics Letters B 540 (2002) 185–198 197

6. Summary

Three-particle, as well as two-particle Bose–Einstein correlations of like-sign charged pions have been measured in hadronic Z decay. Genuine three-particle BE correlations are observed. The correlation functions are better parametrized by an Edgeworth expansion of a Gaussian than by a simple Gaussian. Combining the two- and three- particle correlations shows that the data are consistent with a fully incoherent production mechanism of pions.

Acknowledgements

Clarifying discussions with T. Csörgo˝ are gratefully acknowledged.

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Effect of E1 decay in the population of superdeformed structures

G. Benzoni a, A. Bracco a,F.Cameraa,S.Leonia, B. Million a,A.Majb, A. Algora c, A. Axelsson d,M.Bergströme,N.Blasia, M. Castoldi f, S. Frattini a, A. Gadea c, B. Herskind e,M.Kmiecikb,G.LoBiancog,J.Nybergd, M. Pignanelli a, J. Styczen b, O. Wieland a, M. Zieblinski b, A. Zucchiatti f

a Dipartimento di Fisica, Universitá di Milano and INFN, via Celoria 16, 20133 Milano, Italy b Niewodniczanski Institute of Nuclear Physics, 31-342 Krakow, Poland c Laboratori Nazionali di Legnaro, via Romea, Legnaro (PD), Italy d Department of Neutron Research, Uppsala University, S-75121 Uppsala, Sweden e The Niels Bohr Institute, Blegdamsvej 15-17, 2100, Copenhagen, Denmark f INFN sezione di Genova, Genova, Italy g Dipartimento di Fisica, Universitá di Camerino e INFN Perugia, Camerino, Italy Received 1 February 2002; received in revised form 13 June 2002; accepted 21 June 2002 Editor: V. Metag

Abstract Spectra of the yrast and excited superdeformed bands, forming the E2 quasi-continuum, are measured with the EUROBALL 143 array for the nucleus Eu, in coincidence with high-energy γ -rays (Eγ > 3 MeV). It is found that the intensity population of the superdeformed states is enhanced by a factor of ≈ 1.6 when a coincidence with a γ -ray with energy > 6MeVis required, in reasonable agreement with the increase of the line shape of the Giant Dipole Resonance built on a superdeformed conﬁguration. This result shows that when an high energy E1 γ -ray is involved in the decay it is more likely connected with a SD rather than a ND nucleus. In addition, the analysis of the rotational quasi-continuum suggests the presence of a superdeformed component. The data are also compared and found consistent with simulation calculations of the relative intensities of the SD states, including the E1 decay of superdeformed nature.  2002 Elsevier Science B.V. All rights reserved.

PACS: 21.10.Tg; 21.10.Re; 21.60.Ka; 23.20.En; 23.20.L

The study of nuclear structure at the extreme [3]. While the nature of SD bands has been inter- limits of rotational stress and large shape changes preted in terms of the occupancy of a number of has been a topic extensively investigated in the last high-N intruder orbitals (N being the major oscilla- years [1,2]. Gamma spectroscopy measurements have tor quantum number) only in a small number of cases identiﬁed superdeformed (SD) rotational bands in [4–6] it has been possible to make ﬁrm assignments of more than 50 nuclei in several different mass regions spins, parities and excitation energies from observed single-step connections with normal deformed (ND) yrast states, as a consequence of the highly fragmented E-mail address: [email protected] (S. Leoni). decay-out of the SD states.

Another relevant open question concerning the not clear whether such discrepancy is due to a differ- problem of superdeformation is the understanding of ent mechanism than the originally proposed E1 decay the population mechanisms of the SD states, having (such as, for example, an enhanced neutron emission important consequences on the size of the intensities probability due to smaller level density in the superde- of the measured transitions of the SD bands. In fact, it formed nucleus), or to the experimental conditions. has been found that the typical measured intensities of The latter could be related to the choice of the bom- the SD bands are approximately 2 order of magnitude barding energy and to the efficiency of the apparatus, stronger than those measured at the same highest which were not optimal to address the problem. There- spins in normally deformed nuclei [7,8]. It has been fore, the population mechanism of the SD states still suggested that such intense population of the SD band represents an interesting open problem in the study of at high spins could be related to the E1 feeding of these superdeformation. states, which is expected to be strongly affected by the In this Letter the role of the E1 decay in the popu- shape of the giant dipole resonance (GDR) strength lation of superdeformed structures is investigated with function. As a consequence of the large nuclear prolate the more sensitive EUROBALL set up and in the case deformation, the GDR strength function is predicted of another nucleus of similar mass, 143Eu. This nu- to display a low energy component (corresponding to cleus has been chosen not only because it has a rather the dipole oscillation along the long symmetry axis) strong SD yrast line (of the order of 2% of the total nu- at an energy close to the neutron binding energy and cleus population), but also rotationally correlated tran- exhausting one third of the total energy weighted sum sitions forming a rather intense quasi-continuum struc- rule strength [9]. This enhanced E1 strength together ture of superdeformed character [14–16]. The aim of with the reduced level density of the SD states, as this work is to measure SD transitions (yrast and quasi- compared to the case of normal deformed nuclei, are continuum) in coincidence with high-energy γ -rays expected to enhance the population of SD rotational and to study the dependence of their intensities as a bands. This mechanism has been originally proposed function of the γ -ray energies in the interval 3–8 MeV. in Ref. [10] in connection with the understanding of In this region one can probe the low energy tail of the the measured intensities of the SD yrast band of the GDR strength function that in a spherical nucleus is nucleus 152Dy. characterized by a Lorentzian shape centered at 14– For a full understanding of the population mecha- 15 MeV, while in the case of a superdeformed nu- nism of the SD configuration one would need to mea- cleus one third of the strength is expected to be around sure the absolute population probability for γ and par- 10.5 MeV [9]. At variance from previous works on Gd ticle feeding as a function of particle and γ -ray energy. isotopes, the bombarding energy of the projectile has In the present case we focus only on the E1 feeding by been chosen to be 5 MeV larger than in the study of the testing whether or not when an high energy E1 γ -ray SD yrast line [17], in order to have an entry distribu- is involved in the decay it is more likely connected tion for the population of the 143Eu nucleus extending with a SD rather than a ND nucleus. In particular, the at higher temperature. This increases the phase space population of the SD structures is expected to increase for the decay with high energy γ -rays. In addition, with γ -ray energies, as a consequence of the GDR line this work is also intended to provide a further veri- shape built on a superdeformed nucleus. This is in gen- fication of the superdeformed nature of the E2 quasi- eral a difficult experimental task, due to both the expo- continuum by determining whether or not the intensity nential decrease of the yield of the high-energy γ -rays of the spectral structures behave as a function of high- and to the weak intensity of the SD transitions, being energy γ -rays similarly to the yrast line. at most of the order of few percent for the yrast band. The experiment was performed at the Tandem Ac- However, due to the interest to this problem, already celerator Laboratory of Legnaro (Padova). The 37Cl a decade ago, when the available detection arrays were beam, at the incident energy of 165 MeV, impinged at the limit to address this question, few experimental on a target of 110Pd (97.3% pure and 950 µg/cm2 attempts were made in connection with superdeformed thick) with a Au backing of 15 mg/cm2. The cho- Gd isotopes [11–13], which did not show the behav- sen bombarding energy represents a good compro- iour predicted for the 152Dy case [10]. Presently, it is mise for a good population of the SD band and for G. Benzoni et al. / Physics Letters B 540 (2002) 199–206 201

E1 emission from the final residual nucleus at exci- Doppler corrected on event-by-event basis according tation energies around the binding energy. The com- to the fractional Doppler shift of the superdeformed pound nucleus 147Eu was formed at an excitation en- yrast band [21], and gated on the SD yrast and on the ergy of 79 MeV. The maximum angular momentum is high-energy γ -rays. The single spectra are the sum predicted to be 68 h¯ by the heavy-ion grazing model, of one-dimensional (1D) projections of such matri- in which the excitation of collective modes is taken ces (background subtracted with the Radware method into account in the fusion process [18]. This exceeds [20]), obtained by gating in addition on the cleanest the angular momentum limit where fission becomes SD lines (namely 732, 794, 973, 1032, 1091, 1149, dominant. High-energy γ -rays were detected in the 8 1208, 1266, 1325, 1384, 1444, 1502, 1564, 1624, and BaF2 scintillators of the HECTOR array [19] placed 1684 keV). Since the SD gates used in the construc- at 30 cm from the target center, shielded by lead to at- tion of the spectra have been placed all over the SD tenuate low-energy γ -rays. In addition, 4 small BaF2 band, the extracted intensity of the SD yrast can only detectors were placed with their front face at the outer be used to study its relative behaviour and not to de- surface of the target chamber to measure the time of duce an absolute value. To show the three spectra of flight with good resolution, information needed to re- Fig. 1 on the same scale, the data in panels (a) and (b) ject neutron events. The time of flight of the Ge detec- have been scaled down by a factor giving the same tors was also measured relative to these small BaF2 counts in the 917 keV low spin transition between detectors. The gain of each BaF2 detector has been spherical states, as collected with the high-energy gat- monitored continuously during all the experiment using condition Eγ =7.3 MeV (panel (c)). In all three ing a LED source and small shifts were corrected dur- cases the SD yrast transitions of interest are marked ing the off-line analysis. The calibration of the HPGe by diamonds and can be identified even when the sta- EUROBALL detectors was obtained using standard tistics is largely reduced as in the case of the gating radioactive sources while the large volume BaF2 de- condition 6 3 MeV) measured in the than 8 MeV. As one can see from the figure (with the BaF2 detectors and γ -rays detected in HPGe detectors, help of the horizontal lines at 100 and 200 counts), with the condition that at least 3 Ge detectors (with- the intensities of the SD peaks grow with increasing out Compton suppression) had fired. The coincidence energy of the gating γ -rays. The increase of the SD fold distribution for Ge detectors measured in the ex- yrast population as function of the high-energy tran- periment has an average value at ≈ 3, after Compton sition was evaluated by summing the intensity (after suppression. subtraction of the background between the peaks and The effect of the E1 population of the superde- efficiency correction) of the SD lines marked in the fig- formed yrast band by the γ -decay of the giant di- ure. The deduced values were normalized to that cor- pole resonance built on a superdeformed nucleus is in- responding to the 3.4 MeV high-energy gating condi- vestigated by measuring the relative intensity of the tion and the error bars have been evaluated taking into SD band at different values of the high-energy gat- account both the statistical error and the uncertainty ing γ -rays. In Fig. 1 the spectrum of the superde- related to different low lying ND transitions which formed yrast transitions is shown for three different can be used to normalize the spectra. The resulting in- gating conditions on the high-energy γ -rays measured crease between the lowest and the highest gating con- in the BaF2 scintillator detectors, namely 3

143 Fig. 1. Spectra of the SD yrast band of Eu, gated by high-energy γ -rays with average energies Eγ =3.4, 5.4 and 7.3 MeV (panel (a), (b) and (c), respectively). The spectra are normalized to the intensity of the low spin 917 keV line with the gating condition Eγ =7.3MeV. The horizontal lines at 100 and 200 counts help the comparison of the SD lines intensities in the different spectra.

ﬁed ridge structure in γ –γ spectra. In Fig. 2 one- lated events constructed using the COR treatment of dimensional projections on a 60 keV wide strip per- Ref. [22], with a COR reduction factor of 0.95, em- pendicular to the main diagonal of the Eγ 1 × Eγ 2 ma- phasizing the high spin region. The spectra are nor- trix at the average transition energy (Eγ 1 + Eγ 2)/2 = malized to the intensity collected in the valley region 1140 keV are shown, for average gating transitions of of the spectrum gated by Eγ =6.2MeV.Thesu- Eγ =3.4, 4.4 and 6.2 MeV (panel (a), (b) and (c), perdeformed nature of the ridge structure is conﬁrmed respectively), corresponding to the energy intervals by the distance between the two most inner ridges 2 (2) 3

B(E2) strength out of each state [14,15]. These transitions form a continuous distribution of E2 character (E2 bump), which in the speciﬁc case of 143Eu is ob- 143 served at 1200

Fig. 4. The bottom part of the figure shows the intensity of the SD yrast (squares), of the SD ridge (filled triangles) and of the E2 bump (filled circles), as function of the energy of the gating transition, relative to the corresponding values measured at Eγ =3.4MeV. The cross symbols give the enhancement observed by using the well define 973 keV SD yrast line only. For comparison, the relative intensities of spherical (open squares) and triaxial (open triangles) low spin transitions are also given. The dashed line corresponds to the ratio of the strength functions of the GDR built on a SD and a ND nucleus, while the full line gives the predicted values of the relative intensity of the SD yrast, as obtained from the model of Refs. [10, 15]. In the top part of the figure the total average multiplicity of the γ -cascades leading to 143Eu is shown for both the experimental data (circles) and the model, as function of the energy of the gating transition.

Fig. 3. Spectra of 143Eu collecting the entire decay flow (ND) evaluated the relative increase of the E2 bump using and the triaxial contribution only (TD), gated by high-energy as a reference the same TD spectrum corresponding to = transition with Eγ 3.4, 4.5 and 7.3 MeV (panel (a), (b) and the 3.4 MeV gate (shown in panel (a) of the figure). (c), respectively). The intensity of the E2 bump observed in the ND spectra, obtained as a difference between the ND and TD spectra This gives an enhancement of the E2 continuum of and normalized to the intensity of the 553 keV low spin spherical ≈ 1.3 between the lowest and the highest gating con- transition at the highest energy gating condition, is shown in the top ditions, as shown by full circles in Fig. 4. The lowest part of the figure. limit for the relative increase of the E2 bump has been G. Benzoni et al. / Physics Letters B 540 (2002) 199–206 205 obtained by changing by 20% the COR background the SD nucleus 143Eu [15], giving a good account subtraction. for the experimental data. The model is based on the The intensity of the SD yrast, of the SD ridges and level densities of both ND and SD states, together with of the E2 bump, relatively to those measured at Eγ = the E1 and E2 transition probabilities characteristic 3.4 MeV, are shown in Fig. 4 with square, triangles of the two deformed shapes. In the case of ND and filled circles, respectively. Other experimental states the level density is described by the Fermi- points corresponding to the average over the lower gas expression of Ref. [23], with a level density spin transitions of spherical shape (of energy 159, 176, parameter aND = A/10, while the level density of 293, 728 and 955 keV, open circles) and triaxial shape SD states is taken from the cranking + band mixing (of energy 853, 867, 884 and 936 keV, open triangles), calculations of Ref. [24] (corresponding roughly to obtained with the same procedure used for the SD a level density parameter aSD = A/18.6). In the code, yrast lines, are shown in the figure for comparison. the two deformations are separated by a barrier, the While in the case of transitions of the spherical and tunneling through which allows the mixing between triaxially deformed configurations an approximately SD and ND states. In particular, the statistical E1 constant behaviour is found, with a value of ≈ 1, an strength entering into the simulation is described as increase with γ -ray energy is instead measured in the tail of the giant dipole resonance of Lorentzian the case of the transitions of superdeformed nature. shape, with centroids and widths for the SD and ND As shown in the top part of Fig. 4 by open circles, configurations as given above. The parameters used the total average multiplicity of the γ -cascades is in the calculations are the same as discussed in Ref. found to decrease with increasing energy of the gating [15], with the only difference for the entry excitation transitions, ruling out the possibility that the enhanced energy, which has been increased by 5 MeV to match population of the SD structures could be due to a spin the new experimental condition. This gives a good effect. On the other hand, if the increased population reproduction of the population of the low spin ground of the SD states is caused by the E1 feeding from state transition at 917 keV in 143Eu, for high-energy excited states of the residual nucleus which has also γ -rays ranging from 3 to 8 MeV. an electric dipole vibration, one would expect a trend As one can see in Fig. 4, the increase in the popu- which follows the low energy tail of the GDR strength lation of the SD yrast, as calculated by the simulation function. In Fig. 4 the ratio of the strength function code (solid line), displays a trend similar to the sim- of the superdeformed GDR (E1GDR = 10.5MeV, ple estimate previously considered (dashed line) and Γ1GDR = 3 MeV with 33% of EWSR strength and in rather good agreement with the experimental data. E2GDR = 17 MeV, Γ2GDR = 6.5 MeV with 66% of In addition, as shown by the solid line in the top panel EWSR) with the spherical GDR (EGDR = 14.5MeV, of Fig. 4, the total average multiplicity of the simulated ΓGDR = 5 MeV, 100% of EWSR), normalized to cascades is also found to decrease with increasing en- the data at 3.4 MeV, is shown by the dashed line. ergy of the gating transitions. Concerning spin effects, This represents the lowest limit expected for the it has been found that an increase of the entry spin dis- situation in which the gating high-energy transition tribution produces a better population of the highest has a superdeformed nature and it is feeding directly spin transitions of the SD yrast, ridge and E2 bump. the superdeformed structure. Indeed, it is found that Therefore, we can rule out spin effects as a possible the experimental data display the same trend as given explanation of our experimental observation, since our by this simple estimate, although with slightly higher gating conditions correspond to a lowering of the spin values. In order to understand such difference, which distribution. On the other hand, since we have a sizable could be related to the complex γ -decay flow from spin decrease (≈ 8 units) at the highest-energy gating the entry distribution of the residual nucleus down to conditions, we have checked via simulation this effects the yrast line, we have performed schematic Monte on the population of SD yrast low spin transitions. It Carlo simulation calculations. The adopted model, has been found that the total relative increase of the originally developed to study the feeding of the SD SD yast population is of the order of 1–2% only. nucleus 152Dy [10], has earlier been used to describe The finding that the feeding intensity of the su- the populations of the various spectral components of perdeformed yrast states is approximately 1.6 larger 206 G. Benzoni et al. / Physics Letters B 540 (2002) 199–206 when gating on high energy γ -rays (with Eγ > References 6 MeV), as compared to the ungated case, is the relevant result of this work. This is consistent with the fact [1] M. Carpenter, R.V.F. Janssens, Nucl. Phys. A 583 (1995) 183c. that when an high-energy E1 γ -rayisinvolvedinthe [2] P.J. Twin, Nucl. Phys. A 583 (1995) 199c. [3] B. Singh, R.B. Firestone, S.Y.F. Chu, Report No. LBL-38004, decay it is more likely connected with a SD rather than 1997. a ND nucleus. However, in order to obtain a full under- [4] G. Hackman et al., Phys. Rev. Lett. 79 (1997) 4100. standing of the feeding mechanism of the SD struc- [5] T.L. Khoo et al., Phys. Rev. Lett. 76 (1996) 1583. tures one needs a comparison with other mass regions [6] A. Lopez-Martens et al., Phys. Lett. B 380 (1996) 18. and to investigate the competition with particle feed- [7] K. Schiffer, B. Herskind, J. Gascon, Z. Phys. A 332 (1989) 17. [8] J. Simpson et al., Phys. Rev. C 62 (2000) 024321. ing. [9] M. Gallardo, M. Diebel, T. Døssing, R.A. Broglia, Nucl. Phys. A 443 (1985) 415. [10] B. Herskind et al., Phys. Rev. Lett. 59 (1987) 2416. Acknowledgements [11] B. Hass et al., Phys. Lett. B 245 (1990) 13. [12] P. Taras et al., Phys. Rev. Lett. 61 (1988) 1348. [13] L.H. Zhu et al., Phys. Rev. C 55 (1997) 1169. We wish to acknowledge the support from INFN, [14] S. Leoni et al., Phys. Rev. Lett. 76 (1996) 3281. Italy, from the Danish Natural Science Research Coun- [15] S. Leoni et al., Phys. Lett. B 409 (1997) 71. cil, from the Polish Scientific Committee (KBN Grant [16] S. Leoni et al., Phys. Lett. B 498 (2001) 137. No. 2 P03B 118 22) and the EU Access to Large Scale [17] A. Atac et al., Phys. Rev. Lett. 70 (1993) 1069. Facilities-Training and Mobility of Research Pro- [18] A. Winther, Nucl. Phys. A 594 (1995) 203. [19] A. Maj et al., Nucl. Phys. A 571 (1994) 185. gram Contract No. ERBFMGECT980110, for INFN- [20] D.C. Radford, Nucl. Instrum. Methods A 361 (1995) 297. Laboratori Nazionali di Legnaro. [21] S.A. Forbes et al., Nucl. Phys. A 584 (1995) 149. [22] O. Andersen et al., Phys. Rev. Lett. 43 (1979) 687. [23] S. Aberg, Nucl. Phys. A 477 (1988) 18. [24] K. Yoshida, M. Matsuo, Nucl. Phys. A 612 (1997) 26. Physics Letters B 540 (2002) 207–212 www.elsevier.com/locate/npe

Evidence of kaon nuclear and Coulomb potential effects on soft K+ production from nuclei

M. Nekipelov a,b,M.Büschera,W.Cassingc, M. Hartmann a, V. Hejny a,V.Klebera, H.R. Koch a,V.Koptevb,Y.Maedaa,R.Maiera,S.Merzliakovd, S. Mikirtychiants b, H. Ohm a,A.Petrusd,D.Prasuhna, F. Rathmann a,Z.Rudya,e,R.Schleicherta, H. Schneider a,K.Sistemicha, H.J. Stein a,H.Ströhera, K.-H. Watzlawik a,C.Wilkinf

a Institut für Kernphysik, Forschungszentrum Jülich, D-52425 Jülich, Germany b High Energy Physics Department, Petersburg Nuclear Physics Institute, 188350 Gatchina, Russia c Institut für Theoretische Physik, Justus Liebig Universität Giessen, D-35392 Giessen, Germany d Laboratory of Nuclear Problems, Joint Institute for Nuclear Research, Dubna, 141980 Dubna, Moscow Region, Russia e M. Smoluchowski Institute of Physics, Jagellonian University, Reymonta 4, PL-30059 Cracow, Poland f Physics Department, UCL, Gower Street, London WC1 6BT, England, UK Received 28 February 2002; received in revised form 17 May 2002; accepted 17 June 2002 Editor: V. Metag

Abstract + The ratio of forward K production on copper, silver and gold targets to that on carbon has been measured at proton beam energies between 1.5 and 2.3 GeV as a function of the kaon momentum pK using the ANKE spectrometer at COSY-Jülich. The strong suppression in the ratios observed for pK < 200–250 MeV/c may be ascribed to a combination of Coulomb and + + nuclear repulsion in the K A system. This opens a new way to investigate the interaction of K -mesons in the nuclear medium. + 0 ≈ Our data are consistent with a K A nuclear potential of VK 20 MeV at low kaon momenta and normal nuclear density. Given 0 the sensitivity of the data to the kaon potential, the current experimental precision might allow one to determine VK to better than 3 MeV.  2002 Elsevier Science B.V. All rights reserved.

PACS: 13.60.Le; 13.75.Jz; 14.40.Aq; 24.40.-h

Keywords: Kaon production; Coulomb suppression

Final state interactions of K+ mesons in nuclei are great importance to learn about either cooperative nu- generally considered to be rather small, due to their clear phenomena or high momentum components in strangeness of S =+1. As a consequence, the produc- the nuclear many-body wave function. This is partic- tion of K+-mesons in proton–nucleus collisions is of ularly the case since the production of kaons, being relatively heavy as compared to pions, requires strong medium effects. E-mail address: [email protected] (M. Nekipelov).

Several groups have made experimental and the- spectrometer, which is currently the only device that oretical studies of total and doubly-differential K+- is able to measure K+ mesons with momenta down to production cross sections over a wide range of pro- ≈ 150 MeV/c. ton beam energies [1–12]. These studies show that Measurements of K+ momentum spectra resulting a two-step reaction mechanism, involving the produc- from proton–nucleus collisions have been performed tionofanintermediate∆ or π, dominates below the with the ANKE spectrometer [21] at the COooler threshold of the elementary pN → K+ΛN reaction SYnchrotron COSY-Jülich. A detailed description of (Tp = 1.58 GeV). A strong target mass dependence of the kaon detection system is given in Ref. [22]. The the production rate may be a good indicator for the criteria for the kaon identification and the proce- dominance of such secondary mechanisms. At higher dure of measurements are briefly described as fol- energies the role of the secondary effects decreases, lows: The COSY proton beam, with an intensity of especially in the high momentum part of the kaon (2–4) × 1010 protons and a cycle time of ∼ 60 s, was spectra, where direct production dominates [13]. accelerated to the desired energy in the range Tp = It is, however, clear that the repulsive Coulomb po- 1.5–2.3 GeV on an orbit below the target. The targets tential in the target nucleus will distort the soft part of were thin strips of C, Cu, Ag or Au with a thickness of the momentum spectrum. Furthermore, since the K+ (40–1500) µg/cm2. Over a period of ∼ 50 s, the beam nuclear potential, though small, is also repulsive [14], was slowly brought up to the target by steerers, keep- with a strength rather similar to that of the Coulomb ing the trigger rate in the detectors nearly constant for a heavy nucleus, this distortion will be reinforced. at (1000–1500) s−1, a level that could be handled by For this reason there have been several publications the data acquisition system with a dead time of less which have stressed the importance of including the than 25%. Ejectiles with horizontal angles in the range effects of Coulomb and nuclear potentials on the prop- ±12◦, vertical angles up to ±7◦, and momenta be- agation of mesons in the nuclear medium [14–16]. tween 150 and 600 MeV/c, were deflected by the Such effects can change the interpretation of the shape ANKE dipole magnet, passed one of 23 plastic scin- of the K+ spectrum as well as of the mass dependence tillation counters and 6 planes of 2.5 mm wire-step of the cross sections; therefore they have to be taken MWPCs. They were then focussed onto one of the into account in the interpretation of the experimental 15 kaon range telescopes, each consisting of three results. plastic scintillator counters (stop, E and veto) and The influence of final state rescattering effects two degraders. The 10 cm wide telescopes, placed at in meson production can be investigated experimen- the focal surface of the ANKE spectrometer, defined tally for high momentum mesons by measuring two- ∼ 10% momentum bites in ejectile momenta. The mo- body reactions, e.g., pA → (A + 1)∗π+ for pions or mentum ranges covered by each telescope were kept ∗ + pA → Λ(A + 1) K for kaons, using high precision constant for the different beam energies by operat- spectrometers. However, just as for β±-decay, much ing ANKE at constant magnetic field strength and stronger Coulomb effects are expected in the very low maintaining the relative target-dipole-detector geom- momentum part of the meson spectrum, but there have etry. The thicknesses of the scintillators and degraders as yet been no direct experimental tests, at least for in the telescopes were chosen so as to stop kaons in kaons. A reliable way of studying this phenomenon is the degrader in front of the veto counters. The kaons by measuring directly ratios of cross sections for dif- subsequently decay with a mean life time of ≈ 12.4ns ferent nuclei, since many of the possible systematic and the products of this decay (pions or muons) are errors cancel out. Measurements of cross section ra- detected by the veto counters with a delay of more tios for mesons of different charges, e.g., π+/π− or than 1.3 ns with respect to the signals from the corre- K+/K− are, as a rule, clouded by the differences in sponding stop counters. The combination of the time- reaction mechanisms. Since most of the measurements of-flight between the start and stop counters, energy were carried out for high pion and kaon momenta, the losses in all the scintillators, delayed particle signals, existing experimental data [17–20] are not very infor- and information from MWPCs resulted in clean kaon mative regarding both Coulomb and kaon potentials. spectra, with a background of less than 10% for all This is changing with the commissioning of the ANKE beam energies of 1.5 GeV and above. The MWPC M. Nekipelov et al. / Physics Letters B 540 (2002) 207–212 209 track reconstruction allowed us also to vary the angu- the target to an accuracy of 2% using stop counters lar and momentum acceptances of the individual tele- 2–5 in four-fold coincidence directly looking at the scopes. target [22], thereby selecting ejectiles, produced in the

The ratio R(A/C)pK of the kaon production cross target by hadronic interactions, which bypassed the sections from heavy (A) to carbon (C) targets for spectrometer dipole. Pion production cross sections a given kaon momentum pK can be calculated from in proton–nucleus reactions have been measured by the observed number of kaons n(K+) in the individual several groups in the forward direction in the 0.73– telescopes as 4.2 GeV energy range [18,19,23]. The combined + analysis of these data showed that, to within 10%, nA(K ) LC the ratios of the pion production cross sections can R(A/C)p = × · (1) K n (K+) L 1/3 C pK A be scaled with the target mass number as A [24], whichisusedinEq.(2). L and L denote the integrated luminosities during C A The absolute values of the doubly-differential cross data taking with a particular target. The luminosity + sections can be obtained from the numbers of kaons ratio could be obtained from the number n(π ) of n(K+) identified by each telescope, after correction 500 MeV/c pions, measured during pion calibration for luminosities, detection efficiencies in the scintilla- runs for every energy and for each target tors and MWPCs, kaon decay between the target and 1/3 + the telescopes, and angle-momentum acceptances. For LC = A × nC(π ) · + (2) the cross section ratios, used in our present analysis, L C n (π ) = A A pπ 500 MeV/c absolute values are not needed and many uncertain- All numbers of detected pions and kaons in Eq. (1) ties of the efficiency corrections cancel out. The pre- and Eq. (2), nC and nA, were individually normalized sentation of normalized cross sections is deferred until to the relative luminosities during the corresponding a later publication. runs. This relative normalization was obtained by The gold/carbon ratio is shown in Fig. 1(a) for monitoring the interaction of the proton beam with proton beam energies of 2.3, 1.75, and 1.5 GeV. This

+ + Fig. 1. Ratios of K (π ) production cross sections for Au and C measured at different beam energies (left figure) and kinematic conditions (right figure) as functions of the laboratory meson momentum. (a) All the symbols correspond to kaons measured in the full ANKE acceptance ◦ (θ<12 ). (b) The open circles are as in the left figure, whereas the closed circles correspond to kaons measured in the restricted angular ◦ + interval (θ<3 ). The π -production cross-section ratios of Au to C at Tp = 2.3 GeV are designated by crosses. 210 M. Nekipelov et al. / Physics Letters B 540 (2002) 207–212 ratio has a broadly similar shape at all three energies, atic uncertainty of about 1% in the absolute value of with clear maxima for pmax ≈ 245 MeV/c coinciding momentum as well as the using of different functional within 2 MeV/c. For higher kaon momenta the ratios forms to fit the points near the maxima. decrease monotonically with pK and in this region It is clear from Fig. 1 that the suppression of + the K production in gold is relatively stronger at R(Au/C) at low pK is largely independent of beam 2.3 GeV than at lower energies, reflecting changes in energy and of the angular acceptance of the spectrom- the production mechanism with bombarding energy. eter, suggesting that the phenomenon is principally For low kaon momenta one sees a dramatic fall in the due to the interaction of the K+ with the residual nu- ratio R(Au/C). To ensure that this phenomenon is not cleus. On the other hand, Fig. 2 shows that the position an artefact of the ANKE detection system, the 2.3 GeV of the maximum in R(A/C) increases with A. The sit- run was repeated with a reduced dipole magnetic field, uation has a parallel in the well-known suppression resulting in a change in the values of the momenta of β+ emission in heavy nuclei at low positron mo- that are focussed onto individual range telescopes. The menta arising from the repulsive Coulomb field. Thus + low pK suppression remained unchanged. Identical a K produced at rest at some radius R in the nu- + ratios were also obtained when the polar K emission cleus would, in the absence of√ all other interactions, angles in data analysis were restricted to lie below acquire a momentum of pmin = 2mK VC(R).Taking ◦ ϑK = 3 with the help of the MWPC information (see R to be the nuclear edge, this purely classical argu- Fig. 1(b)). ment leads to a minimum K+ momentum for Au of Ratios of kaon-production cross sections for cop- pmin ≈ 130 MeV/c. per, silver and gold targets measured at 2.3 GeV are It is, moreover, known from K+ elastic scattering presented in Fig. 2. All data exhibit similar shapes, ris- experiments at higher energies [25] that the K+A ing steadily with decreasing kaon momenta, passing potential is mildly repulsive, and this is in accord with a maximum and falling steeply at low momenta. The one-body optical potentials based upon low-energy position of the maximum varies with the nucleus, a fit K+N scattering parameters [26]. At normal nuclear −3 + to the data results in pmax(A/C) = 245 ± 5, 232 ± 6, density, ρ0 ≈ 0.16 fm , the predicted repulsive K A ± 0 ≈ and 211 6MeV/c for Au, Ag, and Cu, respectively. potential of strength VK 20–25 MeV [14] would The error bars include contributions from a system- shift pmin to higher values. In order to see whether the observed low momentum suppression is compatible with such a combination of Coulomb and nuclear repulsion, we performed calculations in the framework of the coupled channel transport model [13,27]. In this approach the different mechanisms for the kaon production and the influence of average Coulomb and nuclear potentials, as well as hadron rescattering effects, which can cause a sudden change of the kaon trajectory when kaon comes close to a nucleon, can be taken into account using realistic density distributions. Results of the calculations for the R(Au/C) ratio are shown in Fig. 3. Without including the Coulomb and kaon potentials (dashed line in Fig. 3(a)) the ratio exhibits a smooth momentum dependence with a steady increase towards low momenta resulting from the stronger K+ rescattering processes for the Au target. A behaviour of this type was observed in π+ production, which was also measured + Fig. 2. Ratios of the K production cross sections on Cu, Ag, and in a short test experiment (see Fig. 1(b)). In this case Au measured at Tp = 2.3 GeV as a function of the laboratory kaon the influence of the Coulomb potential is expected to momentum. show up below pπ ≈ 80 MeV/c, which was below our M. Nekipelov et al. / Physics Letters B 540 (2002) 207–212 211

A much larger change is observed in the calculations when a repulsive kaon nuclear potential is also considered, giving a 40 and 80 MeV/c shift with a potential strength of 20 and 40 MeV, respectively. When 0 = a kaon potential of VK 20 MeV is used in the calculations, a reasonable agreement with the experiment is achieved, with a maximum close to the experimental value of 245 ± 5MeV/c (solid line in Fig. 3(b)). To take into account the absorption of the incident proton, a baryon potential has also been included. The latter does not change the position of the maximum, but makes the agreement with the experimental data better. A precision of 5 MeV/c in the position of the maximum, taken together with the sensitivity of the data to kaon potential shown above, might allow one 0 to determine the strength of VK to better than 3 MeV. In summary, we have observed a strong suppression of the ratio of K+ production by protons on heavy nuclei to that on carbon at low kaon momenta. The independence of this effect from beam energy and the variation of the structure with A provides clear evidence for the influence of the K+ Coulomb and nuclear interaction potentials. The sensitivity found within our model suggests that a careful study of this region will provide a new way to investigate the K+A optical potential at low momenta. For this to be successful, more extensive transport calculations or other phenomenological descriptions have to be developed. Our preliminary analysis suggests that the K+ nuclear potential at normal nuclear matter + density is of the order of 20 MeV, which is in line Fig. 3. Ratios of K production cross sections for Au/C at + Tp = 2.3 GeV as a function of the kaon momentum. (a) The with K elastic scattering experiments [25] and low- + dash-dotted line is obtained from transport calculations including energy K N scattering parameters [26]. Data on only the Coulomb potential, the dotted line corresponds to calcula- K+ production from heavy-ion reactions at GSI also tions with the addition of a kaon potential of 20 MeV at ρ0, whereas point towards K+ nuclear potentials of about the the dashed-double-dotted line shows the result where a kaon potential of 40 MeV has been used. The broken line corresponds to sim- same strength [28,29]. We hope that the accuracy of ulations without Coulomb and nuclear kaon potentials. In all cases our data will stimulate further development of model + considered here, K rescattering in the nucleus has been taken into calculations to provide a better description of the account. (b) The open circles are the experimental data. The solid momentum spectra. If this is done we could expect the line shows the result of transport calculations starting from the dot- strength of the kaon potential to be extracted with an ted line in the top figure with a baryon potential added. accuracy better than 3 MeV. acceptance limit. For kaons the pure Coulomb interaction leads to a distortion of the momentum spec- Acknowledgements trum and provides a maximum at pK ≈ 200 MeV/c (dashed-dotted line in Fig. 3(a)). It has been checked We wish to acknowledge the assistance we received that a 10% change in the charge radius would move from the COSY staff when performing these measure- the maximum by less than ±2MeV/c. ments at ANKE. Financial support from the follow- 212 M. Nekipelov et al. / Physics Letters B 540 (2002) 207–212 ing funding agencies was invaluable for our work: [9] Yu.T. Kiselev et al., J. Phys. G 25 (1999) 381. Georgia (Department of Science and Technology), [10] E.Ya. Paryev, Eur. Phys. J. A 5 (1999) 307; Germany (BMBF: grants WTZ-POL-001-99, WTZ- E.Ya. Paryev, Eur. Phys. J. A 9 (2000) 521. [11] V. Koptev et al., Phys. Rev. Lett. 87 (2001) 022310. RUS-649-96, WTZ-RUS-666-97, WTZ-RUS-685-99, [12] M. Büscher et al., Phys. Rev. C 65 (2002) 014603. WTZ-POL-007-99; DFG: 436 RUS 113/337, 436 RUS [13] Z. Rudy et al., nucl-th/0201069. 113/444, 436 RUS 113/561, State of North-Rhine [14] A. Sibirtsev, W. Cassing, Nucl. Phys. A 641 (1998) 476. Westfalia), Poland (Polish State Committee for Scien- [15] A. Ayala, J. Kapusta, Phys. Rev. C 56 (1997) 407. tific Research: 2 P03B 101 19), Russia (Russian Min- [16] S. Teis et al., Z. Phys. A 359 (1997) 297. [17] D.R.F. Cochran et al., Phys. Rev. D 6 (1972) 3085. istry of Science, Russian Academy of Science: 99- [18] J.F. Crawford et al., Phys. Rev. C 22 (1980) 1184. 02-04034, 99-02-18179a) and European Community [19] V.V. Abaev et al., J. Phys. G 14 (1988) 903. (INTAS-98-500). [20] I. Chemakin et al., Phys. Rev. C 65 (2002) 024904. [21] S. Barsov et al., Nucl. Instrum. Methods A 462 (3) (2001) 364. [22] M. Büscher et al., Nucl. Instrum. Methods A 481 (2002) 378. References [23] J. Papp et al., Phys. Rev. Lett. 34 (1975) 601. [24] S. Barsov et al., Acta Phys. Pol. B 31 (2000) 2159. [25] D. Marlow et al., Phys. Rev. C 25 (1982) 2619. [1] V. Koptev et al., Sov. Phys. JETP 67 (1988) 2177. [26] B.R. Martin, Nucl. Phys. B 94 (1975) 413. [2] S. Schnetzer et al., Phys. Rev. C 40 (1989) 640. [27] W. Cassing, E. Bratkovskaya, Phys. Rep. 308 (1999) 65. [3] W. Cassing et al., Phys. Lett. 238 (1990) 25. [28] F. Laue et al., Eur. Phys. J. A 9 (2000) 397; [4] W. Cassing et al., Z. Phys. A 349 (1994) 77. F. Laue et al., Phys. Rev. Lett. 82 (1999) 1640. [5] A.A. Sibirtsev, M. Büscher, Z. Phys. A 347 (1994) 191. [29] Y. Skin et al., Phys. Rev. Lett. 81 (1998) 1576. [6] M. Büscher et al., Z. Phys. A 335 (1996) 93. [7] M. Debowski et al., Z. Phys. A 356 (1996) 313. [8] A. Badalà et al., Phys. Rev. Lett. 80 (1998) 4863. Physics Letters B 540 (2002) 213–219 www.elsevier.com/locate/npe

Probing the isospin dependent in-medium nucleon–nucleon cross section by nucleon emissions

Jian-Ye Liu a,b, Wen-Jun Guo b, Yong-Zhong Xing b,c,WeiZuoa,b, Xi-Guo Lee a,b, Zeng-Hua Li b

a Center of Theoretical Nuclear Physics, National Laboratory of Heavy Ion Accelerator, Lanzhou 730000, PR China b Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000, PR China c Department of Physics, Tianshui Normal College, Gansu Tianshui, 741000, PR China Received 19 March 2002; received in revised form 20 May 2002; accepted 20 June 2002 Editor: W. Haxton

Abstract The effects of the symmetry potential and the isospin dependent in-medium nucleon–nucleon (NN) cross section on the number of proton(neutron) emissions Np(Nn) are studied respectively within an isospin-dependent quantum molecular dynamics (IQMD) model. The isospin dependent in-medium NN cross section is found to have a strong inﬂuence on Np(Nn) but Np(Nn) is not sensitive to the symmetry potential for the neutron-deﬁcient colliding system at relatively high energies. We propose to make use of the Np(Nn) as a probe to extract information on the isospin dependent in-medium NN cross section.  2002 Elsevier Science B.V. All rights reserved.

PACS: 25.70.pg; 02.70.Ns; 24.10.Lx

Keywords: Isospin effect; Nucleon emission; Nucleon–nucleon cross section; Symmetry potential

1. Introduction 16] and theoretically, e.g., [17–25] over the last few years. Bao-An Li et al. [19,25] have found that in- With the rapid advance in radioactive beam physics formation on the symmetry potential can be extracted a better understanding of the isospin degree of free- by studying the neutron to proton ratio of preeqilib- dom in nuclear collision dynamics may provide us rium emissions in the relative low energy heavy ion with useful hints on how to extract reliable informa- collisions (HIC). In contrast, in this Letter our stud- tion about the in-medium NN cross section and the ies within IQMD found that the numbers of pro- symmetry potential [1–5]. To obtain this information ton(neutron) emissions Np(Nn) depend sensitively on several interesting isospin effects in heavy ion col- the isospin dependent in-medium NN cross section lisions have been explored both experimentally [6– and weakly on the symmetry potential in the relatively high beam energy region for the neutron-deﬁcient systems. In this case, Np(Nn) is a probe for extracting in- E-mail address: [email protected] (J.-Y. Liu). formation about the isospin dependence of in-medium

NN cross section in the relative high beam energy re- U Pauli is the Pauli potential, gion for the neutron-deficient systems. 3 2 h¯ (ri −rj ) U Pauli = V exp − p 2 p0q0 2q 0 2 2. Theoretical model (pi −pj ) − δp p , 2p2 i j The dynamics of intermediate energy heavy ion 0 collisions described by QMD contains two ingredi- = 1, for neutron–neutron or proton–proton, δpi pj ents: density dependent mean field and in-medium 0, for neutron–neutron. (5) NN cross section. To describe isospin effects ap- sym propriately, QMD should be modified properly: the U is the symmetry potential. In the present calcu- density dependent mean field should contain the cor- lation, two different density dependent symmetry po- sym = rect isospin terms including symmetry potential and tentials [2,3,29] are used, i.e., U1 cF1(u)δτz and sym = [ + 1 2] = Coulomb potential, the in-medium NN cross sec- U2 cF2(u) δτz 2 δ ,whereτz 1 for neutron 2 tion should be different for neutron–neutron (proton– and τz =−1 for proton, F1(u) = u and F2(u) = u , − proton) and neutron–proton collisions in which Pauli u ≡ ρ/ρ . δ is the relative neutron excess δ = ρn ρp = 0 ρn+ρp blocking should be counted by distinguishing neutrons ρ −ρ n p .Herec is the strength of symmetry potential, from protons. In addition, the initial condition of the ρ taking the value of 32 or 0 MeV (the c = 0.0 case is ground state of two colliding nuclei should also con- sym denoted by U ). ρ, ρ , ρ and ρ are total density tain isospin information. The main physics ingredients 0 0 n p and its normal value, neutron density and proton den- and their numerical realization in the IQMD model can sity, respectively. It is worth mentioning that the re- be found in Refs. [3,28,30,31,33].In the IQMD model, cent studies on collective flow in HIC at intermediate the density distributions of colliding nuclei were from energies have indicated a reduction of in-medium NN the calculations of the Skyrme–Hatree–Fock with pa- cross sections. An empirical expression of the density rameter set SKM* [32] and the initial code of IQMD dependent in-medium NN cross section [26] is given was used to determine the ground state properties of by the colliding nuclei, such as the binding energies and rms radii which are the same as the experimental data. med = + ρ free In the presence calculations the interaction potential in σNN 1 α σNN, (6) ρ0 the IQMD were determined as follows: where the parameter α ≈−0.2 has been found to free U(ρ)= U Sky + V Coul + U sym + V Yuk reproduce the flow data. σNN is the experimental NN cross section [27]. The parameters of the interaction MDI Pauli + U + U , (1) potentials are given in Table 1. The free neutron–proton cross section σ free is about U Sky is the density-dependent Skyrme potential, NN a factor of 3 times larger than the free proton–proton ρ ρ γ or free neutron–neutron cross section below about U Sky = α + β . (2) ρ ρ 400 MeV (in the Lab). It is worth to mentioning that 0 0 the relationship between the neutron–proton cross sec- Yuk Vc is Coulomb potential. U is the Yukawa poten- tion and neutron–neutron (proton–proton) cross sec- tial, tion depends also on the modification of the nuclear density distributions during reactions. We construct | −| | −| Yuk r1 r2 r1 r2 clusters by means of the isospin-dependent modified U = t3 exp . (3) m m coalescence model [33], in which particles with relative momentum smaller than p0 = 300 MeV/c and U MDI is the momentum dependent interaction, relative distance smaller than R0 = 3.5fmareused. We make use of the restructured aggregation model MDI = 2 − 2 + ρ U t4 ln t5(p1 p2) 1 . (4) [34] to avoid the nonphysical clusters after construct- ρ0 J.-Y. Liu et al. / Physics Letters B 540 (2002) 213–219 215

Table 1 The parameters of the interaction potentials −2 α (MeV) β (MeV) γt3 (MeV) m (fm) t4 (MeV) t5 (MeV ) VP (MeV) p0 (MeV/c) q0 (fm) − −390.1 320.3 1.14 7.5 0.8 1.57 5 × 10 4 30 400 5.64

ing the clusters, until there are not any nonphysical ate mass fragment multiplicity Nimf and the charged clusters to be produced. particle multiplicity Nc. The solid (open) circles represent the experimental data for the reaction 58Ni + 58Ni (58Fe + 58Fe) at E = 75 MeV/u and the solid line 3. Results and discussions (dot line) denotes the IQMD results for 58Ni + 58Ni (58Fe + 58Fe). It is clear that the present IQMD pre- 3.1. Checking the IQMD model dictions are in satisfactory agreement with general features of the experimental data which means that IQMD with the above parameters is a reasonable transfer the- In order to check the IQMD code with the above parameters, the multiplicity of the intermediate mass oretical model for simulating the dynamical process in 58 58 intermediate energy heavy ion collisions. fragments Nimf for the reactions Fe + Fe and 58Ni + 58Ni at the beam energy E = 75 MeV/uhas been calculated by using the IQMD code with the 3.2. A probe of isospin dependent in-medium NN above parameters. The multiplicity of the intermedi- cross section by nucleon emissions ate mass fragments (IMFs) is deﬁned as the number of fragments with charge numbers from 3 to 18. The The isospin effects of the in-medium NN cross sec- calculated results are compared with the experimental tion on the physical quantities arise from the differ- data [35] on the same scale in Fig. 1 which gives the ence between isospin dependent in-medium NN cross iso correlation between the mean value of the intermedi- section denoted by σ in which σnp σnn = σpp and isospin independent NN cross section denoted noiso by σ in which σnp = σnn = σpp .Hereσnp , σnn and σpp are the neutron–proton, neutron–neutron and proton–proton cross sections, respectively. Here Np(Nn) includes all of protons (neutrons) emitted during the nuclear reaction. To identify free nucleons, a phase-space coalescence method has been used at 200 fm/c (when Np(Nn) becomes nearly a constant) after the initial contact of the two nuclei. A nucleon is considered as free if it is not correlated with other nucleon within a spatial distance of r = 3fm and a momentum distance of p = 300 MeV/c.Oth- erwise, it is bound in a cluster. In addition to the nucleon emissions we also calculated all of the fragments during the same reaction. Fig. 2 shows the time evolutions of the Nn (top windows) and Np (bottom windows) for the colliding systems 76Kr + 40Ca, 74Kr + 74Se, 76Kr + 76Kr and 74Se + 74Se with neutron– Fig. 1. The correlation between the mean intermediate mass frag- proton ratios 1.04, 1.06, 1.11 and 1.18 respectively at ment multiplicity Nimf and the charged particle multiplicity Nc. = = Filled (unﬁlled) circles represent the experimental data [35] for the impact parameter b 4.0 fm and beam energy E reactions 58Ni + 58Ni (58Fe + 58Fe) at E = 75 MeV/uandthe 150 MeV/nucleon. To identify the isospin effects of solid line (dot line) indicates the IQMD results for 58Ni + 58Ni the two-body collision on the Nn(Np) we compare the 58 + 58 ( Fe Fe). The charge number of Nimf is taken from 3 to 18. cases when the isospin dependence of in-medium NN 216 J.-Y. Liu et al. / Physics Letters B 540 (2002) 213–219

76 40 74 74 Fig. 2. The time evolutions of the Nn (top windows) and Np (bottom windows) for the colliding systems Kr + Ca, Kr + Se, 76Kr + 76Kr and 74Se + 74Se with neutron–proton ratios 1.07, 1.11, 1.11 and 1.18 respectively at impact parameter b = 4.0 fm and beam energy E = 150 MeV/nucleon for five cases (see text). cross section is either turned on (σ iso) or off (σ noiso). the neutron-deficient systems. This situation is con- Turning on the isospin-dependent NN cross section trary to the neutron–proton ratio of the preeqilibrium is seen to enhance the momentum dissipation as ex- emissions in relatively low energy regions, where the pected, leading to a larger number of proton(neutron) neutron–proton ratio of the preeqilibrium emissions emissions. The isospin effects of the one-body dis- depends sensitively on the symmetry potential and sipation and two-body collision on the Nn(Np) are weakly on the isospin dependence of the in-medium sym + iso identified by using five cases: U1 σ (solid NN cross section [19]. With decreasing beam energy sym + iso sym + iso the role of two-body collision is reduced. In partic- lines), U0 σ (dot lines), U2 σ (dashed sym + noiso sym + noiso ular the symmetry potential is repulsive for the neu- lines), U1 σ (dashed-dot lines), U2 σ (dashed-dot-dot lines). From Fig. 2 it is clear to see trons and attractive for the protons which tends to that the variations among the values of Np(Nn) with make more neutrons than protons unbound, i.e., the the same σ iso or the same σ noiso but different sym- role of symmetry potential on the nucleon emissions metry potentials are smaller but the gaps between at low energies results in the sizeable differences be- iso noiso the Np(Nn) with σ and Np(Nn) with σ are tween neutron emissions and proton emissions, but the larger, i.e., the values of Np(Nn) depends sensitively two-body collision produces about the same probabil- on the isospin dependence of in-medium NN cross ity for gaining enough energy to become unbound for section and weakly on the symmetry potentials for the neutrons and protons. As a result, the isospin de- J.-Y. Liu et al. / Physics Letters B 540 (2002) 213–219 217

Fig. 3. The impact parameter-averaged values of Nn (top window) and Np (bottom window) as a function of the beam energies for the systems 76Kr + 40Ca and 76Kr + 76Kr in the five cases as the same as Fig. 2. pendence of in-medium NN cross section has a small gion from about 50 to 400 MeV/nucleon. But as the affect on the neutron–proton ratio of the preeqilibrium beam energy is decreased to about 50 MeV/nucleon emissions but the role of the symmetry potential on the Np(Nn) depends on both the isospin dependence it is strengthened in regions of relatively low beam of in-medium NN cross section and the symmetry energy. But with increasing beam energy the role of potential. It is worth mentioning that the minimum two-body collisions increases while the role of the energies for remaining above property are small differ- mean field is reduced, i.e., in the relatively high en- ent for the systems with the variation of the neutron– ergy region the two-body collision is dominant. Espe- proton ratios and mass asymmetries of the colliding cially in this Letter we calculated the neutron emis- systems. sions and the proton emissions not the ratio of them. The above results show strongly that Np(Nn) Finally we can get the results as in Fig. 2. In Fig. 3 is during the neutron-deficient nuclear reaction can be shown the impact parameter-averaged asymptotic val- used as a sensitive probe for extracting information on ues of Nn (top window) and Np (bottom window) as a the isospin dependent in-medium NN cross section. function of beam energy for the systems 76Kr + 40Ca Here the above behaviour has been shown for the and 76Kr + 76Kr in the five cases considered as the neutron emissions and proton emissions separately, of same as Fig. 2. From Fig. 3 it is very clear to see course, there is also the same behaviour as a probe for the same conclusion as Fig. 2 in the beam energy re- the sum of them. 218 J.-Y. Liu et al. / Physics Letters B 540 (2002) 213–219

It is worth mentioning that in general, the correc- Acknowledgements tion effect of the sequential decays on the dynamical process of HIC and the number of the nucleon emis- This work was supported in part by the Major sions should be considered. For example, M.B. Tsang State Research Development Project in China under et al. studies show that the apparent temperatures mea- Contract No. G2000077400, 100 person Project of the sured with double ratios of fragment isotope yields Chinese Academy of Sciences, the National Natural display fluctuations that can be attributed to the se- Foundation of China under Grants No. l0175080, quential decay of heavier particle unstable nuclei [36]. No. 10004012 and The CAS Knowledge Innovation However the number of nucleon emissions in our cal- Project No. KJCX2-SW-N02. culations is the final total number of nucleon emissions after colliding system has reached at equilibrium which includes the original nucleon emissions and se- References quential decays leading above conclusion. Namely the final total number of nucleon emissions depends sen- [1] M.S. Hussein, R.A. Rego, C.A. Bertulani, Phys. Rep. 201 sitively on the isospin effect of in-medium NN cross (1993) 279. [2] B.A. Li, C.M. Ko, W. Bauer, Int. J. Mod. Phys. E 7 (1998) 147. section and insensitively on the symmetry potential [3] B.A. Li, W. Udo Schröder, Isospin Physics in Heavy-Ion as a probe for extracting information on the isospin Collisions at Intermediate Energies, Nova Science, New York, dependent in-medium NN cross section in HIC at 2001. the relatively high beam energies for neutron-deficient [4] H.A. Bethe, Rev. Mod. Phys. 62 (1990) 801. colliding systems. Even though the dynamical mech- [5] C.J. Pethick, D.G. Ravenhall, Ann. Rev. Nucl. Part. Sci. 45 (1995) 429. anisms for the original nucleon emissions and the se- [6] R. Wada et al., Phys. Rev. Lett. 58 (1987) 1829. quential decays are different which produces the cor- [7] S.J. Yennello et al., Phys. Lett. B 321 (1994) 14; rection effect on the total number of nucleon emissions S.J. Yennello et al., Nucl. Phys. A 681 (2001) 317c, and but this difference and the correction effect of sequen- references therein. tial decays on the total number of nucleon emissions [8] R. Pak et al., Phys. Rev. Lett. 78 (1997) 1022; R. Pak et al., Phys. Rev. Lett. 78 (1997) 1026. do not influence our final conclusion. [9] G.D. Westfall, Nucl. Phys. A 630 (1998) 27c; G.D. Westfall, Nucl. Phys. A 681 (2001) 343c. [10] G.J. Kunde et al., Phys. Rev. Lett. 77 (1996) 2897. [11] M.L. Miller et al., Phys. Rev. Lett. 82 (1999) 1399. [12] H. Xu et al., Phys. Rev. Lett. 85 (2000) 716; M.B. Tsang et al., Phys. Rev. Lett. 86 (2001) 5023. 4. Summary and conclusions [13] W. Udo Schröder et al., Nucl. Phys. A 681 (2001) 418c, and references therein. [14] L.G. Sobotka et al., Phys. Rev. C 55 (1994) R1272; L.G. Sobotka et al., Phys. Rev. 62 (2000) 031603. In summary, within the IQMD we studied the [15] F. Rami et al., Phys. Rev. Lett. 84 (2000) 1120. [16] W.P. Tan et al., MSUCL-1198, Phys. Rev. C (2001), in press. isospin effects of one-body dissipation and two-body [17] M. Farine, T. Sami, B. Remaud, F. Sebille, Z. Phys. 339 (1991) collision on the number of protons (neutrons) emitted 363. during the nuclear reaction. The calculated results [18] H. Müller, B.D. Serot, Phys. Rev. C 52 (1995) 2072. show strongly that the isospin-dependent in-medium [19] B.A. Li et al., Phys. Rev. Lett. 76 (1996) 4492; NN cross section has a much stronger influence on B.A. Li et al., Phys. Rev. Lett. 78 (1997) 1644; B.A. Li et al., Phys. Rev. Lett. 85 (2000) 4221. Np(Nn) but the effects of the symmetry potential on [20] G. Kortmeyer, W. Bauer, G.J. Kunde, Phys. Rev. C 55 (1997) them are smaller for the neutron-deficient systems in 2730. the relatively high energy region. Studies of Np(Nn) [21] M. Colonna et al., Phys. Lett. B 428 (1998) 1; during the nuclear reactions are proposed to extract V. Baran et al., Nucl. Phys. A 632 (1998) 287; information on the isospin-dependent in-medium NN M. Di Toro et al., Nucl. Phys. A 681 (2001) 426c, and references therein. cross section in HIC based on systematic comparisons [22] J. Pan, S. Das Gupta, Phys. Rev. C 57 (1998) 1839. between the theoretical simulations and experimental [23] Ph. Chomaz, F. Gulminelli, Phys. Lett. B 447 (1999) 221. studies. [24] A. Hombach et al., Eur. Phys. J. A 5 (1999) 77. J.-Y. Liu et al. / Physics Letters B 540 (2002) 213–219 219

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Fading out of J/ψ color transparency in high energy heavy ion peripheral collisions

L. Frankfurt a,M.Strikmanb,M.Zhalovc

a School of Physics and Astronomy, Raymond and Beverly Sackler, Faculty of Exact Science, Tel Aviv University, Ramat Aviv 69978, Tel Aviv, Israel b Pennsylvania State University, University Park, PA 16802, USA c Petersburg Nuclear Physics Institute, Gatchina 188350, Russia Received 26 November 2001; accepted 19 June 2002 Editor: W. Haxton

Abstract We provide predictions for the J/ψ coherent production in the peripheral heavy ion collisions at LHC and RHIC using the leading twist model of nuclear shadowing based on the QCD factorization theorem for diffraction and the HERA hard diffraction data. We demonstrate that for LHC kinematics this model leads to a bump-shape distribution in rapidity which is suppressed overall as compared to the expectations of the color transparency regime by a factor ∼ 6. This is a signiﬁcantly larger suppression than that expected within the impact parameter eikonal model. Thus we show that the interaction of spatially small wave package for which the total cross section of interaction with nucleons is small is still strongly shadowed by nuclear medium in high energy processes.  2002 Elsevier Science B.V. All rights reserved.

1. Introduction amplitude is proportional to number of nucleons times the nuclear form factor. Possibility to approximate Interaction of small size color singlet objects with projectile heavy quarkonium as colorless dipole of hadrons is one of the most actively studied issues heavy quarks can be formally derived from QCD →∞ in high-energy QCD. In perturbative QCD (similar within the limit when mass of heavy quark mQ = 2 to QED) the total cross section of the interaction of but xBj 4mQ/ν is fixed and not extremely small [2]. such systems with hadrons is proportional to the area In this kinematics the size of heavy quarkonium is occupied by color within projectile hadron [1] leading sufficiently small to justify applicability of PQCD. to the expectation of a color transparency phenomenon Recently the color transparency (CT) phenomenon for various hard processes with nuclei. In the case was observed at FNAL by E791 experiment [3] which of incoherent processes cross sections are expected studied the coherent process of dissociation of a to be proportional to the number of nucleons in the 500 GeV pion into two jets off the nuclei. The nuclei while in the case of coherent processes the measurement has confirmed a number of predictions of [4] including the A-dependence, and the transverse and longitudinal momentum distributions of the jets. E-mail address: [email protected] (M. Strikman). Previously the color transparency type behaviour of

0370-2693/02/$ – see front matter  2002 Elsevier Science B.V. All rights reserved. PII:S0370-2693(02)02154-8 L. Frankfurt et al. / Physics Letters B 540 (2002) 220–226 221 the cross section was observed also in the coherent Currently the theory of photoinduced processes in J/ψ photoproduction at Eγ =120 GeV [5]. AA collisions is well developed, for the recent re- A natural question is whether the color trans- view see [9]. Hence we can combine it with our pre- parency will hold for arbitrary high energies? Two vious studies of the coherent photo(electro)production phenomena are expected to work against CT at high of vector mesons to make predictions for production of energies leading to onset of a new regime which we J/ψ in the process (1). Typical transverse momenta refer to as the color opacity regime. One is the lead- which are exchanged between two nuclei in the pe- ing twist gluon shadowing. Indeed the QCD factor- ripheral collisions which leave nuclei intact are much ization for hard exclusive coherent processes with smaller than the typical transverse momenta in the co- ∗ + → + nuclei like γL A Vector meson A implies herent photoproduction of vector mesons. As a result that the cross sections are proportional to the square for the cross section integrated over the momentum of 2 of the gluon parton density GA(x, Q ) at small x the nucleus which emits the quasireal photons we can which is screened in nuclei as compared to the nu- use the standard Weizsacker–Williams approximation. 2 2 cleon: GA(x, Q )/AGN (x, Q )<1. This obviously Hence the cross section of the vector meson pro- should lead to a gradual disappearance of color trans- duction integrated over the transverse momenta of the parency [4,6]. Another mechanism for violation of CT nucleus which emitted a photon can be written in the at high energies is the increase of the small dipole- convoluted form: ∝ 2 nucleon cross section with energy GN (x, Q ).For dσ(AA→ J/ψAA) sufficiently large energies this cross section becomes dk comparable to the meson–nucleon cross sections and 2 n(k, b) hence one may expect a significant suppression of = 2 d bTAA(b) σγA→J/ψA(k). (2) the hard exclusive diffractive processes like DIS dif- k fractive production of vector meson and photoproduc- Here k = γk3 is the photon momentum in the colliding tion of heavy quarkonium states as compared to the frame (k3—momentum of photon in the rest system of CT scenario. However it seems that this phenomenon emitting nucleus and γ —Lorentz factor). is beyond the kinematics achievable for the photo- For the quantity n(k, b) presenting the flux of production of J/ψ mesons at RHIC (x ≈ 2 × 10−2, photons with momentum k in the collider frame we Q2 ≈ 10 GeV2) and probably even at LHC. used the simplest approximated form [9] It was suggested in [2,7] to look for color opac- Z2α 1 1 ity phenomenon using J/ψ (photo)electroproduction. n(k, b) = X2 K2(X) + K2(X) , (3) 2 2 1 0 This however requires energies much larger than those π b γ available at the fixed target facilities and would re- where K0(X) and K1(X) are modified Bessel func- = bk quire use of electron–nucleus colliders. At the same tions with argument X γ and b is the impact pa- time estimates of the counting rates performed within rameter distance between centres of colliding nuclei. the framework of the FELIX study [8] have demon- The factor TAA(b) accounts for inelastic interactions strated that the effective photon luminosities generated of the nuclei at impact parameters b 2RA. It can be in peripheral heavy ion collisions at LHC would lead approximately calculated as to significant rates of coherent photoproduction of vec- 2 tor mesons including Υ in reaction TAA(b) = d b1 TA(b1)TA(b − b1), (4) A + A → A + A + V. (1) ∞ where TA(b) = −∞ dzρA(z, b) is the usual profile As a result it would be possible to study at LHC pho- function of the nucleus. In our calculations we use toproduction of vector mesons in Pb–Pb and Ca–Ca the nuclear matter density ρA(z, b) obtained from collisions at energies much higher than the range the mean field Hartree–Fock–Skyrme (HFS) model, Wγp 17.3 GeV covered at the fixed target experi- which describes many global properties of nuclei ment at FNAL [5]. Note that even current experiments as well as the intermediate and high energy elas- at RHIC (Wγp 25 GeV) should also exceed this tic proton–nucleus scattering and nucleus electromag- limit. netic form factors. This indicates that the HFS model 222 L. Frankfurt et al. / Physics Letters B 540 (2002) 220–226 provides a good description of both proton and neutron distribution in nuclei and takes into account a small difference between the matter distribution and the charge distribution. The main subject of our interest in this Letter is estimating the cross section of the process γA → J/ψA dσ(γA → VA)(s,t)˜ σ → (k) = dt , (5) γA J/ψA dt where Fig. 1. Correspondence between diagrams for hard gluon induced s˜ = 4EN k = 4γkmN (6) diffraction off nucleon and shadowing for the vector meson produc- is invariant energy for γ − N scattering (EN = γmN tion. is the energy per nucleon in the c.m. of the nucleus– =−| V |2 nucleus collisions), t pt is square of the vector bov theory of inelastic shadowing [11] and the QCD meson transverse momentum. factorization theorem for the hard diffraction [12]. An important discovery of HERA is that hard diffraction is indeed dominated by the leading twist contribution 2. Coherent photoproduction of J/ψ off nuclei and gluons play a very important role in the diffraction (this is loosely referred to as gluon dominance of the Let us discuss the photoproduction amplitude γ + pomeron). Analysis of the DESY diffractive data indi- A → J/ψ + A in more details. We are interested here cates that in the gluon induced processes probability of in the Wγp range which can be probed at RHIC and the diffraction is much larger than in the quark induced LHC. In this situation interaction of cc¯ whichinthe processes [7]. The recent H1 data on diffractive di- final state forms J/ψ is still rather far from the black jet production [13] provide an additional confirmation body limit in which cross section can be calculated in of this observation. Large probability of diffraction in the model independent way [10]. Several mechanism the gluon induced hard processes could be understood of coherent interaction with several nucleons were in the s-channel language as formation of color octet suggested for this process. We focus here on the dipoles of rather large size which can elastically scat- the leading twist mechanism of shadowing.There ter with a rather large cross section. The strength of exist qualitative difference between the mechanism of this interaction can be quantified using optical theo- interaction of a small dipole with several nucleons and rem and introducing the case of a similar interaction of an ordinary hadron. dσ (x, Q2)/dt(t = 0) Let us for example consider interaction with two g = diff σeff 16π 2 (7) nucleons. The leading twist contribution is described σtot(x, Q ) by the diagrams where two gluons are attached to the for the hard process of scattering of a virtual photon dipole. To ensure that nucleus remains intact in such off the gluon field of the nucleon. An important a process we need to attach colorless lines to both feature of this mechanism of coherent interaction is nucleons. These diagrams are closely related to the that it is practically absent for x 0.02–0.03 and may diagrams corresponding to the gluon diffractive parton rather quickly become important with decrease of x. densities which are measured at HERA (see Fig. 1) The gluon virtuality scale which is relevant for the and hence to the similar diagrams for the gluon nuclear J/ψ photoproduction is 3–4 GeV2 with a significant shadowing [7]. fraction of the amplitude due to smaller virtualities As a result it was possible to express the quark and [2,14]. Hence we will take the gluon shadowing in gluon nuclear shadowing for the interaction with two the leading twist at Q2 = 4GeV2. Taking a smaller nucleons in a model independent way through the cor- value of Q2 would result in even larger shadowing g 2 = responding diffractive parton densities using the Gri- effect. We present numerical values of σeff(Q L. Frankfurt et al. / Physics Letters B 540 (2002) 220–226 223

Fig. 3. Leading twist diagrams for the production of J/ψ off two and three nucleons.

g 2 = 2 Fig. 2. The quantity σeff for Q 4GeV as a function of the Bjorken x for H1 (solid line) and Alvero et al. [20] (dashed line) parameterizations of the gluon diffractive density.

4GeV2) for two current models of the diffractive gluon densities (Fig. 2) which practically cover the Fig. 4. Typical diagrams for the higher twist eikonal interactions of a small dipole with two nucleons. range of parameterizations available in the literature. The H1 parameterization leads to a more graduate onset of the contribution of the double interactions carried by exchanged gluons and x = x1 − x2.Inthe because in this model diffraction into masses with DIS limit, or for mQ →∞, one finds [14] x2 x 2 2 M /Q 1 is smaller. The dijet data of H1 prefer this and hence xeff ≈ x/2. However in the case of J/ψ scenario though it seems that further measurements photoproduction Fermi motion effects lead to x2/x ∼ will be necessary to clarify the issue. So we keep 0.3–0.5 and hence to xeff ≈ x. both models for the further analysis. For a more In principle the multiple eikonal type rescatterings detailed discussion of the current models of diffraction (at fixed transverse separations) due to multiple gluon g and of the resulting values of σeff see [15]. The exchanges—see Fig. 4 (the impact parameter eikonal effective cross section σeff can be used to estimate rescattering model ) could also result in suppression relative importance of the interactions with N 3[7], of the vector meson production. Though validity of which corresponds to account of diagrams of Fig. 3 this approximation is hard to justify in QCD the in the quasieikonal approximation. As a result the model calculations suggest that this effect is not t-dependence of the photoproduction turns out about small numerically [2]. However, it is still significantly the same as for the case of Glauber scattering of smaller than the leading twist shadowing (at least a projectile with cross section of interaction with a for x 0.001) which we find in our calculations. nucleon equal to σeff. The ratio of the photoproduction Note in passing that if one would consider the gluon cross sections off nucleus and nucleon is expressed shadowing using phenomenological models [16,17] in the leading twist through the ratio of the skewed where shadowing for gluons was assumed to be equal gluon parton densities. In the case of J/ψ production to that for quarks at low normalization scale one they are pretty close to the gluon density calculated at would find comparable suppressions due to the leading xeff = (x1 +x2)/2wherex1,x2 are light cone fractions twist gluon shadowing and the eikonal rescatterings. 224 L. Frankfurt et al. / Physics Letters B 540 (2002) 220–226

This would make extraction of the gluon shadowing from measurements of photoproduction of J/ψ highly problematic.

3. Numerical results and discussion

Having build the quasieikonal model for the amplitudes of the interaction of cc¯ pair with several nucleons we can now calculate the amplitude of scattering off nuclei. We have demonstrated in [7] that the amplitude in this approximation has the same structure of the rescattering terms as the Glauber model with σtot substituted by σeff. Hence we can use the optical limit of the Glauber model [18] to calculate the cross section of J/ψ photoproduction dσγA→VA(s,t)˜ dt dσγN→VN(s,t˜ = 0) Fig. 5. The rapidity distribution for the J/ψ production in lead–lead = peripheral collisions at LHC. Solid line—production by two-side dt beams, dashed—production by one-side beam only. Calculations J/ψN · were performed with σ based on Alvero et al. parameteriza- × 2 iqt b iqlz eff d bdze ρ(b,z)e tion of the gluon diffractive density. ∞ 2 1 × exp − σ (M2 /s)˜ ρ(b,z )dz . 2 eff V z (8) = 2 Here the exponential factor with ql mV /2k accounts for finite longitudinal distances in the transition γ → V (finite longitudinal momentum transfer). The forward elementary cross section for photoproduction of J/ψ meson on nucleon was taken using the fit to experimental data presented in [19] (this is preferable to using the theoretical calculations which for photoproduction of J/ψ have theoretical uncertainty of the order of two). We focus here on the distributions over rapidity V 1 EV − p 2k y = ln 3 = ln . (9) + V 2 EV p3 mV In Figs. 5, 6, and 7 we present the differential cross sections both including effects of gluon shadowing and without gluon shadowing (impulse approximation) for lead–lead peripheral collisions at LHC and Fig. 6. The rapidity distributions for the LHC lead–lead peripheral gold–gold collisions at RHIC. J/ψN collision J/ψ coherent production calculated with σeff based on One can see that on the top of the overall suppres- H1 (solid line) and Alvero et al. (dashed line) parameterizations of sion of the cross section the gluon nuclear shadowing gluon density and in the impulse approximation (dot-dashed line). L. Frankfurt et al. / Physics Letters B 540 (2002) 220–226 225

At LHC the total cross section calculated with accounting for the gluon shadowing effects is σ(Pb Pb → J/ψ + Pb Pb) ≈ 14 mb and the value σ(Pb Pb → J/ψ + Pb Pb) ≈ 85 mb was obtained in the impulse approximation. Hence, we have found a strong suppression of the J/ψ yield for this case as it was predicted on the base of rough estimates in [7]. In kinematic of RHIC the effect of suppression due to the gluon shadowing is rather small and σ(Au Au → J/ψ + Au Au) ≈ 0.320 mb while in the impulse approximation σ(Au Au → J/ψ + Au Au) ≈ 0.360 mb. Let us compare our results with two other calculations of the reaction (1). The first rather detailed calculation of the coherent process AA → A + V + A has been reported in Ref. [21]. To evaluate nuclear shadowing effects in the total J/ψA cross section the vector dominance model, classical mechanics formulae (accounting for the elastic rescatterings of vector me- Fig. 7. The same as in Fig. 6 but for gold–gold collisions at RHIC. son only) have been used in [21]. On the contrary our calculation uses eikonal approximation where inelas- g tic shadowing effects dominate. Really σeff derived leads to a significant modification of the shape of the from the diffractive gluon densities includes both the rapidity distribution as compared to the impulse ap- elastic and inelastic shadowing. It is also assumed in proximation. Bumps near the edges of rapidity distrib- Ref. [21] that the t-dependence of the cross section is 2 ution are due to a sharp increase of the effective cross ∝|FA(t)| (where FA(t) is the nuclear form factor) section found in the calculation based on the Alvero while account for the rescattering effects (Eq. 8) leads et al. model of the gluon diffractive density in the re- to somewhat steeper t-dependence. For the RHIC en- gion of Bjorken x close to x ≈ 10−2. A bump in the ergies whenever comparison is possible the results of center region of y arises due to the drop of the pho- Ref. [21] are pretty close to ours. This is because nu- ton flux as can be seen from Fig. 5. These effects are clear shadowing effects are a small correction for the weakened for the H1 parameterization which leads to photoproduction of J/ψ in the kinematics of RHIC. a more gradual increase of σeff and they disappear in For the LHC kinematics we obtained cross section for the impulse approximation. It is also of interest that in the coherent J/ψ production significantly below the the LHC kinematics we are sensitive to the cross sec- value of [21] (a factor of two for lead–lead collisions). tion of photoproduction at Wγp up to a factor of three The difference is because approach used in [21] sig- larger than Wγp corresponding to production at y = 0. nificantly underestimates the strengths of multiple in- Hence the measurements will actually probe the J/ψ teraction of cc¯ pair with the nucleus. Also rapidity dis- photoproduction at the energies beyond those reach- tributions for lead–lead collisions for which we find an able at HERA in electron-nucleus mode. interesting shape were not considered in Ref. [21]. In the RHIC kinematics we find even more nice After this study was nearly completed a report has picture in the case of gold–gold collision. The decrease appeared [22] where it has been suggested to use the of cross section as a function of rapidity due to the coherent J/ψ production in the peripheral ion–ion shadowing is combined with drop of the photon flux in collisions to measure shadowing of gluon densities in the same region of rapidities. This results in a narrow nuclei. The analysis in [22] is based on the factoriza- dip at y = 0 which is very sensitive to pattern of onset tion theorem of [4,6], the t-dependence of the coherent of the gluon shadowing. The test of this prediction will γA→ J/ψA cross section has been approximated as be feasible at RHIC since the rates of J/ψ production F 2(t) and three sets of the gluon distributions has been are pretty high [21]. used. Calculations with the GRV gluon distribution is 226 L. Frankfurt et al. / Physics Letters B 540 (2002) 220–226 effectively equivalent to the impulse approximation. [6] S.J. Brodsky, L. Frankfurt, J.F. Gunion, A.H. Mueller, Two others model accounts for the nuclear shadowing. M. Strikman, Phys. Rev. D 50 (1994) 3134, hep-ph/9402283. The distribution of [17] assumes the same shadowing [7] L. Frankfurt, M. Strikman, Eur. Phys. J. A 5 (1999) 293, hep- ph/9812322. for gluons as for quarks which is in variance with dif- [8] E. Lippmaa et al., CERN-LHCC-97-45, LHCC-I10, August fractive data from HERA. The second model [23] at- 1997, J. Phys. to be published. tempts to account for nonlinear QCD evolution in the [9] G. Baur, K. Hencken, D. Trautmann, Prog. Part. Nucl. Phys. 42 gluon density and in this case application of the factor- (1999) 357, nucl-th/9810078. ization approximation is hardly justified. It would be [10] L. Frankfurt, V. Guzey, M. McDermott, M. Strikman, Phys. Rev. Lett. 87 (2001) 192301, hep-ph/0104154; reasonable to expect that at least for J/ψ production L. Frankfurt, V. Guzey, M. McDermott, M. Strikman, Phys. in the kinematics at RHIC where the shadowing effects Rev. 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Nucleon–nucleon scattering observables in large-Nc QCD

Thomas D. Cohen, Boris A. Gelman

Department of Physics, University of Maryland, College Park, MD 20742-4111, USA Received 15 April 2002; received in revised form 11 June 2002; accepted 19 June 2002 Editor: W. Haxton

Abstract

Nucleon–nucleon scattering observables are considered in the context of the large Nc limit of QCD for initial states with moderately high momenta (p ∼ Nc). The scattering is studied in the framework of the time-dependent mean-field approximation. We focus on the dependence of those observables on the spin and isospin of the initial state which may be computed using time-dependent mean-field theory. We show that, up to corrections, all such observables must be invariant under simultaneous spin and isospin flips (i.e., rotations through π/2 in both spin and isospin) acting on either particle. All observables of this class obtained from spin unpolarized measurements must be isospin independent up to 1/Nc corrections. 2  Moreover, it can be shown that the leading correction is of relative order 1/Nc rather than 1/Nc. 2002 Elsevier Science B.V. All rights reserved.

The strong interaction between two nucleons is the imated by a Hartree-type mean-field treatment. This basic ingredient of nuclear physics. We wish to ex- picture works cleanly when dynamical gluons are in- plore qualitative features of the nucleon–nucleon integrated out. For the case of light quarks, explicitly teraction which may be understood from QCD. In this deducing the Hartree equations of motion is not a context, it may be useful to consider the interaction in tractable problem with present techniques. However, the limit when the number of colors, Nc,ofQCDbe- the Nc dependence of various baryon properties can comes large [1,2], and to treat 1/Nc as an expansion still be deduced. Thus, for example, the baryon mass parameter. Certain aspects of QCD can be deduced in is of order Nc and the baryon size is of order unity [2]. this limit in a model-independent way. For example, Note, when the large Nc scaling of an observable is the spin-flavor structure of certain amplitudes in the discussed all dimensional scales are implicitly fac- single baryon sector may be fixed [3]. The large Nc tored out. These dimensional scales are proportional limit can also be used to determine the leading spin- to an appropriate power of a typical meson mass. isospin dependence of certain nucleon–nucleon scat- Witten also considered the baryon–baryon interac- tering observables. tion. He argued that the strength of this interaction Baryons in large-Nc QCD were first discussed by is of order Nc and the proper framework is the time- Witten, [2], who argued that they are well approx- dependent mean-field theory (TDMFT). The argument is similar to the single baryon case: each quark moves in an average potential due to the other 2 Nc −1 quarks E-mail address: [email protected] (B.A. Gelman). which in this context is time-dependent. The mean-

field nature of the nucleon–nucleon scattering in the In TDMFT treatments of baryon–baryon scattering, large Nc limit imposes constraints on spin–isospin de- the initial conditions are two well-separated baryons pendence of certain observables. One of the limita- moving towards each other along some axis, nˆ (which tions of such an approach is that only certain observ- in an experimental situation is the beam direction), off- ables can be calculated in TDMFT. This limitation can set by an impact parameter b. The initial baryon states be quite severe. For example, both the elastic and total are (rotated) hedgehogs. The initial states can be para- scattering cross sections are not calculable in TDMFT. meterized by two sets of rotation parameters—one as- Witten pointed out that baryon–baryon scattering sociated with the orientation of each initial hedgehog. observables are expected to possess a smooth limit When a quantity, O, is calculated in TDMFT its value only for momenta of order Nc. For this reason we will depend on the relative orientation of the two ini- will focus on this regime. The regime of momenta tial hedgehogs and b.Upto1/Nc corrections quanti- of order N0 has been considered elsewhere where the ties which are amenable to calculation in TDMFT can c stress has been on the nucleon–nucleon potential [4]. be written as O(A1,A2, b,n,ˆ E),whereA1 and A2 Although we do not know the TDMFT equations represent the initial orientations of the two hedgehog for large Nc QCD, we know its general form and baryons, nˆ is the beam direction, E is the initial kinetic can deduce from this the leading order spin–isospin energy of each baryon (in the center-of-mass frame) dependence of certain observables. As this form may and b is the impact parameter (the projection perpen- not be immediately familiar, it is useful to also dicular to nˆ of the initial displacement of the two parti- derive results in the context of TDMFT for the cles from each other). The orientations, A1 and A2,are Skyrme model. The Skyrme model describes nucleons associated with particular linear combinations of spin as topological solitons of a chiral field U and is and isospin components of the baryons states. By vary- believed to capture all of the model-independent ing these orientations one can extract the part of the re- large Nc results of QCD [2,5,6]. It has the virtue sult which is associated with the various spin–isospin that the mean-field theory is trivially equal to the configurations of the initial nucleon states. This allows classical field theory. In both the Skyrme model and us to deduce our principal result: all observables which the Hartree treatment the mean-field solution for a can be deduced from TDMFT must be invariant un- single baryon is a “hedgehog” form which breaks der simultaneous spin and isospin flips (i.e., rotations both rotational and isorotational invariance. At the through π/2 in both spin and isospin) acting on either mean-field level this means that there exist manifolds particle up to 1/Nc corrections. of degenerate baryon states corresponding to rotated To compute an observable from TDMFT requires hedgehogs. These manifolds are parameterized by the that at the QCD level the observable can be expressed three independent variables which specify a rotation. directly in terms of an expectation value of some One can specify these as being the three Euler Heisenberg-picture operator (which we will denote angles; an alternative parameterization is to spec- as O) in an initial plane wave state of two nucleons. ify the rotation in terms of a matrix A given by All sensitivity to the future time evolution is contained A ≡ a0 +a ·τ ∈ SU(2),wherea0,a1,a2 and a3 are in the operator. First one argues that in the p ∼ Nc 2 + real collective variables satisfying a constraint a0 regime the expectation value in these plane-wave a ·a = 1. The standard interpretation of these states states can be expressed in terms of integrals of at the full quantum level is that the mean-field states expectation values of initial states consisting of well correspond to linear combinations of nearly degen- localized wave packets of two (rotated) hedgehog erate states—i.e., states whose masses are split by baryons heading towards each other and off-set by an O(1/Nc). The quantized states can be viewed as col- impact parameter b. The integrations are over b and lective wave functions associated with the variable the parameters specifying the initial rotations of the parameterizing the rotations of the hedgehog; up to hedgehogs. normalization constants, these collective wave func- The integrations over b is quite standard. For tions are simply the Wigner D matrices [7]: Ψ(A)= p ∼ Nc, the characteristic momentum is much larger (J + 1/2)1/2π−1DI=J (A),whereJ and I are spin than the inverse size of the region of interaction. m,mI and isospin of the quantized state. Only a small part of the interaction region is probed T.D. Cohen, B.A. Gelman / Physics Letters B 540 (2002) 227–232 229

coherently [8]. The variable b is essentially classical and, in the large Nc limit, is irrelevant during reactions and the various types of cross sections we will use taking place during time scales of order unity. as observables can be written as an integral over the The operators obtained from TDMFT are of the impact parameter which specifies the section of the form O(A1,A2, b,n,ˆ E).ThevariablesA1,2, E and b interaction region being probed. are fixed by initial conditions in the TDMFT; b will The integration over the hedgehog orientations fol- be integrated over in the standard manner. The ob- lows the strategy of Ref. [9]. The mean-field theory, servables which we focus are expectation values in which is essentially classical in nature, accurately de- states with fixed initial nucleon spins and isospins. In scribes all of the variables except the collective ones— experiments this is obtained by creating the given ini- i.e., those degrees of freedom which at the classical tial state multiple times and averaging over measure- level can be excited with no energetic cost. These vari- ments. To compute nucleon matrix elements of oper- ables must be quantized in order to project onto physi- ators which are functions of A1,2, one integrates the cal states with the correct quantum numbers. The vari- operator weighted by the appropriate collective wave ables associated with the collective isorotations of the functions Ψ(A): the expectation value of an operator widely-separated hedgehogs are collective variables. f(A1,A2) in a two-nucleon state with spin projections The way the quantization is realized is simple: one along an arbitrary axis m(1,2) and isospin projections (1,2) expresses the observable at the mean-field level in mI is ˙ terms of the A’s and their time derivatives A.TheA1,2 ˙ f (1) (1); (2) (2) and A1,2 are dynamical variables which can be quan- m ,mI m ,mI tized. = 1/2 2 dA1 dA2 D (1) (1) (A1) To proceed further, one must analyze the time scale m ,mI of the interaction. The interaction takes place in a × 1/2 2 0 D (2) (2) (A2) f(A1,A2), (1) time of order Nc : the size of the baryons is order m ,mI N0 and in this regime the velocity is also N0. Thus c c where dA is the SU(2) invariant measure normal- it takes a time of order N0 for the two baryons to c ized so that dA = 2π2. substantially overlap and interact. This in turn, implies The observable is a cross section associated with that the observable as a function A and A˙ should 1,2 1,2 the O for nucleons with fixed initial spins and isospins, be independent of Nc. Once the observable is written O ˙ σ (1) (1) (2) (2) . It is obtained by an integration of as a function of A and A , the next step is to make m mI m mI 1,2 1,2 O(b,n,ˆ E) (1) (1) (2) (2) b a Legendre transformation to express the observable m mI m mI over : as a function of the A’s and their conjugate momenta. O The conjugate momenta can be expressed in terms of σ (1) (1) (2) (2) m mI m mI the isospin and the intrinsic isospin in the body-fixed 2 = b O(b,n,ˆ E) (1) (1) (2) (2) , frame which in hedgehog models is the spin. However, d m ,mI ;m ,mI (2) in the process of making this Legendre transformation, ˙ where corrections are higher order in 1/N . any variables A1,2 get mapped into I1,2/I and S1,2/I. c The central result of our paper is contained in Note, that I, the moment of inertia, is O(Nc) as can be seen in explicit model calculations [6], and from the Eqs. (1) and (2). The dependence on the initial spin model independent analysis [10]. Since in our initial and isospin projections of the individual nucleons of 0 the cross section comes about entirely from the Wigner states the isospin and spin are of order Nc , it is clear D matrices inside the integrals of Eq. (1). Note, how- that if one does a 1/Nc expansion of the Legendre transformed operator, the leading order terms can ever, that what is relevant is the square modulus of depend on the A’s but not on the spins or isospins. the D matrices. There is a well-known property of the Wigner D matrices (DJ )∗ = (−1)m−nDJ Thus at leading order in the 1/Nc expansion we can m,n −m,−n | 1/2 |2 =| 1/2 |2 neglect the spin and isospins and treat the operator as a which implies Dm,n(A) D−m,−n(A) .This diagonal operator in A1,2. A simple way to understand means that up to 1/Nc corrections the observed cross this is that because of the large moment of inertia the section will be unchanged if one simultaneously flips characteristic rotational period of the hedgehogs is Nc both the spin and the isospin of either particle 1 or par- 230 T.D. Cohen, B.A. Gelman / Physics Letters B 540 (2002) 227–232 ticle 2. This strongly constrains the spin and isospin the S-matrix it can only give sensible information dependence of the cross section. about quantities which average over many S-matrix el- To illustrate this constraint, consider some observ- ements. Such observables might, for example, be asso- able which averages over the direction of all detected ciated with various kinds of collective flow. Moreover, particles. Thus, apart from the spins, the only vector such flow must be expressible in terms of operators left in the problem is the beam axis nˆ. Suppose further which do not encode correlations beyond those built that the observable in question is time reversal, par- into the mean-field trial states; one can follow the ity even and isoscalar. Then from general invariance flows of various quantum numbers but not of partic- properties one has ular correlations of individual hadrons. It is not hard to determine which experimental O (1) (2) (1) (2) σ = A0 + B0 σ ·σ + C0 nˆ ·σ nˆ ·σ observables are accessible in mean-field theory. One simply does, or imagines doing, a TDMFT calcu- + A + B σ (1) ·σ (2) I I lation. Any quantity which one both knows how to (1) (2) + CI nˆ ·σ nˆ ·σ compute and to relate to an experimentally accessi- × τ (1) · τ (2) , (3) ble quantity is fair game. To make this concrete we should consider the following experimental observ- where the A, B and C coefficients are functions able. If we scatter two nucleons at moderately high of energy. The leading order amplitude must be energy (E ∼ Nc), one will generally produce a fi- invariant under a simultaneous spin and isospin flip of nal state with a number of mesons along with two either of the two particles. This implies that AI = 0, baryons (and at high enough energies possibly addi- B0 = 0, C0 = 0 at leading order, since the terms they tional baryon–anti-baryon pairs). One can do the fol- multiply change signs under this transformation. For lowing experiment—fix a detector at angle, θ relative a more general observable the same scheme may be to the beam axis and measure whether an outgoing employed: one finds all vectors in the problem and baryon (or anti-baryon) hits the detector. By dividing constructs the most general form the amplitude can the number of baryons striking the detector minus the take contracting the initial spins into all vectors in all number of anti-baryons in solid angle dΩ per unit time ways consistent with the symmetry. One then imposes by the incident flux, one obtains a semi-inclusive par- B the large Nc rule to eliminate all terms which change tial differential cross section, dσ /dΩ. Note that this signs under a simultaneous spin and isospin flip of quantity is semi-inclusive; while we know the position either particle. One immediate consequence of this of one baryon, it does not specify its correlations with analysis is that all observables in this regime which the other baryon, nor does it specify correlations with are obtained from unpolarized measurements must be outgoing mesons. As such this quantity averages over isoscalars. multiple final states and is not associated with any sin- There are several fundamental limitations in the gle S-matrix element. (In fact, with the kinematics re- use of TDMFT to directly compute scattering observ- stricted to p ∼ Nc, the total amplitude for producing ables [11]. TDMFT is a treatment of average quanti- anti-baryons is both exponentially suppressed in Nc ties rather than any particular scattering channel. Thus, [2] and is experimentally small so one can simply use one cannot directly compute S-matrix elements since the semi-inclusive baryon differential cross section in S-matrix elements are defined for particular chan- place of the difference of the baryon from the anti- nels. Moreover, the form of the initial and final states baryon. At higher energies one would have to use the are constrained by the form of the trial wave func- difference of the two.) tion which are not rich enough to include the cor- It is easy to see how to compute this in TDMFT. relations necessary to get the translationally invari- First consider using the Skyrme model at the classical ant time-independent plane-wave asymptotic states of level. While the details of the Skyrme model need not the full quantum theory. Thus the initial states are reproduce QCD, it is thought that the Skyrme model time-dependent solutions of two lumps moving to- captures all of the model independent large Nc results wards each other while the final states are even more from QCD. The model is given in terms of the dy- complicated. Clearly, since TDMFT cannot compute namics of a chiral field U(r,t) ∈ SU(2) with baryons T.D. Cohen, B.A. Gelman / Physics Letters B 540 (2002) 227–232 231 treated as solitons. We can fix initial conditions U(r,0) time-dependent Hartree calculation (were that tech- and U(r,˙ 0) corresponding to widely separated two nically possible). And thus the form obtained is, in hedgehogs with orientations A1 √and A2 moving with fact, model-independent. The semi-inclusive differen- B a velocity of ±vnˆ (with v = 2MN E ) and with tial cross section, dσ /dΩ, is given by the general- their centers offset from the nˆ axis by ±b/ 2. Solving ization of Eq. (3). The spin averaged semi-inclusive the equations of motion gives U(r,t; A1,A2, b,n,ˆ E) differential cross section then is given by A0 and is where A1,A2, b,n,ˆ E parameterize the initial state. In isospin independent. the Skyrme model the baryon current is given by the From this example it is clear how to construct other standard topological current, observables accessible in TDMFT. One can construct a variable measuring the fraction of total energy flow- "µναβ −1 ν −1 α −1 β Bµ = Tr U ∂ U U ∂ U U ∂ U . ing out at a fixed angle. However, the most com- 24π2 (4) monly used observables to describe nucleon–nucleon scattering—the total cross section and the total and ; ˆ Using U(r,t A1,A2, b,n, E) gives the baryon current differential elastic cross sections—are not amenable as a function of time and space as parameterized by to mean-field treatments. Elastic cross sections are ; ˆ the initial conditions: Bµ(r,t A1,A2, b,n, E).Now not calculable since they are directly associated with consider a large sphere of radius R centeredonthe particular S-matrix elements. Moreover, it seems im- nominal classical collision point of the two solitons possible to imagine any way in mean-field theory (which we will take to be the origin). The radius R can to extract the elastic cross section. The total cross be taken to be large compared to all distance scales in section includes forward scattering and, as is well the problem. The average baryon number per unit solid known, semi-classical treatments fail for forward scat- angle flowing at fixed direction (specified by polar tering [8]. angle θ and azimuthal angle φ) out of a single collision Next we consider possible corrections to the spin– is simply given by isospin structure of Eq. (3) and its generalizations. ∞ One difficulty with following Witten’s strategy and be- B dn (θ, φ) − ginning with TDMFT as giving the leading order re- = R 2 dt r(θ,φ)ˆ dΩ sult is that we have not systematically formulated the −∞ full 1/Nc expansion. Nevertheless, it seems apparent × B Rr(θ,φ),tˆ ; A1,A2, b,n,Eˆ . (5) that while generic 1/Nc terms can affect the values of B the various coefficients in Eq. (3), they are unlikely Integration of dn (θ, φ)/dΩ over A1,A2 and b as in to change the spin–isospin structure. The spin–isospin Eqs. (1) and (2) yields the physical observable structure came about when we neglected terms sen- dσB (θ, φ) sitive to the initial rotational velocities of the separated hedgehogs which in turn implied that each ex- dΩ (1) (1) (2) (2) m mI m mI plicit power of I or S came with a factor of 1/N .If c = 2 1/2 2 we are considering observables associated with opera- d b dA1 dA2 D (1) (1) (A1) m ,mI tors of good parity and isospin, the spin or isospin can × 1/2 2 only come in as pairs: e.g., there are no terms propor- D (2) (2) (A2) m ,mI tional to I (1) +I (2) but only to I (1) ·I (2). This suggests ∞ 1 that all corrections to Eq. (3) and its generalizations × dt r(θ,φ)ˆ will be of order 1/N2. R2 c −∞ It is not clear whether the data on a semi-inclusive differential cross sections is readily accessible in a × B Rr(θ,φ),tˆ ; A ,A , b, n,ˆ E (6) 1 2 form that can be compared to Eq. (3). Presumably such plus corrections higher order in Nc. While the pro- data has been collected at some point. Comparisons ceeding derivation was done in the context of the with Eq. (3) would be very interesting. It would also Skyrme model it is clear that we would get the same be of interest to see if data on other observables structure in any mean-field theory including a direct computable from TDMFT can be found. 232 T.D. Cohen, B.A. Gelman / Physics Letters B 540 (2002) 227–232

Acknowledgements R. Dashen, E. Jenkins, A.V. Manohar, Phys. Rev. D 51 (1995) 3697. [4] D.B. Kaplan, M.J. Savage, Phys. Lett. B 365 (1996) 244; This work is supported by the US Department of D.B. Kaplan, A.V. Manohar, Phys. Rev. C 56 (1997) 76; Energy grant DE-FG02-93ER-40762. M.K. Banerjee, T.D. Cohen, B.A. Gelman, Phys. Rev. C 65 (2002) 034011; T.D. Cohen, A.V. Belitsky, hep-ph/0202153. [5] E. Witten, Nucl. Phys. B 223 (1983) 433. References [6] G.S. Adkins, C.R. Nappi, E. Witten, Nucl. Phys. B 228 (1983) 552. [7] M.P. Mattis, Phys. Rev. Lett. 56 (1986) 1103. [1] G. ’t Hooft, Nucl. Phys. B 72 (1974) 461. [8] L.D. Landau, E.M. Lifshitz, Quantum Mechanics: Non- [2] E. Witten, Nucl. Phys. B 160 (1979) 57. relativistic Theory, 3d ed., Pergamon, Elmsford, NY, 1977. [3] J.L. Gervais, B. Sakita, Phys. Rev. Lett. 52 (1984) 87; [9] T.D. Cohen, W. Broniowski, Phys. Rev. D 34 (1986) 3472. J.L. Gervais, B. Sakita, Phys. Rev. D 30 (1984) 1795; [10] E. Jenkins, Phys. Lett. B 315 (1993) 441. C. Carone, H. Georgi, S. Osofsky, Phys. Lett. B 322 (1994) [11] J.J. Grifﬁn, M. Dworzecka, P.C. Lichtner, K.-K. Kan, Nucl. 227; Phys. A 435 (1985) 205; M. Luty, J. March-Russell, Nucl. Phys. B 426 (1994) 71; J.J. Grifﬁn, M. Dworzecka, Phys. Lett. B 93 (1980) 235. R. Dashen, E. Jenkins, A.V. Manohar, Phys. Rev. D 49 (1994) 4713; Physics Letters B 540 (2002) 233–240 www.elsevier.com/locate/npe

0 0 Bd,s–Bd,s mass-differences from QCD spectral sum rules Kaoru Hagiwara a, Stephan Narison b, Daisuke Nomura c

a KEK Theory Group, Tsukuba, Ibaraki 305-0801, Japan b Laboratoire de Physique Mathématique, Université de Montpellier II Place Eugène Bataillon, 34095, Montpellier Cedex 05, France c Institute for Particle Physics Phenomenology, University of Durham, Durham DH1 3LE, UK Received 12 May 2002; received in revised form 10 June 2002; accepted 14 June 2002 Editor: G.F. Giudice

Abstract We present the ﬁrst QCD spectral sum rules analysis of the SU(3) breaking parameter ξ and an improved estimate of the 0 0 renormalization group invariant (RGI) bag constant BB both entering into the B –B mass-differences. The averages of q d,s d,s ± ≡ the results from the Laplace and moment sum rules to order αs are fB BB (247 59) MeV and ξ fBs BBs fB BB (1.18 ± 0.03), in units where fπ = 130.7 MeV. Combined with the experimental data on the mass-differences Md,s, one 2 obtains the constraint on the CKM weak mixing angle |Vts/Vtd| 20.0(1.1). Alternatively, using the weak mixing angle −1 from the analysis of the unitarity triangle and the data on Md , one predicts Ms = 18.6(2.2) ps in agreement with the present experimental lower bound and within the reach of Tevatron 2.  2002 Elsevier Science B.V. All rights reserved.

1 1. Introduction × 0 O 0 Bq q (x) Bq , (1) 2MBq 0 0 = O B(s) and B(s) are not eigenstates of the weak where the B 2 local operator q (x) is deﬁned as Hamiltonian, such that their oscillation frequency is O ≡ ¯ ¯ governed by their mass-difference Mq . The mea- q (x) (bγµLq)(bγµLq), (2) surement by the UA1 Collaboration [1] of a large value with L ≡ (1 − γ5)/2andq ≡ d,s. S0,ηB and CB (ν) of Md was the ﬁrst indication of the heavy top quark are short distance quantities calculable perturbatively. mass. In the SM, the mass-difference is approximately (Here we are following the notation of Ref. [2].) given by [2] 0 |O | 0 On the other hand, the matrix element Bq q Bq requires non-perturbative QCD calculations, and is G2 m2 F 2 | ∗ |2 t usually parametrized for as Mq 2 MW VtqVtb S0 2 ηB CB(ν) 4π MW f 2 0 0 4 Bq 2 B Oq B = M BB . (3) q q 3 2 Bq q E-mail addresses: [email protected] (K. Hagiwara), = [email protected] (S. Narison), fBq is the Bq decay constant normalized as fπ [email protected] (D. Nomura). 130.7MeV,andBBq is the so-called bag parameter

= which is BBq 1 if one uses a vacuum saturation of one includes the contributions of radiative corrections the matrix element and equal to 3/4inthelargeNc due to non-factorizable diagrams. These perturbative limit. From Eq. (1), it is clear that the measurement of (PT) radiative corrections due to factorizable and non- Md provides one of the CKM mixing angles |Vtd| if factorizable diagrams have been already computed in one uses |Vtb|1. One can also extract this quantity Ref. [7] (referred as NP), where it has been found that from the ratio the factorizable corrections are large while the nonfactorizable ones are negligibly small. NP analysis has M V 2 M B0|O |B0 s = ts Bd s s s confirmed the estimate in Ref. [4] from lowest order 0 0 Md Vtd MBs B |Od |B d d calculations, where under some assumptions on the V 2 M contributions of higher mass resonances to the spec- ≡ ts Bd 2 ξ , (4) tral function, the value of the bag parameter B has Vtd MB B s been found to be since in the SM with three generations and unitarity | || | constraints, Vts Vcb .Here B 4M2 (1 ± 0.15). (7) Bd b gs fBs BBs = ξ ≡ ≡ √ . (5) This value is comparable with the one BBd 1 from gd fB BB the vacuum saturation estimate, which is expected to The great advantage of Eq. (4) compared with the be a quite good approximation due to the relative former relation in Eq. (1) is that in the ratio, different high-scale of the B-meson mass. Equivalently, the systematics in the evaluation of the matrix element corresponding RGI quantity is tends to cancel out, thus providing a more accurate −1 ± prediction. However, unlike Md = 0.479(12) ps , BBd (1.5 0.2), (8) which is measured with a good precision [3], the wherewehaveusedtherelation determination of Ms is an experimental challenge 0 0 due to the rapid oscillation of the Bs –Bs system. At γ0/β1 5165 αs −1 BB = BB (ν)α 1 + , (9) present, only a lower bound of 13.1 ps is available q q s 12696 π at the 95% CL from experiments [3], but this bound with γ = 1 being the anomalous dimension of the already provides a strong constraint on |Vtd|. 0 operator Oq and β1 =−23/6 for 5 flavours. ν is the subtraction point. The NLO corrections have been 2. Two-point function sum rule obtained in the MS scheme [2]. We have also used the value of the bottom quark pole mass [6,8] Ref. [4] has extended the analysis of the K0–K0 = ± systems of Ref. [5], using two-point correlator of the Mb (4.66 0.06) GeV. (10) four-quark√ operators into the analysis of the quantity In the following, we study (for the first time), from the 0 0 fB BB which governs the B -B mass difference. QCD spectral sum rules (QSSR) method,2 the SU(3) The two-point correlator defined as breaking effects on the ratio ξ defined previously in Eq. (5), where a similar analysis of the ratios of the 2 ≡ 4 iqx |TO O †| ψH q i d xe 0 q (x) q (0) 0 , (6) decay constants has given the values [10] is built from the B = 2 weak operator given in fDs fBs Eq. (2). The two-point function approach is very con- 1.15 ± 0.04, 1.16 ± 0.04. (11) fD fB venient due to its simple analytic properties which are not the case of approach based on three-point func- We shall also improve the previous result of Ref. [4, 1 ∗ ∗ ∗ tions. However, it involves non-trivial QCD calcu- 7] on BBd by the inclusion of the Bq Bq and Bq Bq lations which become technically complicated when resonances into the spectral function.

1 For detailed criticisms, see [6]. 2 Our preliminary results have been presented in [9]. K. Hagiwara et al. / Physics Letters B 540 (2002) 233–240 235

3. Inputs for the sum rule analysis (OPE) including non-perturbative condensates [11].3 The massless (mq = 0) expression for the lowest We shall be concerned here with the two-point perturbative and gluon condensate contributions has correlator defined in Eq. (6). The hadronic part of been obtained in Ref. [4]. Radiative factorizable and the spectral function can be conveniently parametrized non-factorizable corrections to the perturbative graphs using the effective realization [4] in the massless light quark case have been obtained in NP [7]. eff 1 0 µ 0 O = gq ∂µB ∂ B +···, (12) q 3 q q ··· ≡ 2 × where indicates higher resonances and gq fB 4. SU(3) breaking contributions ∗ ∗ ∗ q BBq . Retaining the BB, BB and B B resonances and parametrizing the higher resonances with the The lowest order perturbative contribution for QCD continuum contribution, it gives ms = 0 to the two-point correlator is 1 1 pert eff Imψs (t) Im ψq (t) π π = θ t − 4(M + m )2 2 2M2 2 4M2 b s 2 gq 2 Bq Bq √ √ √ = t 1 − 1 − (1− δ− δ )2 (1− z)2 9 8π t t t4 × 6 dz duzu 2 1536π √ √ √ √ × θ t − M 4 Bq ( δ+ δ )2 ( δ+ δ )2 2 4 1/2 1/2 δ δ 1/2 δ δ MB∗ MB∗ × λ (1,z,u)λ 1, , λ 1, , + 1 − 4 q + 12 q z z u u t t2 δ δ δ δ 2 × 4M ∗ 4f , f , × − Bq − 2 z z u u 1 θ t 4MB∗ t q δ δ δ δ 2 − 2 M ∗ 2f , g , MB B z z u u + 2λ3/2 1, q , q t t δ δ δ δ − 2g , f , z z u u 2 × θ t − (MB + MB∗ ) (13) q q (1 − z − u)2 δ δ δ δ + g , g , . (16) zu z z u u below the QCD continuum threshold tc. The function λ(x,y,z) is a phase space factor, ≡ 2 ≡ 2 Here δ Mb /t and δ ms /t, respectively. The λ(x,y,z) ≡ x2 + y2 + z2 − 2xy − 2yz− 2zx. (14) functions f(x,y) and g(x,y) are defined by f(x,y)≡ 2 − x − y − (x − y)2, (17) We have used, to a first approximation, the large Mb and vacuum saturation relations: g(x,y) ≡ 1 + x + y − 2(x − y)2. (18) 2 2 O ≡ ∗ ≡ ∗ ∗ We include the (αs ) correction from factorizable gq fB BBq gq fB BB (15) q q q diagrams by using the results in the MS scheme for among the couplings. The results fB ≈ fB∗ have been the two-point correlators of currents [15].4 This can also obtained from QCD spectral sum rules [6], while the vacuum saturation relation B ∗ 1 B is a Bq Bq 3 posteriori expected to be a good approximation as We shall not include here the effects of tachyonic gluon mass [12] or some other renormalon-like terms which give small indicated by the result obtained later on in this Letter effects in various examples [13,14]. The short distance expression of the spectral function 4 We shall neglect the nonfactorizable corrections in our analysis is obtained using the Operator Product Expansion to the results in NP [7] obtained in a slight variant of the MS scheme. 236 K. Hagiwara et al. / Physics Letters B 540 (2002) 233–240 be done using the convolution formula, 5. The sum rule analysis 1 αs Im ψs (t) For the sum rule analysis, we shall work like [4,6] π with the moments 2 = θ t − 4(Mb + ms) (q) √ √ √ tc (1− δ− δ )2 (1− z)2 2 (n) n 1 t M = dt t Im ψq (t), (25) × dz du q π 4 2 6π √ √ √ √ 4(Mb+mq ) ( δ+ δ )2 ( δ+ δ )2 and with the Laplace sum rule × λ1/2(1,z,u) (q) × 0 αs µν tc Im Πµν (zt) Im Π (ut) L = −tτ 1 + αs 0µν (τ)q de Im ψq (t). (26) ImΠµν (zt) Im Π (ut) . (19) π 2 4(Mb+mq ) 0 2 αs 2 Here Πµν (q ) and Πµν (q ) are, respectively, the lowest and the next-to-leading order QCD contribution In so doing, in addition to the pQCD input parameters 2 to the two point correlator Πµν (q ) defined by given previously, we shall need the values of the QCD condensates and SU(3) breaking parameters, which

Π q2 ≡ i d4xeiqx we give in Table 1. We show in Fig. 1 the moment sum µν rules analysis of fB BB for different values of rq × | ¯ ¯ | (s) (s) 0 T b(x)γµLs(x) s(0)γνLb(0) 0 . and n. As one can see from Fig. 1a, b, the stability (20) regions of the quantity, The quark condensate contribution reads ˆ ≡ gd fBd BBd (27) 1 ¯ Im ψ ss (t) = π s versus the number n of moments and the continuum threshold 2 1 θ t − 4(Mb + ms) ms ¯ss 384π3 (d) √ tc 2 ≡ ( t −Mb) rd , (28) 4M2 2 2 ∂ b × dq λ1 4 + 2q 1 ∂q2 are obtained for large ranges ending with an extremum + 2 (Mb ms ) for 2 2 Mb q1 2 × λ0 λ1 1 + − q n −30,rd 1.13. (29) q2 q2 1 These range of values of the sum rule parameters are in 2 2 Mb q1 2 2 2 good agreement with previous results in Ref. [4] and + f1 1 − + q − M − q . (21) q2 q2 b 1 NP [7]. Analogous values of n and r stabilities are s also obtained in the analysis of gˆ ≡ f B (see Here λ0, λ1,andf1 are deﬁned by s Bs Bs Fig. 1c, d), with q2 M2 λ ≡ λ 1, 1 , b , (22) − 0 q2 q2 n 26,rs 1.17. (30) One can notice that the stabilities in the continuum for M2 m2 ≡ b s gˆ and gˆ differ slightly as a reﬂection of the SU(3) λ1 λ 1, 2 , 2 , (23) d s q1 q1 breakings, which one can parametrize numerically as M2 m2 (M2 − m2)2 f ≡ 1 + b + s − 2 b s . (24) √ m 2 1 2 2 4 + s ≈ + q1 q1 q1 rs rd rd 0.05. (31) Mb K. Hagiwara et al. / Physics Letters B 540 (2002) 233–240 237

Table 1 Different sources of errors in the estimate of fB BB and ξ in units where fπ = 130.7 MeV. The box marked with – means that the error is zero or negligible 2 Sources (fB BB ) [MeV] ξ × 10 Moments Laplace Moments Laplace −n (30–10) 8.3– 1.5– − τ (0.1–0.31) GeV 2 –12.4– 1.7 rd 1.06–1.17 7.97.81.02.5 = +25 Λ5 (216−24) MeV [3,16] 0.40.40.10.1 ν = Mb–2Mb 8.79.10.20.2 2 αs : geometric PT series 43.045.00.60.7 Mb = (4.66 ± 0.06) GeV [8] 34.637.10.91.2 2 4 αs G =(0.07 ± 0.01) GeV [17] 1.31.6– – ¯uu (2) =−(254 ± 15)3 MeV3 [6,8,18] – – 0.20.3 ¯ss / ¯uu =0.7 ± 0.2 [6,8,18] – – 0.30.4 m¯ s (2) = (117 ± 23) MeV [6,8,18] – – 1.51.7 Total 57.160.82.63.8

Similar analysis is done with the Laplace sum rules. and deduce the ratio We show in Fig. 2 the predictions of fB(s) BB(s) for fBs BBs different values of r and τ , where an extremum is ξ ≡ 1.174 ± 0.026, (34) q f B obtained for B B where the errors come almost equally from n, rq ,ms , −2 2 τ 0.3GeV . (32) Mb and the αs term. As expected, we have smaller errors for the ratio ξ due to the cancellation of the systematics. We proceed in the same way with the Laplace sum 6. Results and implications on |V /V |2 and ts td rules where we take the range of τ values from 0.1 M − s to 0.37 GeV 2 (see Fig. 2a, b) in order to have a conservative result. Then, we deduce We take as a conservative result for gˆd , from the moments sum rule analysis, the one from a large range fB BB (249 ± 61) MeV, of n =−10 to −30 and for r = 1.06 to 1.17. Adding d ± quadratically the different sources of errors in Table 1, ξ 1.187 0.038. (35) we obtain As a final result, we take the arithmetic average from the moments and Laplace sum rules results. Then, we fB BB (245 ± 57) MeV, (33) deduce in units where fπ = 130.7 MeV. The most relevant er- fB BB (247 ± 59) MeV, rors given in Eq. (33) come from M and the trun- b ξ 1.18 ± 0.03, (36) cation of the PT series. We have estimated the latter 2 by assuming that the coefficient of the αs contribution in the unit where fπ = 130.7 MeV. These results can comes from a geometric growth of the PT coefficients. be compared with different lattice results f B The other parameters n, r ,Λ, ¯qq , α G2 and ν B B d s (230 ± 32) MeV, ξ 1.14 ± 0.06, and global-fit of (subtraction point) induce smaller errors as given in ± Table 1. We proceed in a similar way for gˆs . Then, we the CKM mixing angles giving fB BB (231 take the range n =−10 to −26 and rs = 1.10 to 1.21, 15) MeV quoted in Ref. [19,21]. By comparing our 238 K. Hagiwara et al. / Physics Letters B 540 (2002) 233–240

Fig. 1. Moment sum rules analysis of f B for different values of r and n: f B versus: (a) r at n =−30, (b) n at r = 1.13; B(s) B(s) q B B d d =− = fBs BBs versus: (c) rs at n 26, (d) versus n at rs 1.17. Dotted curve: lowest order perturbative contribution; dashed curve: lowest order 2 perturbative +ms ¯ss [only for (c) and (d)] + αsG condensates; solid curve: total contribution to order αs .

5 results Eq. (36) with the one of fBs /fBd in Eq. (11), and we have assumed that the error from fB com- one can conclude (to a good approximation) that pensates the one in Eq. (36). The result is in excellent agreement with the previous result of Ref. [7] in BB ≈ BB (1.65 ± 0.38) Eqs. (7) and (8), and agrees within the errors with the s d 2 lattice estimates [19,20]. Using the experimental val- ⇒ BB 4M (1.1 ± 0.25), (37) d,s b ues indicating a negligible SU(3) breaking for the bag parameter. For a consistency, we have used the estimate −1 Md = 0.479(12) ps , to order αs [22] −1 Ms 13.1ps (95% CL), (39) fB (1.47 ± 0.10)fπ (192 ± 19) MeV, (38) one can deduce from Eq. (4)

5 One can notice that similar strengths of the SU(3) breakings → ∗ → 2 have been obtained for the B K γ and B Klν form ≡ Vts factors [21]. ρsd 20.0(1.1). (40) Vtd K. Hagiwara et al. / Physics Letters B 540 (2002) 233–240 239

− Fig. 2. Laplace sum rules analysis of f B for different values of r and τ : f B versus: (a) r at τ = 0.31 GeV 2,(b)τ at B(s) B(s) q B B d = = −2 = rd 1.13; fBs BBs versus: (c) rs at τ 0.26 GeV , (d) versus τ at rs 1.20. The curves are the same as in Fig. 1.

Alternatively, using in agreement with the present experimental lower bound and within the reach of Tevatron run 2 exper- 1 ρsd 28.4(2.9) (41) iments. λ2[(1 −¯ρ)2 +¯η2] with [19] λ 0.2237(33), 7. Conclusions λ2 ρ¯ ≡ ρ 1 − 0.223(38), 2 We have applied QCD spectral sum rules for ex- λ2 tracting ( for the ﬁrst time) the SU(3) breaking para- η¯ ≡ η 1 − 0.316(40), (42) meter ξ, and for improving the estimate of the quantity 2 f B . Our predictions are given in Eq. (36). The λ,ρ and η being the Wolfenstein parameters, we B B deduce phenomenological consequences of our results for the 0 0 CKM mixing angle and Bd,s–Bd,s mass-differences −1 Ms 18.6(2.2) ps , (43) are given in Eqs. (40) and (43). 240 K. Hagiwara et al. / Physics Letters B 540 (2002) 233–240

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+ → + Searching for new physics in the rare decay B Ds φ Rukmani Mohanta

School of Physics, University of Hyderabad, Hyderabad-500 046, India Received 31 May 2002; accepted 20 June 2002 Editor: H. Georgi

Abstract + + The rare decay B → Ds φ can occur only via annihilation type diagram in the standard model. The small branching ratio predicted in the standard model makes this channel sensitive to new physics contributions. We analyze this decay mode, both in the standard model and in several extensions of it. The models considered are minimal supersymmetric model with R-parity violation and two Higgs doublet model. The experimental veriﬁcation of our ﬁndings of large branching ratio and/or nonzero CP asymmetry may signal the presence of new physics.  2002 Elsevier Science B.V. All rights reserved.

Recent results from the ongoing experiments in Thus the rare B meson decays are suggested to B-physics at BABAR and BELLE have attracted a lot give good opportunities for discovering new physics of attention. The main objective of these B experi- beyond the SM. Since their branching ratios are ments is to explore in detail the physics of CP viola- small in the SM, they are very sensitive to new tion, to determine many of the flavor parameters of the physics contributions. In the last few years, different standard model (SM) at high precision and to probe for experimental groups have been accumulating plenty of possible effects of new physics beyond the SM [1–3]. data for the rare B decay modes. Some of them have The intensive search for physics beyond the standard already been measured by the B factories in KEK and model is performed now a days in various areas of par- SLAC. ticle physics. The B system thus offers a complemen- In this context it is interesting to analyze the rare + → + tary probe to the search for new physics. In B experi- decay B Ds φ, which is a pure annihilation type ments, new physics beyond the SM may manifest itself decay in the SM. The four valence quarks in the final in the following two ways: states Ds and φ are different from the ones in the parent B meson, i.e., there is no spectator quark in (i) decays which are expected to be rare in the SM this decay. In the usual factorization approach, this and are found to have large branching ratios; decay mode can be described as the b¯ and u quarks in (ii) CP violating asymmetries which are expected to the initial B meson annihilating into vacuum and the vanish or to be very small in the SM are found to final Ds and φ mesons are produced from the vacuum be significantly large. afterwards. The dynamics of exclusive hadronic B decays occurring via the W-exchange or annihilation diagrams, is not yet understood. The decay rates for E-mail address: [email protected] (R. Mohanta). such transitions are argued to be negligibly small due

0370-2693/02/$ – see front matter  2002 Elsevier Science B.V. All rights reserved. PII:S0370-2693(02)02173-1 242 R. Mohanta / Physics Letters B 540 (2002) 241–246 to the suppression of helicity and (or) form factors. the matrix elements of the two currents. Thus in However, a solid justification of this argument is the factorization approximation one can write the necessary in both theory and experiments. This decay corresponding transition amplitude as mode has been recently studied in perturbative QCD − + → + approach with a branching ratio 3 × 10 7 [4], which Amp B Ds φ G is far below the current experimental upper limit [5] = √F ∗ + ¯ µ − VubVcs a1 Ds φ cγ (1 γ5)s 0 + + − 2 B → D φ < . × 4. Br s 3 2 10 (1) ¯ + × 0 bγµ(1 − γ5)u B . (6) Therefore it provides an appropriate testing ground for physics beyond the SM. The main uncertainties in evaluating the transition + |× In this Letter we intend to study the decay mode amplitude are due to the matrix element like Ds φ + + µ 2 2 → (cγ¯ (1 − γ )s)|0 with (pD + pφ) M .Toget B Ds φ, both in the standard model and in several 5 s B extensions of it. The models considered here are the an idea about the contributions from annihilation Two Higgs Doublet Model (2HDM) [6] and minimal diagrams, one can assume single pole dominance for supersymmetric model with R-parity violation (RPV) the matrix element and relate it to the crossed channel |¯ µ − | [7]. φ sγ (1 γ5)c Ds . We see that the annihilations do We first consider the contributions arising from the not give a large contributions here: with single pole SM, where the effective Hamiltonian describing the dominance, form factors are suppressed by a factor Mp2/(M2 −Mp2) because the transferred momentum decay mode is given as Ds B Ds 2 = p G Q MB is large with respect to pole mass MD . H = √F ∗ + s eff VubVcs C1(µ)O1(µ) C2(µ)O2(µ) , (2) The annihilation contribution could be larger if there 2 were other pseudoscalar (cs)¯ resonators heavier than where the pole mass Mp , which would enhance the form Ds factors. The matrix element of the pseudoscalar and O = bγ¯ (1 − γ )u cγ¯ µ(1 − γ )s , 1 µ 5 5 vector meson is usually decomposed as [9] ¯ µ O2 = bγµ(1 − γ5)s cγ¯ (1 − γ5)u (3) µ φ(q,) sγ¯ (1 − γ5)c Ds (p) and C1,2(µ) are Wilson coefficients evaluated at the 2V(Q2) renormalization scale µ. In the next to leading loga- = µναβ∗p q + ν α β rithmic approximation their values are evaluated at the MDs Mφ b quark mass scale as [8] 2M − i( · Q) φ Qµ A Q2 − A Q2 Q2 3 0 C1(mb) = 1.082 and C2(mb) =−0.185. (4) + i(M + M ) Ds φ The corresponding transition amplitude is given as × µ∗ 2 + A1 Q Amp B → D φ s ∗ · Q − (p + q)µA Q2 , = G√F ∗ + ¯ µ − + 2 2 (7) VubVcs a1 Ds φ cγ (1 γ5)s (MDs Mφ) 2 + where Q = (p − q) is the momentum transferred × bγ¯ (1 − γ )u B , (5) µ 5 during the transition process. Using the decay constant where a1 = C1 + C2/Nc, with Nc is the number of relation colors. The evaluation of the matrix element of four | ¯ µ + = µ fermion operator from the first principles of QCD 0 bγ γ5u B (Q) ifBQ (8) is an extremely demanding challenge. To have some the transition amplitude in the SM is given as idea of the magnitudes of matrix element, one can still use the factorization method, factorizing the four Amp(B → Ds φ) quark operators relevant to nonleptonic B decays into G =−√F ∗ 2 ∗ · VubVcs a1fB 2MφA0 MB Q (9) the product of two currents and evaluating separately 2 R. Mohanta / Physics Letters B 540 (2002) 241–246 243 and the corresponding decay width as the final state mesons and therefore cannot be computed self consistently in the context of hard scattering → Γ(B Ds φ) approach. One possible way to go around the problem 3 pc ∗ 2 is to treat the end point divergence arising from differ- = Amp(B → Ds φ)/ · Q , (10) 8πM2 ent sources as different phenomenological parameters φ [12]. The corresponding price one has to pay is the in- where pc is the c.m. momentum of the decay particles. troduction of model dependence and extra numerical The value of the form factor A0(0) at zero momentum uncertainties. In this work we will follow the treatment transfer for the transition Ds → φ is given as [9] of [12] and express the weak annihilation amplitude = 2 + → + A0(0) 0.7andtheQ dependence of the form factor for B Ds φ as is given as + + G ∗ Amp B → D φ = √F V V f f f b (D ,φ). 1 s ub cs B Ds φ 1 s A Q2 = , (11) 2 0 − 2 2 (14) 1 Q /MP The annihilation parameter b1(Ds ,φ)is given as where MP is the pole mass with value MP = 1.97 GeV [9]. Thus we get C b (D ,φ)= F C Ai (D ,φ), (15) 1 s N 1 1 s 2 =− c A0 MB 0.113. (12) 2− Nc 1 where the color factor CF = and Nc = 3. The Using a1 = 1.04 [10], which is extracted from the 2Nc ∗ i experimental data on B → D (π, ρ), fB = 190 MeV, function A1(Ds,φ)is given as the CKM matrix elements from [5] and τ = 1.653 × B 1 1 10−12 s, we obtain the branching ratio in the SM as i = A1(Ds ,φ) παs dx dy ψDs (x)ψφ(y) + + − Br B → D φ = 1.88 × 10 6. (13) s 0 0 1 1 Thus one can see that the obtained branching ratio × + , (16) in the SM using the factorization assumption is far y(1 − xy)¯ x¯2y + → + below the experimental value Br(B Ds φ) < where ψD (x) and ψφ(y) are the light cone distribu- −4 s 3.2 × 10 . Furthermore, it should be noted here that tion amplitudes (LCDA) for the final mesons and x is the direct CP asymmetry for this decay mode is zero the longitudinal momentum fraction of c quark in Ds in the SM since it receives contribution only from the and y¯ is the momentum fraction of s¯ in φ.Usingthe single annihilation diagram. assumption that the LCDAs of the mesons Ds and φ We now proceed to evaluate the branching ratio in are symmetric, one can parameterize the weak annihi- the QCD Improved factorization method, which has lation contribution as [13] been developed recently [11] to study the hadronic B 2 decays. This method incorporates elements of naive i π A (Ds ,φ) 18παs XA + − 4 , (17) factorization approach as its leading term and pertur- 1 3 bative QCD corrections as subleading contributions = 1 and thus allowing one to compute systematic radia- where XA 0 dx/x parameterizes the end point tive corrections to the naive factorization for hadronic divergence as B decays. This method is expected to give a good es- 1 timate of the magnitudes of the hadronic matrix ele- dx MB −iθ XA = = ln + ρe , (18) ments in non leptonic B decays. However, in the QCD x Λ¯ factorization approach the weak annihilation contribu- 0 tions are power suppressed as ΛQCD/mb and hence do ρ varies from 0 to 6 and θ is an arbitrary phase 0 <θ< ◦ ¯ iθ not appear in the factorization formula. Besides power 360 .UsingΛ = ΛQCD = 200 MeV and ρe = iπ as suppression they also exhibit end point singularities default values along with fB = 190 MeV, fφ = 233 = = = even at twist two order in the light cone expansion of MeV, fDs 280 MeV, C1(mb) 1.082 and αs(mb) 244 R. Mohanta / Physics Letters B 540 (2002) 241–246

0.221, we obtain the branching ratio in QCD improved So the total amplitude in 2HDM including SM contri- factorization approach as butions + → + = × −6 ASM+2HDM = ASM − Br B Ds φ 0.67 10 . (19) (1 R1), (25) We now proceed to calculate the branching ratio for where this decay mode in type II of Two Higgs Doublet 2 β m m M2 = 1 tan b s B Model [6]. In type II 2HDM, the up-type quarks get R1 2 . (26) a1 M (mb + mu)(ms + mc) mass from one doublet, while down-type quarks and H charged leptons from the other doublet. Charged Higgs The free parameters of the 2HDM namely tan β,and Yukawa couplings are controlled by the parameter MH are not arbitrary, but there are some semi quantitative restrictions on them using the existing experi- tan β = v2/v1, the ratio of vacuum expectation values of the two doublets, normally expected to be of order mental data. The most direct bound on charged Higgs ± boson mass comes from the top quark decays, which mt /mb. For our concern the H effectively induce the four fermion interaction as yield the bound MH > 147 GeV for large tan β [15]. ∗ Furthermore, there are no experimental upper bounds G V V H2HDM =−√F ub cs on the mass of the charged Higgs boson, but one gen- eff 2 m2 H erally expects to have MH < 1 TeV in order that per- × u¯ m X(1 + γ ) + m Y(1 − γ ) b turbation theory remains valid [16]. For large tan β the b 5 u 5 most stringent constraint on tan β and MH is actually × s¯ mcY(1 + γ5) + msX(1 − γ5) c , on their ratio, tan β/MH . The current limits come from (20) the measured branching ratio for the inclusive decay where mH denotes the mass of the lightest charged −1 B → Xτν¯,givingtanβ/MH < 0.46 GeV [14]. Us- scalar particle, mq ’s denote the constituent quark ing the constituent quark masses as mb = 4.88 GeV, masses and X 1/Y = v2/v1 = tan β.Intheabove ms = 0.5 GeV [17] and current quark masses at the b equation the terms proportional to Y can be safely quark mass scale as mb = 4.34 GeV, mc = 0.95 GeV, neglected as X is generally taken as large [14]. To ms = 90 MeV and mu = 3.2 MeV [18], the branching evaluate the matrix elements we use the equation of ratio in 2HDM is found to be motion to transform the (S − P )(S + P) currents to + → + = × −6 corresponding (V − A)(V + A) form as Br B Ds φ 8.0 10 . (27) 2HDM ∂µ(q¯ γ q ) ∂µ(q¯ γ γ q ) We now analyze the possibility of observing direct q¯ ( ± γ )q = i 1 µ 2 ∓ 1 µ 5 2 , 1 1 5 2 − + CP violation in this decay mode since it now receives mq2 mq1 mq2 mq1 (21) contributions both from the SM and 2HDM. We can write the decay amplitude (25) as where the quark masses are current quark masses. SM+2HDM SM i(γ+δ ) i(φ+δ) Thus we obtain the expressions for matrix elements as A = A e 1 1 −|R1|e , (28) ∗ if M2 where γ = arg(V ) and δ are the weak and strong | ¯ + =− B B ub 1 0 (bγ5u) B (22) phases of standard model amplitude. φ and δ are the mu + mb relative weak and strong phases between 2HDM and and SM amplitudes. Thus the direct CP asymmetry for the 2Mφ 2 φ|¯sγ5c|Ds =i( · Q) A0 Q . (23) decay mode is given as ms + mc Br(B+ → D+φ) − Br(B− → D−φ) Using the above relations, one can obtain the transition a = s s CP + → + + − → − amplitude in 2HDM as Br(B Ds φ) Br(B Ds φ) 2|R | sin φ sin δ + = 1 . Amp B → Ds φ 2 (29) 2HDM 1 +|R1| − 2|R1| cosφ cosδ G 2 β m m M2 = = = √F ∗ tan b s B If we set φ δ π/2, the maximum possible value of VubVcs 2 fB 2 M (mb + mu)(ms + mc) direct CP violation in 2HDM is found to be H × · 2 + → + ( Q)2MφA0 Q . (24) aCP B Ds φ 2HDM 59%. (30) R. Mohanta / Physics Letters B 540 (2002) 241–246 245

We now analyze the decay mode in minimal super- RPV model, symmetric model with R-parity violation. In the su- + + Amp B → D φ persymmetric models there may be interactions which s RPV violate the baryon number B and the lepton number λ λ∗ f M2 = 2i2 i13 B B L generically. The simultaneous presence of both L 2 (m + m )(m + m ) 4Me˜ b u s c and B number violating operators induce rapid pro- Li 2 ton decay which may contradict strict experimental × ( · Q)2MφA0 Q (33) bound. In order to keep the proton lifetime within ex- and the total amplitude as perimental limit, one needs to impose additional sym- SM+RPV SM metry beyond the SM gauge symmetry to force the A = A (1 − R2), (34) unwanted baryon and lepton number violating inter- where √ actions to vanish. In most cases this has been done ∗ 2 1 λ λ M2 by imposing a discrete symmetry called R-parity de- R = 2i2 i13 B . = − (3B+L+2S) 2 ∗ 2 + + fined as Rp ( 1) ,whereS is the intrinsic GF VubVcsa1 4M˜ (mb mu)(ms mc) eLi spin of the particles. Thus the R-parity can be used to (35) =+ distinguish the particle (Rp 1) from its superpart- Using λ λ∗ = 2.88 × 10−3 [19], we obtain the =− 2i2 i13 ner (Rp 1). This symmetry not only forbids rapid branching ratio in RPV model as proton decay, it also render stable the lightest super- + → + = × −4 symmetric particle (LSP). However, this symmetry is Br B Ds φ 3.06 10 . (36) ad hoc in nature. There is no theoretical arguments in Thus we found that the branching ratio in RPV model support of this discrete symmetry. Hence it is interest- is quite large in comparison to SM prediction. So if the ing to see the phenomenological consequences of the experimental value is found to be in this range, it will R B breaking of -parity in such a way that either and definitely be a signal of new physics beyond the SM. L number is violated, both are not simultaneously vi- Replacing R2 in place of R1 in Eq. (29), the direct olated, thus avoiding rapid proton decays. Extensive + → + CP asymmetry in the decay mode B Ds φ in RPV studies has been done to look for the direct as well as model is found to be indirect evidence of R-parity violation from different + → + processes and to put constraints on various R-parity aCP B Ds φ RPV 14%. (37) R violating couplings. The most general -parity and To conclude, in this Letter we have calculated the lepton number violating super-potential is given as branching ratio of the two body hadronic decay mode + → + 1 c c B Ds φ in the standard model as well as in the WL/ = λij k Li Lj E + λ Li Qj D , (31) 2 k ij k k two Higgs doublet model and RPV model. We found where i, j, k are generation indices, Li and Qj are that the branching ratio in the RPV model is quite c large in comparison to the SM prediction, whereas the SU(2) doublet lepton and quark superfields and Ek , c 2HDM prediction is approximately one order higher Dk are lepton and down type quark singlet superfields. Further, λij k is antisymmetric under the interchange of than the SM value. The direct CP asymmetry aCP, the first two generation indices. Thus the relevant four which is expected to be zero in the SM, is found to be fermion interaction induced by the R-parity and lepton nonzero and large in both RPV and 2HDM analyses. number violating model is From our analyses it follows that the new physics ∗ contribution is quite large and significant. Therefore λ λ + → + H =− 2i2 i13 ¯ − ¯ + the rare decay mode B Ds φ provides an ideal R/ 2 u(1 γ5)b s(1 γ5)c, (32) 4M˜ testing ground to look for new physics. eLi where the summation over i = 1, 2, 3 is implied. It should be noted that the RPV Hamiltonian has the References same form as the 2HDM Hamiltonian except the couplings. So using the Eqs. (22) and (23) one can [1] P.F. Harrison, H.R. Quinn (Eds.), The Babar Physics Book, + → + easily obtain the amplitude for the B Ds φ in SLAC-R-504, 1998. 246 R. Mohanta / Physics Letters B 540 (2002) 241–246

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No-go for detecting CP violation via neutrinoless double beta decay

V. Barger a,S.L.Glashowb, P. Langacker c,D.Marfatiab

a Department of Physics, University of Wisconsin, Madison, WI 53706, USA b Department of Physics, Boston University, Boston, MA 02215, USA c Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, PA 19104, USA Received 1 June 2002; accepted 21 June 2002 Editor: H. Georgi

Abstract We present a necessary condition on the solar oscillation amplitude for CP violation to be detectable through neutrinoless double beta (0νββ) decay. It depends only on the fractional uncertainty in the νe–νe element of the neutrino mass matrix. We demonstrate that even under very optimistic assumptions about the sensitivity of future experiments to the absolute neutrino mass scale, and on the precision with which nuclear matrix elements that contribute to 0νββ decay are calculable, it will be impossible to detect neutrino CP violation arising from Majorana phases.  2002 Elsevier Science B.V. All rights reserved.

If neutrinos are Majorana, the potentiality of detecting CP violation using neutrinoless double beta decay exists [1]. We consider this in a scenario wherein there are exactly three left-handed neutrino states with Majorana masses. We derive a necessary condition that involves the solar oscillation amplitude and the uncertainty in the νe–νe element of the neutrino mass matrix that must be satisﬁed for CP violation to be detectable. Assuming that this condition is satisﬁed and allowing for experimental and theoretical uncertainties that are unrealistically small in some cases, we show that it will not be possible to detect neutrino CP violation through this process. The charged-current eigenstates are related to the mass eigenstates by a unitary transformation −iδ νe ν1 c2c3 c2s3 s2e ν1 iδ iδ νµ = UV ν2 = −c1s3 − s2s1c3e c1c3 − s2s1s3e c2s1 V ν2 , (1) iδ iδ ντ ν3 s1s3 − s2c1c3e −s1c3 − s2c1s3e c2c1 ν3 φ φ i 2 i( 3 +δ) where si and ci are the sines and cosines of θi,andV is the diagonal matrix (1,e 2 ,e 2 ). In Eq. (1), φ2 and φ3 are the Majorana phases that are not measurable in neutrino oscillations and which are either 0 or π if CP is conserved. The solar neutrino data allow two solutions at the 3σ C.L. [2]:

2 −5 −4 2 0.56 sin 2θ3 0.95, 2.0 × 10 ∆s 2.3 × 10 eV (LMA), (2)

E-mail address: [email protected] (D. Marfatia).

2 −8 −7 2 0.80 sin 2θ3 0.94, 9.0 × 10 ∆s 2.0 × 10 eV (LOW) (3) of which the LMA is the only solution at the 99% C.L. No solution with θ3 π/4(cos2θ3 0) is allowed at the 2 −3 2 −3 2 5σ C.L. [2]. Atmospheric neutrino data imply sin 2θ1 0.85 and 1.1 × 10 eV ∆a 5 × 10 eV at the 2 99% C.L. [3]. The CHOOZ reactor experiment imposes the constraint sin 2θ2 0.1 at the 95% C.L. [4]. ∆s and ∆a are the mass-squared differences relevant to solar and atmospheric neutrino oscillations, respectively. We choose the mass ordering m1

Here, the mass of the electron neutrino me is what tritium beta decay experiments seek to measure. It is related to the sum of neutrino masses (Σ = Σmi ) obtainable from data on weak lensing of galaxies and the CMB via = + 2 ± Σ 2me me ∆, (10) where the plus sign applies to the normal hierarchy and the minus sign to the inverted hierarchy. Using Eqs. (6)–(9), the Majorana phase in either case is given by 2 = − 2 − Mee cosφ 1 2 1 2 . (11) sin 2θ me A. Barger et al. / Physics Letters B 540 (2002) 247–251 249

The second term on the right-hand side quantifies the deviation of φ from 0. If Mee/me = 1, φ = 0andif Mee/me = cos 2θ, φ = π. Thus, M cos2θ ee 1, (12) me with the boundaries of the interval corresponding to CP conservation. A measurement of Mee/me that excludes the boundaries constitutes a detection of CP violation. A larger solar amplitude (a wider interval) is therefore more conductive to such a measurement. Let us evaluate the minimum sin2 2θ for which CP violation can be detected assuming a measurement Mee(1 ± x),wherex is obtained by summing the theoretical uncertainty in the 0νββ nuclear matrix elements and the experimental uncertainty in quadrature. Then, for CP violation to be detectable the necessary condition is cos2θ M 1 < ee < , (13) 1 − x me 1 + x or 1 − x 2 sin2 2θ>1 − . (14) 1 + x The current solar data require sin2 2θ to be smaller than 0.95 at the 3σ C.L. Thus, x must be smaller than 0.63. It is +2.0 a difficult task to reduce the factor of 3 uncertainty on the nuclear matrix elements (corresponding to Mee(1−0.7)) to such a degree especially since a reliable method for estimating the uncertainty does not exist1 [7]. Conversely, for a realistically achievable improvement in x, it is unlikely that the solar amplitude is sufficiently large so as to satisfy Eq. (14). In what follows, we consider the remote possibility that Eq. (14) is in fact satisfied, and show that it is not sufficient to detect CP violation. We work under the following assumptions about the experimental and theoretical developments that might occur by 2020.

(1) Experiments like GENIUS [8] and EXO [9] are sensitive to Mee above 0.01 eV with a 25% experimental uncertainty [7] and are therefore not sensitive to the hierarchical neutrino mass spectrum. Also, since Mee me, we can draw meaningful conclusions only for me 0.01 eV or equivalently Σ 0.08 eV. (2) A breakthrough in the evaluation of the nuclear matrix elements has allowed an estimate of their uncertainty. The factor of 3 uncertainty that has plagued the matrix elements is reduced so that the combined theoretical 2 and experimental uncertainty on Mee is x = 0.5. Then Eq. (14) is satisﬁed for sin 2θ>0.89. 2 (3) Tritium beta decay experiments like KATRIN are sensitive to me above 0.35 eV with an uncertainty of 0.08 eV 2 on me [10]. (4) Weak lensing of galaxies by large scale structure in conjunction with CMB data can measure Σ to an uncertainty of 0.04 eV [11]. (5) The KamLAND (Borexino) experiment has determined the solar oscillation amplitude to be 0.95 ± 0.04 where the precision in the LMA (LOW) region was estimated in Ref. [12] ([13]). (6) The JHF-Kamioka neutrino project has measured ∆ to be 3 × 10−3 eV2 to within 3% and has constrained 2 −3 sin θ2 to be smaller than 2 × 10 [14]. (7) The neutrino mass hierarchy is determined either from a superbeam experiment or from supernova neutrinos.

While some of these assumptions are overly optimistic, they serve well to show once and for all that it is not possible to detect CP violation from 0νββ decay in the foreseeable future.

1 The often quoted factor of 3 uncertainty represents the range of calculated values of the matrix elements available in the literature. 250 A. Barger et al. / Physics Letters B 540 (2002) 247–251

Fig. 1. “1σ” bands in the (Mee/me)–cos φ plane for three possible measurements assuming the normal hierarchy: (a) Σ = 0.24 ± 0.04 eV, Mee = 0.03(1 ± 0.5) eV with the central value of Mee/me = 0.4, (b) Σ = 0.51 ± 0.04 eV, Mee = 0.10(1 ± 0.5) eV with the central value of Mee/me = 0.6, (c) Σ = 1.12 ± 0.04 eV, Mee = 0.30(1 ± 0.5) eV with the central value of Mee/me = 0.8. Results are shown for the best possible solar amplitude (allowed by present solar data at 3σ) for the detection of CP violation, sin2 2θ = 0.95 ± 0.04. ∆ is ﬁxed at − 3 × 10 3 eV2.

Our assumptions clearly suggest that cosmology will not only probe smaller neutrino masses, but also with greater precision than tritium beta decay experiments. However, to have the possibility of an independent conﬁrmation from a table-top experiment (in the regime of common sensitivity) is invaluable. In our quantitative analysis, we assume the precision on Σ expected from cosmology. We ﬁx ∆ = 3 × 10−3 eV2 and assign no uncertainty to its value. We suppose that precise measurements of Σ and Mee will be made such that the central value of Mee/me lies in the interval of Eq. (12). One expects the extent to which CP violation can be detected to be dependent upon how close the central value of Mee/me is to the middle of the interval. A. Barger et al. / Physics Letters B 540 (2002) 247–251 251

For hypothetical measurements of Σ and Mee, the regions allowed by scans within the 1σ uncertainties assumed for each of the measurements are shown in Fig. 1 for the normal hierarchy. We have chosen the central values of Mee/me to be 0.4, 0.6 and 0.8 so that they are not too√ close to either cos 2θ (= 0.22) or 1 for which cosφ =−1and cos φ = 1, respectively, are unavoidable. For Σ ∆, the regions for both hierarchies are almost identical. For the case in which Σ = 0.24 eV, the regions for the inverted hierarchy do not extend to as high values of Mee/me as for the normal hierarchy, but qualitatively they are the same in that cosφ =−1 is allowed. We emphasize that while CP violation is not detectable via 0νββ decay, if the solar amplitude is found to be larger than current solar data suggest and if the precision on the various measurements and the reﬁnement of the calculation of the nuclear matrix elements assumed by us is achieved, it might be possible to determine if φ is closer to 0 or to π (see Fig. 1).

Acknowledgements

We thank S. Dodelson for drawing our attention to Ref. [11]. This work was supported in part by the NSF under grant No. NSF-PHY-0099529, in part by the US Department of Energy under grant Nos. DE-FG02-91ER40676, DE-FG02-95ER40896 and DOE-EY-76-02-3071, and in part by the Wisconsin Alumni Research Foundation.

References

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Radion and Higgs signals in peripheral heavy ion collisions at the LHC

S.M. Lietti a,C.G.Roldãob

a Instituto de Física da USP, C.P. 66.318, São Paulo, SP 05389-970, Brazil b Instituto de Física Teórica, Universidade Estadual Paulista, Rua Pamplona 145, CEP 01405-900 São Paulo, Brazil Received 27 May 2002; accepted 17 June 2002 Editor: M. Cveticˇ

Abstract We investigate the sensitivity of the heavy ion mode of the LHC to Higgs boson and Radion production via photon–photon fusion through the analysis of the processes γγ → γγ, γγ → bb¯,andγγ → gg in peripheral heavy ion collisions. We suggest cuts to improve the Higgs and Radion signal over standard model background ratio and determine the capability of LHC to detect these particles production.  2002 Elsevier Science B.V. All rights reserved.

1. Introduction vances in string theories have revolutionized our per- spectives and understanding of the problems, namely, The standard model (SM) has been very success- the Planck, grand unification, and string scales can be ful in accounting for almost all experimental data. brought down to a TeV range with the help of ex- The Higgs boson is the only particle in the SM that tra dimensions, compactified or not. Arkani-Hamed has not yet been confirmed experimentally. It is re- et al. [2] proposed that using compactified dimensions sponsible for the mass generation of fermions and of large size (as large as mm) can bring the Planck gauge bosons. The search for the Higgs boson is the scale down to TeV range. Randall and Sundrum [3] main priority in high energy experiments and hints proposed a 5-dimensional space–time model with a of its existence may have been already seen at LEP nonfactorizable metric to solve the hierarchy problem. The Randall–Sundrum model (RSM) has a four- [1] at around mH ∼ 115 GeV. Nevertheless, the SM can only be a low energy limit of a more fundamen- dimensional massless scalar, the modulus or Radion. tal theory because it cannot explain a number of the- The most important ingredients of the above model oretical issues, one of which is the gauge hierarchy are the required size of the Radion field such that it problem between the only two known scales in parti- generates the desired weak scale from the scale M ≈ cle physics—the weak and Planck scales. Recent ad- ( Planck scale) and the stabilization of the Radion field at this value. A stabilization mechanism was proposed by Goldberger and Wise [4]. As a consequence E-mail address: [email protected] (S.M. Lietti). of this stabilization, the mass of the Radion is of or-

0370-2693/02/$ – see front matter  2002 Elsevier Science B.V. All rights reserved. PII:S0370-2693(02)02148-2 S.M. Lietti, C.G. Roldão / Physics Letters B 540 (2002) 252–262 253 der of O(TeV) and the strength of coupling to the 2. Effective Lagrangian for the Radion couplings SM fields is of order of O(1/TeV) [5]. Therefore, the detection of this Radion will be the first signature of In order to describe the interactions of the RSM the RSM and the stabilization mechanism by Gold- Radion with the SM particles, we follow the notation berger and Wise. of Ref. [7]. These interactions are model-independent Higgs and Radion can be produced in various types and are governed by 4-dimensional general covari- of accelerators. Several papers have been published ance, and thus given by the following Lagrangian in order to study the possibility of detection of the R Higgs particle in e+e−, µ+µ−, pp¯, pp and γγ col- L = T µ(SM), (1) int Λ µ liders [6]. Recently, the phenomenology of the Radion R + − µ particle has been also studied for e e , pp and γγ where ΛR =R is of order TeV, and Tµ is the trace colliders [7]. In this Letter we explore the possibil- of SM energy–momentum tensor, which is given by ity of an intermediate-mass Higgs boson or Radion µ = ¯ − 2 + −µ − 2 µ scalar be produced in peripheral heavy ion collisions Tµ (SM) mf ff 2mW Wµ W mZZµZ f through photon–photon interactions [8,9]. The reason + 2 2 − µ +··· to choose photon–photon fusion in peripheral heavy 2mH H ∂µH∂ H , (2) ion collisions resides in the fact that the production where ··· denotes higher order terms. The couplings mode is free of any problem caused by strong interac- of the Radion with fermions and W, Z and Higgs tions of the initial state, which make these processes bosons are given in Eq. (1). Note that the couplings cleaner than pomeron–pomeron or pomeron–photon of the Radion with fermions, W,andZ are similar to fusions. In the context of the SM, the Higgs boson the couplings of the Higgs to these particles, the only has been explored in detail in the literature [10,11], difference resides in the coupling constants where v, with the general conclusion that the chances of find- the vev of the Higgs field, is replaced by Λ . ing the SM Higgs in the photon–photon case are mar- R The coupling of the Radion to a pair of gluons (pho- ginal. On the other hand, a study of Radion produc- tons) is given by contributions from 1-loop diagrams tion in peripheral heavy ion collisions has not yet been with the top-quark (top-quark and W) in the loop, sim- made. ilar to the Higgs boson couplings to the same pair. The Higgs couplings considered in this Letter are However, for the Radion case, there is another contri- given by the usual SM Lagrangian while the Ra- bution coming from the trace anomaly for gauge fields, dion effects can be described by effective opera- that is given by tors involving the spectrum of the SM and the Ra- dion scalar field. The Radion couplings to the SM βa(ga) T µ(SM)anom = F a F aµν. (3) particles are similar to the Higgs couplings to the µ g µν a 2 a same particles, except from a factor involving the Higgs and the Radion vacuum expectation values For the coupling of the Radion to a pair of gluons, =− = − (vev’s), as can be seen in Section 2. In Section 3 βQCD/2gs (αs/8π)bQCD,wherebQCD 11 = we present the strategy to evaluate photon–photon fu- 2nf /3 with nf 6. Thus, the effective coupling of sion processes in peripheral heavy ion collisions and Rg(p1,µ,a)g(p2,ν,b), including the 1-loop diagrams of in Section 4 we explore the capabilities of peripheral top-quark and the trace anomaly contributions is given heavy ion collisions in detecting Higgs and Radion by productions by analyzing the processes γγ → γγ, iδabαs + + − bb¯,andgg. After simulating the signal and back- bQCD yt 1 (1 yt )f (yt ) 2πΛR ground, we find optimal cuts to maximize their ra- × (p1p2gµν − p2 p1 ), (4) tio. We show how to use the invariant mass spectra µ ν ¯ = 2 of the final state γγ, bb,andgg pairs in order to where yt 4mt /2p1p2. improve the SM Higgs boson and RMS Radion sig- TheeffectivecouplingofRγ(p1,µ)γ (p2,ν),in- nals. Finally, in Section 5 we draw our final conclu- cluding the 1-loop diagrams of the top-quark and sions. W boson, and the trace anomaly contributions is given 254 S.M. Lietti, C.G. Roldão / Physics Letters B 540 (2002) 252–262 by where σ(ˆ s)ˆ is the cross section of the subprocess γγ → X. iαem b + bY − 2 + 3yW + 3yW (2 − yW )f (yW ) We choose to use the conservative and more realis- 2πΛ 2 R tic photon distribution of Cahn and Jackson [11], in- 8 cluding a prescription proposed by Baur [9] for re- + yt 1 + (1 − yt )f (yt ) 3 alistic peripheral collisions, where we must enforce × − (p1p2gµν p2µ p1ν ), (5) that the minimum impact parameter (bmin) should be larger than R + R ,whereR is the nuclear radius of where y = 4m2/2p p , b = 19/6andb =−41/6. 1 2 i i i 1 2 2 Y the ion i. A useful fit for the two-photon luminosity In the above, the function f(z)is given by  is: − 2  1 √1 2 2 sin z ,z1, dL Z α 16 f(z)= √ = ξ(z), (8)  − 1 1+√1−z − 2 dτ π 3τ 4 log − − iπ ,z<1. √ 1 1 z where z = 2MR τ, M is the nucleus mass, R its ra- Eqs. (1)–(5) give all necessary couplings to perform dius and ξ(z) is given by calculations on decays and production of the Radion. 3 In order to perform calculations on the decays and pro- − ξ(z)= A e bi z, (9) duction of the Higgs, we consider its SM couplings, i = widely discussed in the literature. i 1 which is a fit resulting from the numerical integration of the photon distribution, accurate to 2% or better 3. Simulations for 0.05

In this Letter we consider electromagnetic processes efficiency of reconstruction for H → qq¯ or gg of 80%. of peripheral Ar–Ar and Pb–Pb collisions in order to Taking all these efficiencies into account, the cross produce a Higgs and/or Radion scalar via photon– sections are evaluated with a total efficiency factor of photon fusion since the pomeron contributions are 70(32)[80]% for the decay H or R → γγ(bb)¯ [gg]. negligible for subprocesses with center of mass en- The results are presented in Table 1 for a Higgs and ergy close to the Higgs mass. According to Ref. [15], Radion masses of 115 GeV, with ΛR = 4v ≈ 1TeV,in 40 208 the total center of mass energy for 18Ar ( 82Pb) is peripheral Ar–Ar and Pb–Pb collisions at LHC. Re- equal to 7 (5.5) TeV/nucleon and an average luminos- sults for Ca–Ca collisions at LHC can also be ob- × 29 × 26 −2 −1 ity of 5.2 10 (4.2 10 ) cm s , which implies tained, according to Eq. (8), by simply multiplying Z 4 an effective photon–photon luminosity for mγγ = the results for Ar–Ar collisions by the factor Ca = × 28 × 26 −2 −1 ZAr 115 GeV equals to 2 10 (8 10 ) cm s 20 4 − − ≈ 1.524. [0.63 (0.0025) pbarn 1 year 1] at LHC, as can be 18 In order to improve the Higgs and Radion sig- seen in Fig. 1, which was extracted from Ref. [15]. nal over SM background, i.e., all other Feynman di- We will also consider the optimistic possibility of agrams that contribute to the process considered, we Ca–Ca collisions [16,17], where the total center of have studied several kinematical distributions of the fi- mass energy for 40Ca is equal to 7 TeV/nucleon 20 nal state particles. Since the Higgs and Radion interac- and an average luminosity of 5 × 1030 cm−2 s−1, tions occur mainly when these particles are produced which implies an effective photon–photon luminosity on-shell, the most promising one is the invariant mass for m = 115 GeV equals to 1.92 × 1029 cm−2 s−1 γγ of the final particles. (6 pbarn−1 year−1). The behavior of the normalized invariant mass distribution of the final state particles is plotted in Fig. 2 for the process γγ → bb¯ with a Higgs mass and 4. Results

In our analyses, we computed the cross sections for the Higgs and Radion production via photon– photon fusion in peripheral heavy ion collisions at LHC, with the subsequent decay of the Higgs and/or Radion into γγ, bb¯ and gg pairs. The main sources of background for these processes are the box diagram for the process γγ → γγ, the usual electromagnetic tree level diagrams for the process γγ → bb¯,and the box diagram γγ → gg and the usual tree level diagrams γγ → qq¯,whereq = u, d, s, c,forthe process γγ → gg. We begin our analyses using similar cuts and efﬁciencies as the ones ATLAS Collaboration [18] applied in their studies of Higgs boson searches. Our initial results are obtained imposing the following acceptance set of cuts: γ(b)[g] | | pT > 25 GeV, ηγ(b)[g] < 2.5,

4Rγγ(bb)[¯gg] > 0.4, (11) Fig. 2. Normalized invariant mass distribution of the bb¯ pair and taking into account an efficiency for reconstruc- with a Higgs mass and a Radion mass equal to 115 GeV and tion and identification of one photon of 84%, an effi- λ = v ≈ ¯ R 4 1 TeV. The full line corresponds to the SM background ciency of reconstruction for H → bb of 90% with a discussed in the text while the dashed (dotted) line corresponds to b-tagging of 60% per each quark b [18], and finally an the Higgs (Radion) contribution. 256 S.M. Lietti, C.G. Roldão / Physics Letters B 540 (2002) 252–262

Table 1 Cross section in pb for the process γγ → final state with mH = mR = 115 GeV and ΛR = 4v = 984 GeV ≈ 1TeVinheavyion collisions at LHC with the acceptance set of cuts of Eq. (11). σBackground stands for the SM background, σHiggs stands for the contribution γγ → H → final state and σRadion stands for the contribution γγ → R → final state

Ion considered Final state σBackground (pb)σHiggs (pb)σRadion (pb) 40 × −2 × −4 × −5 18Ar γγ 1.961 10 1.346 10 3.020 10 40 ¯ × 0 × −1 × −3 18Ar bb 6.927 10 1.038 10 8.982 10 40 × 2 × −3 × −1 18Ar gg 5.682 10 2.874 10 1.334 10 208 × 0 × −3 × −4 82Pb γγ 1.160 10 3.913 10 8.627 10 208 ¯ × 2 × 0 × −1 82Pb bb 3.919 10 3.023 10 2.666 10 208 × 4 × −2 × 0 82Pb gg 3.179 10 8.344 10 3.883 10

Table 2 Cross section in pb for the process γγ → final state with mH = mR = 115 GeV and ΛR = 4v = 984 GeV ≈ 1 TeV in heavy ion collisions at LHC with the refined set of cuts of Eqs. (11) and (12). σBackground stands for the SM background without Higgs and/or Radion diagrams, σHiggs stands for the contribution γγ → H → final state and σRadion stands for the contribution γγ → R → final state

Ion considered Final state σBackground (pb)σHiggs (pb)σRadion (pb) 40 × −3 × −4 × −5 18Ar γγ 2.050 10 1.349 10 3.030 10 40 ¯ × −1 × −1 × −3 18Ar bb 8.486 10 1.038 10 9.254 10 40 × 1 × −3 × −1 18Ar gg 7.170 10 2.875 10 1.338 10 208 × −2 × −3 × −4 82Pb γγ 6.589 10 4.103 10 9.222 10 208 ¯ × 1 × 0 × −1 82Pb bb 2.721 10 3.159 10 2.818 10 208 × 3 × −2 × 0 82Pb gg 2.298 10 8.753 10 4.072 10

a Radion mass equal to 115 GeV and λR = 4v ≈ Considering the effective photon–photon luminosi- 1 TeV. For instance, if we impose an additional cut ties given by Fig. 1 and Refs. [15,16], we note | − | → ¯ ≈ of mbb¯ mH < 15 GeV in the process γγ bb, that the Ar–Ar (Ca–Ca) luminosity is 250 (2500) the value for the SM background cross section in pe- times greater than the Pb–Pb luminosity. On the other ripheral Ar–Ar collisions is reduced from 6.927 pb hand, Table 2 shows that the Pb–Pb cross sections to 0.8486 pb, while the value for the Higgs (Radion) are ≈ 30 (20) times greater than the Ar–Ar (Ca–Ca) cross section (γγ → H(R) → bb¯) is almost unaf- cross sections. Taking into account both luminos- fected, varying from 0.1038 (1.923 × 10−2)pbto ity and cross section behavior for each mode of the 0.1038 (1.919 × 10−2) pb when the invariant mass heavy ion LHC accelerator, one can realize that the cut is imposed. Similar behavior is observed in the total number of events in Ar–Ar (Ca–Ca) collisions processes γγ → γγ and γγ → gg, as can be seen is ≈ 8 (125) times greater than in Pb–Pb collisions, in Table 2. Therefore we collected ﬁnal states γγ, bb¯ which shows that Pb–Pb collisions is less indicated and gg events whose invariant masses fall in bins of than Ar–Ar (Ca–Ca) collisions for photon–photon fu- size of 30 GeV around the Higgs (Radion) mass sion processes with a typical center of mass energy of O(100) GeV. Therefore, the Pb–Pb mode will not be m − 15 GeV

¯ Fig. 3. Cross sections for the processes (a) γγ → γγ,(b)γγ → bb,and(c)γγ → gg,forλR = 4v ≈ 1 TeV, considering events whose invariant masses fall in bins of size of 30 GeV around the mass M, as in Eq. (12). The full line corresponds to the SM background discussed in the text while the dashed (dotted) line corresponds to the Higgs (Radion) contribution. photon luminosity and the invariant mass of the the invariant mass of the initial γγ pair. Similar initial γγ pair, which indicates that a discovery conclusion can be obtained from Fig. 3, where the of a SM Higgs or a RSM Radion production via behavior of the cross sections of the processes γγ → photon–photon fusion is favored for low values for γγ, γγ → bb¯,andγγ → gg, for events whose 258 S.M. Lietti, C.G. Roldão / Physics Letters B 540 (2002) 252–262

¯ Fig. 4. Cross sections for the processes (a) γγ → γγ,(b)γγ → bb,and(c)γγ → gg in terms of the ratio of the vev’s of the Radion (ΛR )and the Higgs (v) ﬁelds. The mass of the Higgs and/or Radion is equal to 115 GeV and the set of cuts given by Eqs. (11) and (12) was applied. The full line corresponds to the SM background discussed in the text while the dashed (dotted) line corresponds to the Higgs (Radion) contribution. invariant masses fall in bins of size of 30 GeV around lower than 200 GeV. Since the latest hints from the mass M used to impose the cut in Eq. (12), the LEP Higgs search [1] experiment indicates a is presented for λR = 4v ≈ 1 TeV. Higher values light Higgs (mH ∼ 115 GeV) and considering that for the cross sections are obtained for masses M the Radion mass may be suppressed relative to the S.M. Lietti, C.G. Roldão / Physics Letters B 540 (2002) 252–262 259

Table 3 Total integrated luminosity needed for a 95% C.L. Higgs signal for the process γγ → final state with mH = 115 GeV in heavy ion collisions at LHC with the refined set of cuts of Eqs. (11) and (12). It is also presented the number of years needed for a 95% C.L. Higgs signal considering − − a luminosity of 0.63(6) pb 1 year 1 for the Ar–Ar (Ca–Ca) mode as discussed in the text − Ion considered Final state Integrated luminosity (pb 1) Years 40 × 5 × 5 18Ar γγ 4.612 10 7.321 10 40 ¯ × 2 × 2 18Ar bb 3.396 10 5.390 10 40 × 7 × 7 18Ar gg 3.333 10 5.290 10 40 × 5 × 4 20Ca γγ 3.026 10 5.044 10 40 ¯ × 2 × 1 20Ca bb 2.228 10 3.713 10 40 × 7 × 6 20Ca gg 2.186 10 3.644 10 warped Planck scale [19], being in the range of a few signal is obtained when S = 1.96 for Gaussian dis- GeV’s, we will only consider in our analysis, from tributions. The results presented in Table 2 for the 40 this point on, a SM Higgs and a RSM Radion mass 18Ar mode show that the SM background cross sec- of 115 GeV. tions are at least one order of magnitude higher Another point considered in our analyses is the than the Higgs or Radion signals. Note that if one dependence between the cross sections for the Radion has one event identified as a Higgs or Radion ex- contribution of the three processes and the ratio of the change, than there will be at least ten SM background vev’s of the Radion (ΛR) and the Higgs (v)fields. events, fact that justifies a Gaussian distribution ap- Fig. 4 shows the behavior of the cross sections in the proach. ΛR range 0.5 v 4. Note that Fig. 4(a) and (b) show Therefore, it is possible to evaluate the integrated that the SM Higgs contribution is greater than the luminosity needed for a 95% C.L. Higgs signal by RSM Radion contribution in the processes γγ → γγ taking in Eq. (13) S = 1.96 and the cross sections ¯ and γγ → bb, while Fig. 4(c) shows that the RSM presented in Table 2, with σTotal givenbythesum Radion contribution is greater than the SM Higgs (σBackground + σHiggs) since the interference effects are contribution in the process γγ → gg. Therefore, the negligible, as checked in our MadGraph/Helas code. process γγ → gg is the most sensitive for a Radion The results are presented in Table 3, where the number search while the other two processes are most sensitive of years needed to establish a 95% C.L. Higgs signal for a Higgs search. is also shown when we consider the accelerator lumi- In order to identify a 95% C.L. signal of a SM nosity given by L = 0.63 pb−1 year−1, as discussed Higgs or a RSM Radion production at the heavy ion above in the text. Table 3 also shows the results for mode of the LHC, let us consider the significance (S) the Ca–Ca mode of the accelerator, with luminosity of a signal given by the equation given by L = 6pb−1 year−1. Analogously, a 95% C.L. √ Higgs plus Radion signal can be considered by simply NTotal − NBackground σTotal − σBackground S = √ = √ L, taking σTotal = (σBackground +σHiggs +σRadion),andthe NTotal σTotal results for this case is presented in Table 4. (13) The results in Table 3 indicate that the process ¯ where N is the number of events, L is the inte- γγ → bb is the best choice to search the Higgs boson grated luminosity of the accelerator, σ is cross sec- because the integrated luminosity needed for a 95% − tion of the process considered. The subscript Back- C.L. signal (≈ 250 pb 1) is three orders of magnitude ground stands for the SM background contribution smaller than the luminosity needed for the process without any Higgs and/or Radion diagrams, and the γγ → γγ and five orders of magnitude smaller than subscript Total stands for the total contribution, in- the luminosity needed for the process γγ → gg. cluding Higgs and/or Radion diagrams. A 95% C.L. However, this integrated luminosity is still very high 260 S.M. Lietti, C.G. Roldão / Physics Letters B 540 (2002) 252–262

Table 4 Total integrated luminosity needed for a 95% C.L. Higgs plus Radion signal for the process γγ → ﬁnal state with mH = mR = 115 GeV and ΛR = 4v = 984 GeV ≈ 1 TeV in heavy ion collisions at LHC with the reﬁned set of cuts of Eqs. (11) and (12). It is also presented the number − − of years needed for a 95% C.L. Higgs plus Radion signal considering a luminosity of 0.63(6) pb 1 year 1 for the Ar–Ar (Ca–Ca) mode as discussed in the text − Ion considered Final state Integrated luminosity (pb 1) Years 40 × 5 × 5 18Ar γγ 3.118 10 4.950 10 40 ¯ × 2 × 2 18Ar bb 2.890 10 4.588 10 40 × 4 × 4 18Ar gg 1.477 10 2.345 10 40 × 5 × 4 20Ca γγ 2.046 10 3.410 10 40 ¯ × 2 × 1 20Ca bb 1.896 10 3.161 10 40 × 3 × 3 20Ca gg 9.693 10 1.615 10

Table 5 Cross section in pb for the process γγ → final state with mH = mR = 115 GeV and ΛR = 4v = 984 GeV ≈ 1 TeV in heavy ion collisions at LHC with the refined set of cuts of Eqs. (11) and (14). σBackground stands for the SM background without Higgs and/or Radion diagrams, σHiggs stands for the contribution γγ → H → final state and σRadion stands for the contribution γγ → R → final state

Ion considered Final state σBackground (pb)σHiggs (pb)σRadion (pb) 40 × −4 × −4 × −5 18Ar γγ 6.433 10 1.346 10 3.024 10 40 ¯ × −1 × −1 × −3 18Ar bb 2.682 10 1.036 10 9.252 10 40 × 1 × −3 × −1 18Ar gg 2.268 10 2.874 10 1.334 10

Table 6 Total integrated luminosity needed for a 95% C.L. Higgs signal for the process γγ → final state with mH = 115 GeV in heavy ion collisions at LHC with the refined set of cuts of Eqs. (11) and (14). It is also presented the number of years needed for a 95% C.L. Higgs signal considering − − a luminosity of 0.63(6) pb 1 year 1 for the Ar–Ar (Ca–Ca) mode as discussed in the text − Ion considered Final state Integrated luminosity (pb 1) Years 40 × 5 × 5 18Ar γγ 1.649 10 2.618 10 40 ¯ × 2 × 2 18Ar bb 1.331 10 2.112 10 40 × 7 × 7 18Ar gg 1.055 10 1.675 10 40 × 5 × 4 20Ca γγ 1.082 10 1.804 10 40 ¯ × 1 × 1 20Ca bb 8.731 10 1.455 10 40 × 6 × 6 20Ca gg 6.922 10 1.154 10 compared to the luminosity expected for both Ar–Ar of magnitude smaller than the one of Table 3. The and Ca–Ca mode, tens of years being needed for a 95% reason for this change is that the Radion contribution C.L. signal detection. is greater than the Higgs contribution only in the The results in Table 4 include the RMS Radion in process γγ → gg, as can be seen in Figs. 3(c) the analysis. There are small changes for the γγ → and 4(c). bb¯ and γγ → γγ processes. The main difference In order to improve the results, one could collected appears in the γγ → gg process, where the integrated final states γγ, bb¯ and gg events whose invariant luminosity needed for a 95% C.L. signal is three orders masses fall in bins of size of 10 GeV around the Higgs S.M. Lietti, C.G. Roldão / Physics Letters B 540 (2002) 252–262 261

Table 7 Total integrated luminosity needed for a 95% C.L. Higgs plus Radion signal for the process γγ → ﬁnal state with mH = mR = 115 GeV and ΛR = 4v = 984 GeV ≈ 1 TeV in heavy ion collisions at LHC with the reﬁned set of cuts of Eqs. (11) and (14). It is also presented the number − − of years needed for a 95% C.L. Higgs plus Radion signal considering a luminosity of 0.63(6) pb 1 year 1 for the Ar–Ar (Ca–Ca) mode as discussed in the text − Ion considered Final state Integrated luminosity (pb 1) Years 40 × 5 × 5 18Ar γγ 1.143 10 1.814 10 40 ¯ × 2 × 2 18Ar bb 1.149 10 1.824 10 40 × 3 × 3 18Ar gg 4.720 10 7.492 10 40 × 4 × 4 20Ca γγ 7.496 10 1.249 10 40 ¯ × 1 × 1 20Ca bb 7.541 10 1.257 10 40 × 3 × 2 20Ca gg 3.097 10 5.161 10

(Radion) mass through the analysis of the processes γγ → γγ, bb¯ and gg in peripheral heavy ion collisions. − + mH(R) 5GeV

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Double diffractive higgs and di-photon production at the Tevatron and LHC

Brian Cox a,JeffForshawa, Beate Heinemann b

a Department of Physics and Astronomy, University of Manchester, Manchester M13 9PL, UK b Oliver Lodge Laboratory, University of Liverpool, Liverpool, L69 7ZE, UK Received 29 May 2002; accepted 17 June 2002 Editor: P.V. Landshoff

Abstract

We use the POMWIG Monte Carlo generator to predict the cross-sections for double diffractive higgs and di-photon production at the Tevatron and LHC. We ﬁnd that the higgs production cross-section is too small to be observable at Tevatron energies, and even at the LHC observation would be difﬁcult. Double diffractive di-photon production, however, should be observable within one year of Tevatron Run II.  2002 Elsevier Science B.V. All rights reserved.

1. Introduction matically accessible, at least in principle. The missing mass method can be used in the case where higgs production is exclusive, that is the process p + p → With the start of Run II at the Tevatron the possi- p + gap + H + gap + p. Unfortunately, recent cal- bility of detecting a light higgs boson has of course culations of this rate indicate that it is too low to be been the focus of much attention. One suggested ex- observable at the Tevatron [2]. The inclusive process, perimentally clean search strategy is to look in double p + p → p + gap + H + X + gap + p, however, was diffractive collisions in which the ﬁnal state contains recently estimated by Boonekamp et al. [3] to be large only intact protons, which escape the central detec- enough to be detectable. If this turns out to be the case, tors, and the decay products of the higgs, in this case then the double diffractive channel might still provide two b quark jets. The protons would be tagged in ro- a relatively clean environment in which to produce man pot detectors a long way down stream of the in- the higgs. In this Letter, we use the POMWIG Monte teraction point, allowing a very precise determination Carlo [4] to predict the inclusive rate. POMWIG imple- of the higgs mass, using the so-called missing mass ments an Ingelman–Schlein type model of diffraction method [1]. With such proton tagging, the available [5] which has proved successful in describing a range center of mass energy of up to 200 GeV means that of diffractive processes in DIS at HERA [6]. We de- the preferred higgs mass of around 115 GeV is kine- scribe the POMWIG physics model in Section 2. We also consider two other double diffractive E-mail addresses: [email protected] (B. Cox), processes: dijet production and di-photon production. [email protected] (B. Heinemann). The CDF Collaboration has measured double dif-

Fig. 1. Di-photon production in POMWIG. Fig. 2. Diffractive higgs production in POMWIG. fractive dijet production rates in Run I [7], and we can therefore compare our predictions to these data. refer to as “pomeron exchange” (P) and “reggeon ex- Ingelman–Schlein type models such as POMWIG typ- change” (R). In the current version of POMWIG it is ically predict rates higher than those observed at the not possible to mix the two exchanges. For particu- 1 Tevatron. Our strategy in this Letter is to attribute this lar details of the POMWIG default parameterisations difference to rapidity gap survival probability: interac- (which we use in this Letter) see [4]. tions between spectator partons in the beam protons Fig. 2 illustrates the mechanism by which we can destroy a rapidity gap produced in a diffractive in- generate higgs boson production in POMWIG.In teraction [9]. We assume that the factor necessary to all cases, the higgs is accompanied by “pomeron scale our dijet predictions to data is to a good approx- remnants”. This whole system is separated by rapidity imation independent of the hard subprocess, and use gaps from the outgoing hadrons due to the requirement it to rescale our higgs and di-photon production cross- that the energy fraction lost by the incoming hadrons, section predictions. ξ, is small (the transverse momentum transfer to the We expect the di-photon process to be an interest- outgoing hadrons is also small). More specifically, ing process to measure in the first years of Run II the total cross-section for producing higgs bosons in since, as we shall see, the rates are almost certainly events where the incoming hadrons lose a fraction of high enough to be observed. Di-photon production their momenta less than ξmax (assuming the top quark is dominated by the quark–quark diagram of Fig. 1. is sufficiently heavy) is This is in contrast to the dijet and higgs production processes which are dominated by gluon exchange. 1 G α2 dx ≈ F √s 2 2 The di-photon process therefore provides complemen- σH τ g1 x,mh g2 τ/x,mh , (1) 288π 2 x tary information to the dijet process about the structure τ of the diffractive exchange in a very clean experimen- √ = 2 tal environment. where τ mh/s ( s is the hadron–hadron centre- of-mass energy) and the diffractive gluon density for hadron i in pomeron exchange is 2. Double diffraction in POMWIG ξmax 2 2 POMWIG generates diffractive hard scattering gi x,Q = dξi fP/i(ξi )gP x/ξi,Q . (2) events in the spirit of the Ingelman–Schlein model. x That is to say it assumes that the production cross- fP (ξ ) is referred to as the pomeron flux factor section factorises into a product of regge flux fac- /i i and gP(β, Q2) is the gluon density of the pomeron, tor and corresponding parton distribution function. At where β is the fractional momentum of the pomeron present, two regge exchanges are included which we carried by the gluon. Similar expressions exist for the subleading exchange which we generically refer to 1 For a recent review see [8] and references therein. as “reggeon”. We use the diffractive parton densities B. Cox et al. / Physics Letters B 540 (2002) 263–268 265

Fig. 3. The inclusive and double diffractive higgs production cross-sections at Tevatron and LHC energies as a function of higgs mass are shown in (a). In (b) the ratio of the double diffractive to inclusive higgs production cross-sections is shown. as extracted from diffractive deep-inelastic scattering to the limited phase space for high mass resonance measurements by the H1 Collaboration [6]. production forced by the requirement that ξ<0.1. The relevant POMWIG subprocesses for di-photon No such dramatic fall is present at LHC energies. The production are shown in Fig. 1. The hard subprocess effect is visible in Fig. 3(b), where we show the ratio of is as in HERWIG 6.1 [10]. double diffractive to inclusive higgs production. The phase space constraints are clearly visible in Figs. 4(a) and (b), where we show the ξ and β distributions 3. Results for a higgs mass of 115 GeV. β and ξ are forced to high values since the center of mass energy of In Fig. 3(a) we show the inclusive higgs production the two gluons participating in the higgs production ˆ ≡ 2 and double diffractive higgs production cross-sections process s sξ1ξ 2β1β2 >mh. In Figs. 5(a) and (b) we at Tevatron and LHC energies as a function of mh.We show the corresponding distributions at the LHC. Put require that ξ<0.1 for the diffractive cross-sections. simply, diffraction as implemented in POMWIG looks The non-diffractive cross-section was generated using very much like a hadron–hadron collision at center the HERWIG 6.1 Monte Carlo and the diffractive cross- of mass energy s ξ 1ξ 2s, which in the case of the section was generated using the POMWIG default Tevatron is of order 200 GeV. It is extremely difficult parameters for both the P and R contributions.2 to produce resonances of 100 GeV or more given this The steep fall in the double diffractive cross-section picture. with increasing higgs mass at the Tevatron is due In Table 1 we show the cross-sections for higgs production at the Tevatron for mh = 115 GeV using two different H1 fits for the pomeron structure function, as 2 Prior to HERWIG 6.3, HERWIG overestimated the higgs pro- described in [6]. Fit 2 is the POMWIG default, whilst duction cross-section by a factor of two. Actually this is quite close to the enhancement typical of NLO QCD corrections [11] and so we fit 3 has a gluon distribution that peaks at high β.Both use the uncorrected results from HERWIG 6.1. distributions were found to fit the H1 measurement 266 B. Cox et al. / Physics Letters B 540 (2002) 263–268

Fig. 4. In (a) the ξ distribution is shown for double diffractive higgs production with higgs mass mh = 115 GeV at the Tevatron. The β distribution is shown in (b).

Fig. 5. In (a) the ξ distribution is shown for double diffractive higgs production with higgs mass mh = 115 GeV at the LHC. The β distribution is shown in (b). B. Cox et al. / Physics Letters B 540 (2002) 263–268 267

Table 1 Predictions for Tevatron Process σ/fb σ/fb σ/fb σ/fb σ/fb (inclusive) PR(P + R)(×S2) ¯ H → bb, mh = 115 GeV (Fit 2) 726.50.189 0.024 0.21 0.02 ¯ H → bb, mh = 115 GeV (Fit 3) 726.50.642 0.024 0.67 0.07 H → WW, mh = 160 GeV 260 0.003 0.0002 0.0032 2 γ , Et > 12 GeV, η<2 72477 265.6 1009.8 1275.4 128 2 γ , Et > 20 GeV, η<1 6345 19.663.282.88.3 2 γ , Et > 7 GeV (CDF cuts) 684 567 1251 125 6 6 6 6 2jetsEt > 7 GeV (CDF cuts) 243 × 10 87 × 10 329.6 × 10 33 × 10

Table 2 Predictions for LHC Process σ/fb σ/fb σ/fb σ/fb (inclusive) PR(P + R) ¯ H → bb, mh = 115 GeV (Fit 2) 37 660 176 100 276 ¯ H → bb, mh = 115 GeV (Fit 3) 37 660 254 100 354 H → WW, mh = 160 GeV 19 050 97 36 133

of the diffractive structure function equally well, al- where the subleading component is significantly re- though measurements of diffractive dijet production at duced. Our prediction of 125 events/fb suggests that HERA suggest that fit 2 is favoured [12]. We also show this process should be observed in the first year of Run the cross-section for mh = 160 GeV, where the domi- II. nant decay mode is to WW. In Table 2 we show our predictions for LHC ener- In order to take gap survival effects into account, gies. In this case, we omit the gap survival factor since we have simulated double diffractive dijet production we cannot estimate it from data. We note the recent in the kinematic range measured by the CDF Collab- estimate of S2 ∼ 0.01 at LHC energies [2]. oration [7], namely, 0.035 <ξp¯ < 0.095, 0.01 <ξp < 0.03, −4.2 <ηjets < 2.4. The jets were found using a cone algorithm with radius 0.7. CDF measured the cross-section to be 43.6 ± 4.4(stat) ± 21.6(syst) nb, 4. Summary a factor of approximately 10 lower than our result. We therefore estimate that the gap survival factor S2 = We have used the POMWIG Monte Carlo generator 0.1. This compares favourably with theoretical esti- to estimate the double diffractive higgs boson and di- mates [2,13,14]. Our final cross-section predictions photon production rates at the Tevatron and LHC. We including our estimate of the gap survival factor are conclude that the higgs production rate is too small to × 2 shown in the column labelled S . be observable at the Tevatron. Double diffractive di- Table 1 also contains our predictions for double dif- γ photon production, an interesting process in its own fractive di-photon production for three choices of Et , γ right, should be observable within the first year of η and ξ cuts. Due to the dominance of the quark initi- Tevatron Run II. The situation at the LHC is less clear, ated subprocess, the diphoton production cross-section primarily due to the uncertainty in the gap survival fac- is dominated by the poorly constrained subleading ex- tors at 14 TeV.If S2 is not much smaller than that at the changes at the ξ values which will be available at Run Tevatron, then we would expect of order 10 events/fb II. We also show the production cross-section in the for mh = 115 GeV, which is a sufficiently high rate to kinematic range used in the CDF dijet measurement see a signal if the background can be controlled. 268 B. Cox et al. / Physics Letters B 540 (2002) 263–268

Acknowledgements [6] C. Adloff et al., H1 Collaboration, Z. Phys. C 76 (1997) 613. [7] T. Affolder et al., CDF Collaboration, Phys. Rev. Lett. 85 We thank Mike Albrow, Valery Khoze, Misha (2000) 4215. Ryskin and Mike Seymour for helpful discussions. [8] B.E. Cox, K. Goulianos, L. Lönnblad, J.J. Whitmore, J. Phys. G 26 (2000) 667. [9] J.D. Bjorken, Phys. Rev. D 47 (1993) 101. [10] G. Marchesini, B.R. Webber, G. Abbiendi, I.G. Knowles, References M.H. Seymour, L. Stanco, Comput. Phys. Commun. 67 (1992) 465. [1] M.G. Albrow, A. Rostovtsev, FERMILAB-PUB-00-173, hep- [11] M. Spira, Fortsch. Phys. 46 (1998) 203. ph/0009336. [12] C. Adloff et al., H1 Collaboration, Eur. Phys. J. C 20 (2001) [2] V.A. Khoze, A.D. Martin, M.G. Ryskin, Eur. Phys. J. C 14 29. (2000) 525; [13] R.S. Fletcher, T. Stelzer, Phys. Rev. D 48 (1993) 5162. V.A. Khoze, A.D. Martin, M.G. Ryskin, Eur. Phys. J. C 19 [14] E. Gotsman, E. Levin, U. Maor, Phys. Lett. B 438 (1998) 229; (2001) 477. E. Gotsman, E. Levin, U. Maor, Phys. Rev. D 60 (1999) [3] M. Boonekamp, R. Peschanski, C. Royon, hep-ph/0107113. 094011. [4] B.E. Cox, J.R. Forshaw, hep-ph/0010303. [5] G. Ingelman, P.E. Schlein, Phys. Lett. B 152 (1985) 256. Physics Letters B 540 (2002) 269–277 www.elsevier.com/locate/npe

Neutrino masses in the left–right supersymmetric model

M. Frank

Department of Physics, Concordia University, 1455 De Maisonneuve Blvd., W. Montreal, QC, H3G 1M8 Canada Received 6 May 2002; received in revised form 17 May 2002; accepted 18 June 2002 Editor: M. Cveticˇ

Abstract We show that in a left–right supersymmetric model with a Higgs structure that supports the see-saw mechanism, the neutrinos get additional contributions to their masses at one loop level. The mechanism responsible is analogous to the Grossman–Haber see-saw mechanism, but the additional mass terms are proportional to the mass difference of the right-handed sneutrinos. We show that the data on both the solar and the atmospheric neutrinos can be accommodated by either two almost degenerate right- handed sneutrinos, or two heavy sneutrino with different, but still relatively small, mass splittings. We discuss the implications of this result for the masses and mixings of the heavy sneutrinos, and the soft-breaking parameters of the left–right supersymmetric model.  2002 Elsevier Science B.V. All rights reserved.

PACS: 13.35.Bv; 12.60.Jv; 14.60.Ef

1. Introduction

The discovery of neutrino oscillations [1] has provided a serious dent in experimental conﬁrmation of the structure of the Standard Model. To accommodate neutrino masses, the Standard Model is most often extended by inclusion of Majorana neutrinos that violate lepton number by two units. This insertion is most natural in a left–right symmetric version of the model. Models extending the Standard Model to account for light Higgs bosons usually include supersymmetry. Some authors have considered a fully supersymmetric extension of the Left–Right Symmetric Model [2–4]. Originally seen as a natural way to suppress rapid proton decay and as a mechanism for providing small neutrino masses [4], this model includes the see-saw mechanism in a supersymmetric context. Unlike its non-supersymmetric counterpart, in this model the neutrino masses are simply obtained by the canonical see-saw [5]. In previous works, neutrino masses were given at tree level. The purpose of this Letter is to analyse the neutrino masses to one-loop level, and show that, because of the left–right couplings, new contributions arise. These contributions are not present in the non-supersymmetric version of the model and could offer some additional insight into the parameters in the scalar neutrino sector.

E-mail address: [email protected] (M. Frank).

2. The left–right supersymmetric model

Left–right supersymmetric models (LRSUSY) are based on the symmetry group SU(2)L × SU(2)R × U(1)B−L, which would then break spontaneously to SU(2)L × U(1)Y [2]. Besides being a plausible symmetry itself, the LRSUSY model has the added attractive feature that it can be embedded in a supersymmetric grand unified theory such as SO(10) [6]. Left–right supersymmetry also arises through building realistic brane worlds from Type I strings [7]. The LRSUSY model has matter doublets for both left- and right-handed fermions and the corresponding left- and right-handed scalar partners (sleptons and squarks) [4]. In the gauge sector, corresponding to SU(2)L and +,− 0 +,− 0 SU(2)R, there are triplet gauge bosons (W ,W )L, (W ,W )R and a singlet gauge boson V corresponding to U(1)B−L, together with their superpartners. The Higgs sector of this model consists of two Higgs bi-doublets, 1 1 1 1 Φu( 2 , 2 , 0) and Φd ( 2 , 2 , 0), which are required to give masses to the up and down quarks. The spontaneous symmetry breaking of the group SU(2)R × U(1)B−L to the hypercharge symmetry group U(1)Y is accomplished by the vacuum expectation values of a pair of Higgs triplet fields ∆L(1, 0, 2) and ∆R(0, 1, 2), which transform as the adjoint representation of SU(2)R. The choice of the triplets (versus four doublets) is preferred, because with this choice one could generate a large Majorana mass for the right-handed neutrino, and a small one for the left-handed neutrino (the see-saw mechanism) [3]. As in the standard model, in order to preserve U(1)EM gauge invariance, only the neutral Higgs fields acquire non-zero vacuum expectation values (VEVs). These values are: 00 00 κ1,2 0 ∆L= , ∆R= , Φu.d. = , vL 0 vR 0 0 κ1,2 where we neglect the possible CP-violating phase in Φ. The Higgs fields acquire non-zero VEVs to break both parity and SU(2)R. In the first stage of breaking, the right-handed gauge bosons, WR and ZR, acquire masses proportional to vR and become much heavier than the left-handed neutral gauge bosons WL and ZL, which pick up masses proportional to κ1 and κ2 at the second stage of breaking. ˜ ˜ ˜ ˜ ˜− ˜− The supersymmetric sector of the model has six singly-charged charginos: λL, λR, φu, φd , ∆L ,and∆R .The ˜ ˜ ˜ ˜0 ˜0 ˜0 ˜0 ˜0 ˜0 ˜0 ˜0 model also has eleven neutralinos: λZ , λZ , λV , φ1u, φ2u, φ1d , φ2d , ∆L, ∆R, δL,andδR. The superpotential for the LRSUSY is: = (i) T c + (i) T c + T + cT c W hq Q τ2Φi τ2Q hl L τ2Φi τ2L i hLRL τ2∆LL hLRL τ2∆RL + + + T MLR Tr(∆LδL ∆RδR) µij Tr τ2Φi τ2Φj . (1)

Here hu, hd , hν and he are the Yukawa couplings for the up and down quarks and neutrino and electron, respectively, and hLR is the coupling for the triplet Higgs bosons.

3. Neutrino states and tree-level masses

After symmetry breaking, the neutrino mass Lagrangian is:

− L =¯c +¯c ∗ c 2 mass nLMν nR nRMν nL. (2)

There are 6 weak neutrino eigenstates, and the symmetric 6 × 6 mass matrix Mν is given by: D = 0 mα Mνα D† , (3) mα MR M. Frank / Physics Letters B 540 (2002) 269–277 271

D = 1 + 2 = with mα hl κ1 hl κ2,andMR 2hLRvR. The mass of the α-flavor neutrino is given by the canonical see-saw formula: =− D 2 mνα mα /MR (4) where we took ∆L=vL = 0, so that the see-saw is not modified by additional terms proportional to ∆L,which would then be subjected to fine-tuning [5]. To find the neutrino eigenstates, we perform a unitary transformation: T nR = UnˆR, such that U Mν U = Mν is a diagonal, positive mass matrix, and

1 ¯ c 1 L =− nˆ Mν nˆR + h.c. ≡− NMν N, (5) mass 2 L 2 =ˆ +ˆc ≡ˆ + ¯ˆT with N nR nL nR CnR,and U ∗ U = L , (6) UR = ˆ c ≡ = ˆ ≡ where νL ULνL UL(PLN),andνR URνR UR(PRN). Similarly, the charged lepton mass matrix is = 1 + 2 l† l = Ml hl κ1 hl κ2, and is diagonalized by the unitary transformation: UL MlUR Ml , with Ml a diagonal, positive × = l ˆ × 3 3 mass matrix. If we denote the physical lepton fields by lL,R UL,RlL,R, we can define the 6 3 leptonic charged current interaction matrices (the leptonic Cabibbo–Kobayashi–Maskawa matrices) as: CKM = † l CKM = † l KL ULUL,KR URUR. (7) The above formulas can easily be generalized to three flavours. For instance, the effective light Majorana mass matrix in the νe,νµ,ντ basis becomes:  √ √  m m m m m √ νe νe νµ √ νe ντ M =  m m m m m  (8) νανβ √ νe νµ √ νµ νµ ντ mνe mντ mνµ mντ mντ with entries defined by (4).

4. One-loop neutrino masses

Having defined the neutrino states, we are in a position to estimate loop corrections to the neutrino masses. At one-loop level, radiative corrections would contribute to neutrino Majorana masses through self-energy diagrams involving sneutrino Majorana masses. The sneutrino masses are generated through the soft-symmetry breaking potential. The form of the SUSY-breaking terms for the Higgs bosons and lepton sector in LRSUSY is given by: i (i) ˜ T ˜ c ˜ T ˜ cT ˜ c (ij)2 † Lsoft = A h L τ2Φi τ2L + ALRhLR L τ2∆LL + L τ2∆RL + m Φ Φj l l Φ i + 2 ˜† ˜ + 2 ˜† ˜ − 2 + + −[ + ] mL ij lLilLj mR ij lRilRj MLR Tr(∆RδR) Tr(∆LδL) h.c. Bµij Φi Φj h.c. , (9) where the A-matrices (Au, Ad , Aν , Ae and ALR) are of similar form to the Yukawa couplings and provide additional sources of flavor violation, and B is a mass term. The full mass for left- and right-handed sneutrino has a complicated 12 × 12 matrix structure [8]:   2 + ∗ T ∗ T A∗ ∗ mL mDmD 0 mDM ν mD  2 † ∗ †   0 m + mDm A mD mDM  m2 =  L D ν  , (10) νÑ˜  ∗ T † A∗ 2 + ∗ T + ∗ † ∗ ∗  M mD mD ν mR M M mDmD ALRM A T ∗ 2 + † + † ν mD MmD ALRMmR MM mDmD 272 M. Frank / Physics Letters B 540 (2002) 269–277

(a) (b)

Fig. 1. One-loop contributions to the neutrino masses in LRSUSY. (a) The left- and right-handed zino contributions, and (b) the left–right higgsino contribution. where Aν = Aν + µ cotβ and mD and M are, respectively, the Dirac and the Majorana mass matrices. At one-loop level, the sneutrinos contribute to the neutrino masses through the Grossman–Haber mechanism [9], where here both the left-handed and the right-handed zinos can participate, Fig. 1(a). But, additionally, this mechanism can proceed through the bi-doublet higgsino, and the neutrinos can get their mass from the mixing of the right-handed sneutrinos, Fig. 1(b). This process cannot occur in the non-supersymmetric version of the model, or in MSSM with additional right-handed neutrinos, and is due entirely to the Higgs structure of the LRSUSY. It would occur in any generic left–right supersymmetric model and does not depend on the choice of triplets versus doublets for symmetry breaking. From Fig. 1(a), the contribution to the neutrino masses comes from the internal loop with neutralinos and sneutrinos, with a .L = 2 left-sneutrino Majorana mass insertion, and it becomes, for LRSUSY: 2 11 | |2 | |2 (1) g ZiZL ZiZR cos2θW m = .m˜ f(x ) + (11) ναβ 2 ναβ i 2 2 32π cos θW cos θW i=1 where Zij is the neutralino mixing matrix [10], and: Z = cos θ Z − cos 2θ tan θ Z − sin θ Z , iZL √ W i2 W W i1 W i3 √ cos2θ xi[xi − 1 − ln(xi)] Z = W Z − tan θ Z , where f(x ) = . (12) iZR i3 W i1 i 2 cosθW (1 − xi) = 2 2 = + ≈ Here xi mν˜ /m˜ 0 and mν˜ (mν˜α mν˜β )/2 is the average light sneutrino mass. As before, the function f(xi) χi 0.2–0.57. This term by itself does not bring anything significantly new to the Grossman–Haber mechanism, even for low ZR, given the unitarity of the neutralino mixing matrix Zij . The light sneutrino Majorana mass in LRSUSY is: 2 ALR .m˜ ≈ Aν − µ cotβ − .mν ν (13) ναβ 2 α β where, as in MSSM with right-handed neutrinos, the parameter ALR can be large, ALR MZL , but this does not generate sufficient sneutrino mixing to significantly enhance the loop contribution over the tree-level one. There is, however, a new left–right contribution given by the graph from Fig. 1(b): 2 11 mDmD ( ) g α β m 1 = f(y ) |Z |2.m ναβ 2 i 2 i4 Nαβ 8π (2MW sin β) i=1 3 mlρ mlσ CKM † CKM† + |Z |2KCKMK KCKMK .m (14) 2 i6 L R L R Nρσ (2MW cosβ) ρ,σ=1 M. Frank / Physics Letters B 540 (2002) 269–277 273

(a) (b)

(c)

Fig. 2. Tree-level contributions to the heavy scalar neutrino masses. Here ν˜ represent the light sneutrino states and N the heavy sneutrino states. = − We denote Vβ κ2λβ (Aνβ µ cot β).

= 2 2 CKM CKM where mlα are the ordinary lepton masses, Nα are the heavy sneutrino states, yi m/m˜ 0 and KL ,KR are N χi the left- and right-handed CKM matrices in the lepton sector. The first term corresponds to the typical higgsino coupling, which is proportional to neutrino masses and thus negligible. The second term is the flavor changing term, which appears in left–right theories only. We assume the leptonic mass matrices to be approximately diagonal, and CKM = CKM consider the case of manifest left–right symmetry, in which KLαj KRj α . The two neutralinos entering this loop are the left–right doublet higgsinos, and the result connects the light neutrino masses with the right-handed 2 (heavy) sneutrino Majorana mass. We estimate .m by calculating the tree-level heavy sneutrino mass [11]. Nαβ The graphs contributing to the sneutrino masses are given in Fig. 2(a), (b) and (c). Note that the contribution from Fig. 2(c) is zero, since we have taken ML = 2HLRvL = 0. The sum from the contributions of Fig. 2(a) and (b) is: mDmD 2 ∼ 1 α β .m = MR − (ALR)αβ + (Aν − µ cotβ) . (15) Nαβ 2 2 mν˜ As expected, the Majorana mass splitting of the right-handed sneutrinos is proportional to MR and potentially very large. In most cases, we expect the first term in the bracket to dominate the second term, which is, in O D 2 −2 = 10 = general of ((m ) /mν˜L ) 10 GeV. Taking as typical values, mR MR 10 GeV, m ˜ 0 500 GeV, we get χ6 −8 f(y6) = 5 × 10 . The light neutrino-charged lepton CKM matrix is obtained from the phenomenological ansatz whereby lepton flavor mixing is consistent with the atmospheric neutrino oscillation data, favoring the large-angle (LMA MSW) solution to the solar neutrino problem. Several parametrizations exist, all including two large angles, such as the democratic [12], the bimaximal [13], or the tri-bimaximal [14]. We will choose the last, and take:  √ √  √2/3 √1/30√ CKM =  −  Kαj √ 1/6 √1/3 √1/2 , (16) 1/6 − 1/3 1/2 although the particular choice of matrix does not affect our estimation significantly, as long as there are two large angles. There are several parameters involved in (14). We will analyse two simple scenarios:

• (I) Single sneutrino dominance. If we assume that, for a given ﬂavor, neutrino masses are dominated by the same ﬂavor heavy sneutrino mass splitting, we obtain: (1) ∼ −21 m = 3 × 10 .m (GeV), (17) νe Ne ( ) ∼ − m 1 = 2.8 × 10 16.m (GeV), (18) νµ Nµ ( ) ∼ − m 1 = 8.2 × 10 15.m (GeV). (19) ντ Nτ 274 M. Frank / Physics Letters B 540 (2002) 269–277

Fig. 3. Values of (ALR)ττ as a function of the right-handed neutrino mass parameter MR (both in GeV), for scenario I (single sneutrino = = = dominance). The dashed curve is for m ˜ 0 100 GeV; the solid curve for m ˜ 0 200 GeV. We take in both cases tan β 5. χ6 χ6

This gives, assuming that the loop contributions alone saturate the experimental bounds consistent with the atmospheric [15] and solar [16] LMA solutions to neutrino oscillations, that the heavy sneutrino mass splitting are of order:

∼ .m ≈ .m = 3 × 104 GeV = 30 TeV. (20) Nµ Nτ ∼ ∼ This is achievable if (ALR)µµ = (ALR)ττ = 200 MeV. • (II) Sneutrino democracy. If we allow neutrino masses to receive contributions from all diagonal entries in the heavy sneutrino mass, we obtain:

(1) ∼ −17 m = 5.5 × 10 .m (GeV), (21) νe Nµ (1) ∼ −17 −15 m = 2.8 × 10 .m ˜ + 2 × 10 .m (GeV), (22) νµ Nµ Nτ ( ) ∼ − − m 1 = 4.2 × 10 17.m + 2 × 10 15.m (GeV). (23) ντ Nµ Nτ M. Frank / Physics Letters B 540 (2002) 269–277 275

= 9 Fig. 4. Values of the trilinear mixing parameter (ALR )ττ as a function of the higgsino mass m ˜ 0 (both in GeV), for ﬁxed MR 10 GeV χ6 10 (dashed curve) for scenario I; and MR = 10 GeV (solid curve). Here we took tan β = 5, too.

Again, if these values are consistent with the atmospheric and solar LMA solutions to neutrino oscillations if the heavy sneutrino mass splitting are of order: ∼ ∼ .m = 106 GeV,.m = 3 × 104 GeV. (24) Nµ Nτ ∼ ∼ This is achievable if (ALR)µµ = 200 GeV and (ALR)ττ = 200 MeV.

It appears that in both of these scenarios (ALR)ττ must be unnaturally small. This would mean that the two heavy τ sneutrinos are almost degenerate in mass, but the mass splitting in the e,µ heavy sneutrino sectors is allowed to be large. With such small values, even an interference between ALR and the light sneutrino mass splitting parameters in (15) is possible, which might drive the mass splitting even lower. −10 8 The function f(yi) is very sensitive to the value of MR and drops to 10 for MR = 10 m ˜ 0 . We have assumed χ6 mν˜ ≈ Aν ≈ µ, but there could be an order of magnitude difference between these quantities. The values of the ALR parameters are quite sensitive to the mass scale of the neutralinos (both in the gaugino and higgsino sectors), and also depend on the values of tan β. For further illustration, we plot in Fig. 3 the dependence of the trilinear ˜ 0 parameter ALR on the mass of the right-handed τ neutrino in scenario I, in the case in which the χ6 neutralino 9 mass is 100 GeV (dashed curve), and 200 GeV (solid curve). We set the lower bound MR 10 GeV, in agreement with bounds from leptogenesis [17]. It is apparent from the figure that, for fixed m ˜ 0 , ALR varies almost linearly χ6 276 M. Frank / Physics Letters B 540 (2002) 269–277 with MR; and the value of ALR is unnaturally small compared to MR throughout all of the parameter space. This is even more evident in Fig. 4, where we plot, also for scenario I, the value of the trilinear mixing parameter ALR as 9 10 a function of the higgsino mass m ˜ 0 (both in GeV), for fixed MR = 10 GeV (dashed curve); and MR = 10 GeV χ6 (solid curve). The values of the parameter ALR remain in the O(100) MeV region for all of the parameter space. We have set tan β = 5 in both of these graphs. In conclusion, we have shown than in generic left–right supersymmetric models, there exists a new contribution to the neutrino masses arising from loops involving the left–right higgsino. This higgsino is the partner of the bi-doublet Higgs boson needed to break the SU(2)R gauge group. The neutrino mass becomes proportional to the heavy sneutrino mass splitting, and can, in general, be quite large. Restricting the heavy sneutrino mass splitting to generate correct values for the solar and atmospheric neutrino oscillations sets bounds on the product between the right-handed neutrino mass and the trilinear (triplet) scalar coupling, MRALR. Although an exact value for heavy sneutrino masses and splittings depends on several parameters of the model, only small regions of the parameter space for MRALR survive. In particular, it appears that (ALR)ττ must be comparatively (with respect to vR and MR) quite small for all values of MR in all scenarios.

Acknowledgement

This work was funded by NSERC of Canada (SAP0105354).

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Constraints on light bottom squarks from radiative B-meson decays

Thomas Becher a, Stephan Braig b, Matthias Neubert b, Alexander L. Kagan c,1

a Stanford Linear Accelerator Center, Stanford University, Stanford, CA 94309, USA b Newman Laboratory of Nuclear Studies, Cornell University, Ithaca, NY 14853, USA c Fermi National Accelerator Laboratory, Batavia, IL 60510, USA Received 31 May 2002; accepted 20 June 2002 Editor: H. Georgi

Abstract ˜ ∼ ∼ The presence of a light b squark (mb˜ 4 GeV) and gluino (mg˜ 15 GeV) might explain the observed excess in b-quark production at the Tevatron. Though provocative, this model is not excluded by present data. The light supersymmetric particles can induce large ﬂavor-changing effects in radiative decays of B mesons. We analyze the decays B → Xsγ and B → Xsg in this scenario and derive restrictive bounds on the ﬂavor-changing quark–squark–gluino couplings.  2002 Elsevier Science B.V. All rights reserved.

1. Motivation

The measured b-quark production cross section at hadron colliders exceeds next-to-leading order (NLO) QCD predictions by more than a factor of two. While it is conceivable that this discrepancy is due to higher-order corrections, the disagreement is surprising since NLO calculations have been reliable for other processes in this energy range. Berger et al. have analyzed b-quark production in the context of the Minimal Supersymmetric Standard Model and find that the excess in the cross section could be attributed to gluino pair-production followed by gluino decay into pairs of b quarks and b˜ squarks, if both the gluino and the b˜ squark are sufficiently light [1]. In order to reproduce the transverse-momentum distribution of the b quarks, the masses of the gluino and light ˜ = = b-squark mass eigenstate should be in the range mg˜ 12–16 GeV and mb˜ 2–5.5 GeV. The masses of all other supersymmetric (SUSY) particles are assumed to be large, of order several hundred GeV, so as to have evaded detection at LEP2. Berger et al. have further observed that a light b˜ squark could have escaped direct detection. For example, the additional contribution to the e+e− → hadrons cross section at large energy would only be about 2% and hence difficult to disentangle. The pair-production of light scalars would alter the angular distribution of hadronic jets in

E-mail address: [email protected] (T. Becher). 1 On leave from: Department of Physics, University of Cincinnati, Cincinnati, Ohio 45221, USA.

0370-2693/02/$ – see front matter  2002 Elsevier Science B.V. All rights reserved. PII:S0370-2693(02)02172-X T. Becher et al. / Physics Letters B 540 (2002) 278–288 279 e+e− collisions, but the present data are not sufficiently precise to rule out the existence of this effect [1]. On the other hand, there are important Z-pole constraints on the parameters of this model. Most importantly, production of the light b˜ squark at the Z pole has to be suppressed, which implies a stringent constraint on the mixing angle θ relating the sbottom mass and weak eigenstates [2]. More recently, several authors have studied loop effects of the light SUSY particles on electroweak precision measurements [3–5]. Potentially large contributions may exist, in particular, to the quantity Rb. However, a conflict with existing data can be avoided by having some of the superpartner masses near current experimental bounds, or by allowing for a new CP-violating phase in the SUSY sector [5]. The null result of a CLEO search for the semileptonic decays B → D(∗)lπ and B → D(∗)lχ˜ 0 of sbottom hadrons implies that the branching ratios for the decays b˜ → cl induced by R-parity violating couplings, or b˜ → clχ0 with an ultra-light neutralino χ0, must be highly suppressed [6]. However, a light b˜ squark would be allowed to decay promptly via hadronic R-parity violating couplings in the modes b˜ →¯cq¯ or b˜ →¯uq¯ (with q = u, s). Alternatively, it could be long-lived, forming b˜-hadrons. An interesting consequence of hadronic R-parity violating decays would be the abundant production of light baryons. This could significantly alter the thrust-axis angular distribution for continuum events containing baryons at the B factories. A striking manifestation of the light b˜-squark scenario would be the production of like-sign charged B mesons at hadron colliders, because the Majorana nature of the gluino allows for the production of bbb˜∗b˜∗ and b¯b¯b˜b˜ final states [1]. Another potential signature at hadron colliders is an enhanced yield of ttb¯ b¯ events [7]. It has also been ˜ ˜∗ ˜ ˜∗ pointed out that sbottom pairs would be copiously produced in Υ(nS)→ bb and χbJ → bb decays [8,9]. Precise measurements of bottomonium decays could lead to new constraints on the squark and gluino masses. The presence of light SUSY particles alters the running of αs , and it is often argued that this would exclude the existence of light gluinos. This argument is incorrect. First, a gluino with mass mg˜ ∼ 15 GeV would have a relatively small effect on the evolution of αs . Taking, for instance, αs (mb) = 0.21 (a value in agreement with all low-energy determinations of the QCD coupling) and including the contribution of the gluino octet to the β function above the scale µg˜ = mg˜ yields αs (mZ) = 0.126, which is about three standard deviations higher than the canonical value αs (mZ) = 0.118±0.003. However, considering that at leading order only virtual gluino pairs contribute to the β function, a more realistic treatment would include the gluino contribution above a scale µg˜ = 2mg˜ ∼ 30 GeV, in which case αs (mZ) = 0.121, in good agreement with the standard value. Secondly, it is important to realize that even a value of αs (mZ) significantly above 0.118 would not rule out the model, the reason being that the characteristic scale µ inherent in all determinations of αs (µ) is typically much smaller than the total energy. This is true, in particular, for the determinations based on event-shape variables. In practice, the measurements fix αs (µ) somewhere between a fraction of the Z mass down to several GeV, where the gluino contribution to the β function is negligible. Using these determinations to quote values of αs(mZ) (as is routinely done) assumes implicitly that the coupling runs as predicted in the SM. Finally, a careful analysis of the running of αs in the presence of light SUSY particles would have to include, for each observable, the modifications in the theoretical formulae due to virtual and real emissions of the new particles. These corrections could be significant, and could partially compensate effects arising from the modification of the β function. If we are to take the possibility of a light b˜ squark and light gluinos seriously, then the theoretical study of their impact must be extended to the phenomenology of weak decays of the b quark. New sources of flavor violation arise from s–b˜–g˜ and d–b˜–g˜ couplings. The overall scale of SUSY flavor-changing interactions originating from 2 2 ∼ 2 2 gluino exchange is set by the factor gs /mg˜, which is much larger than the corresponding factor GF gW /mW for weak decays in the Standard Model (SM). Consequently, the new flavor-changing couplings must be much smaller than the CKM mixing angles in order for this model to be phenomenologically viable. The most stringent bounds ˜ arise from the radiative decay B → Xs γ , which we discuss in the present work. (Contributions of light b squarks to kaon decays, K–K mixing, and D–D mixing are strongly suppressed.) The presence of such tight bounds implies stringent constraints on model building. If the light b˜ squark is sufficiently light to be pair-produced in b decays, new unconventional decay channels would be opened up, which could affect the phenomenology of B mesons and beauty baryons. Examples of 280 T. Becher et al. / Physics Letters B 540 (2002) 278–288 potentially interesting consequences include modifications of beauty lifetime ratios, an enhancement of the semileptonic branching ratio of B mesons via production of charmless final states containing b˜ squarks, an ˜ ˜ enhancement of %Γ (B) and of the semileptonic CP asymmetry ASL, and wrong-sign kaon production via b →¯sbb transitions allowed by the Majorana nature of the gluino. The phenomenology of such effects will be discussed elsewhere. If the light b˜ squark is too heavy to be pair-produced, it would still give rise to potentially large virtual effects in B decays. Their study is the main purpose of this Letter.

2. The low-energy effective B = 1 Hamiltonian ˜ We denote by di with i = 1,...,6 the down-squark mass eigenstates, and by q˜L and q˜R with q = d,s,b the interaction eigenstates (the superpartners of the left-handed and right-handed down quarks). They are related by a ˜ = L† ˜ ˜ = R† ˜ ˜ unitary transformation qL Γqi di and qR Γqi di. We identify d3 with the light sbottom mass eigenstate and R = L = ˜ define a sbottom-sector mixing angle θ through Γb3 cos θ and Γb3 sin θ. The fact that the light b squarks are not produced in Z decays implies m2 ≈± 2 ≈± 2 − W sin θ sin θW 1 2 . (1) 3 3 mZ ˜ ˜ The phenomenologically favored range for the Zd3d3 coupling is | sin θ|=0.3–0.45 [2], meaning that the light sbottom is predominantly the superpartner of the right-handed bottom quark. In our numerical analysis we will assume a vanishing tree-level coupling to the Z and thus use sin θ =±0.395. The flavor-changing couplings involving the light b˜ and g˜ fields can be parameterized by dimensionless quantities AB = A† B AL = AR *qb Γq3 Γb3 (with *qb *qb tan θ), (2) = = → → AB where A,B L,R,andq s or d for b s or b d transitions, respectively. In general the parameters *qb are complex, which can lead to new CP-violating effects. These parameters are invariant under a phase redefinition ˜ ∗ = of the light b-squark state, and they transform in the same way as the products ViqVib (with i u, c, t)ofCKM AB CD matrix elements under a phase redefinition of the down-type quark fields. It follows that ratios of the type *qb /*qb AB ∗ and *qb /(ViqVib) are invariant under phase redefinitions, and thus can carry an observable, CP-violating phase. Flavor-changing hadronic processes in the model with a light gluino and a very light b˜ squark are most transparently described by means of an effective “weak” Hamiltonian. If we neglect effects that are suppressed by inverse powers of the heavy SUSY scale, the relevant energy scales are the electroweak scale, at which the usual SM flavor-changing operators are generated by integrating out the top quark and the W and Z bosons, and the scale mg˜ , at which new flavor-changing operators are generated by integrating out the gluinos. We start by discussing the construction of the effective theory below the gluino scale, focusing on the new interactions proportional AB to *qb induced by gluino exchange, as illustrated in Fig. 1. SUSY modifications of the renormalization-group (RG) evolution of the standard weak-interaction operators will be discussed later. The remaining light degrees of freedom in the low-energy theory are the quarks u, d, s, c, b, the photon and gluons, and the light b˜ squark. Operators in the effective Hamiltonian can be organized in an expansion in inverse powers of the gluino mass. For mg˜ ≈ 15 GeV mb, it is a good approximation to keep the leading terms in this expansion, which have mass dimension five. These operators comprise the usual electromagnetic and chromomagnetic dipole operators, and new operators containing two scalar b˜ fields. The effective Hamiltonian for b → s transitions is (here and below, mg˜ ≡ mg˜(mg˜ ) denotes the running gluino mass at the gluino matching scale) 4πα (m ˜ ) SUSY = s g C LR LR + ↔ + 2 Heff i(µ) *sb Oi (µ) (L R) O 1/mg˜ , (3) m ˜ g i T. Becher et al. / Physics Letters B 540 (2002) 278–288 281

Fig. 1. Examples of b → s transitions induced by gluino exchange. The diagrams on the right show the corresponding contributions in the effective theory where the gluinos are integrated out. where

LR =¯ ˜ ˜∗ LR =¯ ˜ ˜∗ O1 sLtabb tabR,O2 sLbb bR, LR e µν LR gs µν O =− s¯Lσµν F bR,O=− s¯Lσµν G bR. (4) 7 16π2 8 16π2 µ µ µ µ We work with the covariant derivative iD = i∂ + eQd A + gs Aa ta,whereQd =−1/3 is the electric charge of a down-type (s)quark. (To facilitate comparison with the literature, which usually adopts the opposite sign convention for the couplings, we have included a factor of −1 in the definition of the dipole operators O7 and O8.) c ˜∗ ˜∗ In addition, there are dimension-five fermion-number violating interactions of the form s¯ (1 ± γ5)b b b ,which mediate b →¯sb˜b˜ transitions. They are irrelevant to our discussion here. The Wilson coefficients at a scale µ ∼ mg˜ are obtained by matching the effective theory to the full theory. At leading order we find

N2 − 1 N2 + 1 C (m ˜) = 2, C (m ˜) = 0, C (m ˜ ) = Qd , C (m ˜) =− , (5) 1 g 2 g 7 g 4N 8 g 4N where N = 3 is the number of colors. In order to use the effective Hamiltonian for calculating B-decay amplitudes, we compute the values of the Wilson coefﬁcients at a low scale µ ∼ mb by solving the RG equation (d/d ln µ − γ T)C(µ) = 0. At leading order, the anomalous dimension matrix γ receives contributions from the one-loop mixing of the operators (O1,O2) and (O7,O8) among themselves, and from the two-loop mixing of O1,2 into O7,8. This is analogous to the case of the SM, in which one needs to consider the two-loop mixing of the current-current operators into the dipole operators at leading order [10]. In our case, only the three two-loop diagrams shown in Fig. 2 give a nonvanishing contribution. All other graphs vanish after their subdivergences are removed. The calculation of the UV divergences of these diagrams can be reduced to the evaluation of massive tadpole integrals [11]. The resulting anomalous dimension matrix in the operator basis (O1,O2,O7,O8) reads   − + 9 − 3 + 3 1 − 1 − 1 − 1 6N 2 Qd 2 2  N 2 2N 4 4N 8 4N   −6 −3N + 3 Q − N + 1 − N + 1  = αs  N d 2 2N 4 2N  + 2 γ   O αs . (6) 4π  00N − 1 0  N − 4 − 5 00Qd 4N N N N 282 T. Becher et al. / Physics Letters B 540 (2002) 278–288

O1 → O7 O2 → O7 O1 → O8 O2 → O8 − 1 + 3 − 1 + 1 N − 1 1 N − 1 D1 2 16* Qd 4 2 Qd 2 2N 2 2 2N 8* 4N 4N − 1 + 3 − 1 + 1 N − 1 1 + 1 − 1 D2 2 16* Qd 4 2 Qd 2 2N 4 2 2N 8* 4N 4N D 1 − 1 Q − 1 + 1 Q N − 1 1 + 1 N − 1 3 4*2 8* d 4 4N2 d 2 2N 8 4N2 4 2N

Fig. 2. Two-loop diagrams relevant to the mixing of O1,2 into O7,8, and corresponding results, in units of αs /4π, after the subtraction of subdivergences. Mirror-symmetric graphs with the gluon attached to the s-quark line give identical contributions. Results in the ﬁrst column of the attached table have to be multiplied by the color and charge factors in the remaining columns.

Table 1 Results for the Wilson coefﬁcients and the running b-quark mass for different values of µ. Input parameters are mb(mb) = 4.2GeV, mt (mW ) = 174 GeV, mg˜ (mg˜ ) = 15 GeV, and αs (mb) = 0.21. In the upper portion of the table the gluino is integrated out at µ = mg˜ ,in the lower portion at µ = 2mg˜ . If two signs are shown, the upper (lower) one refers to positive (negative) mixing angle θ

Scale mb(µ) [GeV] C1 C2 C7 C8 C2 C7γ C8g mW 3.17 ∓ 0.42––––1−0.195 −0.097 mg˜ 3.59 2 0 −0.222 −0.833 1.040 −0.255 ± 0.028 −0.124 ± 0.014 mb 4.20 2.691 0.066 −0.264 −0.804 1.104 −0.313 ± 0.023 −0.143 ± 0.011 mW 3.13 ∓ 0.25––––1−0.195 −0.097 mg˜ 3.59 2.274 0.025 −0.241 −0.821 1.042 −0.255 ± 0.015 −0.124 ± 0.008 mb 4.20 3.064 0.104 −0.280 −0.791 1.106 −0.313 ± 0.013 −0.143 ± 0.006

The scale dependence of the Wilson coefficients is now readily obtained by solving the RG equation. Setting N = 3, we find 16 −8 2 −7/2 8 −8 −7/2 C1(µ) = η + η , C2(µ) = η − η , 9 9 27 4 −8 4 −7/2 438 2/3 1224 4/3 C7(µ) = Qd − η − η + η − η , 273 145 65 203 1 − 1 − 219 C (µ) = η 8 − η 7/2 − η2/3 . (7) 8 39 60 260 1/β (5,0,1) Here η =[αs (mg˜)/αs (µ)] 0 ,and 11N 2 2N 1 β (nf ,ng,ns ) = − nf − ng − ns (8) 0 3 3 3 6 is the first coefficient of the generalized QCD β function in the presence of nf light Dirac fermions, ng light gluino octets, and ns light complex scalars. Numerical results for the coefficients Ci(µ) will be given in Table 1. The scale dependence of the Wilson coefficients below the scale mg˜ arises mainly from the mixing of O1 with O2 and O7 with O8. The mixing of O1 and O2 into the dipole operators turns out to be small numerically. T. Becher et al. / Physics Letters B 540 (2002) 278–288 283

The presence of light SUSY particles also affects the RG evolution of the SM contributions to the effective weak Hamiltonian below the electroweak scale. We will now discuss these effects for the operators of relevance to radiative B decays.

3. The radiative decay B → Xsγ

The inclusive radiative decay B → Xs γ is one of the most sensitive probes of physics beyond the SM. Indeed, LR RL we will see that this decay provides very stringent bounds on the ﬂavor-changing couplings *sb and *sb .The SM prediction for the B → Xs γ decay rate is known at NLO [12–15] and, within errors, agrees with the data. The change of this prediction due to the light SUSY particles present in our model is fourfold.

(1) The main effects are the genuine SUSY ﬂavor-changing interactions due to quark–squark–gluino couplings. For µ

The last two effects change the evolution of the Wilson coefficients of the SM operators between the electroweak scale and the scale µ ∼ mg˜ , where the gluino degrees of freedom are integrated out. (Beyond leading order, the anomalous dimensions of the SM operators are also changed due to internal loops involving SUSY particles.) The effective weak Hamiltonian governing B → Xs γ decays in the SM is G weak =−4√ F ∗ Heff VtsVtb Ci (µb)Qi (µb). (9) 2 i The operators relevant to our calculation are =¯i j ¯j µ i =¯ ¯ µ = LR = LR Q1 sLγµcLcLγ bL,Q2 sLγµcLcLγ bL,Q7γ mbO7 ,Q8g mbO8 . (10) To an excellent approximation, the contributions of other operators can be neglected. To obtain the values of the corresponding Wilson coefficients Ci (µb) in our model, we first evolve them from the electroweak scale µ = mW down to a scale µg˜ ∼ mg˜ , and in a second step from µg˜ to a scale µb ∼ mb. Above the gluino scale, there are SUSY contributions to the wave-function renormalization constants of left- and right-handed b-quark fields from gluino–squark loops. At one-loop order, we obtain the gauge-independent results

CF αs 2 CF αs 2 δZ (bL) =− sin θ, δZ (bR) =− cos θ. (11) 2 4π* 2 4π* Next, by calculating the self-energies of b quarks and gluinos we find that their masses mix under renormalization. The corresponding anomalous dimension matrix in the basis (mb,mg˜), defined such that dmi/d ln µ =−(γm)ij mj , 284 T. Becher et al. / Physics Letters B 540 (2002) 278–288 reads 5N − 5 − 1 αs 2 2N N N sin 2θ 2 γm = + O α . (12) 4π − 1 s sin 2θ 6N 2 Note that the off-diagonal entries are sensitive to the sign of the mixing angle θ. The RG evolution of the operators Qi in (9) is complicated by the effects of mass mixing. To compute the resulting modifications of the Wilson LR LR LR LR coefficients we work in the extended operator basis (Q1,Q2,mbO7 ,mbO8 ,mg˜ O7 ,mgÕ8 ). Using (11), (12), and the well-known anomalous dimensions of the SM operators [10], we obtain for the anomalous dimension = αs (0) + 2 = =− matrix γQ 4π γQ O(αs ), where (setting N 3andQd 1/3)   4 2 − 3 sin θ 260300  4 2 416 70   6 sin θ − 2 00  3 81 27   32 − 4 2 θ 8 θ  (0)  003 3 sin 0 3 sin 2 0  γ =   . (13) Q  00− 32 28 − 4 sin2 θ 0 8 sin 2θ   9 3 3 3   43 − 4 2  00sin2θ 0 2 3 sin θ 0 − 32 121 − 4 2 00 0sin2θ 9 6 3 sin θ The solution of the RG equation in this basis yields coefficients (c1,c2,c3,c4,c5,c6) at a scale between mW and mg˜ . Their initial values at the electroweak scale are given by (0, 1,C7γ (mW ), C8g(mW ), 0, 0). The relevant β-function coefficient in this range is β0(5, 1, 1). From these solutions, we obtain the SM Wilson coefficients at the scale µg˜ ∼ mg˜ by means of the relations C1,2(µg˜) = c1,2(µg˜ ) and

mg˜(µg˜ ) mg˜(µg˜) C7γ (µg˜ ) = c3(µg˜) + c5(µg˜ ), C8g(µg˜) = c4(µg˜ ) + c6(µg˜). (14) mb(µg˜ ) mb(µg˜)

The sign of the coefficients c5,6 depends on the sign of the mixing angle θ. At leading order, the running b-quark 4/β (5,0,1) mass at the gluino scale is obtained from mb(µg˜ ) = mb(mb)[αs(µg˜ )/αs(mb)] 0 . Once we have determined the SM contributions to the Wilson coefficients at the scale µg˜ , their evolution down to lower scales is governed by the well-known evolution equations of the SM. The corresponding 4 × 4 anomalous dimension matrix coincides with the upper left 4 × 4 corner of the extended matrix in (13) evaluated at θ = 0. The resulting formulae are more complicated than in the SM, because in our case the coefficient C1(µg˜) does not vanish at the matching scale (whereas C1(mW ) = 0 for the standard evolution). Table 1 shows the results for the Wilson coefficients at different values of the renormalization scale. (The coefficient C1 does not enter the B → Xs γ branching ratio and is omitted here.) The values of C7γ and C8g depend on the sign of the mixing angle θ, although this effect is numerically small. For comparison, the values obtained at µ = mb in the SM (using αs (mZ) = 0.118) are C2 1.12, C7γ −0.32 and C8g −0.15. They are very close to the values found in the presence of the light SUSY particles. In addition, there are the extra contributions proportional to the new coefficients Ci . The second column in the table shows the running b-quark mass at the various scales. Note that the value of mb above the gluino scale is very sensitive to the sign of θ. The result mb(mW ) = 2.75 GeV corresponding to positive θ appears to be favored by the DELPHI measurement mb(mW ) = (2.67 ± 0.50) GeV obtained from three-jet production of heavy quarks at LEP [16]. However, here a similar comment as in our discussion of the running of αs applies, namely that the DELPHI analysis implicitly assumes that mb runs as predicted in the SM. We are now ready to present our results for the B → Xsγ decay rate, including both the SM and the new SUSY flavor-changing contributions. It is convenient to define new coefficients √ √ LR RL 2 πα (m ˜) * C 2 πα (m ˜) * C LR = − s g sb 7(µ) RL =− s g sb 7(µ) C7 (µ) C7γ (µ) ∗ ,C7 (µ) ∗ , (15) GF mg˜ VtsVtb mb(µ) GF mg˜ VtsVtb mb(µ) T. Becher et al. / Physics Letters B 540 (2002) 278–288 285

LR RL and analogous coefficients C8 and C8 . These expressions exhibit the general features of our model√ as described 2 πα (m ˜ ) earlier. The SUSY contributions are enhanced relative to the SM contributions by a large factor s g ≈ 103, GF mg˜ mb AB ∗ meaning that the ratio of flavor-changing couplings, *sb /(VtsVtb), must be highly suppressed so as not to spoil the successful SM prediction for the branching ratio. The resulting leading-order expression for the B → Xsγ decay rate is 2 3 2 GF αMb mb(mb) ∗ 2 LR 2 RL 2 Γ(B→ Xs γ)= V Vtb C (µb) + C (µb) , (16) 32π4 ts 7 7 where µb ∼ mb is the renormalization scale. Mb is a low-scale subtracted quark mass, which naturally enters the theoretical description of inclusive B decays once the pole mass is eliminated so as to avoid bad higher-order perturbative behavior. It is well known that NLO corrections have a significant impact on the B → Xsγ decay rate in the SM, which is largely due to NLO corrections to the matrix elements of the operators Qi in the effective weak Hamiltonian. (NLO corrections to the Wilson coefficient C7γ have a much smaller effect.) In order to capture the bulk of these corrections, we include the O(αs ) contributions to the matrix elements but neglect SUSY NLO corrections to LR RL ˜ the coefficients C7 and C7 . We also neglect two-loop contributions to the matrix elements involving b-squark loops. This is justified because of the relatively large mass of the b˜ squark, and because our two-loop anomalous dimension calculation has shown that there is very little mixing of the squark operators into the dipole operators. At NLO our results become sensitive to the precise definition of the mass parameter Mb, which we identify with the so- 1S = ± called Upsilon mass [17], for which we take the value mb 4.72 0.06 GeV [18]. (Up to a small nonperturbative 1S min = 1 − 1S contribution, mb is one half of the mass of the Υ(1S) resonance.) We also introduce a cutoff Eγ 2 (1 δ)mb on the photon energy in the B-meson rest frame, which is required in the experimental analysis of radiative B decays. We then obtain G2 α(m1S)3m2(m ) → = F b b b ∗ 2 Br(B Xsγ) min τB VtsVtb KNLO), (17) Eγ >Eγ 32π4 where KNLO) is obtained from the formulae in [14] by obvious modifications to include the effects of the new SUSY contributions, and by a change in some of the NLO terms due to the introduction of the Upsilon mass in (16).2 The dependence of the branching ratio on the SUSY flavor-changing couplings can be made explicit by writing → = −4 + LR + LR2 + RL2 Br(B Xsγ) min 10 B0) 1 A1) Re * A2) * * . (18) Eγ >Eγ sb sb sb

In Table 2, we give results for the coefficients B0 and A1,2 including the dominant theoretical uncertainties. Following [15], we use a running charm-quark mass in the penguin-loop diagrams rather than the pole mass. This is justified, because the photon-energy cut imposed in the experimental analysis prevents the intermediate charm- quark propagators from being near their mass shell. Specifically, we work with the mass ratio mc(µ)/mb(µ),where the running masses are obtained from mc(mc) = (1.25 ± 0.10) GeV and mb(mb) = (4.20 ± 0.05) GeV. In the left-hand plot in Fig. 3, we confront our theoretical result for the B → Xs γ branching ratio with the −4 CLEO measurement Br(B → Xs γ)= (3.06 ± 0.41 ± 0.26) × 10 obtained for Eγ > 2 GeV [19]. (This result actually corresponds to the sum of B → Xsγ and B → Xd γ decays. However, the suppression of the exclusive

2 Speciﬁcally, the expression for the quantity k77,µ b) in Eq. (13) of [14] must be replaced by αs(µ ) m 2 λ − 9λ αs(µ ) k ,µ ) = S(δ) 1 + b r + γ ln b + α2(µ ) + 1 2 + b f ), 77 b 2π 7 77 µ 3 s b 1S 2 π 77 b 2(mb ) 2 2 where −λ1 ≈ (0.25 ± 0.15) GeV and λ2 ≈ 0.12 GeV are hadronic parameters. 286 T. Becher et al. / Physics Letters B 540 (2002) 278–288

Table 2 Results for the coefficients B0 and A1,2 for the SM (first row), and for the SUSY scenarios with positive (middle portion) and negative (lower portion) mixing angle θ,forEγ > 2 GeV. The quoted errors refer to the variations of the theoretical parameters within the ranges specified ∗ =− = in the text. The renormalization scale is varied between 2.5 and 7.5 GeV. Other input parameters are VtsVtb 0.04, αs(mb) 0.21 for the SUSY scenario, and αs(mb) = 0.225 for the SM 1S 3 2 Default % mb mb %(mc/mb)%µb SM ± ∓ +0.09 B0 3.44 0.21 0.11 −0.16 ± ∓ +0.06 B0 2.93 0.18 0.09 −0.13 −4 ± +0.12 10 A1 3.60 0 0.05 −0.02 −8 ± +0.28 10 A2 3.12 0 0.10 −0.12 ± ∓ +0.04 B0 3.68 0.23 0.11 −0.11 −4 ± +0.06 10 A1 3.19 0 0.04 −0.01 −8 ± +0.18 10 A2 2.48 0 0.07 −0.08

Fig. 3. Allowed regions (at 95% c.l.) for the SUSY ﬂavor-changing parameters obtained from the CLEO measurements of the B → Xs γ (left) and B → Xsg (right) branching ratios, using central values for all theory input parameters. The shaded regions correspond to the SUSY model with positive mixing angle θ, the dashed lines refer to negative θ.

B → ργ decay with respect to the B → K∗γ mode implies that the dominant contribution to the inclusive decay → AB must come from b sγ transitions. In the context of our model, it follows that the couplings *db must obey even AB AB −4 tighter constraints than the *sb .) It follows that the maximum allowed values of the parameters *sb are 10 ,as −5 is already obvious from the magnitude of the coefﬁcients A1,2 in Table 2. However, values larger than 5 × 10 LR would require a ﬁne-tuning of the phase of *sb and are thus somewhat unnatural. Note that the ratio shown on the − RL = vertical axis in the plot is bound to lie between 1 and 1, and in the limit *sb 0 corresponds to cosϑLR,where LR ϑLR denotes the CP-violating phase of *sb . The right-hand plot in Fig. 3 shows a similar constraint arising from the inclusive charmless decay B → Xsg. → 4 At leading order, the decay rate for this process is obtained from (16) by the replacements α 3 αs (µb) and AB → AB → C7 C8 . The allowed region corresponds to the CLEO upper bound of 8.2% (at 95% c.l.) for the B Xsg branching ratio [20]. There are, however, potentially large theoretical uncertainties in this result, because we neglect T. Becher et al. / Physics Letters B 540 (2002) 278–288 287

NLO corrections to the branching ratio. We therefore refrain from combining the two plots to reduce the allowed parameter space. In the SM, the direct CP asymmetry in the inclusive decay B → Xs γ is very small, below 1% in magnitude [21]. LR In the SUSY scenario, on the other hand, the phase of the coupling *sb could lead to a large asymmetry. In the approximation where one neglects the SUSY contributions to the CP-averaged decay rate in (16), which is justiﬁed in view of the good agreement of the SM prediction with the data, the formulae in [21] yield the prediction ≈− × 4 LR ACP 50% 10 Im(*sb ), where we have neglected the small contribution from the charm-quark loops and the ˜ = ∗ yet smaller contribution from b-squark loops. (We use the standard phase convention where λt VtsVtb is real. LR In general, the CP asymmetry depends on Im(*sb /λt ).) It follows that even within the very restrictive bounds shown in Fig. 3 there can be a potentially large contribution to the CP asymmetry, which would provide a striking manifestation of physics beyond the SM. In fact, the CLEO bounds −27%

4. Conclusions

New supersymmetric contributions to b-quark production at hadron colliders can account for the long-standing discrepancy between the measured cross sections and QCD predictions if there is a light b˜ squark with mass in the range 2–5.5 GeV, accompanied by a somewhat heavier gluino [1]. In this Letter, we have explored the phenomenology of rare B decays in such a scenario and have found tight bounds on the flavor-changing parameters controlling supersymmetric contributions to b → s and b → d FCNC transitions. The most restrictive constraints ˜ arise from virtual effects of light b squarks in B → Xsγ decays. We have analyzed this process by constructing a low-energy effective Hamiltonian, in which the gluinos are integrated out, while the b˜ squarks remain as dynamical RL LR −5 degrees of freedom. We find that the flavor-changing couplings *sb and *sb must be of order few times 10 or less. (Even tighter constraints hold for the analogous b → d couplings.) This implies that certain off-diagonal entries of the down-squark mass matrix must be suppressed by a similar factor compared to the generic squark- mass squared. Even with such tight constraints on the couplings, this model allows for interesting and novel New Physics effects in weak decays of B mesons and beauty baryons. As an example, we have discussed the direct CP asymmetry in B → Xs γ decays, which could be enhanced with respect to its Standard Model value by an order of magnitude. Other possible effects include an enhanced B → Xsg decay rate. We have not considered here the possibility of b˜-squark pair-production, which would be kinematically allowed for very light squark masses. The ˜ ˜∗ ˜ ˜ new decay modes b → sbb and b →¯sbb would affect the decay widths of B mesons and Λb baryons differently, and hence might explain the anomaly of the low Λb lifetime. We will report on this interesting possibility elsewhere.

Acknowledgements

S.B. and M.N. are supported by the National Science Foundation under Grant PHY-0098631. T.B. is supported by the Department of Energy under Grant DE-AC03-76SF00515, and A.K. under Grant DE-FG02-84ER40153.

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Orbifolded SU(7) and uniﬁcation of families

Kyuwan Hwang, Jihn E. Kim

School of Physics, Seoul National University, Seoul 151-747, South Korea Received 27 May 2002; accepted 18 June 2002 Editor: H. Georgi

Abstract A5DSU(7) family unification model with two spinor representations of SO(14) is presented. The fifth dimension is 1 × compactified on S /Z2 Z2. The orbifolding is used to obtain 4D SO(10) chiral fermions. The 4D grand unification group is the flipped SU(5) × U(1). The doublet–triplet splitting through the missing partner mechanism is achieved. Also, fermion mass matrices are considered.  2002 Elsevier Science B.V. All rights reserved.

PACS: 12.10.-g; 11.30.Hv; 11.25.Mj; 11.30.Ly

Keywords: Family uniﬁcation; Orbifold; SU(7) grand uniﬁcation; Mass matrix

The idea of grand unified theories (GUTs) is probably the most influential one in particle physics in the last three decades [1]. It was so attractive that some obstacles in simple GUT models are expected to be resolved in a more complete theory. One of the problems is the proton decay problem. In the SU(5) model, the proton lifetime 4 is predicted to be of order MGUT in units of GeV. The current experimental upper bound on the partial decay rate + 0 33 −1 15 into the e π decay mode is (1.6 × 10 yr) , which implies a huge MGUT > 10 GeV. It is consistent with the significant separation of the coupling constants of the strong, weak, and electromagnetic interactions. This was considered as one of the successes of GUTs. But this huge mass MGUT led to the so-called gauge hierarchy problem, which in turn led to the developments of technicolor, supersymmetry, and superstring in the last two decades. Another problem in this huge MGUT is the doublet–triplet splitting problem in the quintet (5H ) Higgs that the Standard Model doublet Higgs boson is light (∼ 100 GeV) while the accompanying color triplet boson is needed to be supermassive. In most GUT models, one needs a fine-tuning to achieve this doublet–triplet splitting. Because of the dramatic success of GUTs in the unification of coupling constants, the flavor problem (or the family problem), which is the most important problem in the Standard Model, has been expected to be resolved with the GUT idea [2]. Let us call this kind of unification the grand unification of families (GUF). There have been attempts toward flavor unification in larger GUT groups such as SU(7) GUF [3], SU(8) GUF [4], etc., but the predictions given in any of these models have not been confirmed. Therefore, it is fair to say that the GUF attempts along this line has not led to any convincing theory so far. On the other hand, in the heterotic superstring models the representation 248 of E8 is so large that the known three families are believed to be contained in 248. Indeed, the

E-mail addresses: [email protected] (K. Hwang), [email protected] (J.E. Kim).

0370-2693/02/$ – see front matter  2002 Elsevier Science B.V. All rights reserved. PII:S0370-2693(02)02150-0 290 K. Hwang, J.E. Kim / Physics Letters B 540 (2002) 289–294 superstring compactifications led to phenomenologically interesting multi generation models [5–7]. In particular, the Z3 orbifold compactification has been very attractive since they give the family number as multiples of 3. Also, it has been noted that the doublet–triplet splitting problem is resolved in some orbifold compactifications [7]. The orbifold compactification is one of the efficient and simple ways to break down the huge heterotic string × group E8 E8 [6]. However, the ten-dimensional (10D) superstring world is too far separated away from our low energy four-dimensional (4D) world. Therefore, the field theoretic orbifold compactification (FTOC) [8] in five dimensions (5D) has attracted a great deal of attention recently because of its simplicity, requiring only the field theoretic information. In a sense, the FTOC is a bottom-up approach. In this Letter, we consider the FTOC even though a more fundamental theory is based on the string-theoretic orbifold compactification (STOC) [6]. The initiation of FTOC started from the observation that the doublet–triplet splitting can be understood by making the color triplet boson superheavy, while the doublet Higgs boson can be made a Kaluza–Klein (KK) zero mode by appropriately choosing the charges of the discrete group in consideration. As noted in STOC, the orbifold is known to have the mechanisms both for the doublet–triplet splitting [7] and for the unification of flavor [6,7]. In this regard, it is not unreasonable to attempt the flavor unification also in FTOC as first tried in [9]. Along this FTOC line, we attempt to understand the flavor problem in a 5D extended GUT, compactified on the 1 × 1 orbifold S /Z2 Z2. The group SU(6) cannot unify the flavor since 15 of SU(6) contains only one 10 of SU(5). The simplest GUT unifying the flavor is SU(7).TheSU(7) model of Ref. [3] contains two standard families and two non-standard families2 among which one lepton family becomes standard, but the others are unfamiliar ones. Alas, due to the orbifolding in 5D instead of twisting the group, all the unfamiliar families can be made ⊕ familiar ones which can be removed or kept depending on the Z2 charge. We note that the 10 5 of SU(5) [1] and 35 ⊕ 21 ⊕ 7 of SU(7) [3] models are basically the SO(10) and SO(14) models with the spinor representations for fermions, breaking down to SU(5) and SU(7), respectively. Thus, the family unification hints toward the chain SU(2n + 1) or SO(4n + 2). In this Letter, we choose the simplest generalization and construct a GUF model in 5D SU(7) gauge group with the spinor representation(s) as the matter assignment. In this Letter, SO(14) is considered interchangeably with SU(7) up to a singlet [3],

ABC A 64 = ψ + ψAB + ψ + 1, (1) where the multi-indices imply the antisymmetric combinations, and A = 1, 2,...,7. When we say an SU(7) spinor, it is meant Eq. (1) without the singlet.

Orbifold compactiﬁcation

1 In 5D, the fifth dimension y = Rx5 is compactified on the circle S : x5 ≡ x5 + 2π. Points on S1 are identified →− → − under the Z2(x5 x5) and Z2(x5 π x5). Let any fermion in SU(7) tensor representation has the following parity symmetry:

: AB... − = A B ··· AB... ≡ Z2 ψ ( x5) λψ γ5PA PB ψ (x5), P diag(I5,I2), (2) : AB... − = A B ··· AB... ≡ − Z2 ψ (π x5) λψ γ5PA PB ψ (x5), P diag(I5, I2), (3) where In is the n-dimensional identity matrix, and λ and λ are either +1or−1. Due to the non-commuting boundary conditions given by P in the group space, the gauge group breaks down to

SU(7) −→ SU(5) × SU(2)F × U(1), (4)

1 Our motivation for FTOC is different from that of Ref. [8] in that we want to unify the families while Ref. [8] tries to split the SU(5) quintet Higgs boson. 2 In Ref. [3], it was possible to have chiral families due to the twisting of the electromagnetic charges of two right-handed families. K. Hwang, J.E. Kim / Physics Letters B 540 (2002) 289–294 291 where SU(2)F plays the role of family symmetry. Because of the SU(2)F , we expect that light two generations and the third heavy generation are discriminated. Since we start with a group containing SU(5), there exists a possibility that U(1)-electromagnetism contains an SU(5) singlet piece [10] which is called the flipped SU(5). The flipped SU(5) was extensively studied in fermionic construction of 4D string models [11]. The merit of the flipped SU(5) in string models is that one does not need an αβ adjoint representation of SU(5) for breaking SU(5) down to the standard model (SM). The ψ (10) has a Qem = 0 element ψ67 = νc which can have a GUT scale vacuum expectation value (VEV), hence breaks the unified group to the SM. At the same time, this VEV gives a large mass to the color triplet Higgs fields through the missing partner mechanism as discussed below [12]. Note that orbifolding is not needed for the doublet–triplet splitting. Therefore, let us choose the matter representation and the Z parity assignment λ so that SU(5) × U(1) (the 2 flipped SU(5)) is the GUT group. Under this choice of Z2 eigenvalues, the resulting zero modes automatically form an anomaly free combination of SO(10) spinors. The 4D chiral anomaly depends not only on the bulk matter but also on the Z2 parity assignment [13]. However, our selection of Z2 parity will give no anomaly since the zero mode fermions form SO(10) spinors. This property may be understood better if we consider the connection between the two symmetry breaking chains

SO(10) × SU(2) × SU(2) F × × × SO(14) SU(5) SU(2)F U(1) U(1) (5) SU(7) × U(1)

Matter content

A spinor of SO(14) under the breaking chain of Eq. (5) is

ABC A Ψ ⊕ ΨAB ⊕ Ψ ⊕ Ψ = 16 ⊗ 2F ⊕ 16 ⊗ 2 , (6) where the RHS is the decomposition into SO(10) × SU(2) × SU(2) and the anti-symmetrization of the indices are assumed. Since we are dealing with SO(4n + 2) groups, the models considered do not have the anomaly problem. A5DSO(14) spinor has four left-handed and four right-handed 4D SO(10) spinors. Under the torus compactiﬁcation, these eight SO(10) spinors form four massive Dirac spinors and are removed from the low energy spectrum. But twisting can allow some zero modes. Let the Z2 action in Eq. (2) makes the right-handed component of a 5D spinor heavy (breaking one supersymmetry if there was). In other words, only 4 left-handed SO(10) spinors (one left-handed SU(7) spinor) in 4D remain as zero modes. It is represented under SU(5) × SU(2) × U(1) as:

ABC αβγ αβi αij Ψ = ψ (10, 1)6 ⊕ ψ (10, 2)−1 ⊕ ψ (5, 1)−8, ΨAB = ψαβ (10, 1)−4 ⊕ ψαi (5, 2)3 ⊕ ψij (1, 1)10, A = α ⊕ i Ψ ψ (5, 1)2 ψ (1, 2)−5, (7) where the total number of 10 and 10 is four which is the number of massless SO(10) spinor zero modes. Here, the upper-case Roman letters A,B,C,... are the SU(7) indices (1, 2,...,7), the lower-case Greek letters α,β,γ,... are the SU(5) indices (3, 4,...,7), and the lower case Roman letters i, j are the SU(2)F indices 1, 2. We can assign =− ABC A λ 1totheZ2 parity of the whole SU(7) spinor (Ψ ,ΨAB ,Ψ ), leaving the following zero modes

(10, 2)−1,(5, 2)3,(1, 2)−5, (8) which is exactly the anomaly free combination of the flipped SU(5) model [10]. Thus, this consistent choice of Z2 parity picks up one irreducible representation of 16 ⊗ 2 of SO(10) × SU(2) in 4D among the full spinor of SO(14) 292 K. Hwang, J.E. Kim / Physics Letters B 540 (2002) 289–294 shown in Eq. (6). The reason for this consistent selection is in that a spinor of SO(4n + 2) can be decomposed into the sum of alternating totally antisymmetric tensors of SU(2n + 1) as shown in Eq. (6).3 The 5D SU(7) model presented above has two families, neatly unified in a doublet of SU(2)F in Eq. (8). We need to introduce the third family. A simple choice is that the third family is a singlet under SU(2)F . We can put ⊕ ⊕ × this SU(2)F singlet, (10, 1)−1 (5, 1)3 (1, 1)−5 under SU(5) U(1), at the asymmetric fixed point. Then we need to put Higgs fields with the gauge charges 10−1, 101, 52, 5−2 at the asymmetric fixed brain also. 10 and 10 are required to break SU(5) × U(1) → SU(3)c × SU(2)L × U(1)Y . 5 and 5 contain the doublet Higgs for the SU(2)L × U(1)Y breaking into U(1)em. In the remainder of this Letter, however, we study a more interesting case that the third family is also a member of an SU(2)F doublet. In addition, let us extend to the supersymmetric case so that the discussion on the Higgs multiplets is neat. Put the same SU(7) combination of Eq. (6) in the bulk again, from which we obtain the additional zero modes given in Eq. (8). Below the SU(2)F breaking scale, one set of the SU(2)F doublet becomes the third family fermions. The superpartners of the remaining SU(2)F doublet can be Higgs multiplets: H(10−1), ¯ ¯ h(53), φ(1−5).However,h(53) in the flipped SU(5) does not have a color triplet with Qem =−1/3; hence the 3 ¯ component of H(10−1) with Qem =+1/3 does not have a partner in h(53), and the doublet–triplet problem is not solved. To solve this doublet–triplet splitting problem, we introduce 52 and 5−2 which have color triplets with the needed electric charge. These may come from 7 ⊕ 7 of SU(7),or14 of SO(14).

Missing partner mechanism

We introduced two SU(2)F -doublet spinors of SU(7). For the Higgs ﬁelds, let us introduce 52 and 5−2 in the bulk, and in addition {101 ⊕ 5−3 ⊕ 15} at the asymmetric ﬁxed point, which are SU(2)F -singlets. Toward a detail discussion on the mass matrices of light fermions and the doublet–triplet splitting mechanism, let us name two SU(2)F -doublets of SO(14) spinor as

c c Ti(10−1), Fi(53), Ei (1−5) and Ti (10−1), Fi (53), Ei (1−5), (9) where the family indices i = 1, 2andSU(2)F -singlets as ¯ H(101), h (5−3), ϕ(15) and h(52), h(5−2), (10) and the components of each multiplet as dc q uc D D 10− : , 5 : , 5− : , 5+ : , (11) 1 qνc 3 - 2 h+ 2 h− where D and h+ carries the hypercharge 1/3and1/2, respectively. { 1 2}= In order to break the unified gauge group, we need two additional SU(2)F -doublet fields χi ,χi 2(1, 2)0 at the asymmetric fixed point. The superpotential relevant to the GUT symmetry breaking and the masses of the third generation fermions, written in the asymmetric fixed brain, are given by

¯ ¯ c 2 c 1 WH = HHh + T T h + T F h + F E h + F h χ + E ϕχ . (12) This superpotential contains the most general cubic terms of the singlet ﬁelds in Eq. (10) and the primed doublet ﬁelds in Eq. (9) consistent with the following two discrete symmetries χ : 1 →− 1 →− H : →− Z2 χ χ ,ϕ ϕ, Z2 H H (13)

3 One might want to unify all 3 generations in SU(8). However, our mechanism does not work for SU(2n) models, since an SO(4n) spinor always branches to a vectorlike representation of SU(2n). K. Hwang, J.E. Kim / Physics Letters B 540 (2002) 289–294 293

χ H ¯ while the other fields are invariant under Z2 and Z2 . We do not allow hh term in the superpotential, which is anticipated in the superstring models. By the development of VEV along the D-flat (and F -flat) direction (T Hχ 1)(χ1χ2), c = ¯c = √1 1 = 2 = νT ν χ2 χ1 MG, (14) 1 H 2 both q and q are either eaten by the heavy gauge bosons or made heavy by the supersymmetric Higgs T1 H c ¯c c mechanism. From the superpotential terms in Eq. (12) the components d ,Dh, d , Dh¯ , F2,E1 and one linear T2 H combination of h+ and h become massive after the symmetry breaking, while h− and the other linear combination of h+ and h remain massless and fulfill the doublet–triplet splitting. The rest massless components c c c {d ,qT ,u ,-F ,E2} form the third generation family. T1 2 F1 1

Mass matrices

In order to reproduce the realistic fermion masses and mixing angles, we need an additional global symmetry which prevents the light generation doublets T,F,E c from acquiring the same large mass as the third generation c ones T , F ,E . Here, as a simplest option available, we just try an anomalous global U(1)F symmetry. Like the models with U(2)F family symmetry in the literature [14], if we break the SU(2)F × U(1)F in two steps

2 2 SU(2)F × U(1)F −→ U(1) −→ { e}, (15) where 2 ∼ 0.02 and 2 ∼ 0.004 in units of a UV cutoff scale are the order parameters for each step, we can suppress light generation masses by small parameters 2 and 2 . For a model construction, let us assign U(1)F charge +1 to unprimed SU(2)F -doublet fields, and 0 to the other fields. In addition, let us introduce an SU(2)F − 1,2 − singlet φ( 1) and triplets S{ij }( 2) (ij symmetric) with the U(1)F charges indicated inside the parenthesis. The relevant superpotential terms are given by4 2 2 1 a φ a ¯ φ a c WY = S TTh+ + S T F h + + S FE h M∗ M∗ M∗ a=1,2 φ + TT h + (T F + T F) h¯ + (FE c + F Ec)h , (16) M∗ where M∗ is the UV cutoff scale. Requiring the VEVs of the ‘flavon’ fields φ,S1,2 to be ∼ 1 ∼ 2 ∼ φ 2M∗, S{22} 2M∗, S{12} 2 M∗, (17) the mass matrices look like Mu,d 0 2 0 Me 0 2 2 ≈ 2 22, ≈ 2 2 0 . (18) u,d Me M33 2 01 33 0 2 1 This form of mass matrices gives the qualitatively correct mass spectrum and CKM mixing matrix elements. If we let the two symmetry breaking steps in Eq. (15) occur with a single triplet S{ij } instead of two different triplets 1,2 = = S{ij },theSU(2)F symmetry would enforce the unrealistic relation mu/mc md /ms me/mµ precisely, as long as the mixing between light two generations and the third generation remains small. In our model, however, the

4 The superpotential WH achieves the missing partner mechanism presented in the preceding paragraph. Both WH and WY can be obtained by introducing a Z3 discrete symmetry instead of a global U(1)F . 294 K. Hwang, J.E. Kim / Physics Letters B 540 (2002) 289–294 discrepancy between mu/mc, md /ms and me/mµ as well as mc/mt , ms /mb and mµ/mτ can be accounted for by the numerical coefficients of tolerable size, since the up-type quark, down-type quark and lepton masses come from different superpotential terms. In this Letter, we constructed a 5D SU(7) (or SO(14)) GUF model with two spinors of SO(14), with the orbifold × compactification S1/Z2 Z2, which realizes the three families of fermions in the flipped SU(5) and the doublet– triplet splitting of Higgs multiplet. Out of four SO(10) spinors, one is used in the split multiplet mechanism. We introduced 5+2 and 5−2,anSO(10) vector arising from the SO(14) vector 14. There may be a deep reason for two SO(14) spinors we introduced. In the ZN orbifold compactification, the bulk field appears in the multiple of N.So, a successful model with Z2 orbifold can render two spinors in the bulk.

Acknowledgements

This work is supported in part by the BK21 program of Ministry of Education, the KOSEF Sundo Grant, and by the Ofﬁce of Research Affairs of Seoul National University.

References

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Exceptional coset spaces and uniﬁcation in six dimensions

T. Asaka, W. Buchmüller, L. Covi

Deutsches Elektronen-Synchrotron DESY, Hamburg, Germany Received 1 May 2002; received in revised form 5 June 2002; accepted 19 June 2002 Editor: P.V. Landshoff

Abstract

The coset spaces E8/SO(10) × HF allow complex structures which can account for three quark–lepton generations including right-handed neutrinos. We show that in the context of supersymmetric SO(10) gauge theories in 6 dimensions they also provide the Higgs ﬁelds which are needed to break the electroweak and B − L gauge symmetries, and to generate small neutrino masses via the seesaw mechanism.  2002 Elsevier Science B.V. All rights reserved.

The standard model gauge groups of electroweak The group E8 also appears in the ten-dimensional and strong interactions are naturally unified in the sim- Yang–Mills supergravity theory where a cancellation ple gauge group SU(5) [1]. With the increasing exper- of all gauge and gravitational anomalies is obtained by imental evidence for neutrino masses and mixings the means of the Green–Schwarz mechanism for the group larger gauge group SO(10) [2] with the Pati–Salam E8 ×E8 [7]. Compactification to four dimensions [8,9] subgroup SU(4) × SU(2) × SU(2) [3] appears par- can yield low energy effective theories with unbroken ticularly attractive, since all quarks and leptons of a N = 1 supersymmetry and chiral fermions, similar to single generation, including the right-handed neutrino, the structure of the standard model. The number of are then also unified in a single multiplet. families is determined by the Euler characteristic of The unification groups SU(5) = E4,SO(10) = E5 the compact manifold. and E6 [4] belong to the sequence of exceptional Further, the group E8 has been considered in groups En which terminates at E8. This largest excep- attempts to relate quarks and leptons to a coset space tional group is attractive for unification [5,6] since its G/H, where G is an appropriate simple group and smallest multiplet, the 248-dimensional adjoint repre- H contains the standard model gauge group [10]. By sentation, is large enough to accommodate all three pairing the scalar degrees of freedom of G/Hinto generations of quarks and leptons. The theory is nat- complex fields, which become the superpartner of urally supersymmetric. However, in addition to the quarks and leptons, the problem of mirror families three known quark–lepton generations the theory also can be avoided [11]. Particularly attractive are coset predicts three light mirror generations, contrary to ob- spaces E8/(SO(10) × HF )whereHF is a subgroup servation. of SU(3) × U(1) [11–14]. In the case HF = SU(3) × U(1) the representation of chiral multiplets is unique, = + ∗ + ∗ + E-mail address: [email protected] (L. Covi). Ω (16, 3)1 16 , 1 3 10, 3 2 (1, 3)4 . (1) 0370-2693/02/$ – see front matter  2002 Elsevier Science B.V. All rights reserved. PII:S0370-2693(02)02153-6 296 T. Asaka et al. / Physics Letters B 540 (2002) 295–300

− − + −1 Hence, in addition to three quark–lepton generations PGGV(x, y, z πR6/2)PGG contained in the three 16’s of SO(10), one mirror gen- = ηGGV(x,y,z+ πR6/2). (5) eration, 16∗, occurs, again in contrast to observation. 2 3 = For HF = SU(2) × U(1) and HF = U(1) various Here PI I, the matrices PPS and PGG are given in complex structures are possible [13], which contain ei- Ref. [19], and the parities are chosen as ηI = ηPS = ∗ ∗ ther three 16’s and one 16 ,ortwo16’s and two 16 ’s ηGG =+1. The extended supersymmetry is broken in addition to three 10’s and SO(10) singlets. by choosing in the corresponding equations for Σ Orbifold compactifications [9] have recently been all parities ηi =−1. At the fixpoints in the extra di- applied to GUT field theories. The breaking of the mensions, O = (0, 0), OPS = (πR5/2, 0) and OGG = GUT symmetry then automatically yields the required (0,πR6/2) the unbroken subgroups are SO(10),GPS doublet–triplet splitting of Higgs fields [15]. Several and GGG, respectively. In addition, there is a fourth SU(5) models have been constructed in 5 dimensions fixpoint at Ofl = (πR5/2,πR6/2) [20], which is ob- [15–18], whereas 6 dimensions are required for the tained by combining the three discrete symmetries Z2, PS GG breaking of SO(10) [19,20]. The N = 2 supersymmet- Z2 and Z2 defined above ric SO(10) theory in 6 dimensions can also be used as − P V(x,−y + πR /2, −z + πR /2)P 1 a starting point to obtain a SU(5) GUT with three gen- fl 5 6 fl erations [21]. =+V(x,y+ πR5/2,z+ πR6/2). (6) In the following we want to point out that the com- The unbroken subgroup at the fixpoint Ofl is flipped plex structure (1), when combined with the orbifold SU(5), i.e., Gfl = SU(5) × U(1) . The physical region compactification of the 6d SO(10) theory, yields in a is obtained by folding the shaded regions in Fig. 1 simple way the supersymmetric standard model with along the dotted line and gluing the edges. The result right-handed neutrinos. The three 16’s, which contain ∗ is a ‘pillow’ with the four fixpoints as corners. The quarks and leptons, are brane fields; the 16 and the unbroken gauge group of the effective 4d theory is three 10’s are bulk fields. Vacuum expectation values given by the intersection of the SO(10) subgroups × of bulk fields break the symmetries SU(2) U(1)Y at the fixpoints. In this way one obtains the standard and U(1)X, which are left unbroken by the orbifold = model group with an additional U(1) factor, GSM compactification. SU(3) × SU(2) × U(1)Y × U(1)X. The difference of Consider the SO(10) gauge theory in 6d with baryon and lepton number is the linear combination N = 1 supersymmetry. The gauge fields VM (x,y,z), with M = µ,5, 6, x5 = y, x6 = z, and the gauginos λ1, λ2 are conveniently grouped into vector and chiral multiplets of the unbroken N = 1 supersymmetry in 4d,

V = (Vµ,λ1), Σ = (V5,6,λ2). (2) Here V and Σ are matrices in the adjoint representation of SO(10). Symmetry breaking is achieved 2 I × PS × by compactification on the orbifold T /(Z2 Z2 GG Z2 ). The discrete symmetries Z2 break the extended supersymmetry in 4d; they also break the SO(10) gauge group down to the subgroups SO(10),GPS = SU(4) × SU(2) × SU(2) and GGG = SU(5) × U(1)X, respectively, at three different fixpoints − − −1 = P IV(x, y, z)PI ηIV(x,y,z), (3) − + − −1 P PSV(x, y πR5/2, z)PPS 2 I × PS × GG Fig. 1. Orbifold T /(Z2 Z2 Z2 ) with the fixpoints O, OPS, = ηPSV(x,y+ πR5/2,z), (4) OGG,andOfl. T. Asaka et al. / Physics Letters B 540 (2002) 295–300 297 √ √ B − L = 16/15Y − 8/5 X. The zero modes of the of H and H are opposite. In the following we denote vector multiplet V form the gauge fields of GSM . by ηi the parities of the first 4d chiral multiplet, and The vector multiplet V is a 45-plet of SO(10) we choose ηI =+1. PS which has an irreducible anomaly in 6 dimensions. It is The discrete symmetry Z2 implies automatically related to the irreducible anomalies of hypermultiplets a splitting between the SU(2) doublets and the SU(3) in the fundamental and the spinor representations by triplets contained in the 10-plets. The choice ηPS =+1 (cf. [23]) leads to massless SU(2) doublets and massive colour triplets (cf. Table 1). Choosing further ηGG =+1for a(45) =−2a(10), c H1 and ηGG =−1forH2, selects the doublet H from = ∗ =− ∗ a(16) a(16 ) a(10). (7) the SU(5) 5 -plet contained in H1, and the doublet Hence, the anomaly of the vector multiplet can be can- H from the SU(5) 5-plet of H2 (cf. Table 1). The c celed by adding two 10 hypermultiplets, H1 and H2. doublets H and H have the quantum numbers of The cancellation of the reducible anomalies can be the Higgs fields Hd and Hu in the supersymmetric achieved by means of the Green–Schwarz mechanism standard model. [7]. Quarks and leptons can be incorporated by adding For these hypermultiplets we have to define parities 16-plets, additional 10-plets etc. in the bulk and on the with respect to the discrete symmetries fixpoints. Without any constraint on the multiplicity of these fields there are many possibilities [20,22]. − − = PIH(x, y, z) ηIH(x,y,z), (8) It is remarkable that for the complex structure Ω PPSH(x, −y + πR5/2, −z) given in Eq. (1), the requirement of SO(10) anomaly cancellations in the bulk determines the distribution of = ηPSH(x, y + πR5/2,z), (9) multiplets uniquely. As already mentioned, two 10’s − − + PGGH(x, y, z πR6/2) are required to cancel the anomaly of the 45 vector ∗ = ηGGH(x,y,z+ πR6/2), (10) multiplet. Hence, the (10, 3 ) have to be bulk fields. with η =±1(i = I, PS, GG). All hypermultiplets The anomaly of the third 10-plet can only be canceled i by the (16∗, 1), which leaves (16, 3) as brane fields. split under the extended 6d supersymmetry into two Note, that in general a bulk field contains two 4- N = 1 4d chiral multiplets, H = (H, H ).Theyhave dimensional N = 1 chiral multiplets which transform the 6d superpotential interactions [24], as complex conjugates of each other with respect to √ 6 2 the gauge group. Hence, in our SO(10) model in S6 = d x d ΘH ∂ + 2 gΣ H + h.c. , 6 dimensions the content of 4d chiral multiplets is (11) larger than the complex structure (1). where ∂ = ∂5 − i∂6 and g is the 6d gauge coupling. We now have to choose the parities for H3, the third c ∗ c Invariance of the action then requires that the parities 10-plet, and for Φ ,the16 -plet. The presence of Φ

Table 1 c = = Parity assignment for the bulk 10 hypermultiplets. H1 Hd and H2 Hu SO(10) 10

GPS (1, 2, 2)(1, 2, 2)(6, 1, 1)(6, 1, 1) ∗ ∗ GGG 5−2 5+2 5−2 5+2

H c HGc G PS GG PS GG PS GG PS GG Z2 Z2 Z2 Z2 Z2 Z2 Z2 Z2

H1 ++ +− −+ −− H2 +− ++ −− −+ H3 −+ −− ++ +− 298 T. Asaka et al. / Physics Letters B 540 (2002) 295–300

Table 2 ∗ Parity assignment for the bulk 16 hypermultiplet ∗ SO(10) 16 ∗ ∗ GPS (4 , 2, 1)(4 , 2, 1)(4, 1, 2)(4, 1, 2) ∗ ∗ GGG 10+1 5−3 10+1 5−3, 1+5

Qc Lc Uc,Ec D,N PS GG PS GG PS GG PS GG Z2 Z2 Z2 Z2 Z2 Z2 Z2 Z2

Φc −− −+ +− ++

Table 3 normalized. The first three terms are familiar from Charge assignments for the symmetries U(1)R and U(1)X ordinary SO(10) GUTs whereas the last three terms c H1 H2 ψi Φ H3 are additional couplings among bulk fields. R 00102 It is instructive to consider in Eq. (12) just the X −2a −2aa−a 2a zero mode interactions and their couplings to a single heavy field in the bilinear and cubic terms. In standard notation, with ψ = (q, uc,ec,l,dc,nc), one obtains, offers the possibility to break U(1) − spontaneously B L = c + c c + c + c by a vacuum expectation value of its SU(5) singlet W4 hd qd le H1 hu qu ln H2 component N (Table 2). To have N as zero mode hN c 2 hN c 2 hN c c c = =− + n N + d D + d Dn N fixes the parities of Φ to be ηPS ηGG 1. Then M∗ M∗ M∗ an additional colour triplet, D, appears as zero mode. c c c + λ1DNG + M1H H3 + M2H2H As we shall see, D can acquire a Dirac mass together 3 1 3 c + h quc + lnc H with another colour triplet, G , which can be chosen d 1 c c c c as zero mode of the third 10-plet. The corresponding + hd qq + u e + d n G1 =− =+ parities of H3 are ηPS 1, ηGG 1. The parities + + c c c + c + c c c hd ql u d G1 hu qd le H2 of all components of Hi and Φ are listed in Tables 1 + h qq + ucec + dcnc G and 2. The SO(10) singlets contained in the complex u 2 structure Ω could be brane fields or bulk fields. Since + + c c c; hu ql u d G2 (13) we have no constraints from anomaly cancellation or from phenomenology on their properties, we do not here the three couplings hN , hN and hN correspond to discuss them further. the three SO(10) invariants which can be formed from c c Knowing the parities of all fields we can now ψψΦ Φ . = c = discuss the superpotential. We consider the three brane Vacuum expectation values v1 H1 , v2 H2 and v = N yield the mass terms, fields 16, ψi ,ontheSO(10) symmetric fixpoint N O. To restrict the number of terms we require R- c c c c Wm = muuu + mν nn + md dd + meee invariance and an additional global U(1) symmetry X c c (cf. Table 3). + mN n n , (14) The most general superpotential up to quartic terms with mass matrices mu = mν = huv2, md = me = is then given by, = 2 hd v1 and mN hN vN /M∗. With v1,2 vN this gives, to first approximation, a good description of the hN c c W4 = hd ψψH1 + huψψH2 + ψψΦ Φ M∗ observed properties of quarks and leptons. Since there c c are two Higgs doublets, H1 and H2, the unwanted + M1H1H3 + M2H2H3 + λ1Φ Φ H3, (12) relation of minimal SO(10) models, mu = md ,is where we choose M∗ > 1/R5,6 to be the cutoff of avoided. The evidence for small neutrino masses, the 6d theory, and the bulk fields have been properly together with the seesaw mechanism [25], suggests T. Asaka et al. / Physics Letters B 540 (2002) 295–300 299

Table 4 Parity assignment for the bulk 10 and 16 hypermultiplets SO(10) 10

GPS (1, 2, 2)(1, 2, 2)(6, 1, 1)(6, 1, 1) ∗ ∗ GGG 5−2 5+2 5−2 5+2

H c HGc G PS GG PS GG PS GG PS GG Z2 Z2 Z2 Z2 Z2 Z2 Z2 Z2

H4 −− −+ +− ++

SO(10) 16 ∗ ∗ GPS (4, 2, 1)(4, 2, 1)(4 , 1, 2)(4 , 1, 2) ∗ ∗ GGG 10−1 5+3 10−1 5+3, 1−5

QLU, EDc, Nc PS GG PS GG PS GG PS GG Z2 Z2 Z2 Z2 Z2 Z2 Z2 Z2

Φ −− −+ +− ++

that U(1)B−L is broken near the unification scale, i.e., fixpoint OGG. This will be discussed in more detail vN ∼ 1/R5,6, which implies large Majorana masses elsewhere [27]. c [26]. This also leads to a large mass term, mT DG3, Alternatively, one can avoid the occurrence of c for the colour triplet zero modes of Φ and H3, with anomalies by ‘partial doubling’, familiar from super- mT = λ1vN . The vacuum expectation value N = symmetric σ -models where one associates an entire × vN breaks U(1)X U(1)X to the diagonal global chiral multiplet with some degrees of freedom of the subgroup, leaving U(1)R unbroken. As a consequence, coset space G/H (cf. [30]). Let us then add two bulk c integrating out the heavy fields G1,2,G1,2 does not fields, a 16-plet, Φ, and a fourth 10-plet, H4,which lead to a dimension-5 operator for proton decay. For have no irreducible 6d anomaly. The choice of pari- c =− =− the same reason, integrating out H3 and H3 does ties ηPS 1andηGG 1 for both fields leads to c c c not generate a µ-term H1 H2. Finally, the electroweak the zero modes D , N and G (cf. Table 4). The zero scale v1,2 may be induced together with the µ-term modes of all bulk fields now form a real, and thereby supersymmetry breaking. The resulting low energy fore anomaly free, representation of GSM . Note, that effective theory is just the supersymmetric standard adding a 16-plet in the bulk is equivalent to adding model with right-handed neutrinos. a 16∗-plet. The ‘partial doubling’ then corresponds What determines the vacuum expectation value N to giving a larger multiplicity to some bulk fields, which breaks U(1)X and therefore U(1)B−L?Note, which may be related to an extended supersymmetry that these U(1) symmetries are anomalous, since the in higher dimensions. In this case one also has to con- c 2 zero modes of Φ and H3 lead to 4d anomalies sider HF = SU(2) × U(1) rather than HF = SU(3) × 2 × 3 ∗ = + SU(3) U(1)X and U(1)X. In general, these can U(1),sothat(10, 3 )2 (10, 2)2,−1 (10, 1)2,2 [30], be compensated by a Chern–Simons term.1 One may to single out one of the 10-plets. then hope to break U(1)X spontaneously by means The additional superpotential terms on the SO(10) of a Fayet–Iliopoulos term on the Georgi–Glashow brane are,

= 2 + c + W4 M S λ2SΦΦ λ3ΦΦH4 1 For recent discussions in 5-dimensional theories and refer- + λ4 + λ5 ences, see [28,29]. SΦΦH1 SΦΦH2 M∗ M∗ 300 T. Asaka et al. / Physics Letters B 540 (2002) 295–300

λ6 c λ7 c + ΦΦ H1H3 + ΦΦ H2H3, (15) [4] F. Gürsey, P. Ramond, P. Sikivie, Phys. Lett. B 60 (1976) 177. M∗ M∗ [5] N.S. Baaklini, Phys. Lett. B 91 (1980) 376. where we have also included one of the SO(10) [6] I. Bars, M. Günaydin, Phys. Rev. Lett. 45 (1980) 859. [7] M.B. Green, J.H. Schwarz, Phys. Lett. B 149 (1984) 117. singlets, S.TheR-andX-charges of the additional [8] P. Candelas, G.T. Horowitz, A. Strominger, E. Witten, Nucl. fields are RΦ = 0, RH = RS = 2, XΦ = a, XH = Phys. B 258 (1985) 46. 4 4 −2a, XS = 0. Without the singlet field S,thereisa [9] L.J. Dixon, J.A. Harvey, C. Vafa, E. Witten, Nucl. Phys. B 261 c bulk D- and F-flat direction, N = N =vN [27]. (1985) 678. The couplings of S on the brane lift this degeneracy, [10] W. Buchmüller, S.T. Love, R.D. Peccei, T. Yanagida, Phys. Lett. B 115 (1982) 233. = = c = − 2 and one has vN N N M /λ2.The [11] C.L. Ong, Phys. Rev. D 27 (1983) 3044; cubic term in (15) gives mass to two zero modes G C.L. Ong, Phys. Rev. D 31 (1985) 3271. c c and D , N − N makes to the U(1)X vector multiplet [12] S. Irié, Y. Yasui, Z. Phys. C 29 (1985) 123. massive, and N +Nc has a common mass term with S. [13] W. Buchmüller, O. Napoly, Phys. Lett. B 163 (1985) 161. Hence, one obtains the same low energy theory as in [14] K. Itoh, T. Kugo, H. Kunitomo, Progr. Theor. Phys. 75 (1986) 386. the anomalous model discussed above. [15] Y. Kawamura, Progr. Theor. Phys. 103 (2000) 613; We have shown that complex structures of coset Y. Kawamura, Progr. Theor. Phys. 105 (2001) 999. spaces E8/SO(10) × HF can provide the starting [16] G. Altarelli, F. Feruglio, Phys. Lett. B 511 (2001) 257. point of a full grand unified model in the context of [17] L.J. Hall, Y. Nomura, Phys. Rev. D 64 (2001) 055003. supersymmetric SO(10) theories in 6 dimensions with [18] A. Hebecker, J. March-Russell, Nucl. Phys. B 613 (2001) 3. [19] T. Asaka, W. Buchmüller, L. Covi, Phys. Lett. B 523 (2001) orbifold compactification. Open questions concern 199. the breaking of supersymmetry, the origin of the [20] L.J. Hall, Y. Nomura, T. Okui, D.R. Smith, Phys. Rev. D 65 brane superpotential and, in particular, the possible (2002) 035008. [21] T. Watari, T. Yanagida, hep-ph/0201086. connection to theories in 10 dimensions with E8 symmetry. [22] N. Haba, T. Kondo, Y. Shimizu, hep-ph/0112132; N. Haba, T. Kondo, Y. Shimizu, hep-ph/0202191. [23] A. Hebecker, J. March-Russell, Nucl. Phys. B 625 (2002) 128. [24] N. Arkani-Hamed, T. Gregoire, J. Wacker, JHEP 0203 (2002) Acknowledgements 055. [25] T. Yanagida, in: Workshop on unified Theories, KEK Report We would like to thank A. Hebecker, H.P. Nilles, 79-18, 1979, p. 95; M. Gell-Mann, P. Ramond, R. Slansky, in: P. van Nieuwen- M. Olechowski and C. Scrucca for helpful discussions. huizen, D. Freedman (Eds.), Supergravity, North-Holland, Amsterdam, 1979, p. 315. [26] E. Witten, Phys. Lett. B 91 (1980) 81. References [27] T. Asaka, W. Buchmüller, L. Covi, in preparation. [28] R. Barbieri, R. Contino, P. Creminelli, R. Rattazzi, C.A. Scrucca, hep-th/0203039. [1] H. Georgi, S.L. Glashow, Phys. Rev. Lett. 32 (1974) 438. [29] S. Groot Nibbelink, H.P. Nilles, N. Olechowski, hep- [2] H. Georgi, in: C.E. Carlson (Ed.), Particles and Fields 1974, th/0203055. AIP, New York, 1975, p. 575; [30] W. Buchmüller, W. Lerche, Ann. Phys. (N.Y.) 175 (1987) 159. H. Fritzsch, P. Minkowski, Ann. Phys. 93 (1975) 193. [3] J.C. Pati, A. Salam, Phys. Rev. D 10 (1974) 275. Physics Letters B 540 (2002) 301–308 www.elsevier.com/locate/npe

Effective supergravity for supergravity domain walls

M. Cveticˇ a,c, N.D. Lambert b

a Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, PA 19104, USA b Department of Physics and Astronomy, Rutgers University, Piscataway, NJ 08854-8019, USA c Isaac Newton Institute for Mathematical Sciences, University of Cambridge, Cambridge, UK Received 5 June 2002; accepted 17 June 2002 Editor: H. Georgi

Abstract We discuss the low energy effective action for the bosonic and fermionic zero-modes of a smooth BPS Randall–Sundrum domain wall, including the induced supergravity on the wall. The result is a pure supergravity in one lower dimension. In particular, and in contrast to non-gravitational domain walls or domain walls in a compact space, the zero-modes representing transverse ﬂuctuations of domain wall have vanishing action.  2002 Elsevier Science B.V. All rights reserved.

1. Introduction broken supersymmetry diverges on the wall. This result was interpreted as an explanation for the absence Supergravity domain walls [1] have been recently of smooth Randall–Sundrum domain walls in a large the subject of much attention from a variety of dif- class of supergravities. However, there are smooth do- ferent points of view (for example, see [1–11]; for an main walls of that type in four-dimensional supergrav- earlier review see [12]). A central area of study has ity [1] for which the problem described in [5] still oc- been that of the so-called Randall–Sundrum domain curs, as it also does in the recent five-dimensional ex- walls [13], which trap gravity to their worldvolumes. ample [16]. In addition, to the extent of our knowl- It was recognized early on [2] that the tuning used edge, the coupling of the zero-modes to the gravity on to stabilize such domain walls is simply a supersym- the wall has not yet been addressed. metry condition. However, it has proven very difficult In this Letter we would like to extend the discussion to obtain smooth four-dimensional Randall–Sundrum of the bosonic and fermionic zero-modes and obtain domain walls from the known supergravities. This led the effective action associated with these modes. In to several no-go theorems [4,5,14,15]. More recently a particular, we will reexamine the dynamics of the zero- five-dimensional supergravity which admits a smooth modes of a supergravity domain wall (of the Randall– Randall–Sundrum domain wall was obtained in [16]. Sundrum type). The wall will be viewed as a solitonic In [5] it was observed that in a Randall–Sundrum object and thus our analysis will be analogous to background the proposed Goldstone fermion from the classic treatment of the low energy motion of monopoles in a gauge theory [17], which was first applied to gravity in [18–20]. We will see that in E-mail addresses: [email protected] (M. Cvetic),ˇ [email protected] (N.D. Lambert). this context there is no problem with fermion zero-

0370-2693/02/$ – see front matter  2002 Elsevier Science B.V. All rights reserved. PII:S0370-2693(02)02146-9 302 M. Cvetiˇc, N.D. Lambert / Physics Letters B 540 (2002) 301Ð308 modes, indeed these modes do not blow up on the diagonal in the i indices. However, supergravities wall. However, the dynamics of Randall–Sundrum of this type have received considerable attention in type domain walls are qualitatively different to that of recent years and the extension of our analysis to other non-gravitational theories, or domain walls in compact supergravities is clear. In particular, for the case of spaces [21–24]. In particular, in contrast to the Higg’s D = 5 a more detailed discussion can be found in [26]. mechanism observed in [21,22] for domain walls on a In even dimensions one expects that there are terms circle, we will see that the zero-modes that represent involving ΓD+1. However, we expect that the analysis the transverse ﬂuctuations are, in a sense, removed presented here is fairly insensitive to the precise form from the physical spectrum. This is caused by an of the supersymmetry. exact cancellation between the positive tension of the An action which is invariant under these supersym- domain wall and the negative energy density of bulk metries, at least to lowest order in the fermions, has anti-de-Sitter space. In effect the domain wall behaves the form [9] as if it were tensionless. 1 D ¯ mnp i The rest of this Letter is organized in the following S = d x exp R + ψmi Γ ∇nψ κD−2 p way. In Section 2 the supersymmetric action (up to D−2 m D−2 ¯ m i bi-linear fermionic terms) in D dimensions is given: − κ ∂mφ∂ φ + κ λi Γ ∇mλ 2 it contains a gravity supermultiplet and a matter − − ∂ W − κD 2V(φ)+ 2(D − 2)κD 2 λ¯ λi supermultiplet, whose real scalar ﬁeld creates a wall. ∂φ2 i There the form of the fermionic and bosonic zero- − − (D − 2)κ2D 4Wλ¯ λi modes is given and their effective action, whose i − − D ¯ mn i prefactors turn out to be zero, is discussed. In Section 3 (D 2)Wκ ψmi Γ ψn we discuss the properties of the effective action and 1 D ¯ n m i ¯ m n i + κ ∂nφ ψmi Γ Γ λ + λi Γ Γ ψ supersymmetry transformations for the theory reduced 2 m on the wall. In Section 4 we conclude with some ∂W + (D − 2)κD ψ¯ Γ mλi − λ¯ Γ mψi remarks, interpreting the results and suggesting further mi i m ∂φ investigations. +··· , (2.2)

2. Domain walls and zero-modes where the ellipsis denotes higher order terms in the fermions. The scalar potential is For simplicity we assume that we are in D dimens- 2 ∂W − D − 1 ions (D>3) with only gravity, one scalar φ and their V = 4(D − 2)2 − κD 2 W 2 . i i ∂φ D − 2 superpartners ψm and λ active. Here i an internal spinor index which we include for generality. We will (2.3) restrict our attention here to supergravities where the This action reproduces the equations of motion used supersymmetries takes the form in [5]. We note that the supersymmetric vacuum conﬁg- n =−¯ n i + δem iΓ ψm c.c., urations are simply AdS spacetimes with φ ﬁxed at i = δφ =¯ iλ + c.c., a critical point of the superpotential, W,sothatV D−2 2 − −4(D − 1)(D − 2)κ W 0. We will use coordi- δψi =∇ i + κD 2WΓ i, m m m nates in which the metric is i 1 m ∂W i δλ = − Γ ∂mφ + (D − 2) , (2.1) 2 = 2 + 2A µ ν 2 ∂φ ds dr e ηµν dx dx , (2.4) =− D−2 = D−1 = where an underlined index refers to the tangent frame, where A 2κ W(φ0)r, r x and µ,ν m, n = 0, 1, 2,...,D− 1andκ is the D-dimensional 0,...,D− 2. The Killing spinors are of the form Planck length. This is certainly not the most general i 1 A i = e 2 η˜+, form that one can imagine. In particular, we have 1 − 1 − 1 i = 2 A − D 2 2 A µ ˜i restricted the supersymmetry transformations to be 2 e 2κ W(φ0)e x Γµ η−. (2.5) M. Cvetiˇc, N.D. Lambert / Physics Letters B 540 (2002) 301Ð308 303

i r i Here η˜± are constant spinors which satisfy Γ η˜± = tions of motion. Instead, they represent an effective de- i ±˜η±. scription, analogous to the way that motion on mono- Our interest here is in supersymmetric domain wall pole moduli space is an effective description of the be- solutions which have the same form for the metric haviour of monopoles. However, we must also check (2.4) but the scalar field φ is not constant and A is that the grr and gµr equations of motion are satisfied not a linear function identically since the effective action we construct does not have any fields that represents them (i.e., they act ∂W − φ = 2(D − 2) ,A=−2κD 2W, (2.6) as constraints on the low energy effective action). ∂φ With this prescription we obtain the (D − 1)- where a prime denotes differentiation with respect to dimensional Lagrangian (we postpone including vari- r. This is merely a gravitational version of a BPS kink ations of the metric on the wall until the next section) solution that interpolates between two supersymmetric 2 (D−3)A vacua. Broken Poincaré invariance implies that it has LB = LW − 4(D − 2) dr e a bosonic zero-mode r˜ corresponding to the location 2 ∂W − D − 3 of the kink, i.e., the general scalar field profile has × − κD 2 W 2 the form φ(r −˜r) for any r˜. A particular class of ∂φ D − 2 domain walls are the so-called Randall–Sundrum type µν × η ∂µr∂˜ ν r,˜ (2.8) [13], where W changes sign between the two vacua. A These walls have the interesting feature that e2 ∼ where LW is the integral over r of the Lagrangian −4κD−2|Wr| →∞ evaluated on the wall solution e as r leading to localized gravity on the wall. For the rest of our discussion we will restrict − L =−8(D − 2)2 dr e(D 1)A our attention to these types of domain walls. W

Next we wish to construct the low energy dynam- ∂W 2 D − 1 ics associated to the zero-mode r˜. From the field the- × − κD−2 W 2 . − (2.9) ory perspective this is achieved by allowing r˜ to de- ∂φ D 2 µ pend on the walls’ coordinates x . One then simply However, we note that, for any d, evaluates the D-dimensional Lagrangian around such 2 a background to lowest order in derivatives. In a the- ∂W − d 2(D − 2) − κD 2 W 2 edA ory with local diffeomorphism invariance one must be ∂φ D − 2 a little more careful. Following [25] we first note that d the transformation = WedA . (2.10) dr δgrr = 0,δgµr = 0, Hence, for a Randall–Sundrum domain wall, due to L = =− ˜ 2A the exponential fall-off of the metric at large r, B δgµν 2A re ηµν , (2.7) LW = 0. Thus the effective action for fluctuations of that corresponds to an infinitesimal but constant shift the wall vanishes, even though these fluctuations are r → r −˜r is simply a diffeomorphism. However, not diffeomorphisms. Note that this integral vanishes if we now let r˜ depend on xµ then (2.7) is not a only if κ = 0. diffeomorphism and therefore we expect it to represent Now we wish to consider the fermionic properties i a physical mode. of a domain wall. One can readily check that 1 is To continue we consider a variation of the form still a Killing spinor of the domain wall background, ˜ µ i (2.7) with an arbitrary fluctuation r(x ). To obtain the whereas 2 is not. The fact that half of the supersym- effective Lagrangian we substitute the domain wall so- metries of the AdS vacuum are preserved by the wall lution back into (2.2), and then integrate over r.Note implies that the bosonic zero-mode has a superpartner, that this procedure does not imply that the full D- so that the preserved supersymmetry is linearly real- dimensional equations of motion are satisfied. Hence, ized. In particular, the broken supersymmetry creates solutions to the effective action equations of motion this fermionic zero-mode. However, in supergravity, or do not lift to full solutions of the D-dimensional equa- any theory with local symmetry, there are an infinite 304 M. Cvetiˇc, N.D. Lambert / Physics Letters B 540 (2002) 301Ð308 number of broken supersymmetries. Any such spinor A better choice is to find a goldstino mode with = i = is a linear combination ∂µ 0butψµ 0. This can be done by simply acting on the domain wall with the supersymmetry generated i i i = F+η˜+ + F−η˜−, (2.11) − 1 A i by e 2 η˜− and yields with arbitrary coefficients F+(xm) and F−(xm).To i ∂W − 1 A i λ = 2(D − 2) e 2 η˜−, continue then let us outline two natural choices for the ∂φ physical goldstino mode. − 1 i = i = D 2 2 A ˜i At first thought we should choose the resulting ψr 0,ψµ 2κ WΓµe η−. (2.15) goldstino to respect the same symmetries as the wall, This solution is much nicer. Indeed, if we evaluate the = i = i.e., ∂µ 0,ψµ 0. This is obtained by acting with effective action for it we find supersymmetry that is preserved by AdS space but 2 (D−3)A broken by the wall: LF = 4(D − 2) dr e 2 F+ = 0, ∂W − D − 3 × − κD 2 W 2 1−D −1 − 1 A D−2 1 A µ ∂φ D − 2 F− = κ W e 2 − 2κ We2 x Γµ (2.12) × ¯˜ µ ˜i and yields η−i Γ ∂µη−. (2.16) Thus we encounter precisely the same, vanishing, 2(D − 2) 1 ∂W − 1 i = 2 A ˜i λ − e η−, integral that we obtained for the bosonic zero-mode. In κD 1 W ∂φ ¯˜ ˜ ˜i addition, the cross terms involving η−i∂ν rη− are total − 2 i 2(D 2) 1 ∂W − 1 A i derivatives and can be discarded. Therefore, (2.15) ψ =− e 2 η˜−, r κD−1 W ∂φ seems to be the correct choice of goldstino that is linearly related to r˜. ψi = 0. (2.13) µ It is instructive to contrast this discussion with the This is precisely the goldstino found in [5] and case of a domain wall a in non-gravitating theory. diverges if the superpotential W changes sign. This Specifically, we take the flat space limit κ = 0andset n = n i = will mean that when we construct the effective action em δm , ψm 0. Thus the original D-dimensional for the fermionic zero-mode, found by letting η˜i action (2.2) simplifies to − become a field which depends on xµ,wefind ∂W 2 =− D m + − 2 2 Sκ=0 d x ∂mφ∂ φ 4(D 2) 2 2−2D (D−3)A 1 ∂W ∂φ LF = 4(D − 2) κ dr e W ∂φ 2 ¯ m i ∂ W ¯ i × ¯˜ µ ˜i +··· − λi Γ ∂mλ − 2(D − 2) λi λ . (2.17) η−iΓ ∂µη− , ∂φ2 (2.14) The supersymmetry transformations are easy determined from (2.1) by setting κ = 0 and they reduce where the ellipsis denotes cross terms involving η¯˜ × −i to the rigid supersymmetry δη˜i = 1 Γ˜ µ∂ r˜ , δr˜ = ∂ rη˜ i . The existence of these terms indicates that − 2 µ ν − −¯˜ ˜i + a field redefinition is needed to put the action into +iη− c.c. However, integrals of terms of the form a standard form. Thus in a Randall–Sundrum type (2.10) no longer vanish and we instead find the effec- ˜ ˜i of domain wall, where W passes through zero, the tive action of a free scalar r and its superpartner η− kinetic term for such a fermionic zero-modes diverges. 2 2 ∂W L = =−8(D − 2) dr This seems contradictory since the kinetic term for κ 0 ∂φ the bosonic zero-mode r˜ is well-behaved, indeed it × + 1 ˜ µ ˜ − 1 ¯˜ µ ˜i vanishes. In [5] this was used as an indication that 1 ∂µr∂ r η−i Γ ∂µη− . (2.18) W cannot change sign in a well-defined supergravity. 2 2 However, it is clear that the divergence is caused by The integral over r is again a total derivative and can i = −1 i − | =∞ − = the choice W 2. be evaluated to be 4(D 2) W(r ) W(r M. Cvetiˇc, N.D. Lambert / Physics Letters B 540 (2002) 301Ð308 305

−∞)|, which is simply the tension of the domain wall. With the form for λi given in (2.15) the δφ and δλi Thus the low energy dynamics of the zero-modes in variations simply reduce to the supergravity domain wall cannot be continuously ˜ =−¯˜ ˜i + reduced to the flat space limit by making κ arbitrarily δr +iη− c.c., small. i 1 µ αA ˜ i δη˜− = Γ ∂µr˜ + e Aµ ˜+. (3.3) 2 Recall that we restrict to terms that are at most 3. Supergravity on the wall quadratic in the Fermi fields. The supersymmetry reduces to an expression that involves only the (D − In this section we are interested in understanding 1)-dimensional fields only if α = 0. That this occurs how the zero-modes found above, which describe the at all is due to the precise form for the λi goldstino fluctuations of the wall, couple to the gravitational in (2.15) and does not occur if we used any other ˜ fields. We will be primarily interested in whether form. The appearance of Aµ is reminiscent of a Higg’s or not the full effective action of the wall can be mechanism where Aµ “eats” ∂µr˜. Thus we see that r˜ − i identified with a (D 1)-dimensional supergravity. and η˜− defined by (2.15) are indeed superpartners on Previous studies have discussed the reduction of the the wall. r supergravity to the wall [6–8], however, these have Next we must ensure that the gauge choice er = r not included the zero-modes, i.e., they treat the wall as 1andeµ = 0 is preserved by the supersymmetries rigid. We also seek to further justify the choice (2.15) ˜i i = generated by +. This implies that ψr 0. However, as the correct fermionic zero-mode. i = to preserve ψr 0wemusthave More precisely, we wish to reduce the D-dimens- n i i − i 1 µν − 3 A i ional supersymmetry involving em ,φ,λ ,ψm to (D 0 = δψ =− F Γ e 2 ˜ , (3.4) i r µν + 1)-dimensional supersymmetry involving r,˜ η˜− and 8 the bulk supergravity fields, suitably dimensionally re- where here, and in what follows, we have set α = 0. duced. In particular, we consider the following stan- Thus to preserve supersymmetry on the wall we must dard ansatz for the bosonic fields set Fµν = 0. Note that this does not necessarily imply A˜ = 0. Another way to see this restriction arises in φ = φ(r −˜r), A= A(r −˜r), µ the physically interesting cases of D 5. It is not hard αA ˜ n = 1 e Aµ to see that the reduction of the Einstein–Hilbert action em A ˜ ν , (3.1) 0 e eµ in D dimensions using the ansatz (3.1) leads to the kinetic term where φ and A continue to satisfy the domain wall Bogomoln’yi equations (2.6) and α is a parameter that 1 − − dr e(D 5)AF 2, (3.5) we will fix shortly. In what follows we use tildes to 4κD−2 denote (D − 1)-dimensional fields and Γ =˜e ν Γ . µ µ ν for the graviphoton. Clearly, in D 5 the integral over Our next task is to determine the correct form for the r is infinite. From the point of view of the theory fermions so as to obtain a supersymmetry acting only on the wall this can be interpreted as saying that on the (D − 1)-dimensional fields living on the wall. the (D − 1)-dimensional electromagnetic coupling First we note that the action (2.2), when expressed constant vanishes and hence Maxwell’s equation is in terms of the new fields, is still invariant under the simply F = 0. supersymmetry (2.1). Although the transformations µν Next, we consider the gravitini ψi . In particular, given in (2.1) are symmetries for any i ,weare µ using (2.15) as a guide, we let only interested in those that preserve the domain wall. − − 1 1 Thus we are led to introduce the (D 1)-dimensional i = D 2 2 A ˜i + 2 A ˜ i + βA ˜ i i ψµ 2κ We Γµη− e ψµ+ e ψµ− supersymmetry generator ˜+ as γA i + e Γµχ˜+, (3.6) 1 i = 2 A ˜i e +, (3.2) i where β and γ are to be determined. Note that χ˜+ i i i where Γr ˜+ =˜ +. and η˜− can be distinguished from each other by their 306 M. Cvetiˇc, N.D. Lambert / Physics Letters B 540 (2002) 301Ð308

˜ i chiralities and the coefficient of ψµ+ will be justified The additional term in the vielbein variation is by the calculations which follow. Considering the puzzling. However, it is in fact rather harmless. To see i − variation of ψµ we find this we may obtain the effective action for the (D 1)- dimensional fields by substituting in the ansatz (3.1), − 1 ˜ i + β 2 A ˜ i = ∇ ˜i ˜ i = δψµ+ e δψµ− µ +,δχ+ 0, (3.7) (3.1) into (2.2). For a Randall–Sundrum domain wall this yields = wherewehaveusedtherestrictionFµν 0. Next we r ˜ 1 (D−3)A ¯˜ µνρ ˜ i substitute this ansatz into the variation of eµ = Aµ to L = ˜ + ∇ D−2 dr e e R ψµ+i Γ ν ψρ+ . find κ (3.12) 1 + 1 + ˜ = 2 β A ¯˜ ˜ i + 2 γ A ¯˜ ˜ i + δAµ e +i ψµ− e +iΓµχ+ c.c. Note that in deriving (3.12) we have discarded several terms whose integral over r vanishes due to the (3.8) asymptotic fall off of the metric. We have also checked To ensure that the right-hand side is independent of r that the g and g equations of motion are satisfied = =− ˜ i rr µr we must set β γ 1/2. Thus χ+ plays the role identically. ˜ = ¯˜ ˜ i ˜ i of a graviphotini since δAµ +i Γµχ+ while ψµ− The action (3.12) is the minimal supergravity La- ˜ i = − is sterile in the sense that (3.7) implies δψµ− 0. grangian in D 1 dimensions. The integral over r in However, as with A˜ above, it is easy to see that the (3.12) is finite and is simply absorbed into Planck’s µ − ˜ ˜ ψ˜ i and χ˜ i kinetic terms in the effective action are constant in D 1 dimensions. In addition r, Aµ and µ− + ˜i infinite if D 5 and hence they must be set to zero. η− have all disappeared from the action, i.e., they do This also follows by supersymmetry for any D since not represent any physical modes of the low energy ef- i fective dynamics. Therefore, without loss of generality we must have Fµν = 0 for all variations ˜+. Thus we we may set A˜ =−∂ r˜ and η˜i = 0. In this case the fi- set ψ˜ i =˜χi = 0. ν ν − µ− + nal supersymmetry on the wall is just that of a minimal Lastly, we can obtain the variation of the vielbein supergravity on the wall ˜ ν =−¯˜ ν ˜ i + ν ¯ ν ˜ i D−2 ¯ νλ i δeµ +i Γ ψµ+ c.c., δe˜µ =− ˜+i Γ ψ + − 2κ W ˜+iΓ η˜−e˜µλ + c.c. µ ˜ i = ∇ ˜i δψµ+ µ +. (3.13) (3.9) We note that for an arbitrary choice of r(x)˜ and Here we find that the variation involves the r-depend- ˜ − ¯ δgµr = Aµ the corresponding variation (2.7) of the ent term 2κD 2W ˜+ Γ νλη˜i e˜ + c.c. To continue let i − µλ wall is not a diffeomorphism. However, in the special us first summarize our calculations so far. We have case that A˜ =−∂ r˜, where the supersymmetry of found the following ansatz for the fermions µ µ transformation rules are simple, then the variation is i ∂W − 1 A i a diffeomorphism. λ = 2(D − 2) e 2 η˜−, ∂φ In a sense the bosonic zero-mode is eaten by − 1 1 the graviphoton, in a manner similar to [21,22] and i = D 2 2 A ˜i + 2 A ˜ i ψµ 2κ We Γµη− e ψµ+, similarly the gravitini has, in a sense, eaten the i = ψr 0, (3.10) fermionic zero-mode. However, the graviphoton and (part of the) gravitini are not really massive but rather which leads to the symmetry have been frozen out all together. Furthermore, and in ν ¯ ν i D−2 ¯ νλ i ˜ =−˜+ ˜ − ˜+ ˜ ˜ + contrast to the Higg’s mechanism, the kinetic terms δeµ i Γ ψµ+ 2κ W iΓ η−eµλ c.c., i for r˜ and η˜− have disappeared. Thus, rather than ¯ i ˜ δr˜ =− ˜+i η˜− + c.c., δAµ = 0, becoming components of some massive fields, the i 1 µ ˜ i zero-modes are completely removed from the physical δη˜− = Γ ∂µr˜ + Aµ ˜+, 2 spectrum. We can understand the vanishing of the i ˜ i i kinetic terms for r˜ and η˜− as an exact cancellation δψ + = ∇µ ˜+, (3.11) µ between the positive tension of the domain wall, given where we have imposed Fµν = 0. by the first term on the left-hand side of (2.10), and the M. Cvetiˇc, N.D. Lambert / Physics Letters B 540 (2002) 301Ð308 307 negative energy density of anti-de-Sitter space, given modes, such as the fluctuations of gravity along the by the second term on the left-hand side of (2.10). wall, are described by the appropriate action in the We would also like to contrast our results to lower dimension and lead to non-trivial dynamics at those obtained in [23,24] where the effective action low energy. However, the transverse fluctuations have for (infinitely thin) Randall–Sundrum domain walls, not been frozen out, nor do they have a kinetic term. where the transverse dimension was a compact Z2 Instead, they disappear from the action regardless of orbifold. In this case prefactor for the a kinetic energy the form they take. In effect, even though these modes term of the modulus is non-zero. Note however, that are not diffeomorphisms in the full theory, they bein this case the transverse direction is finite (due to the have like diffeomorphisms from the perspective of the orbifold compactification) whereas here the vanishing effective theory. of the kinetic terms arises because of the infinite extra Note that the effective action does not contain dimension. In addition, in this case the choice of the the full dynamics of the theory, just those that are boundary conditions seems to remove the bosonic zero relevant to small fluctuations about the soliton. It mode r˜. Consequently, the authors of [23,24] allowed would be interesting to further understand the validity for a more general fluctuation of the metric component of this approximation and determine if there are grr and this may be an interesting direction to further applications to brane-world scenarios where the low explore within our context. We also note that the energy dynamics of our world are insensitive to the action that we have considered should be viewed as presence of the extra-dimension through any of the a truncation of supergravity to the sector most relevant fluctuations of the domain wall. to the domain wall. In general, we expect that there will be other scalars and p-form fields which will contribute to the low energy dynamics. However, we Acknowledgements do not expect that these fields will affect the dynamics of r˜ that we discussed here. We would like to thank N. Arkani-Hamed, H. Liu, R. Myers, J.-L. Lehners, D. Tong and particularly K. Stelle for discussions. N.D.L. would like to 4. Conclusions thank the University of Pennsylvania for its hospitality where this work was initiated and was also par- In this Letter we have evaluated the effective tially supported by a PPARC Advanced Fellowship at action for the bosonic and fermionic zero-modes of a the Department of Mathematics, King’s College Lon- Randall–Sundrum domain wall. Our result was simply don and the Isaac Newton Institute of Cambridge Uni- (3.12), i.e., pure supergravity. In particular, the zero- versity. M.C. is supported by the DOE grant EY-76- mode and its superpartner, that are normally identified 02-3071 and by the UPenn Class of 1965 Endowed with transverse fluctuations of the wall, were found to Term Chair and would like to thank High Energy The- have a vanishing action. ory group of Rutgers University, and the Isaac Newton Given this somewhat surprising result it seems ap- Institute of Cambridge University for hospitality and propriate to mention some aspects of the approxima- support during the course of the work. tion that we have used to obtain the effective action. In particular, we considered slow motion on the moduli space by allowing the domain wall to fluctuate and References then evaluating the action to second order in derivatives. As a consequence some modes, such as the [1] M. Cvetic,ˇ S. Griffies, S.J. Rey, Nucl. Phys. B 381 (1992) 301, graviphoton, are frozen, i.e., their kinetic terms di- hep-th/9201007. verge. 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Optimized Rayleigh–Schrödinger expansion of the effective potential

Wen-Fa Lu a,b,c, Chul Koo Kim a,c, Kyun Nahm d

a Institute of Physics and Applied Physics, Yonsei University, Seoul 120-749, South Korea b Department of Physics, Shanghai Jiao Tong University, Shanghai 200030, People’s Republic of China 1 c Center for Strongly Correlated Materials Research, Seoul National University, Seoul 151-742, South Korea d Department of Physics, Yonsei University, Wonju 220-710, South Korea Received 28 May 2002; accepted 7 June 2002 Editor: T. Yanagida

Abstract An optimized Rayleigh–Schrödinger expansion scheme of solving the functional Schrödinger equation with an external source is proposed to calculate the effective potential beyond the Gaussian approximation. For a scalar field theory whose potential function has a Fourier representation in a sense of tempered distributions, we obtain the effective potential up to the second order, and show that the first-order result is just the Gaussian effective potential. Its application to the λφ4 field theory yields the same post-Gaussian effective potential as obtained in the functional integral formalism.  2002 Elsevier Science B.V. All rights reserved.

PACS: 11.10.-z; 11.10.Lm; 11.15.Tk

Keywords: Effective potential; Functional Schrödinger equation; Rayleigh–Schrödinger expansion; Variational perturbation approach; Non-perturbative quantum ﬁeld theory

1. Introduction

Since the mid 1980s, the effective potential (EP) in quantum field theory (QFT) [1] beyond the Gaussian approximation (GA) has received much attention [2], because it collects merits and removes weakpoints of the conventional perturbation theory and the GA as shown by the so-called variational perturbation theory in other fields [3]. Some schemes have been proposed to calculate such an EP, for example, schemes of designing some non-Gaussian trial wavefunctionals [2a], using the Brueckner–Goldstone formula with a variational basis [2b], rearranging loop diagrams in the functional integral formalism with the background field method [2c], optimized expansions in the functional integral formalism with the steepest-descent method [2d] and with the background field method [2e], and so on. Noting that the well-developed Rayleigh–Schrödinger (RS) expansion in quantum

E-mail address: [email protected] (W.-F. Lu). 1 Permanent address.

0370-2693/02/$ – see front matter  2002 Elsevier Science B.V. All rights reserved. PII:S0370-2693(02)02140-8 310 W.-F. Lu et al. / Physics Letters B 540 (2002) 309–318 mechanics [4] was generalized to QFT [5], recently we have developed a variational perturbation scheme with the RS expansion to calculate the EP as per its variational minimum definition [6]. In the scheme, a free-field Hamiltonian with a mass fixed from the GA was adopted as a solvable part so as to perform the RS expansion [6]. Obviously, this scheme amounts to a series expansion around the Gaussian EP, and so do those schemes in Refs. [2a–c]. In the present Letter, taking a free-field Hamiltonian instead with an arbitrary mass parameter µ as the solvable part, we will apply the RS expansion to solve the functional Schrödinger equation in the presence of an external source so that the vacuum energy functional of the source can be obtained. Then, adopting an alternative, but equivalent definition of the EP in the functional Schrödinger picture [5], we extract the EP through a Legendre transform of the energy functional. In the new scheme, µ will be determined according to the principle of minimal sensitivity [7] and, consequently, the value of µ in an approximation up to one order is different from those up to other orders of the expansion. This way of fixing µ makes the expansion give the Gaussian EP with its lowest orders, but be not an expansion around the Gaussian EP. Furthermore, unlike Ref. [6], the vacuum expectation value ϕ of the field operator will naturally be given in the present scheme. Hereafter, we will call this scheme the optimized RS expansion (ORSE). Since, for the case of scalar field theory, the EP beyond the GA was given only for the λφ4 model except for the φ6 model [2, (1994)] up to now, we use the ORSE in this Letter to give the EP for a class of scalar field models (see Section 3) up to the second order. The resultant formula can easily be used to give the EP beyond the GA for a number of concrete scalar field models, for instance, models with polynomial or/and exponential interactions. We also show that its application to λφ4 field theory gives the same post-Gaussian EP as Refs. [2d,2e], and yields the result in Refs. [2b,2c,6] if one chooses to use the same constraint on ϕ as in Ref. [2b]. Next, for the sake of convenience, we will first give the complete basis set for a free field theory with an external source, which will be employed in the ORSE. In Section 3, the ORSE will be proposed. In Section 4, we will perform the ORSE to obtain the EP for a class of scalar field theories up to the second order, and its application to the λφ4 field theory will be given in Section 5. Conclusions will be made in Section 6 with discussions on some possible extensions and developments of the present work.

2. A free ﬁeld theory with an external source

In this section, we discuss the free-field theory with an external source Jx ≡ J(x) in a time-fixed functional Schrödinger picture. The Hamiltonian is given by J,µ 1 2 1 2 1 2 2 1 1 −1 H = Π + (∂xφx) + µ φ − Jxφx − fxx + Jx h Jy , (1) 0 2 x 2 2 x 2 2 xy x y = 1 2 D ≡ D where x (x ,x ,...,x ) represents a position in D-dimensional space, x d x, µ an arbitrary mass δ parameter and φx ≡ φ(x) the field at x. Πx ≡−i is canonically conjugate to φx with the commutation δφx [ ]= − ≡ − 2 + 2 − −1 = − relation, φx,Πy iδ(x y). In Eq. (1), fxy ( ∂x µ )δ(x y) with z fxzfzy δ(x y),and ≡ − 2 + 2 − −1 = − hxy ( ∂x µ )δ(x y) with z hxzhzy δ(x y). J,µ| ; (0) = (0)[ ]| ; (0) The functional Schrödinger equation for Eq. (1), H0 n J En J n J , is easily solved [5]. (Here, (0) the subscript n in En [J ] is the index of eigenstates, and the superscript “(k)” means “at the kth order of δ”. See the next section.) First, the eigenenergy of the vacuum state vanishes, and the corresponding wavefunctional is a Gaussian-type functional (0) 1 −1 −1 |0; J = N exp − φx − h Jz fxy φy − h Jz , (2) 2 xz yz x,y z z W.-F. Lu et al. / Physics Letters B 540 (2002) 309–318 311

(0) (0) (0) (0) where N is the normalization constant (i.e., J ; 0|0; J = 1). One can show that J ; 0|φx|0; J = −1 z hxz Jz which, unlike Eq. (3) in Ref. [6], is dependent on x. Then, for the above vacuum, the annihilation and creation operators can respectively be constructed as 1/2 1 −ipx −1 Af (p; J)= e f(p) φx − h Jz + iΠx (3) 2(2π)Df(p) xz x z and 1/2 1 − A† (p; J)= eipx f(p) φ − h 1J − iΠ (4) f 2(2π)Df(p) x xz z x x z with [A (p; J),A† (p ; J)]=δ(p − p) and A (p; J)|0 (0) = 0. It is not difﬁcult to verify that H J,µ = f f f 0 D † ; ; = 2 + 2 = D ip(x−y) = d p f (p)Af (p J)Af (p J),wheref(p) p µ arises from fxy d p f (p)e with p (p1,p2,...,pD). Consequently, the eigenwavefunctionals for excited states can be easily written as

1 n |n; J (0) = √ A† (p ; J)|0; J (0),n= 1, 2,...,∞, (5) ! f i n i=1 and the corresponding eigenenergies are n (0)[ ]= En J f(pi ). (6) i=1 (0) (0) (0) (0) Evidently, the eigenwavefunctionals |n; J and |0; J are orthogonal and normalized, J ; m|n; J = δ 1 n δ(p − p ). Here, P (n) represents a permutation of the set {i }={1, 2,...,n} and the mn n! Pi (n) k=1 k ik i k (0) summation is over all Pi(n)’s. |n; J describes a n-particle state with the continuous momenta p1,p2,...,pn. | ; (0) | ; (0) = ∞ J,µ 0 J and n J with n 1, 2,..., constitute the complete set for H0 , and satisfy the closure | ; (0)(0) ; |+ ∞ D D ··· D | ; (0)(0) ; |= 0 J J 0 n=1 d p1 d p2 d pn n J J n 1.

3. Optimized Rayleigh–Schrödinger expansion for the effective potential

The EP for a field system is equivalently defined through the Feynman graphs, the operator representation, the path integral [1], or the minimum expected energy in a set of normalized states [5,8]. They were used to give the loop or Gaussian EP and propose those schemes in Refs. [2,6]. Yet another equivalent definition of the EP is given through the vacuum energy functional of an external source obtained by solving the relevant functional Schrödinger equation [5]. Based on it, we will construct the ORSE in this section. In this and next sections, we work with a scalar field model whose Lagrangian density is [9] 1 L = ∂ φ ∂µφ − V(φ ). (7) 2 µ x x x √dΩ iΩφx In Eq. (7), the model potential is assumed to be written as V(φ x ) = V(Ω)e , at least, in a sense of 2π tempered distributions [10]. It represents several scalar-field models, such as λφ4 model [2, (one exclusion, 1994)], [11], general and special φ6 models [2, (1994)], [12], sine-Gordon and sinh-Gordon models [13], massive and double sine-Gordon models [14], Liouville model [15], as well as two generic models investigated in Ref. [16]. 312 W.-F. Lu et al. / Physics Letters B 540 (2002) 309–318

For the system, Eq. (7), the time-independent functional Schrödinger equation in the presence of an external source Jx is

H − Jx φx |Ψn =En[J ]|Ψn (8) x = [ 1 2 + 1 2 + ] [ ] with the Hamiltonian H x 2 Πx 2 (∂xφx) V(φ x) . Here, the eigenvalue En J is a functional of Jx . For our purpose, Eq. (8) will be modiﬁed. We make a shift φx → φx + Φ (Φ is a constant), and Eq. (8) can equivalently be rewritten as

H(φx + Φ)− Jx (φx + Φ) Ψn[φx + Φ,J]=En[J ; Φ]Ψn[φx + Φ,J] (9) x + = [ 1 2 + 1 2 + + ] with H(φx Φ) x 2 Πx 2 (∂x φx) V(φ x Φ) . This shift is really in the spirit of the background- field method [2c,17]. Further, normal-ordering the Hamiltonian in Eq. (9) with respect to a normal-ordering [ 1 2 2 − 1 2 2] mass M [18], and inserting a vanishing term x 2 µ φx 2 µ φx with µ an arbitrary mass parameter into the Hamiltonian[19], one can have N + − + = J,µ + µ,Φ − M H(φx Φ) Jx(φx Φ) H0 HI C (10) x with µ,Φ 1 2 2 H = − µ φ + NM V(φ x + Φ) (11) I 2 x x and 2 1 1 −1 1 2 M 2 C = − fxx + Jx h Jy + I M − I M + Jx Φ . (12) 2 2 xy 2 0 4 1 x y √ D 2+ 2 Here, the notation N [···] represents normal-ordered form with respect to M, I (Q2) ≡ d p p Q , M n (2π)D (p2+Q2)n Ω2 2 √dΩ iΩ(φx+Φ)+ I1(M ) and NM [V(φ x + Φ)]= V(Ω)e 4 [9]. From now on, we will use the normal-ordered 2π Hamiltonian in Eq. (10) instead of the original one, which will naturally make the EP in (1 + 1) dimensions free of J,µ explicit ultraviolet divergences [9,18]. Noting that C is a constant independent of φx and H0 is an exactly-solved µ,Φ Hamiltonian, we can formally treat HI as a “perturbed” interaction in the RS expansion [4]. To mark the order of µ,Φ the RS expansion, an index factor δ will be inserted in front of HI in Eq. (10). Consequently, Eq. (9) is modified as J,µ + µ,Φ [ + ; ]= [ ; ]+ [ + ; ] H0 δHI Ψn φx Φ,J δ En J Φ,δ C Ψn φx Φ,J δ . (13) Now, applying the RS expansion, one can solve Eq. (13) to get energy eigenvalues, En[J ; Φ,δ],and eigenwavefunctionals, Ψn[φx + Φ,J; δ]. Here, we are interested only in the eigenenergy functional for vacuum [ ; ] [ ; ] (0)[ ; ] state, E0 J Φ,δ . Obviously, the zeroth-order approximation to E0 J Φ,δ , E0 J φ , satisfies (0)[ ; ]+ = (0)[ ]= E0 J φ C E0 J 0 (14) (0)[ ; ] and at the nth-order of δ, the correction to E0 J φ is − 1 n 1 E(n)[J ; φ]= (0) J ; 0|H µ,Φ Q E(1)[J ; φ]−H µ,Φ |0; J (0) (15) 0 I 0 J,µ − (0)[ ] 0 I H0 E0 J W.-F. Lu et al. / Physics Letters B 540 (2002) 309–318 313 = ∞ D D ··· D | ; (0)(0) ; | [ ; ]= (0)[ ; ]+ ∞ n (n)[ ; ] with Qn j=n d p1 d p2 d pj j J J j . Thus, E0 J Φ,δ E0 J φ n=1 δ E0 J φ . With the vacuum energy functional E0[J ; Φ,δ], one can have [ ; ] δE0 J Φ,δ =− D †[ + ; ] + [ + ; ]≡− φΨ0 φx Φ,J δ (φx Φ)Ψ0 φx Φ,J δ ϕx, (16) δJx where we used the Feynman–Hellmann theorem [20]. Evidently, for δ = 1, ϕx = Ψ0|φx|Ψ0 . Then, a Legendre transformation of E0[J ; Φ,δ] yields the static effective action [5]

Γs[ϕ; Φ,δ]=−E0[J ; Φ,δ]− Jx ϕx. (17) x

To calculate EP, one can conveniently take ϕx = Φ in Eq. (16) to fix the arbitrary shifted parameter Φ. In analogy to Appendix A of Ref. [2, (1990)] but with the Feynman–Hellmann theorem [20], one can show that other choices of Φ will give rise to the same EP (when the wavefunction renormalization procedure is not needed). Finally, one can have the EP [5] Γs[ϕ; Φ,δ] V(Φ) ≡− . (18) x ϕx =constant=Φ,δ=1 If the EP is truncated at some order of δ, the extrapolation of δ = 1 should be made after renormalizing the approximated EP. When truncated at a given order of δ, the right side of Eq. (18) will depend upon µ. To obtain an approximated EP up to the same order, one can determine µ according to the principle of minimal sensitivity [7]. That is, µ should be such a value that the approximated EP up to the given order is optimized to be as insensitive to variations in µ as possible. This can be realized by analyzing the vanishing first- or higher-order derivatives of the truncated result with respect to µ [2e,7]. From the next section, one will see that the EP up to the first order is just the Gaussian EP. Thus, the ORSE is a systematic tool of improving the Gaussian EP.

4. Optimized effective potential for a class of models up to the second order

In this section, we carry out the ORSE for the system, Eq. (7), to calculate the EP up to the second order. The matrix elements which appear in Eq. (15) involve only Gaussian integrals except commutators of creation and annihilation operators and, thus, can be readily calculated as follows, (0) (0) n|NM V(φ x + Φ) |m n 1 − + − − + = √ Ci Cm n i (n − i)! 2(2π)D (m n 2i)/2 ! ! n m n m i=0 n m −1/2 m n n −i × − − f(pj ) f(pk) exp i pk pj x δ(pl pl) j=n−i+1 k=n−i+1 k=n−i+1 j=n−i+1 l=1 ∞ dα −α2 (m−n+2i) −1 2 −1 × √ e V α f − I1 M + Φ + h Jz (19) π xx xz −∞ z

k with n m. In Eq. (19), V k(z) ≡ d V(z) . For simplicity, in getting the above results, we have employed the (dz)k permutation symmetry of momenta in Eq. (15) for various products of δ functions. Note that matrix elements 2 of φx are special cases of Eq. (19). 314 W.-F. Lu et al. / Physics Letters B 540 (2002) 309–318

Substituting the above matrix elements into Eq. (15), one can obtain the first- and the second-order corrections (0)[ ; ] to E0 J φ as ∞ 2 2 (1) µ −1 1 −1 dα −α2 −1 2 −1 E [J ; φ]= − h Jz + f + √ e V α f − I1(M ) + Φ + h Jz 0 2 xz 2 xx π xx xz x z −∞ z (20) and 4 D 2 4 D (2) µ d p 1 ipx −1 µ d p 1 E [J ; φ]=− e h Jz − 0 2 (2π)D f 2(p) xz 16 (2π)D f 3(p) xz x ∞ D d p 1 dα 2 2 −ipx1 −1 ipx2 −α + µ e h Jz e √ e (2π)D f 2(p) x1z π x z x −∞ 1 2 − × V (1) α f 1 − I (M2) + Φ + h−1 J x2x2 1 x2z z z ∞ 2 D µ d p 1 dα −α2 (2) −1 2 −1 + √ e V α f − I1(M ) + Φ + h Jz 8 (2π)D f 3(p) π xx xz x −∞ z ∞ j D 1 d pk 1 − k=1 j!2j (2π)jD j j j=0 k=1 f(pk) k=1 f(pk) ∞ 2 j dα 2 ix = pk −α (j) −1 2 −1 × e k 1 √ e V α f − I1 M + Φ + h Jz , (21) π xx xz x −∞ z respectively. Here, “|···|” represents the absolute value. Next, we extract the approximated EP for the system order by order. δE(0)[J ;φ] At the zeroth order, E(0)[J ; φ]=−C from Eq. (14), and so, taking − 0 = h−1J + Φ = ϕ(0) as Φ, 0 δJx y xy y x one has J (0) = 0. Consequently, the EP at the zeroth-order of δ is (0)[ ; ] 2 (0) Γs ϕ Φ,δ 1 1 2 M 2 V (Φ) =− = fxx − I0 M + I1 M . (22) x ϕx =Φ 2 2 4 Up to the first order (hereafter, any Greek-number superscript, such as “I”, “II”, means “up to the order whose number is consistent with the Greek number”), I [ ; ]= (0)[ ; ]+ (1)[ ; ] E0 J Φ,δ E0 J φ δE0 J φ (23) δEI [J ;Φ,δ] and − 0 = ϕI = Φ yields δJx x ∞ −1 −1 2 −1 dα −α2 (1) −1 2 −1 h Jy + δ h µ h Jz − √ e V α f − I1 M + Φ + h Jz = 0. (24) xy xy yz π yy yz y y z −∞ z I When extracting the EP up to first order, Eqs. (17), (18) and (23) imply that only the Jx up to the first order, J ,is (0) I necessary. Owing to J = 0, it is enough to take Jx = 0 for the last term in the left hand of Eq. (24). Thus, J can W.-F. Lu et al. / Physics Letters B 540 (2002) 309–318 315 be solved from Eq. (24) as

∞ I dα −α2 (1) −1 2 J = δ √ e V α f − I1 M + Φ . (25) π xx −∞

Even J I will not be needed to get the EP up to the ﬁrst order of δ, because there exists no linear, but the quadratic I [ ; ]− term of Jx in the zeroth-order term of (E0 J Φ,δ x Jx Φ) as shown in Eq. (12). In fact, to obtain the EP up to the nth order, one need the approximated J only up to the (n − 1)th order. Now one can write down the EP up to the ﬁrst order ∞ I 1 2 1 2 2 1 2 −1 dα −α2 −1 2 V (Φ, δ) = fxx − I0 M + M I1 M − δ µ f + δ √ e V α f − I1 M + Φ . 2 4 4 xx π xx −∞ (26) Obviously, this result will yield nothing but the Gaussian EP [9, (1995, 2002)]. δEII[J ;Φ,δ] Finally, we consider the second order. ϕ = ϕII =− 0 = Φ can be solved for J II. In the present case, x x δJx however, it is enough to use only J I for the EP. Substituting J I into Eq. (18), we obtain the EP for the system, Eq. (7), up to the second order as

∞ II 1 2 1 2 2 1 2 −1 dα −α2 −1 2 V (Φ, δ) = fxx − I0 M + M I1 M − δ µ f + δ √ e V α f − I1 M + Φ 2 4 4 xx π xx −∞ ∞ 2 D 2 µ d p 1 2 dα −α2 (2) −1 2 − δ µ − 2 √ e V α f − I1 M + Φ 16 (2π)D f 3(p) π xx −∞ ∞ j−1 D 1 = d pk 1 − δ2 k 1 j!2j (2π)(j−1)D j−1 j−1 j=2 f k=1 pk k=1 f(pk) ∞ 2 1 dα −α2 (j) −1 2 × √ e V α fxx − I1 M + Φ , (27) j−1 + j−1 π f k=1 pk k=1 f(pk) −∞ where, one should take δ = 1 after renormalizing VII(Φ, δ),andµ is determined from the stationary condition

∂VII(Φ) = 0. (28) ∂µ

Here, VII(Φ) is the EP after VII(Φ, δ) is renormalized. If Eq. (28) has no real solutions, µ can be fixed by 2VII ∂ (Φ) = 0 [7]. Note that in (1+1) dimensions, { 1 [f −I (M2)]+1 M2I (M2)− 1 µ2f −1} and [f −1 −I (M2)] (∂µ)2 2 xx 0 4 1 4 xx xx 1 in Eq. (27) with δ = 1 is finite and, thus, for any (1 + 1)-dimensional theories which make the series in Eq. (27) finite, no renormalization procedure is needed. Similarly, employing Eq. (19), one can obtain higher order corrections to the Gaussian EP from Eq. (15). To conclude this section, we emphasize that Eq. (27) can easily be used to give the EPs for a number of scalar field theories including those discussed in Refs. [2,11–15]. 316 W.-F. Lu et al. / Physics Letters B 540 (2002) 309–318

5. Application to λφ4 ﬁeld theory

In this section, we consider the potential,

1 2 2 4 V(φ x ) = m φ + λφ , (29) 2 x x which was widely studied with several variational perturbation techniques [2,6]. ∞ 2 ∞ 2 α2n+1e−α √dα = n = , ,... α2ne−α × Substituting Eq. (29) into Eq. (27), and noting that −∞ π 0for 1 2 and −∞ √dα = −n · · · ··· ·( n − ) n = , , ,... π 2 1 3 2 1 for 0 1 2 , one can easily obtain the EP for the system, Eq. (29), up to the second order II 1 2 1 2 2 1 2 2 4 1 2 −1 V (Φ, δ) = fxx − I M + M I M + δ m Φ + λΦ − µ f 2 0 4 1 2 4 xx + 1 −1 − 2 2 + 2 + −1 − 2 δ fxx I1 M m 12λΦ 3λ fxx I1 M 4 D 1 d p 1 − − δ2 m2 − µ2 + 12λΦ2 + 6λ f 1 − I M2 2 16 (2π)D f 3(p) xx 1 dDp dDp 1 1 − δ212λ2Φ2 1 2 (2π)2D f(p ) + f(p ) + f(p + p ) f(p )f (p )f (p + p ) 1 2 1 2 1 2 1 2 3 dDp dDp dDp 1 1 − δ2 λ2 1 2 3 . (30) 2 (2π)3D 3 + 3 3 3 f k=1 pk k=1 f(pk) f k=1 pk k=1 f(pk) 2 Discarding terms with In(M )(n= 0, 1), the above result becomes identical to Eq. (2.36) in Ref. [2, (1990)]. (n) This can be verified by carrying out integrations of In(Ω) (n = 0, 1) and I (Ω) (n = 2, 3, 4) in Ref. [2e] over one component of each Euclidean momentum. In (1 + 1) dimensions, taking δ = 1, one finds that Eq. (30) is finite. Because the (1 + 1)-dimensional Gaussian EP in the Coleman’s normal-ordering prescription is consistent [9, (2002)] with that in the Stevenson’s reparametrization scheme [11, (1985)], Eq. (30) with δ = 1andD = 1is 2 = + consistent with the -mB 0 version of Section 4 in Ref. [2, (1991)]. As for the case of (2 1) dimensions, Stancu has performed a renormalization procedure to make VII(Φ) finite [2, (1991)]. Using µ fixed at the GA result for each order of the ORSE will simultaneously imply that up to each order, the vacuum expectation value of the field operator φx is identical to that in the GA. If one chooses to do so, Eq. (30) and also the second-order result in our former work Ref. [6] will yield the result in Refs. [2b,2c]. Finally, we point out that taking µ = m will lead to the conventional perturbation result on the EP.

6. Discussion and conclusion

In this Letter, an optimized RS expansion scheme of solving the functional Schrödinger equation with an external source is proposed to calculate the EP beyond the GA. For the class of scalar field theories, Eq. (7), we obtain a general expression on the EP up to the second order which can be used easily to obtain the EP for several models. Since the RS expansion is a basic tool in quantum physics and the Schrödinger picture can give us some quantum-mechanical intuition in QFT, we believe our investigation is interesting and useful. Some investigations relevant to the present work can be envisioned. To our knowledge, scalar field models in Refs. [12–15] were seldom investigated beyond the GA, and so it is worth applying Eq. (27) to those concrete models as well as to some bosonized models in condensed matter physics. This Letter discussed just ground state solution of Eq. (13), and actually, one can consider excited state solutions of Eq. (13) to give the EP for the excited states [8]. Our work here also implies that it is possible to introduce the optimized procedure to those schemes W.-F. Lu et al. / Physics Letters B 540 (2002) 309–318 317 proposed in Refs. [2a–c]. Furthermore, since Ref. [5] has generalized the RS perturbation theory to the spinor theory, QED and Yang–Mills theory, it should be viable to generalize the ORSE here to those higher-spin field theories. Besides, since an effective action contains complete information of a field system, it will be useful to develop the ORSE here to calculate the effective action [2c]. Finally, Cornwall, Jackiw and Tomboulis developed a generalized EP for composite operators and calculated it with Rayleigh–Ritz procedure [21], and it will be interesting to develop the ORSE here for calculating the generalized EP which will go beyond the variational result.

Acknowledgements

This project was supported by the Korea Science and Engineering Foundation through the Center for Strongly Correlated Materials Research (SNU). Lu’s work was also supported in part by the National Natural Science Foundation of China under the grant No. 19875034.

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On a core instability of ’t Hooft–Polyakov type monopoles

F.A. Bais, J. Striet

Institute for Theoretical Physics, University of Amsterdam, Valckenierstraat 65, 1018XE Amsterdam, The Netherlands Received 4 June 2002; received in revised form 17 June 2002; accepted 19 June 2002 Editor: P.V. Landshoff

Abstract We discuss a core instability of ’t Hooft–Polyakov monopoles in Alice electrodynamics type of models in which charge conjugation symmetry is gauged. The monopole may deform into a toroidal defect which carries an Alice ﬂux and a (non- localizable) magnetic Cheshire charge.  2002 Elsevier Science B.V. All rights reserved.

1. Introduction by some observations that were made in theories with global symmetries [3]. We start by briefly summarizing the main features Since the pioneering work of ’t Hooft and Polyakov of Alice Electrodynamics (AED), then we discuss [1] magnetic monopoles have been studied in detail in the particular tensor model we will use to explicitly many different models. In this Letter we address the establish the core instability and determine some question of stability of the core of the fundamental, region in parameter space where this occurs. spherically symmetric, monopole configuration, a stability which appears to be so obvious that it was never seriously questioned. We will show that in a rather 2. A core deformation in Alice electrodynamics simple model the spherically symmetric unit charge magnetic monopole is not the global minimal energy Alice electrodynamics (AED) is a gauge theory solution for all parameter values in the model. The fact with gauge group H = U(1) Z2 ∼ O(2), i.e., a min- that the core topology is not uniquely determined by imally non-abelian extension of ordinary electrody- the boundary conditions and different core topologies namics. The nontrivial Z2 transformation reverses the can be deformed into each other was already estab- direction of the electric and magnetic fields and the lished earlier [2]. As we will indicate, Alice theories sign of the charges. In other words, in Alice elec- have a special feature which makes it more plausible trodynamics charge conjugation symmetry is gauged. that such a core deformation really may be favored en- However, as this non-abelian extension is discrete, it ergetically. Our interest in this problem was rekindled only affects electrodynamics through certain global (topological) features, such as the appearance of Al- ice fluxes and Cheshire charges [4,5]. The possibility E-mail addresses: [email protected] (F.A. Bais), of a non-localizable magnetic Cheshire charge will be [email protected] (J. Striet). of great importance in our study of the core instability

Fig. 1. A slice of the core deformation of the spherical monopole into an Alice loop, carrying a magnetic Cheshire charge. of the monopole. The topological structure of U(1) metry of the order parameter reflects the charge con- Z2 differs from that of U(1) in a few subtle points. jugation symmetry of the theory. In AED the spherical AED allows topologically stable localized fluxes since monopole can be punctured and be deformed into an Π0(U(1) Z2) = Z2, the so-called Alice fluxes. Note Alice loop, this configuration is consistent with the or- that in this theory this flux is coexisting with the under parameter because of the head–tail symmetry of broken U(1) of electromagnetism and is therefore not the order parameter. In Fig. 1 we show a slice of this an ordinary “magnetic” flux. Just as a U(1) gauge the- core deformation. Note that the order parameter on the ory, AED may contain magnetic monopoles, which right-hand side of Fig. 1 only rotates over an angle follows from the fact that Π1(U(1) Z2) = Z.We π when going around a single flux. This is the hall- note however, that due to the fact that the Z2 and the mark for an Alice flux, i.e., the core deformed spheri- U(1) part of the gauge group do not commute, mag- cal monopole is in fact an Alice loop carrying a magnetic charges of opposite sign belong to the same topo- netic Cheshire charge. logical sector. Alice phases can be generated by spontaneously breaking SU(2) (or SO(3))toU(1)Z2,for example by choosing a Higgs field in a 5-dimensional 3. The tensor model of AED representation of the gauge group. In that case the topological defects, fluxes [6,7] and monopoles [1], To be able to answer stability questions of the will correspond to regular classical solutions. It was spherically symmetric monopole configuration (or the pointed out long ago that there are interesting issues Cheshire charged Alice loop) we consider an explicit concerning the core stability of magnetic monopoles. model. In the remainder we focus on the original Fixing the asymptotics of the Higgs field, the core tensor Alice model [8]. The action of this model is (i.e., the zeros of the Higgs field1) may have dif- given by: ferent topologies, notably that of a ring rather than the conventional point. These core topologies can be 4 1 a,µν a 1 µ smoothly deformed into each other and it is a question S = d x F F + Tr D ΦDµΦ 4 µν 4 of energetics what will be the lowest energy monopole state [2]. We return to this issue in this Letter be- − V(Φ) , (1) cause the core deformation would be accompanied by the rather unusual delocalized version of (magnetic) = ab charge, the so-called Cheshire charge. In the specific where the Higgs field Φ Φ is a real, symmet- × AED model we studied the Higgs field is a symmetric, traceless 3 3 matrix, i.e., Φ is in the five- = ric tensor, whose vacuum expectation value may be dimensional representation of SO(3) and DµΦ − [ ] = a depicted as a bidirectional arrow. The head–tail sym- ∂µΦ ie Aµ,Φ , with Aµ AµTa,whereTa are the generators of SO(3). The most general renormalize- able potential is given by [9]:

1 In fact in AED the Higgs field does not even need to go to zero 1 2 2 1 3 1 2 2 for the ring type solution. V =− µ Tr Φ − γ Tr Φ + λ Tr Φ (2) 2 3 4 F.A. Bais, J. Striet / Physics Letters B 540 (2002) 319–323 321 with γ>0, since (Φ, γ ) = (−Φ,−γ). of the Higgs field. Inserting the previous expressions For a suitable range of the parameters in the potential, for the Higgs field, we find the gauge symmetry of the model will be broken to 1 µ the symmetry of AED. In the “unitary” gauge, where Tr D ΦDµΦ the Higgs field is diagonal, the ground state is (up to 4 1 1 permutations) given by the following matrix: = (∂ φ )2 + D3 φ 2   2 µ 1 2 µ 2 −f 00 9 2 2 =  −  + e2f 2 A1 + A2 +··· (7) Φ0 0 f 0 (3) 2 µ µ 002f 3 = − 3 with Dµ ∂µ i2eAµ. The second term shows that = γ + + 2 2 with f 12λ (1 1 24µ λ/γ ). The full action the φ2 component of the Higgs field carries a charge 2 3 has four parameters, e,µ ,γ,λ, this number can be 2e with respect to the unbroken U(1) component Aµ reduced to two dimensionless parameters by appropri- of the gauge field. The first term describes the usual ate rescalings of the variables. A physical choice for charge neutral Higgs particle and the third term yields these dimensionless parameters is to take the ratio’s of the mass of the charged gauge fields. Thus the relevant the masses that one finds from perturbing around the lowest order action is given by: homogeneous minimum. To determine these, we write 4 1 a a,µν 1 2 1 3 2 the action in the unitary gauge where the massless S = d x F F + (∂µφ ) + D φ 4 µν 2 1 2 µ 2 components of Φ have been absorbed by the gauge fields. The physical components of the Higgs field may − 1 2 2 − 1 2| |2 m1φ1 m2 φ2 be expanded as: 2 2 √ 1 2 2 µ µ − 2 1 + 2 +··· Φ x = Φ0 + 2 φ1 x E1 mA Aµ Aµ (8) √ 2 + µ µ µ T 2φ2 x R3 a x E2R3 a x (4) 2 = 2 + 2 = 2 = 2 2 with m1 4µ 2γf, m2 6γf and mA 9e f . with Thus two degrees of freedom of the five-dimensional   −100 Higgs field are ‘eaten’ by the broken gauge fields, 1   one degree of freedom forms the real neutral scalar E1 = √ 0 −10, 6 field and two degrees of freedom form the complex  002 (doubly charged) scalar field. To specify a point in 100 the parameter space of classical solutions we may, up 1   E2 = √ 0 −10, to irrelevant rescalings, use the dimensionless mass 2  000 ratio’s m1/m2 and mA/m2. 001 1   E3 = √ 000 (5) 2 100 4. The core instability and R are the usual rotation matrices. To second i In this section we will show that the monopole core, order, the potential V(Φ)takes the following form:2 see Fig. 2, becomes meta- or unstable for a certain = + 2 + 2 + | |2 +··· range in the parameter space of the theory. Our strat- V(Φ) const 2µ γf φ1 3γf φ2 egy is as follows. Using a numerical cooling method, (6) we look for the global and local minima of the mono- yielding the two distinct masses of the Higgs modes. pole energy within a class of configurations given by a 1 µ Next we look at the ‘kinetic’ term, 4 Tr(D ΦDµΦ), suitable ansatz. We restrict ourselves to static configurations and the ansatz we use contains the spherically 2 ia symmetric ’t Hooft–Polyakov monopole solution as a It is most convenient to use φ2 for the combination φ2e ,since these two Higgs modes, φ2 and a, combine to form one complex special case [8]. The ansatz is cylindrically symmetric charged field, from now on called φ2. and also has reflection symmetry with respect to the 322 F.A. Bais, J. Striet / Physics Letters B 540 (2002) 319–323

= Fig. 3. A slice of a Cheshire charged Alice loop conﬁgura- Fig. 2. A slice of a magnetic monopole solution at y 0, we plotted = − 2 2 = 2 2 tion at y 0, we plotted 1 (Tr Φ /f ). m1/m2 0.882 and 1 − (Tr Φ /f ). m1/m2 = 0.571 and mA/m2 = 0.0095. mA/m2 = 0.0073. z = 0 plane. Our variational approach does not nec- Table 1 essarily lead to solutions to the equations of motion = = of the model. However since the spherical monopole ρ 0 z 0 = = solution is contained in the variational ansatz we can φ1 ∂ρ φ1 0 ∂zφ1 0 = = = study its stability. The ansatz for the Higgs ﬁeld is: φ2 ∂ρ φ2 φ2 0 ∂zφ2 0 φ3 φ3 = 0 φ3 = 0 A∂ρ A = 0 ∂zA = 0 Φ(z,ρ,θ = 0) = φ1(z, ρ)E1 + φ2(z, ρ)E2

+ φ3(z, ρ)E3 (9) important point is that our ansatz in principle allows and for the possibility of an Alice loop configuration carry- T ing a magnetic Cheshire charge, see Fig. 3. These are Φ(z,ρ,θ) = R3(θ)Φ(z, ρ, θ = 0)R3(θ) . (10) exactly the two configurations that we want to com- The ansatz for the gauge fields is simply given by pare. k eAj =−( x A(z, ρ), very similar to the one for i ij k x2 Using the ansatz, we indeed found configurations the spherically symmetric monopole [1], except that having less energy than the spherically symmetric we allow A(z, ρ) to depend on ρ and z instead of monopole solution, at least in a certain region of only depending on r = ρ2 + z2. The boundary con- the parameter space. We even found that the spheri- ditions for r →∞ are the boundary conditions of cally symmetric monopole is not always locally stable. the spherically symmetric monopole as in [8], i.e., Strictly speaking our non-spherical symmetric con- A(z, ρ) goes to one and the Higgs field to Φ(z,ρ,θ) = figurations, the magnetically Cheshire charged con- T T R3(θ)R2(arccos(z/r))Φ0R2(arccos(z/r)) R3(θ) . figurations, need not be exact solutions and conse- The boundary conditions for ρ = 0andz = 0 follow quently, they only yield an upper bound to the energy by imposing the cylindrical and reflection symmetry of the true solution. Obviously this suffices to show and are given in Table 1. With this ansatz one only the instability of the standard monopole and we do ex- needs to determine the solution for θ = 0, ρ>0and pect that the true solution is very close to this mag- z>0. The rest of the solution follows with the help of netically Cheshire charged Alice loop configuration. the cylindrical and reflection symmetry of the ansatz. In this Letter we only present the results concerning We used a grid of up to 200 × 200 points, 175 points configurations along a specific path in the parameter are used to describe the core and 25 to describe the space of the theory. We refer to an forthcoming pa- long range behavior of the configuration. This spilt up per [10] where we will determine the stability, meta in is due to the big difference in the value of mA with stability and instability regions of both configurations, respect to the values of m1 and m2. within this ansatz for the ‘full’ parameter space of the It is easy to see that the boundary conditions are also model. The path, see Fig. 4, we have considered cov- met by the spherically symmetric monopole ansatz, so ers three regions of the model. In one region, on the it is indeed contained in our more general ansatz. The left of point A, the monopole is the only stable solu- F.A. Bais, J. Striet / Physics Letters B 540 (2002) 319–323 323

5. Conclusion and outlook

In this Letter we showed that monopoles of the ’t Hooft–Polyakov type may exhibit a core instability, depending on the parameters of the theory. In one part of the parameter space, the spherical monopole is the global minimum. In another part it corresponds to a local minimum and there even is a region where it is unstable. We found that the competing configuration is a magnetically Cheshire charged Alice loop. Since Fig. 4. Going from the left to the right: after point A, the thick we worked within a limited ansatz, the regions we line represents the errorbar along the path, the Alice loop becomes found for the monopole global and/or local stability locally stable, between point A and C both the monopole and the are in fact only upper bounds on the stability regions Alice loop are locally stable. After point B the monopole is only of the spherical monopole, and these regions can only meta stable and after point C it is even an unstable solution. become smaller when no (or less) restrictions are put on the configurations one may sample. At the moment tion. In the next region, between point A and C, both we are scanning the ‘full’ parameter space of the the monopole and the Alice loop are locally stable and model. The results obtained as well as more detailed in the last, on the right of point C, the Alice loop is information on the model, the simulations and the the only locally stable configuration. Somewhere half configurations we found, will be published elsewhere way the middle region, point B, the monopole is no [10]. More in general we may expect this type of core longer the global minimum, whereas the Alice loop is, instability of topological defetcs if the theory admits i.e., the monopole is only a meta stable solution. The different types of defects that coexist. Indeed also two extreme sides of the path can be understood as fol- higher-dimensional analogous should be of interest. lows. The two masses m1 and m2 correspond to the energy cost to deviate from the vacuum in two different ways. m1 is the energy cost for deviating in the neu- References tral direction or ‘length’ of the Higgs field, while m2 is the energy cost in deviating in a non-uniaxial direc- [1] G. ’t Hooft, Nucl. Phys. B 79 (1974) 276; tion. Thus in the limit m1/m2 → 0, the deviations in A.M. Polyakov, JETP Lett. 20 (1974) 194. the non-uniaxial directions are suppressed. There one [2] F.A. Bais, P. John, Int. J. Mod. Phys. A 10 (1995) 3241. would expect the uniaxial monopole to be the global [3] R. Rosso, E.G. Virga, J. Phys. A 29 (1996) 4247; stable solution. In the opposite limit, m /m →∞, S. Kralj, E.G. Virga, J. Phys. A 34 (2001) 829; 1 2 S. Mkaddem, E.C. Gartland Jr., Phys. Rev. E 62 (2000) 6694. one would expect an ‘escape’ into the non-uniaxial di- [4] A.S. Schwarz, Nucl. Phys. B 208 (1982) 141. rections and a suppression in the length deviation, sig- [5] M. Alford, K. Benson, S. Coleman, J. March-Russell, naling the meta stability of the uniaxial monopole, as F. Wilczek, Nucl. Phys. B 349 (1991) 414. is the case for the Alice loop configuration. Notice that [6] H.B. Nielsen, P. Olesen, Nucl. Phys. B 61 (1973) 45. the length of the Higgs field never becomes zero3 in [7] J. Striet, F.A. Bais, Phys. Lett. B 497 (2001) 172. [8] R. Shankar, Phys. Rev. D 14 (1976) 1107. − 2 2 the case of the Alice loop, i.e., 1 (Tr Φ /f ) never [9] H. Georgi, S.L. Glashow, Phys. Rev. D 6 (1972) 2977. becomes one in Fig. 3. [10] F.A. Bais, J. Striet, in preparation.

3 Not shown here, but we also ﬁnd that the minimum length of the Higgs ﬁeld in the case of the Alice loop increases for increasing m1/m2 as this argument indicates. Physics Letters B 540 (2002) 324–325 www.elsevier.com/locate/npe Erratum Erratum to: “Hunting the QCD-odderon in hard diffractive electroproduction of two pions” [Phys. Lett. B 535 (2002) 117] ✩

Ph. Hägler a,B.Pireb, L. Szymanowski b,c,O.V.Teryaevd

a Institut für Theoretische Physik, Universität Regensburg, D-93040 Regensburg, Germany b CPhT, École Polytechnique, F-91128 Palaiseau, France c Sołtan Institute for Nuclear Studies, Ho˙za 69, 00-681 Warsaw, Poland d Boguliubov Laboratory of Theoretical Physics, JINR, 141980 Dubna, Russia Received 24 June 2002

Due to a numerical error in the calculations the Figs. 3, 4, 5, 6 have to be replaced by new ones presented below. The d-wave contribution shows now up clearly. The main conclusion about the sizable value of the charge asymmetry remains unchanged.

Fig. 3. Charge asymmetry given by Eq. (28) for a minimal f0 width, Fig. 4. Same as Fig. 3 but for maximal f0 width. with an error band showing the uncertainty comming from different values of αsoft and ΛQCD.

✩ PII of original article: S0370-2693(02)01736-7. E-mail address: [email protected] (B. Pire).

2 2 2 Fig. 5. m2π -dependence of the asymmetry for t =−0.8GeV and Fig. 6. t-dependence of the asymmetry for Q = 3GeV and 2 2 2 for different values of Q :1GeV (dashed line), 3 GeV (dotted m2π = 0.95 GeV; the f0 width has been taken to be 75 MeV. line), 10 GeV2 (solid line); the d-wave contribution at Q2 = 3GeV2 is shown with a dense dotted line; the f0 width has been taken to be 75 MeV. Physics Letters B 540 (2002) 326–371 www.elsevier.com/locate/npe

Cumulative author index to volumes 531–540

Aalseth, C.E., 532,8 Ahmed, S.N., 531,52 Abazov, V.M., 531,52 Ahn, B.S., 538, 11; 540,33 Abbaneo, D., 533, 223; 537,5 Ahn, C., 539, 281; 540, 111 Abbiendi, G., 533, 207; 539,13 Ahn, S.H., 531,9 Abbott, B., 531,52 Aihara, H., 538, 11; 540,33 Abdalla, E., 538, 435 Ainsley, C., 533, 207; 539,13 Abdalla, M.C.B., 536, 114 Airapetian, A., 535,85 Abdallah, J., 533, 243 Aitala, E.M., 539, 218 Abdesselam, A., 531,52 Akaishi, Y., 535,70 Abe, K., 538, 11, 11; 540, 33, 33 Akatsu, M., 538, 11; 540,33 Abe, R., 540,33 Åkesson, P.F., 533, 207; 539,13 Abe, T., 531,9;538, 11; 539, 197; 540,33 Akopov, N., 535,85 Ablikim, M., 536,34 Akopov, Z., 535,85 Abolins, M., 531,52 Albrecht, T., 533, 243 Abramov, V., 531,52 Albright, C.H., 532, 311 Abramowicz, H., 531,9;539, 197 Alcaraz, J., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Abreu, P., 533, 243 540, 43, 185 Abriola, D., 534,45 Alderweireld, T., 533, 243 Achard, P., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; 540, 43, 185 Alekseev, A., 532, 350 Acharya, B.S., 531,52 Alemanni, G., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Achasov, M.N., 537, 201 540, 43, 185 Achasov, N.N., 534,83 Alemany-Fernandez, R., 533, 243 Ackermann, M., 539, 179 ALEPH Collaboration, 533, 223; 537,5 Adachi, I., 538, 11; 540,33 Alexander, G., 533, 207; 539,13 Adam, W., 533, 243 Adamczyk, L., 531,9;539, 197 Alexandre, J., 531, 316 Adams, D.L., 531,52 Alexanian, G., 537, 103 Adams, M., 531,52 Alexeev, G.D., 531,52 Adamus, M., 531,9;539, 197 Algora, A., 540, 199 Added, N., 534,45 Alimonti, G., 535, 43; 537, 192; 540,25 Adler, S., 537, 211 Alishahiha, M., 535, 328; 536, 129; 538, 180 Adler, S.L., 533, 121 Alkofer, R., 536, 177 Adloff, C., 539,25 Allaby, J., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; 540, 43, 185 Adriani, O., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Allison, J., 533, 207; 539,13 540, 43, 185 Allmendinger, T., 533, 243 Adzic, P., 533, 243 Allport, P.P., 533, 243 Afanasiev, S.V., 538, 275 Almehed, S., 533, 243 Affholderbach, K., 533, 223; 537,5 Aloisio, A., 531, 28, 39; 534, 28; 535, 37, 59; 536, 24, 209, 217; Aghuzumtsyan, G., 531,9;539, 197 537, 21; 538, 21; 540, 43, 185 Agostino, L., 535, 43; 537, 192; 540,25 Alton, A., 531,52 Aguilar-Benitez, M., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Alves, G.A., 531,52 540, 43, 185 Alves, J.J.S., 534,45

0370-2693/2002 Published by Elsevier Science B.V. PII:S0370-2693(02)02254-2 Cumulative author index to volumes 531–540 (2002) 326–371 327

Alves, V.S., 531, 289 Arik, E., 539, 188 Alviggi, M.G., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Arkadov, V., 539,25 540, 43, 185 Arleo, F., 532, 231 Amaldi, U., 533, 243 Armenise, N., 539, 188 Amapane, N., 533, 243 Armstrong, S.R., 533, 223; 537,5 Amaral, P., 539,13 Arneodo, M., 531,9;539, 197 Amarian, M., 535,85 Arnoud, Y., 531, 52; 533, 243 Amato, S., 533, 243; 539, 218 Arnowitt, R., 538, 121 Ambjørn, J., 538, 189 Artamonov, A., 539, 188 Ambrosino, F., 535, 37; 536, 209; 537, 21; 538,21 Asahi, K., 534,39 Anagnostou, G., 533, 207; 539,13 Asai, S., 533, 207; 539,13 Anashkin, E., 533, 243 Asaka, T., 540, 295 Anchordoqui, L.A., 535, 302 Asano, Y., 538, 11; 540,33 Anderhub, H., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Aschenauer, E.C., 535,85 540, 43, 185 Ashery, D., 539, 218 Anderson, E.W., 531,52 Ask, S., 533, 243 Anderson, K.J., 533, 207; 539,13 Asman, B., 533, 243 Ando, S., 533,25 Aso, T., 538, 11; 540,33 Andreazza, A., 533, 243 Astone, P., 540, 179 Andreev, O., 534, 163 Astvatsatourov, A., 539,25 Andreev, V., 539,25 Atiya, M.S., 537, 211 Andreev, V.P., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Atwood, D., 533,37 540, 43, 185 Augustin, I., 537,28 Andrieu, B., 539,25 Augustin, J.E., 533, 243 Andringa, S., 533, 243 Augustinus, A., 533, 243 Andryakov, A., 535,37 Aulchenko, V., 538, 11; 540,33 Anjos, J.C., 535, 43; 537, 192; 539, 218; 540,25 Aushev, T., 538, 11; 540,33 Anjos, N., 533, 243 Avakian, H., 535,85 Anjos, R.M., 534,45 Anselmo, F., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Avakian, R., 535,85 540, 43, 185 Avetissian, A., 535,85 Anthonis, T., 539,25 Avetissian, E., 535,85 Anticic, T., 538, 275 Avignone, F.T., 532,8 Antilogus, P., 533, 243 Avila, C., 531, 52; 537,41 Antonelli, A., 533, 223; 535, 37; 536, 209; 537, 5, 21; 538,21 Awunor, O., 533, 223; 537,5 Antonelli, M., 533, 223; 535, 37; 536, 209; 537, 5, 21; 538,21 Axelsson, A., 540, 199 Antonioli, P., 531,9;539, 197 Axen, D., 533, 207; 539,13 Antonov, A., 531,9;539, 197 Ayala Filho, A.L., 534,76 Antonov, D., 535, 236 Azemoon, T., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Antusch, S., 538,87 540, 43, 185 Anzivino, G., 533, 196; 536, 229; 537,28 Aziz, T., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; 540, 43, 185 Aoki, M., 537, 211 Azuelos, G., 533, 207; 539,13 Aoki, S., 539, 188 Azuma, T., 538, 393 Apel, W.-D., 533, 243 Azzurri, P., 533, 223; 537,5 Aphecetche, L., 538,27 Appel, J.A., 539, 218 Baarmand, M.M., 531,52 Apreda, R., 536, 161 Babaev, A., 539,25 Arbuzov, A.B., 535, 378 Babintsev, V.V., 531,52 Arcelli, S., 539,13 Babu, K.S., 532,77 Arcidiacono, R., 533, 196; 536, 229; 537,28 Babukhadia, L., 531,52 Ardebili, M., 537, 211 Babusci, D., 540, 179 Aref’eva, I.Ya., 532, 291; 536, 138 Bacci, C., 535, 37; 536, 209; 537, 21; 538,21 Areﬁev, A., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Bacon, T.C., 531,52 540, 43, 185 Badaud, F., 533, 223; 537,5 Arena, V., 535, 43; 537, 192; 540,25 Baden, A., 531,52 Arhrib, A., 537, 217 Bagliesi, G., 533, 223; 537,5 328 Cumulative author index to volumes 531–540 (2002) 326–371

Bagnaia, P., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Bari, G., 531,9;539, 197 540, 43, 185 Barillère, R., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Bagnasco, S., 533, 237 540, 43, 185 Bahcall, J.N., 534, 120 Baringer, P., 531,52 Bahns, D., 533, 178 Barker, G., 533, 243 Bähr, J., 539,25 Barklow, T., 533, 223; 537,5 Baier, R., 539,46 Barlow, R.J., 533, 207; 539,13 Bailey, D.S., 531,9;539, 197 Barna, D., 538, 275 Bailey, I., 533, 207; 539,13 Baroncelli, A., 533, 243 Bailey, P., 535,85 Barr, G., 533, 196; 536, 229; 537,28 Baillon, P., 533, 243 Barreiro, F., 531,9;539, 197 Bais, F.A., 540, 319 Barrelet, E., 539,25 Bajo, A., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; 540, 43, 185 Barreto, J., 531,52 Baker, W.F., 537,41 Barrow, J.D., 532, 153 Bakich, A.M., 538, 11; 540,33 Barshay, S., 535, 201 Baksay, G., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Barszczak, T., 539, 179 540, 43, 185 Bartalini, P., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Baksay, L., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; 540, 43, 185 540, 43, 185 Bartel, W., 539,25 Balabanski, D.L., 537, 45; 538,33 Bartke, J., 538, 275 Balachandran, A.P., 537, 103 Bartl, A., 538, 59, 137 Baldew, S.V., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Bartlett, J.F., 531,52 540, 43, 185 Barton, R.A., 538, 275 Baldin, B., 531,52 Bartoš, E., 538,45 Baldini, W., 533, 237 Bartsch, D., 531,9;539, 197 Baldini-Ferroli, R., 538,21 Bashkirov, V., 531,9;539, 197 Baldo, M., 533,17 Basile, M., 531, 9, 28, 39; 534, 28; 535, 59; 536, 24, 217; 539, 197; Ballestrero, A., 533, 243 540, 43, 185 Balm, P.W., 531,52 Bassan, M., 540, 179 Bambade, P., 533, 243 Bassett, B.A., 536,9 Bamberger, A., 531,9;539, 197 Bassler, U., 531,52 Ban, Y., 538, 11; 540,33 Batalin, I.A., 534, 201 Banas, E., 538, 11; 540,33 Batalova, N., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Bandyopadhyay, A., 540,14 540, 43, 185 Banerjee, R., 533, 162; 537, 340 Batist, L., 532,29 Banerjee, S., 531, 28, 39, 52; 534, 28; 535, 59; 536, 24, 217; Batley, J.R., 533, 196; 537,28 539, 218; 540, 43, 185 Batley, R.J., 533, 207; 539,13 Banerjee, Sw., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Battaglia, M., 533, 243 540, 43, 185 Battiston, R., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Barakbaev, A.N., 531,9;539, 197 540, 43, 185 Baranov, P., 539,25 Baturin, V., 535,85 Barate, R., 533, 223; 537,5 Baubillier, M., 533, 243 Barbagli, G., 531,9;539, 197 Bauer, D., 531,52 Barberio, E., 533, 207; 539,13 Bauerdick, L.A.T., 531,9;539, 197 Barberis, E., 531,52 Baumgarten, C., 535,85 Barberis, S., 535, 43; 537, 192; 540,25 Bay, A., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; 538, 11; Barbi, M., 531,9;539, 197 540, 33, 43, 185 Barbier, R., 533, 243 Baylac, M., 539,8 Barbot, C., 533, 107 Bazarko, A.O., 537, 211 Barbuto, E., 539, 188 Beacom, J.F., 537, 227 Barczyk, A., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Bean, A., 531,52 540, 43, 185 Beane, S.R., 535, 177 Bardek, V., 531, 311 Beaudette, F., 531,52 Bardin, D., 533, 243 Becattini, F., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Barenboim, G., 534, 106; 537, 227 540, 43, 185 Barger, V., 532, 15, 19; 534, 120; 537, 179; 538, 346; 540, 247 Becher, T., 535, 127; 540, 278 Cumulative author index to volumes 531–540 (2002) 326–371 329

Bechtle, P., 533, 207; 539,13 Berges, P., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; 540, 43, 185 Becker, H.G., 533, 196; 536, 229; 537,28 Berggren, M., 533, 243 Becker, J., 539,25 Berglund, P., 534, 147 Becker, U., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Bergström, M., 540, 199 540, 43, 185 Beri, S.B., 531,52 Beckmann, M., 535,85 Berkelman, K., 533, 223; 537,5 Becks, K.-H., 533, 243 Bernabéu, J., 531,90 Bediaga, I., 535, 43; 537, 192; 539, 218; 540,25 Bernardi, G., 531,52 Bednarek, B., 531,9;539, 197 Berndt, T., 539,25 Beer, G., 535,52 Bernreuther, S., 535,85 Begalli, M., 533, 243 Berns, H.G., 539, 179 Begel, M., 531,52 Berntzon, L., 533, 243 Beglarian, A., 539,25 Bertanza, L., 537,28 Behera, P.K., 538, 11; 540,33 Bertin, P.Y., 539,8 Behner, F., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; 540, 43, 185 Bertolin, A., 531,9;539, 197 Behnke, O., 539,25 Bertolini, M., 540, 104 Behnke, T., 533, 207; 539,13 Bertolucci, S., 535, 37; 536, 209; 537, 21; 538,21 Behrens, U., 531,9;539, 197 Bertram, I., 531,52 Behrmann, A., 533, 243 Bertrand, D., 533, 243 Beier, C., 539,25 Bertucci, B., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Belitsky, A.V., 538, 289 540, 43, 185 Bell, K.W., 533, 207; 539,13 Besancon, M., 533, 243 Bell, M., 531,9;539, 197 Besson, A., 531,52 Bell, P.J., 533, 207; 539,13 Besson, N., 533, 243 Bella, G., 533, 207; 539,13 Betev, B.L., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Bellagamba, L., 531,9;539, 197 540, 43, 185 Bellato, M., 535,93 Betev, L., 538, 275 Belle Collaboration, 538, 11; 540,33 Bethke, S., 533, 207; 539,13 Belleguic, V., 532,29 Bettoni, D., 533, 196, 237 Bellerive, A., 533, 207; 539,13 Beuselinck, R., 531, 52; 533, 223; 537,5 Bellucci, L., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Bevan, A., 536, 229; 537,28 540, 43, 185 Bevan, A.J., 533, 196 Bellunato, T., 533, 243 Bezzubov, V.A., 531,52 Beloborodov, K.I., 537, 201 Bhadra, S., 531,9;539, 197 Belostotski, S., 535,85 Bhat, P.C., 531,52 Belousov, A., 539,25 Bhatnagar, V., 531,52 Belyaev, A., 531,52 Bhattacharjee, M., 531,52 Bencivenni, G., 533, 223; 535, 37; 536, 209; 537, 5, 21; 538,21 Bhattacharya, S., 531, 281 Bender, M., 537,28 Bhuyan, B., 537, 211 Benekos, N., 533, 243 Bialas, A., 532, 249 Benelli, G., 533, 207; 539,13 Białkowska, H., 538, 275 Benen, A., 531,9;539, 197 Bianchi, N., 535,85 Bensalem, W., 538, 309 Bianco, S., 535, 43; 537, 192; 540,25 Benussi, L., 535, 43; 537, 192; 540,25 Biasini, M., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Benvenuti, A., 533, 243 540, 43, 185 Benzoni, G., 540, 199 Biebel, O., 533, 207; 539,13 Berat, C., 533, 243 Bigazzi, F., 536, 161 Berbeco, R., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Bigi, A., 537,28 540, 43, 185 Bigi, I.I., 535, 155 Berdugo, J., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Biglietti, M., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; 540, 43, 185 540, 43, 185 Berdyugin, A.V., 537, 201 Biino, C., 533, 196; 536, 229; 537,28 Berezhiani, Z., 535, 207 Biland, A., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; 540, 43, 185 Bergbusch, P.C., 537, 211 Bilenky, S.M., 533, 191; 538,77 Berger, Ch., 539,25 Bilic,´ N., 535,17 Berger, J.-F., 531,61 Billmeier, A., 538, 275 330 Cumulative author index to volumes 531–540 (2002) 326–371

Billó, M., 536, 121 Bonomi, G., 535, 43; 537, 192; 540,25 Bini, C., 535, 37; 536, 209; 537, 21; 538,21 Boogert, S., 531,9;539, 197 Bizot, J.C., 539,25 Boonekamp, M., 533, 243 Bizzeti, A., 533, 196; 536, 229; 537,28 Boos, E.E., 534,97 Blackmore, E.W., 537, 211 Boos, E.G., 531,9;539, 197 Blair, G.A., 533, 223; 537,5 Booth, C.N., 533, 223; 537,5 Blais, D., 535,11 Booth, P.S.L., 533, 243 Blaising, J.J., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Borcherding, F., 531,52 540, 43, 185 Bordag, M., 533, 182 Blasi, N., 540, 199 Borean, C., 533, 223; 537,5 Blaufuss, E., 539, 179 Borgia, B., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; 540, 43, 185 Blaylock, G., 539, 218 Borisov, G., 533, 243 Blazey, G., 531,52 Borissov, A., 535,85 Blekman, F., 531,52 Borissov, L., 534, 106; 537, 227 Blessing, S., 531,52 Bornyakov, V.G., 537, 291 Bloch, D., 533, 243 Borras, K., 531,9;539, 197 Bloch-Devaux, B., 533, 223; 537,5 Borreani, G., 533, 237 Bloise, C., 535, 37; 536, 209; 537, 21; 538,21 Borremans, D., 537, 45; 538,33 Blom, M., 533, 243 Bos, K., 531,52 Blondel, A., 533, 223; 537,5 Boscherini, D., 531,9;539, 197 Bloodworth, I.J., 533, 207; 539,13 Boschini, M., 535, 43; 537, 192; 540,25 Blume, C., 538, 275 Bose, T., 531,52 Blumenhagen, R., 532, 141 Bossi, F., 533, 223; 535, 37; 536, 209; 537, 5, 21; 538,21 Blumenschein, U., 533, 223; 537,5 Botje, M., 538, 275 Blümer, H., 537,28 Botner, O., 533, 243 Blyth, C.O., 538, 275 Botta Cantcheff, M., 533, 126 Blyth, S.C., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Bottai, S., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; 540, 43, 185 540, 43, 185 Böttcher, H., 535,85 Bobbink, G.J., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Boucrot, J., 533, 223; 537,5 540, 43, 185 Boudry, V., 539,25 Bobkov, K., 537, 155 Bouhali, O., 535,85 Boca, G., 535, 43; 537, 192; 540,25 Bouhova-Thacker, E., 533, 223; 537,5 Boccali, T., 533, 223; 537,5 Boumediene, D., 533, 223; 537,5 Bocci, V., 535, 37; 536, 209; 537, 21; 538,21 Bouquet, B., 533, 243 Bock, P., 533, 207; 539,13 Bourilkov, D., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Bocquet, G., 533, 196; 536, 229; 537,28 540, 43, 185 Bodmann, B., 531,9;539, 197 Bourquin, M., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Boehnlein, A., 531,52 540, 43, 185 Boeriu, O., 533, 207; 539,13 Boutemeur, M., 533, 207; 539,13 Bogdanchikov, A.G., 537, 201 Bouwhuis, M., 535,85 Böhm, A., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; 540, 43, 185 Bowcock, T.J.V., 533, 243 Böhme, J., 539,25 Bowdery, C.K., 533, 223; 537,5 Böhrer, A., 533, 223; 537,5 Bowman, J.D., 534,39 Boimska, B., 538, 275 Boyarsky, A., 532, 357 Boix, G., 533, 223; 537,5 Boyd, S.C., 539, 179 Bojko, N.I., 531,52 Boyko, I., 533, 243 Bokel, C., 531,9;539, 197 Boz, M., 531, 119 Boldizsar, L., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Bozek, A., 538, 11; 540,33 540, 43, 185 Bozhenok, A.V., 537, 201 Bolton, T.A., 531,52 Bozza, C., 539, 188 Bonacorsi, D., 533, 207; 539,13 Braccini, S., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Bonasera, A., 537, 268 540, 43, 185 Bondar, A., 538, 11; 540,33 Bracco, A., 535, 93; 540, 199 Bonesini, M., 533, 243 Bracinik, J., 538, 275 Bonifazi, P., 540, 179 Brack, J., 535,85 Bonissent, A., 533, 223; 537,5 Bracker, S.B., 539, 218 Cumulative author index to volumes 531–540 (2002) 326–371 331

Bracko, M., 533, 243 Brunner, J., 539, 188 Bracko,ˇ M., 540,33 Bruski, N., 539, 188 Bragadireanu, A.M., 535,52 Bryman, D.A., 537, 211 Brahmachari, B., 531, 99; 536, 94, 259 Buchbinder, I.L., 535, 280 Braibant, S., 533, 207; 539,13 Buchholz, D., 531,52 Braig, S., 540, 278 Buchmüller, O., 533, 223; 537,5 Bramm, R., 538, 275 Buchmüller, W., 540, 295 Branchini, P., 535, 37; 536, 209; 537, 21; 538,21 Buehler, M., 531,52 Brandenberger, R.H., 534,1 Buescher, V., 531,52 Brandenburg, A., 539, 235 Buesser, K., 533, 207; 539,13 Brandt, A., 531,52 Bugge, L., 533, 243 Brandt, S., 533, 223; 537,5 Buijs, A., 531,39 Branson, J.G., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Bukin, D.A., 537, 201 540, 43, 185 Bülte, A., 539, 188 Brauksiepe, S., 535,85 Bulten, H.J., 535,85 Braunschweig, W., 539,25 Bulychjov, S.A., 535, 37; 536, 209; 537, 21; 538,21 Bravo, S., 533, 223; 537,5 Bunciˇ c,´ P., 538, 275 Brax, P., 538, 426 Bunkov, Y.M., 538, 257 Brax, Ph., 531, 135 Bunyatyan, A., 539,25 Breedon, R., 531,52 Buontempo, S., 539, 188 Breitweg, J., 531,9 Burchat, P.R., 539, 218 Brenner, R., 533, 243 Burckhart, H.J., 533, 207; 539,13 Breunlich, W., 535,52 Burdin, S.V., 537, 201 Brient, J.-C., 533, 223; 537,5 Bürger, J., 539,25 Brigliadori, L., 533, 207; 539,13 Burger, J.D., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Brihaye, Y., 534, 137 540, 43, 185 Briskin, G., 531,52 Burger, W.J., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Brisson, V., 539,25 540, 43, 185 Brochu, F., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Burnstein, R.A., 539, 218 540, 43, 185 Burrage, A., 539,25 Brock, I., 531,9;539, 197 Burtin, E., 539,8 Brock, R., 531,52 Burtovoi, V.S., 531,52 Broderick, A.E., 531, 167 Büscher, M., 540, 207 Brodet, E., 533, 243 Buschhorn, G., 539,25 Brodzicka, J., 533, 243; 540,33 Buschmann, P., 533, 243 Brodzinski, R.L., 532,8 Büsser, F.W., 539,25 Bröker, H.-B., 539,25 Bussey, P.J., 531,9;539, 197 Brooijmans, G., 531,52 Butler, J.M., 531,52 Brook, N.H., 531,9;539, 197 Butler, J.N., 535, 43; 537, 192; 540,25 Bross, A., 531,52 Butterworth, J.M., 531,9;539, 197 Browder, T.E., 538,11 Buzzo, A., 533, 237 Brown, D.P., 539,25 Bylsma, B., 531,9;539, 197 Brown, R.M., 533, 207; 539,13 Bystritskaya, L., 539,25 Browne, R.E., 540,68 Bruckman, P., 533, 243 Cabibbo, G., 535, 37; 536, 209; 537, 21; 538,21 Brückner, W., 539,25 Cacciapaglia, G., 531, 105 Brugnera, R., 531,9;539, 197 Cacciatori, S., 536, 101 Brüll, A., 535,85 Cai, X.D., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; 540, 43, 185 Brümmer, N., 531,9;539, 197 Calabrese, R., 533, 196, 237 Brun, R., 538, 275 Calaﬁura, P., 537,28 Bruncko, D., 539,25 Caldwell, A., 531,9;539, 197 Brunelière, R., 533, 75, 223; 537,5 Callot, O., 533, 223; 537,5 Brunet, J.M., 533, 243 Calmet, X., 540, 173 Bruni, A., 531,9;539, 197 Caloi, R., 535, 37; 536, 209; 537, 21; 538,21 Bruni, G., 531,9;539, 197 Calvetti, M., 533, 196; 536, 229; 537,28 Brunn, I., 535,85 Calvi, M., 533, 243 332 Cumulative author index to volumes 531–540 (2002) 326–371

Camera, F., 535, 93; 540, 199 Casper, D., 539, 179 Cameron, W., 533, 223; 537,5 Cassing, W., 540, 207 Camino, J.M., 533, 313 Cassol-Brunner, F., 539,25 Cammin, J., 533, 207; 539,13 Castaños, O., 534,57 Campana, P., 533, 223; 535, 37; 536, 209; 537, 21; 538,21 Castilla-Valdez, H., 531,52 Campana, S., 533, 207; 539,13 Castoldi, M., 540, 199 Campbell, A.J., 539,25 Castro, C., 539, 133 Camporesi, T., 533, 243 Castro, N., 533, 243 Canale, V., 533, 243 Cataldi, G., 535, 37; 536, 209; 537, 21; 538,21 Canelli, F., 531,52 Catanesi, M.G., 539, 188 Cao, J., 539,25 Cattaneo, M., 533, 223; 537,5 Capell, M., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Catterall, C.D., 531,9;539, 197 540, 43, 185 Cavallari, F., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Capitani, G.P., 535,85 540, 43, 185 Capon, G., 533, 223; 535, 37; 536, 209; 537, 5, 21; 538,21 Cavallo, F., 533, 243 Capua, M., 531,9;539, 197 Cavallo, N., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Capurro, O.A., 534,45 540, 43, 185 Cara Romeo, G., 531, 9, 28, 39; 534, 28; 535, 59; 536, 24, 217; Cavanaugh, R., 533, 223; 537,5 539, 197; 540, 43, 185 Cavata, C., 539,8 Carboni, G., 535, 37; 536, 209; 537, 21; 538,21 Cawlﬁeld, C., 535, 43; 537, 192; 540,25 Cardoso, V., 538,1 Cebrián, S., 532,8 Carelli, P., 540, 179 Cecchi, C., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Carena, F., 533, 243 540, 43, 185 Cargnelli, M., 535,52 Ceccucci, A., 533, 196; 536, 229; 537,28 Carimalo, C., 533, 243 Cenci, P., 533, 196; 536, 229; 537,28 Carli, T., 531,9;539, 197 Ceradini, F., 535, 37; 536, 209; 537, 21; 538,21 Carlin, R., 531,9;539, 197 Cerny, V., 538, 275 Carlino, G., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Cerrada, M., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; 540, 43, 185 540, 43, 185 Carmona, J.M., 531,71 Cerri, C., 533, 196; 536, 229; 537,28 Carnegie, R.K., 533, 207; 539,13 Cerutti, A., 540,25 Caron, B., 533, 207; 539,13 Cerutti, F., 533, 223; 537,5 Caron, S., 539,25 Cervelli, F., 535, 37; 536, 209; 537, 21; 538,21 Carone, C.D., 538, 115 Cester, R., 533, 237 Carosi, R., 536, 229; 537,28 Cevenini, F., 535, 37; 536, 209; 537, 21; 538,21 Carrillo, S., 535, 43; 537, 192; 540,25 Chaichian, M., 535, 321 Cartacci, A., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Chakraborty, B., 537, 340 540, 43, 185 Chakraborty, D., 531,52 Cartas-Fuentevilla, R., 536, 283, 289 Chamizo, M., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Carter, A.A., 533, 207; 539,13 540, 43, 185 Carter, B., 534,1 Chamseddine, A.H., 537, 383 Carter, J.R., 533, 207; 539,13 Chan, K.M., 531,52 Carter, T., 539, 218 Chandrasekharan, S., 536,72 Cartiglia, N., 531,9;533, 196; 536, 229; 537, 28; 539, 197 Chang, C.Y., 533, 207; 539,13 Cartwright, S., 533, 223; 537,5 Chang, K.H., 540,25 Carvalho, H.S., 539, 218 Chang, P., 540,33 Carvalho, W., 531,52 Chang, Y.H., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Casado, M.P., 533, 223; 537,5 540, 43, 185 Casali, R., 533, 196; 536, 229; 537,28 Chao, Y., 538, 11; 540,33 Casarsa, M., 535, 37; 536, 209; 537, 21; 538,21 Chapin, D., 531,9 Casaus, J., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; 540, 43, 185 Chapkin, M., 533, 243 Casavola, V., 535, 37; 536, 209; 537, 21; 538,21 Charlton, D.G., 533, 207; 539,13 Casey, B.C.K., 538, 11; 540,33 Charpentier, Ph., 533, 243 Casey, D., 531,52 Checchia, P., 533, 243 Casilum, Z., 531,52 Chekanov, S., 531,9;539, 197 Casimiro, E., 535, 43; 537, 192; 540,25 Chekulaev, S.V., 531,52 Cumulative author index to volumes 531–540 (2002) 326–371 333

Chemarin, M., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Cindolo, F., 531, 9, 28, 39; 534, 28; 535, 59; 536, 24, 217; 540, 43, 185 539, 197; 540, 43, 185 Chen, A., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; 540, 43, 185 Cinquini, L., 535, 43; 537, 192; 540,25 Chen, G., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; 540, 43, 185 Cirelli, M., 531, 105 Chen, G.M., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Cirilli, M., 533, 196; 536, 229; 537,28 540, 43, 185 Cirio, R., 531,9;539, 197 Chen, H.F., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Cisbani, E., 535,85 540, 43, 185 Ciulli, V., 533, 223; 537,5 Chen, H.S., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Ciullo, G., 535,85 540, 43, 185 Civitarese, O., 534,57 Chen, M.L., 539, 179 Claes, D., 531,52 Cheng, H.-Y., 533, 271 Clare, I., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; 540, 43, 185 Cheon, B.G., 538, 11; 540,33 Clare, R., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; 540, 43, 185 Chernodub, M.N., 537, 291 Clark, A.R., 531,52 Cheshkov, C., 536, 229; 537,28 Clarke, D., 539,25 Cheskov, C., 533, 196 Clarke, D.P., 533, 223; 537,5 Cheung, H.W.K., 535, 43; 537, 192; 540,25 Clemencic, M., 536, 229 Cheze, J.B., 533, 196; 536, 229; 537,28 Clerbaux, B., 533, 223; 537,5;539,25 Chiang, I.-H., 537, 211 Clifft, R.W., 533, 223; 537,5 Chiarella, V., 533, 223; 537,5 Cloth, P., 531,9;539, 197 Chiefari, G., 531, 28, 39; 534, 28; 535, 37, 59; 536, 24, 209, 217; Coccia, E., 540, 179 537, 21; 538, 21; 540, 43, 185 Cocco, A.G., 539, 188 Chierici, R., 533, 243 Cogan, J., 533, 196; 536, 229; 537,28 Chikawa, M., 539, 188 Cohen, I., 533, 207; 539,13 Chiochia, V., 531,9;539, 197 Cohen, T.D., 540, 227 Chiodini, G., 535, 43; 537, 192; 540,25 Coignet, G., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Chistov, R., 538, 11; 540,33 540, 43, 185 Chiu, T.-W., 538, 298 Colaleo, A., 533, 223; 537,5 Chivukula, R.S., 532, 121 Colangelo, P., 532, 193 Chizhov, M., 539, 188 Colas, P., 533, 223; 537,5 Chliapnikov, P., 533, 243 Coldewey, C., 531,9 Chmeissani, M., 533, 223; 537,5 Cole, J.E., 531,9;539, 197 Chmel, S., 538,33 Colino, N., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Cho, D.K., 531,52 540, 43, 185 Cho, K., 535, 43; 537, 192; 540,25 Collard, C., 539,25 Choi, S., 531,52 Collazuol, G., 533, 196; 536, 229; 537,28 Choi, S.-K., 538, 11; 540,33 Collin, E., 538, 257 Choi, Y., 538, 11; 540,33 Collins, J.C., 536,43 Chollet, J.C., 533, 196; 536, 229; 537,28 Collins, P., 533, 243 Chopra, S., 531,52 Collins-Tooth, C., 531,9;539, 197 CHORUS Collaboration, 539, 188 Combley, F., 533, 223; 537,5 Choubey, S., 531, 99; 536, 94; 540,14 Conetti, S., 535, 37; 536, 209; 537, 21; 538,21 Choudhury, S.R., 535, 289 Coney, L., 531,52 Christenson, J.H., 531,52 Connolly, B., 531,52 Chumney, P., 535,85 Conroy, J.M., 538, 115 Chung, M., 531,52 Contalbrigo, M., 533, 196; 536, 229; 537,28 Chung, S.U., 533, 243 Contin, A., 531,9;539, 197 Chung, Y.S., 535, 43; 537, 192; 540,25 Contreras, J.G., 539,25 Chwastowski, J., 531,9;539, 197 Contri, R., 533, 243 Ciambrone, P., 535, 37; 536, 209; 537, 21; 538,21 Convery, M.R., 537, 211 Cibinetto, G., 533, 237 Cooper, W.E., 531,52 Ciborowski, J., 531,9;539, 197 Cooper-Sarkar, A.M., 531,9;539, 197 Ciesielski, R., 531,9;539, 197 Coppage, D., 531,52 Cieslik, K., 533, 243 Coppens, Y.R., 539,25 Cifarelli, L., 531, 9, 28, 39; 534, 28; 535, 59; 536, 24, 217; Coppola, N., 531,9;539, 197 539, 197; 540, 43, 185 Copty, N.K., 539, 218 334 Cumulative author index to volumes 531–540 (2002) 326–371

Cormack, C., 531,9;539, 197 Cvach, J., 539,25 Cornell, A.S., 535, 289 Cvetic,ˇ M., 534, 172; 540, 301 Corradi, M., 531,9;539, 197 Czy˙z, H., 538,52 Correa, D.H., 534, 185 Corriveau, F., 531,9;539, 197 Dadhich, N., 538, 233 Cosme, G., 533, 243 Dado, S., 533, 207; 539,13 Cossutti, F., 533, 243 Dagan, S., 531,9;539, 197 Costa, M., 531,9;539, 197 D’Agostini, G., 531,9;533, 196; 536, 229; 537, 28; 539, 197 Costa, M.J., 533, 243 Dainton, J.B., 539,25 Costantini, F., 533, 196; 536, 229; 537,28 Dal Corso, F., 531,9;539, 197 Costantini, S., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Dallavalle, G.M., 533, 207; 539,13 540, 43, 185 Dallison, S., 533, 207; 539,13 Cotrone, A.L., 536, 161 Dalmau, J., 533, 243 Coughlan, J.A., 539,25 Dalpiaz, P., 533, 196, 237; 536, 229; 537,28 Coulier, N., 537, 45; 538,33 Dalpiaz, P.F., 535,85 Court, G.R., 535,85 D’Ambrosio, N., 539, 188 Cousinou, M.-C., 539,25 Da Motta, H., 531,52 Coussement, R., 538,33 D’Angelo, P., 535, 43; 537, 192; 540,25 Covi, L., 540, 295 Danilov, M., 538, 11; 540,33 Cowan, G., 533, 223; 537,5 Danilov, P., 531,9;539, 197 Coward, D., 537,28 Dannheim, D., 531,9;539, 197 Cox, B., 540, 263 D’Antonio, S., 540, 179 Cox, B.E., 539,25 Darling, C., 539, 218 Coyle, P., 533, 223; 537,5 Das, A., 531, 289; 533, 146 Cozzika, G., 539,25 Das, P., 531, 187 Cramer, J.G., 538, 275 Das, S.K., 531, 187 Cranmer, K., 533, 223; 537,5 Da Silva, T., 533, 243 Crawley, B., 533, 243 Da Silva, W., 533, 243 Creanza, D., 533, 223; 537,5 Datta, A., 533, 65; 538, 309; 540,97 Cremaldi, L.M., 539, 218 Dau, W.D., 539,25 Creminelli, P., 532, 284 Daugas, J.-M., 537,45 Crennell, D., 533, 243 Daum, K., 539,25 Crépé, S., 537,28 Davenport, T.F., 535, 43; 537, 192; 540,25 Crépé-Renaudin, S., 531,52 David, A., 533, 223; 537,5 Crespo, J.M., 533, 223; 537,5 Davidson, S., 535,25 Cristadoro, G., 531, 105 Davidsson, M., 539,25 Crittenden, J., 531,9;539, 197 Davier, M., 533, 223; 537,5 Cross, R., 531,9 Davies, G., 533, 223; 537,5 Csáki, C., 535,33 Davis, A.-C., 534,1 Csató, P., 538, 275 Davis, A.C., 531, 135 Csilling, A., 533, 207; 539,13 Davis, G.A., 531,52 Csizmadia, P., 531, 209 Dazeley, S., 539, 179 Cuautle, E., 535, 43; 537, 192; 540,25 De, K., 531,52 Cucciarelli, S., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; De Angelis, A., 533, 243 540, 43, 185 De Angelis, G., 535,93 Cuevas, J., 533, 243 De Asmundis, R., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Cufﬁani, M., 533, 207; 539,13 540, 43, 185 Cuhadar-Dönszelmann, T., 533, 196; 536, 229; 537,28 Deb, A., 540,52 Cumalat, J.P., 535, 43; 537, 192; 540,25 De Beer, M., 533, 196; 536, 229; 537,28 Cummings, M.A.C., 531,52 De Boer, W., 533, 243 Cundy, D., 533, 196; 536, 229; 537,28 De Bonis, I., 533, 223; 537,5 Curceanu (Petrascu), C., 535,52 Debreczeni, J., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Curien, D., 535,93 540, 43, 185 Curtil, C., 533, 223; 537,5 Debu, P., 533, 196; 536, 229; 537,28 Cussans, D., 539, 188 Decamp, D., 533, 223; 537,5 Cutts, D., 531,52 De Clercq, C., 533, 243 Cumulative author index to volumes 531–540 (2002) 326–371 335

De Fazio, F., 532, 193 De Oliveira Santos, F., 537,45 De Filippis, N., 533, 223; 537,5 De Palma, M., 533, 223; 537,5 Deger, N.S., 538, 164 De Pasquale, S., 531,9;539, 197 Déglon, P., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; 540, 43, 185 De Paula, L., 533, 243 Degré, A., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; 540, 43, 185 De Poli, M., 535,93 Deguchi, S., 532, 329 De Robertis, G., 535, 37; 536, 209; 537, 21; 538,21 Dehmelt, K., 540, 43, 185 De Roeck, A., 533, 207; 539, 13, 25 De Huu, M., 532, 179 De Rosa, G., 539, 188 Deiters, K., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Derrick, M., 531,9;539, 197 540, 43, 185 Derue, F., 533, 196; 537,28 De Jager, C.W., 539,8 Desai, S., 531, 52; 539, 179 De Jong, M., 539, 188 De Salvo, A., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; De Jong, P., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; 540, 43, 185 540, 43, 185 DeSalvo, R., 537,41 De Jong, S.J., 531,52 De Sanctis, E., 532, 87; 535,85 De la Cruz, B., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; De Sangro, R., 535,37 540, 43, 185 Desch, K., 533, 207; 539,13 Delaere, C., 533, 223; 537,5 De Schepper, D., 535,85 Delagrange, H., 538,27 De Serio, M., 539, 188 Delbar, T., 539, 188 Deshpande, A., 531,9;539, 197 Delcourt, B., 539,25 Deshpande, N.G., 533, 116 De Lellis, G., 539, 188 De Simone, P., 535, 37; 536, 209; 537, 21; 538,21 De Leo, R., 535,85 Desler, K., 531,9 Delerue, N., 539,25 Dessagne, S., 533, 223; 537,5 Delﬁno, G., 536, 169 Devenish, R.C.E., 531,9;539, 197 Delheij, P.P.J., 534,39 Devitsin, E., 535,85 Dell’Agnello, S., 535, 37; 536, 209; 537, 21; 538,21 De Vivie de Régie, J.-B., 533, 223; 537,5 Della Ricca, G., 533, 243 Devmal, S., 539, 218 Della Volpe, D., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Dewald, A., 535,93 540, 43, 185 De Witt Huberts, P.K.A., 535,85 Delmeire, E., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; De Wolf, E., 531,9;539, 197 540, 43, 185 De Wolf, E.A., 533, 207; 539, 13, 25 De Lotto, B., 533, 243 De Zorzi, G., 535, 37; 536, 209; 537, 21; 538,21 Del Peso, J., 531,9;539, 197 Dhamotharan, S., 533, 223; 537,5 DELPHI Collaboration, 533, 243 Dhawan, S., 531,9;539, 197 De Lucia, E., 535, 37; 536, 209; 537, 21; 538,21 D’Hondt, J., 533, 243 De Maria, N., 533, 243 Diaconu, C., 539,25 Demarteau, M., 531,52 Diaz-Torres, A., 533, 265 De Mello Neto, J.R.T., 539, 218 Dibon, H., 533, 196; 536, 229; 537,28 Dementiev, R.K., 531,9;539, 197 Di Capua, E., 539, 188 De Min, A., 533, 243 Di Capua, F., 539, 188 Demina, R., 531,52 Di Ciaccio, L., 533, 243 Demine, P., 531,52 Dick, R., 535, 295 De Miranda, J.M., 535, 43; 537, 192; 539, 218; 540,25 DiCorato, M., 535, 43; 537, 192; 540,25 Demirchyan, R., 539,25 Dicus, D.A., 536,83 De Nardo, L., 535,85 Di Domenico, A., 535, 37; 536, 209; 537, 21; 538,21 Denes, P., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; 540, 43, 185 Di Donato, C., 535, 37; 536, 209; 537, 21; 538,21 Deng, J.R., 536, 203 Diehl, H.T., 531,52 Denig, A., 535, 37; 536, 209; 537, 21; 538,21 Diehl, M., 532,99 Denisenko, K., 539, 218 Diemoz, M., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Denisov, D., 531,52 540, 43, 185 Denisov, S.P., 531,52 Dierckxsens, M., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Denner, A., 533,75 540, 43, 185 DeNotaristefani, F., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Diesburg, M., 531,52 540, 43, 185 Dietl, H., 533, 223; 537,5 D’Enterria, D.G., 538,27 Di Falco, S., 535, 37; 536, 209; 537, 21; 538,21 336 Cumulative author index to volumes 531–540 (2002) 326–371

Di Giacomo, A., 537, 173 Duchesneau, D., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; DiLella, L., 531, 175 540, 43, 185 Dimopoulos, S., 531, 127; 534, 124 Duchovni, E., 533, 207; 539,13 Dimova, T.V., 537, 201 Duckeck, G., 533, 207; 539,13 Di Nezza, P., 535,85 Duclos, J., 533, 196; 536, 229; 537,28 Dingfelder, J., 539,25 Ducros, Y., 531,52 Dini, P., 535, 43; 537, 192; 540,25 Dudko, L.V., 531,52 Dinkelaker, P., 538, 275 Duensing, S., 531,52 Dionisi, C., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Duerdoth, I.P., 533, 207; 539,13 540, 43, 185 Duﬂot, L., 531, 52; 533, 223; 537,5 Di Simone, A., 533, 243 Dugad, S.R., 531,52 Dissertori, G., 533, 223; 537,5 Duinker, P., 531,39 Dittmaier, S., 533,75 Dunne, G.V., 531,77 Dittmar, M., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Dunning, C., 537, 297 540, 43, 185 Duperrin, A., 531,52 Di Vecchia, P., 540, 104 Duprel, C., 539,25 Diwan, M.V., 537, 211 Düren, M., 535,85 Dixon, P., 539,25 Durhuus, B., 539, 277 Doble, N., 533, 196; 536, 229; 537,28 Durkin, L.S., 531,9;539, 197 DØ Collaboration, 531,52 Dusini, S., 531,9;539, 197 Dodonov, V., 539,25 Dutta, B., 535, 249; 538, 121 Doff, A., 537, 275 Dutta, S., 535, 219 Dolan, L., 537, 155 Dyshkant, A., 531,52 Dolgoshein, B.A., 531,9;539, 197 D’Oliveira, A.B., 539, 218 E-811 Collaboration, 537,41 Dong, L.Y., 538, 11; 540,33 E787 Collaboration, 537, 211 Donkers, M., 533, 207; 539,13 Ealet, A., 533, 223; 537,5 Donnachie, A., 533, 277 Earl, M., 539, 179 Doplicher, S., 533, 178 Eartly, D.P., 537,41 Dore, U., 539, 188 Eberl, H., 538, 353 Doria, A., 531, 28, 39; 534, 28; 535, 37, 59; 536, 24, 209, 217; Ebert, D., 537, 241 537, 21; 538, 21; 540, 43, 185 Echenard, B., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Döring, J., 532,29 540, 43, 185 Dornan, P.J., 533, 223; 537,5 Eckardt, V., 538, 275 Doroba, K., 533, 243 Eckerlin, G., 539,25 Dosanjh, R.S., 533, 196; 536, 229; 537,28 Eckstein, D., 539,25 Dos Reis, A.C., 535, 43; 537, 192; 539, 218; 540,25 Edera, L., 535, 43; 537, 192; 540,25 Doucet, M., 539, 188 Edgecock, T.R., 533, 223; 537,5 Doulas, S., 531,52 Edmunds, D., 531,52 Dova, M.T., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Efremenko, V., 539,25 540, 43, 185 Egger, J.-P., 535,52 Dowell, J.D., 539,25 Egli, S., 539,25 Doyle, A.T., 531,9;539, 197 Ehrenfried, M., 535,85 Drees, J., 533, 243 Eichler, R., 539,25 Drees, M., 533, 107 Eidelman, S., 538, 11; 540,33 Dreucci, M., 535, 37; 536, 209; 537,21;538,21 Eigen, G., 533, 243 Drevermann, H., 533, 223; 537,5 Eiges, V., 538, 11; 540,33 Drews, G., 531,9;539, 197 Eisele, F., 539,25 Dris, M., 533, 243 Eisenberg, Y., 531,9;539, 197 Droutskoi, A., 539,25 Eisenhandler, E., 539,25 Drutskoy, A., 540,33 Ejiri, H., 531, 190 Druzhinin, V.P., 537, 201 Ekelof, T., 533, 243 Du, D.-S., 536,34 El-Aidi, R., 539, 188 Duan, L.-M., 538,39 Elbakian, G., 535,85 Dubak, A., 539,25 El Baz, M., 536, 321 Dubbert, J., 533, 207; 539,13 Elfgren, E., 539,13 Cumulative author index to volumes 531–540 (2002) 326–371 337

Eline, A., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; 540, 43, 185 Falaleev, V., 533, 196; 536, 229; 537,28 Ellerbrock, M., 539,25 Falciano, S., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Ellert, M., 533, 243 540, 43, 185 Ellinghaus, F., 535,85 Falkowski, A., 535, 258; 538, 426 Ellis, G., 533, 223; 537,5 Falvard, A., 533, 223; 537,5 Ellis, J., 532, 318; 539, 107 Fang, F., 538,11 Ellison, J., 531,52 Fanourakis, G., 533, 243 Ellsworth, R.W., 539, 179 Fantechi, R., 533, 196; 536, 229; 537,28 El Mamouni, H., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Fantoni, A., 535,85 540, 43, 185 Farilla, A., 535, 37; 536, 209; 537, 21; 538,21 Elsen, E., 539,25 Farnea, E., 535,93 Elsing, M., 533, 243 Fassarella, L., 538, 415 Eltzroth, J.T., 531,52 Fassouliotis, D., 533, 243 Elvira, V.D., 531,52 Faulkner, P.J.W., 539,25 Ely, J., 535,85 Faustov, R.N., 537, 241 Enari, Y., 538, 11; 540,33 Favara, A., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; 540, 43, 185 Engelen, J., 531,9;539, 197 Favart, D., 539, 188 Engelmann, R., 531,52 Favart, L., 539,25 Engh, D., 535, 43; 537, 192; 540,25 Fay, J., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; 540, 43, 185 Engler, A., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; 540, 43, 185 Fayard, L., 533, 196; 536, 229; 537,28 Eno, S., 531,52 Fayolle, D., 533, 223; 537,5 Eppard, M., 533, 196; 536, 229; 537,28 Fechtchenko, A., 535,85 Eppley, G., 531,52 Fedin, O., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; 540, 43, 185 Eppling, F.J., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Fedotov, A., 539,25 540, 43, 185 Fein, D., 531,52 Erba, S., 535, 43; 537, 192; 540,25 Feindt, M., 533, 243 Erdmann, M., 539,25 Felawka, L., 535,85 Erdmann, W., 539,25 Felcini, M., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Ereditato, A., 539, 188 540, 43, 185 Ermolov, P., 531,52 Feld, L., 533, 207; 539,13 Ermolov, P.F., 531,9;539, 197 Felici, G., 535, 37; 536, 209; 537, 21; 538,21 Eroshin, O.V., 531,52 Felst, R., 539,25 Erriquez, O., 535, 37; 536, 209; 537, 21; 538,21 Feng, J.L., 535, 302 Escofﬁer, S., 539,8 Ferbel, T., 531,52 Eskola, K.J., 532, 222 Ferencei, J., 539,25 Eskreys, A., 531,9;539, 197 Ferguson, D.P.S., 533, 223; 537,5 Espirito Santo, M.C., 533, 243 Ferguson, T., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Estrada, J., 531,52 540, 43, 185 Etoh, M., 539, 179 Fermilab E791 Collaboration, 539, 218 Etzion, E., 533, 207; 539,13 Fermilab E835 Collaboration, 533, 237 Evans, H., 531,52 Fernandez, A., 539, 218 Evdokimov, V.N., 531,52 Fernandez, E., 533, 223; 537,5 Everett, L.L., 531, 263 Fernández, F., 538,27 Ewers, A., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; 540, 43, 185 Fernandez, J., 533, 243 Extermann, P., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; 540, 43, 185 Fernandez-Bosman, M., 533, 223; 537,5 Fernández Niello, J.O., 534,45 Fabbri, F., 533, 207; 539,13 Ferrando, J., 531,9;539, 197 Fabbri, F.L., 535, 43; 537, 192; 540,25 Ferrari, A., 535, 37; 536, 209; 537,21;538,21 Fabbri, R., 535,85 Ferrari, P., 533, 207; 539,13 Fabbro, B., 533, 223; 537,5 Ferrer, A., 533, 243 Fafone, V., 540, 179 Ferrer, M.L., 535, 37; 536, 209; 537,21;538,21 Fahland, T., 531,52 Ferrero, M.I., 531,9;539, 197 Fajfer, S., 539,67 Ferro, F., 533, 243 Falagan, M.A., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Ferron, S., 539,25 540, 43, 185 Ferstl, A., 532, 318 338 Cumulative author index to volumes 531–540 (2002) 326–371

Fesefeldt, H., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Forty, R.W., 533, 223; 537,5 540, 43, 185 Foster, B., 531,9;539, 197 Feverati, G., 534, 216 Foster, F., 533, 223; 537,5 Fiandrini, E., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Fouchez, D., 533, 223; 537,5 540, 43, 185 Foudas, C., 531,9;539, 197 Fiedler, F., 533, 207; 539,13 Fourletov, S., 531,9;539, 197 Field, J.H., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Fourletova, J., 531,9;539, 197 540, 43, 185 Fox, B., 535,85 Figiel, J., 531,9;539, 197 Fox, G.F., 539, 218 Filges, D., 531,9;539, 197 Fox, H., 531, 52; 533, 196; 536, 229; 537,28 Filip, P., 538, 275 Fox-Murphy, A., 531,9;539, 197 Filippone, B.W., 535,85 Frabetti, P.L., 533, 196; 536, 229; 537,28 Filthaut, F., 531, 28, 39, 52; 534, 28; 535, 59; 536, 24, 217; Frame, K.C., 531,52 540, 43, 185 Frampton, P.H., 536,79 Finch, A.J., 533, 223; 537,5 Franceschi, A., 535, 37; 536, 209; 537,21;538,21 Finocchiaro, G., 535, 37; 536, 209; 537,21;538,21 Frank, J.S., 537, 211 Fioretto, E., 535,93 Frank, M., 533, 223; 537,5;540, 269 Fiorillo, G., 539, 188 Franke, G., 539,25 Fiorini, L., 533, 196 Frankfurt, L., 537, 51; 540, 220 Fischer, C.S., 536, 177 Franz, J., 535,85 Fischer, G., 536, 229; 537,28 Franzini, P., 535, 37; 536, 209; 537, 21; 538,21 Fischer, H., 535,85 Frattini, S., 540, 199 Fischer, H.G., 538, 275 Frau, M., 540, 104 Fischer, P., 532, 373 Fredenhagen, K., 533, 178 Fisher, P.H., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Freese, K., 540,1 540, 43, 185 Frekers, D., 532, 179; 539, 188 Fisher, W., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; 540, 43, 185 Freudenreich, K., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Fisk, H.E., 531,52 540, 43, 185 Fisk, I., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; 540, 43, 185 Freund, P., 538, 275 Fisyak, Y., 531,52 Frey, A., 533, 207; 539,13 Fitzler, A., 535,93 Fricke, U., 531,9;539, 197 Flagmeyer, U., 533, 243 Friese, V., 538, 275 Flattum, E., 531,52 Fritzsch, H., 540, 173 Fleck, I., 533, 207; 539,13 Fleischer, M., 539,25 Frois, B., 539,8 Fleming, Y.H., 539,25 Frullani, S., 535,85 Fleuret, F., 531,52 Fu, S., 531,52 Flügge, G., 539,25 Fuess, S., 531,52 Foà, L., 533, 223; 537,5 Fuhrmann, H., 535,52 Focardi, E., 533, 223; 537,5 Fujii, M., 538, 107 FOCUS Collaboration, 535, 43; 537, 192; 540,25 Fujikawa, K., 538, 197 Fodor, Z., 534, 87; 538, 275 Fukuda, S., 539, 179 Foeth, H., 533, 243 Fukuda, Y., 539, 179 Foka, P., 538, 275 Fukunaga, C., 538, 11; 540,33 Fokitis, E., 533, 243 Fulda-Quenzer, F., 533, 243 Fomenko, A., 539,25 Funahashi, H., 534,39 Forconi, G., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Funk, W., 533, 196; 537,28 540, 43, 185 Furetta, C., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Ford, C., 540, 153 540, 43, 185 Ford, M., 533, 207; 539,13 Furman, A., 538, 266 Foresti, I., 539,25 Furnstahl, R.J., 531, 203 Formánek, J., 539,25 Fürtjes, A., 533, 207; 539,13 Formica, A., 533, 196; 536, 229; 537,28 Furuta, K., 537, 165 Forshaw, J., 540, 263 Fusayasu, T., 531,9;539, 197 Forti, C., 535, 37; 536, 209; 537, 21; 538,21 Fushimi, K., 531, 190 Fortner, M., 531,52 Fuster, J., 533, 243 Cumulative author index to volumes 531–540 (2002) 326–371 339

Gabareen, A., 531,9;539, 197 Gavrilov, V., 531,52 Gabathuler, E., 539,25 Gay, P., 533, 223; 537,5 Gabathuler, K., 539,25 Gaycken, G., 533, 207; 539,13 Gabyshev, N., 540,33 Gay Ducati, M.B., 533,43 Gadea, A., 535, 93; 540, 199 Gayler, J., 539,25 Gagnon, P., 533, 207; 539, 13, 218 Ga´zdzicki, M., 538, 275 Gago, A., 539, 179 Gazis, E., 533, 243 Gaines, I., 535, 43; 537, 192; 540,25 Geer, S., 532, 311 Gajewski, W., 539, 179 Geich-Gimbel, C., 533, 207; 539,13 Gál, J., 538, 275 Geiser, A., 531,9;539, 197 Galaktionov, Yu., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Gele, D., 533, 243 540, 43, 185 Gelman, B.A., 540, 227 Galea, R., 531,9;539, 197 Genik, R.J., 531,52 Galkin, V.O., 537, 241 Genser, K., 531,52 Gallas, E., 531,52 Genta, C., 531,9;539, 197 Gallo, E., 531,9;539, 197 Gentile, S., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Galyaev, A.N., 531,52 540, 43, 185 Gandelman, M., 533, 243 Georgiev, G., 537, 45; 538,33 Gando, Y., 539, 179 Georgopoulos, G., 538, 275 Ganezer, K.S., 539, 179 Geralis, T., 533, 243 Ganguli, S.N., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Gerber, C.E., 531,52 540, 43, 185 Gerhards, R., 539,25 Ganis, G., 533, 223; 537,5 Gerlich, C., 539,25 Gao, M., 531,52 Gershon, T., 538, 11; 540,33 Gao, Y., 533, 223; 537,5 Gershon, T.J., 533, 196; 536, 229; 537,28 Gaponenko, A., 537,28 Gershtein, Y., 531,52 Gärber, Y., 535,85 Gevorkyan, S.R., 538,45 Garbincius, P.H., 535, 43; 537, 192; 540,25 Geweniger, C., 533, 223; 537,5 Garcia, C., 533, 243 Geyer, B., 535, 349 García, E., 532,8 Gharibyan, V., 535,85 García, G., 531,9 Ghazaryan, S., 539,25 Garcia-Abia, P., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Gherghetta, T., 536, 277 540, 43, 185 Ghete, V.M., 533, 223; 537,5 Garcia-Bellido, A., 533, 223; 537,5 Ghezelbash, A.M., 535, 315; 537, 329 García-Bellido, J., 536, 193 Ghosh, D., 540,52 Gardner, R., 535, 43; 537, 192; 540,25 Ghosh, J., 540,52 Garfagnini, A., 531,9;539, 197 Ghosh, S., 533, 162; 537, 340 Garibaldi, F., 535,85 Giacomelli, G., 533, 207; 539,13 Garmash, A., 540,33 Giacomelli, P., 533, 207; 539,13 Garousi, M.R., 536, 129 Giagu, S., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; 540, 43, 185 Garren, L.A., 535, 43; 537, 192; 540,25 Gialas, I., 531,9;539, 197 Garrido, Ll., 533, 223; 537,5 Giammanco, A., 533, 223; 537,5 Garutti, E., 535,85 Giammarchi, M., 535, 43; 537, 192; 540,25 Garvey, J., 539,25 Gianini, G., 535, 43; 537, 192; 540,25 Gary, J.W., 533, 207; 539,13 Giannini, G., 533, 223; 537,5 Garzoglio, G., 533, 237 Gianoli, A., 533, 196; 536, 229; 537,28 Gassner, J., 539,25 Gianotti, F., 533, 223; 537,5 Gataullin, M., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Giassi, A., 533, 223; 537,5 540, 43, 185 Gibbons, G.W., 534, 172; 537,1 Gates, S.J., 535, 280 Gierlik, M., 532,29 Gatignon, L., 533, 196; 536, 229; 537,28 Giersch, M., 535,52 Gatti, C., 535, 37; 536, 209; 537,21;538,21 Gilmartin, R., 531,52 Gattringer, C., 535, 358 Gilmore, J., 531,9;539, 197 Gauzzi, P., 535, 37; 536, 209; 537, 21; 538,21 Ginsburg, C.M., 531,9;539, 197 Gavillet, Ph., 533, 243 Ginther, G., 531,52 Gavrilov, G., 535,85 Giordano, G., 540, 179 340 Cumulative author index to volumes 531–540 (2002) 326–371

Giovannella, S., 535, 37; 536, 209; 537, 21; 538,21 Göttlicher, P., 531,9;539, 197 Girod, M., 531,61 Gottschalk, E., 535, 43; 537, 192; 540,25 Girone, M., 533, 223; 537,5 Gouge, G., 533, 196 Girtler, P., 533, 223; 537,5 Gounder, K., 531, 52; 539, 218 Giryavets, A.A., 532, 291; 536, 138 Goussiou, A., 531,52 Giudici, S., 533, 196; 536, 229; 537,28 Govender, M., 538, 233 Giunta, M., 533, 207; 539,13 Govi, G., 533, 196; 536, 229; 537,28 Giusti, P., 531,9;539, 197 Goy, C., 533, 223; 537,5 Gladilin, L.K., 531,9;539, 197 Goyal, A., 535, 219 Gladkov, D., 531,9;539, 197 Grab, C., 539,25 Gładysz, E., 538, 275 Grabowska-Bold, I., 531,9;539, 197 Glashow, S.L., 532, 15; 536, 79; 540, 247 Grabski, V., 539,25 Glasman, C., 531,9;539, 197 Graça, E.L., 536, 114 Glozman, L.Ya., 539, 257 Gracey, J.A., 535, 377; 540,68 Glück, M., 540,75 Graciani, R., 531,9 Göbel, C., 535, 43; 537, 192; 540,25 Graf, N., 531,52 Gobel, C., 539, 218 Grafström, P., 533, 196; 536, 229; 537,28 Göckeler, M., 532,63 Graham, M., 533, 237 Godfrin, H., 538, 257 Grancagnolo, F., 535, 37; 536, 209; 537,21;538,21 Goebel, F., 531,9;539, 197 Granier de Cassagnac, R., 533, 196; 536, 229; 537,28 Goerlich, L., 539,25 Grannis, P.D., 531,52 Goers, S., 531,9;539, 197 Grässler, H., 539,25 Gogitidze, N., 539,25 Grasso, M., 535, 103 Gogohia, V., 531, 321 Graugés, E., 533, 223; 537,5 Gokieli, R., 533, 243 Graw, G., 535,85 Goldberg, H., 535, 302 Graziani, E., 533, 243; 535, 37; 536, 209; 537,21;538,21 Goldberg, J., 533, 207; 539, 13, 188 Graziani, G., 533, 196; 536, 229; 537,28 Goldhaber, M., 539, 179 Grebeniouk, O., 535,85 Gollwitzer, K., 533, 237 Green, J.A., 531,52 Golob, B., 533, 243 Green, M.G., 533, 223; 537,5 Golubev, V.B., 537, 201 Green, P.W., 535,85 Golubkov, Yu.A., 531,9;539, 197 Greeniaus, L.G., 535,85 Gomes, P.R.S., 534,45 Greening, T.C., 533, 223; 537,5 Gómez, B., 531,52 Greenlee, H., 531,52 Gomez-Ceballos, G., 533, 243 Greenshaw, T., 539,25 Gonçalo, R., 531,9;539, 197 Goncalves, P., 533, 243 Greenwood, Z.D., 531,52 Gonçalves, V.P., 534,76 Grégoire, G., 539, 188 Goncharov, P.I., 531,52 Greiner, W., 531, 195 Gong, J.-O., 538, 213 Grella, G., 539, 188 Gong, Z.F., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Grenier, G., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; 540, 43, 185 540, 43, 185 Gonidec, A., 533, 196; 536, 229; 537,28 Grigoriev, A., 535, 187 González, O., 531,9;539, 197 Grigoriev, D.Yu., 540, 146 González, S., 533, 223; 537,5 Grijpink, S., 531,9;539, 197 Goodman, J.A., 539, 179 Grimm, O., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Gorbunov, I.V., 531, 255 540, 43, 185 Gorbunov, P., 539, 188 Grindhammer, G., 539,25 Gordon, H., 531,52 Grinstein, B., 533,8 Gorenstein, M.I., 531, 195 Grinstein, S., 531,52 Gorini, B., 533, 196; 536, 229; 537,28 Grinza, P., 536, 169 Gorini, E., 535, 37; 536, 209; 537,21;538,21 Grivaz, J.-F., 533, 223; 537,5 Goss, L.T., 531,52 Groer, L., 531,52 Goswami, S., 540,14 Grojean, C., 535, 258 Gotow, K., 538,11 Groot Nibbelink, S., 536, 270 Gotsman, E., 532,37 Grosdidier, G., 533, 243 Cumulative author index to volumes 531–540 (2002) 326–371 341

Gross, E., 533, 207; 539,13 Hagiwara, K., 540, 233 Gross, F., 531, 161 Hägler, Ph., 535, 117; 540, 324 Grossman, Y., 538, 327 Hagopian, S., 531,52 Gruenewald, M.W., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Hagopian, V., 531,52 540, 43, 185 Hahn, F., 533, 243 Grumiller, D., 532, 373 Hahn, S., 533, 243 Grünendahl, S., 531,52 Haidt, D., 539,25 Grunhaus, J., 533, 207; 539,13 Hain, W., 531,9;539, 197 Grupen, C., 533, 223; 537,5 Haines, T.J., 539, 179 Gruwé, M., 533, 207; 539,13 Hajdu, C., 533, 207; 539,13 Grzelak, G., 531,9;539, 197 Hajduk, L., 539,25 Grzelak, K., 533, 243 Hakobyan, R.Sh., 534, 28; 535, 59; 536, 24, 217; 540, 43, 185 Grzelinska,´ A., 538,52 Hall, L.J., 532, 111; 538, 359 Gu, Y.F., 538,6 Hall, R.E., 531,52 Guaraldo, C., 535,52 Hall-Wilton, R., 531,9;539, 197 Gubarev, F.V., 537, 291 Haller, J., 539,25 Guchait, M., 535, 243 Halley, A., 533, 223 Guida, M., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; 540, 43, 185 Hallgren, A., 533, 243 Guida, R., 533, 196; 537,28 Halling, A.M., 539, 218 Guillian, G., 539, 179 Hamacher, K., 533, 243 Güler, M., 539, 188 Hamaguchi, K., 538, 107 Guliyev, E., 532, 173 Hamann, M., 533, 207; 539,13 Günther, P.O., 533, 207; 539,13 Hamatsu, R., 531,9;539, 197 Guo, R., 538, 11; 540,33 Hamilton, K., 533, 243 Guo, W.-J., 540, 213 Hammer, H.-W., 531, 203 Gupta, A., 531, 52; 533, 207; 539,13 Han, S.W., 535, 37; 536, 209; 537,21;538,21 Gupta, V.K., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Han, T., 538, 346 540, 43, 185 Hanagaki, K., 538,11 Gurtu, A., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; 540, 43, 185 Handa, F., 540,33 Gurzhiev, S.N., 531,52 Handler, T., 535, 43; 537, 192; 540,25 Guss, C., 537,41 Hanke, P., 533, 223; 537,5 Gutay, L.J., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; 540, 43, 185 Hanlon, S., 531,9;539, 197 Gute, A., 535,85 Hansen, J., 533, 243 Gutierrez, G., 531,52 Hansen, J.B., 533, 223; 537,5 Gutierrez, P., 531,52 Hansen, J.D., 533, 223; 537,5 Gutsche, O., 539, 197 Hansen, J.R., 533, 223; 537,5 Güven, R., 535, 309 Hansen, P.H., 533, 223; 537,5 Guy, J., 533, 243 Hansen, S., 531,52 Gwenlan, C., 531,9;539, 197 Hanson, G.G., 533, 207; 539,13 Gyulassy, M., 538, 282 Hara, T., 538, 11; 539, 188; 540,33 Harada, M., 537, 280 H1 Collaboration, 539,25 Harada, Y., 540,33 Haag, C., 533, 243 Harder, K., 533, 207; 539,13 Haas, D., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; 540, 43, 185 Harel, A., 533, 207; 539,13 Haas, T., 531,9;539, 197 Harin-Dirac, M., 533, 207; 539,13 Haba, J., 538,11 Harindranath, A., 536, 250 Haba, N., 531, 245; 532, 93; 535, 271 Harrison, P.F., 535, 163, 229 Habig, A., 539, 179 Hart, J.C., 531,9;539, 197 Hadig, T., 539,25 Hartig, M., 535,85 Hadley, N.J., 531,52 Hartmann, B., 534, 137 Haeberli, W., 535,85 Hartmann, H., 531,9;539, 197 Haﬁdi, K., 535,85 Hartmann, M., 540, 207 Hagen, C.R., 539, 168 Hartner, G.F., 531,9;539, 197 Haggerty, H., 531,52 Hartnoll, S.A., 532, 297 Haggerty, J.S., 537, 211 Harvey, J., 533, 223; 537,5 342 Cumulative author index to volumes 531–540 (2002) 326–371

Hasch, D., 535,85 Henoch, M., 535,85 Hasegawa, T., 539, 179 Henschel, H., 539,25 Haseyama, T., 534,39 Hensel, C., 533, 207; 539,13 Hassouni, Y., 536, 321 Hepp, V., 533, 223; 537,5 Hatakeyama, S., 539, 179 Herce, H.D., 537, 141 Hatakeyama, Y., 539, 179 Heremans, R., 539,25 Hatano, M., 533,1 HERMES Collaboration, 535,85 Hatzifotiadou, D., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Hernandez, H., 535, 43; 537, 192; 540,25 540, 43, 185 Hernández, R., 536, 294 Haug, S., 533, 243 Herr, H., 533, 243 Hauler, F., 533, 243 Herrera, G., 539, 25, 218 Hauptman, J.M., 531,52 Herskind, B., 540, 199 Hauschild, M., 533, 207; 539,13 Herten, G., 533, 207; 539,13 Hauschildt, J., 533, 207; 539,13 Hertenberger, R., 535,85 Hautmann, F., 535, 159 Hervé, A., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; 540, 43, 185 Hawkes, C.M., 533, 207; 539,13 Herynek, I., 539,25 Hawkings, R., 533, 207; 539,13 Hess, J., 533, 223; 537,5 Hay, B., 536, 229; 537,28 Hess, P.O., 534,57 Hayashi, K., 531, 190 Hesselbach, S., 538, 346 Hayashii, H., 538, 11; 540,33 Hesselink, W.H.A., 535,85 Hayato, Y., 539, 179 Heuer, R.D., 533, 207; 539,13 Hayes, M.E., 531,9 Heusch, C., 531,9;539, 197 Hayes, O.J., 533, 223; 537,5 Heusse, Ph., 533, 223; 537,5 Haynes, W.J., 539,25 Heyde, K., 538,33 Hays, C., 531,52 Hidaka, K., 538, 137 Hazumi, M., 538, 11; 540,33 Higuchi, I., 538, 11; 540,33 He, H., 533, 223; 537,5 Higuchi, T., 538,11 He, H.-J., 532, 121; 536,83 Hilaire, S., 531,61 He, J., 536,59 Hildebrandt, M., 539,25 He, W., 532, 345 Hildreth, M.D., 531,52 He, X.-G., 533, 116 Hilger, E., 531,9;539, 197 Heaphy, E.A., 531,9;539, 197 Hilgers, M., 539,25 Heath, G.P., 531,9;539, 197 Hill, J., 539, 179 Heath, H.F., 531,9;539, 197 Hill, J.C., 533, 207; 539,13 Hebbeker, T., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Hill, R.D., 533, 223; 537,5 540, 43, 185 Hiller, B., 539,76 Hebecker, A., 539, 119 Hiller, K.H., 539,25 Hebert, C., 531,52 Hillert, S., 531,9;539, 197 Hedberg, V., 533, 243 Hiraoka, Y., 536, 147 Hedin, D., 531,52 Hirose, T., 531,9;539, 197 Heenan, E.M., 538, 11; 540,33 Hirosky, R., 531,52 Heesbeen, D., 535,85 Hirsch, J.G., 534,57 Hegyi, S., 538, 275 Hirschfelder, J., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Heinemann, B., 539, 25; 540, 263 540, 43, 185 Heinmiller, J.M., 531,52 Hirstius, A., 533, 196; 537,28 Heinsius, F.H., 535,85 Hladký, J., 539,25 Heinson, A.P., 531,52 Hobbs, J.D., 531,52 Heintz, U., 531,52 Hochman, D., 531,9;539, 197 Heinzelmann, G., 539,25 Hodgson, P.N., 533, 223; 537,5 Heister, A., 533, 223; 537,5 Hoeneisen, B., 531,52 Hejny, V., 540, 207 Hofer, H., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; 540, 43, 185 Helbich, M., 531,9;539, 197 Hoffman, K., 533, 207; 539,13 Hemingway, R.J., 533, 207; 539,13 Hoffmann, D., 539,25 Henderson, R.C.W., 539,25 Hoffmann, R., 535, 358 Hengstmann, S., 539,25 Hofman, G., 535,85 Hennecke, M., 533, 243 Hofmann, F., 532, 179 Cumulative author index to volumes 531–540 (2002) 326–371 343

Hohlmann, M., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Hutchcroft, D.E., 533, 223; 537,5 540, 43, 185 Hüttmann, K., 533, 223; 537,5 Höhne, C., 538, 275 Hwa, R.C., 534,69 Hojo, T., 540,33 Hwang, K., 540, 289 Hokuue, T., 538, 11; 540,33 Hyakutake, Y., 539, 153 Holder, M., 533, 196; 536, 229; 537,28 Hölldorfer, F., 533, 223; 537,5 Iacobucci, G., 531,9;539, 197 Hollenberg, L.C.L., 538, 207 Iacopini, E., 533, 196; 536, 229; 537,28 Holler, Y., 535,85 Ianni, A., 540,20 Holm, U., 531,9;539, 197 Iarygin, G., 535,85 Holmgren, S.-O., 533, 243 Iaselli, G., 533, 223; 537,5 Holt, P.J., 533, 243 Iashvili, I., 531,52 Holt, R.J., 535,85 Ibarra, A., 535,25 Holtz, K., 533, 196; 536, 229; 537,28 Ibbotson, M., 539,25 Holzner, G., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Ichikawa, A., 539, 179 540, 43, 185 Ichikawa, Y., 539, 179 Homer, R.J., 533, 207; 539,13 Iconomidou-Fayard, L., 533, 196; 536, 229; 537,28 Hommez, B., 535,85 Ida, D., 535, 315 Honkanen, H., 532, 222 Ieva, M., 539, 188 Horisberger, R., 539,25 Iga, Y., 531,9;539, 197 Horiuchi, T., 536,18 Igaki, T., 538, 11; 540,33 Horsley, R., 532,63 Igarashi, Y., 535, 363; 540,33 Horváth, D., 533, 207; 539,13 Igo, G., 538, 275 Hosack, M., 535, 43; 537, 192; 540,25 Igo-Kemenes, P., 533, 207; 539,13 Hoshi, Y., 538, 11; 540,33 Iijima, T., 538,11 Hoshina, K., 540,33 Iliescu, M., 535,52 Hoshino, K., 539, 188 Illingworth, R., 531,52 Höting, P., 539,25 Imbergamo, E., 533, 196; 536, 229; 537,28 Hou, S.R., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; 538, 11; Inagaki, T., 537, 211; 539, 179 540, 33, 43, 185 Inami, K., 538, 11; 540,33 Hou, W.-S., 538, 11; 540,33 Inami, T., 537, 165 Houlden, M.A., 533, 243 Incagli, M., 535, 37; 536, 209; 537,21;538,21 Hovhannisyan, A., 539,25 Ingrosso, L., 535, 37; 536, 209; 537, 21; 538,21 Howard, R., 533, 207; 539,13 Inoue, K., 539, 179 Hristov, P., 533, 196; 536, 229; 537,28 Inoue, Y., 536,18 Hristova, I.R., 539, 188 Inuzuka, M., 531,9;539, 197 Hsieh, T.-H., 538, 298 Inzani, P., 535, 43; 537, 192; 540,25 Hu, H., 533, 223; 537,5 Iofa, M.Z., 538, 385 Hu, M., 533, 237 Irastorza, I.G., 532,8 Hu, R.-J., 538,39 Irrgang, P., 531,9;539, 197 Hu, Y., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; 540, 43, 185 Ishibashi, M., 538, 197 Huang, C.-S., 538, 301 Ishida, M., 539, 249 Huang, H.-C., 538, 11; 540,33 Ishida, S., 539, 249 Huang, W.-H., 537, 311 Ishihara, K., 539, 179 Huang, X., 533, 223; 536, 209; 537,5 Ishii, K., 533, 207; 539, 13, 40 Huang, Y., 531,52 Ishii, T., 539, 179 Hübel, H., 538,33 Ishikawa, A., 538, 11; 540,33 Huber, S.J., 531, 112 Ishimoto, S., 534,39 Hübsch, T., 534, 147 Ishino, H., 538, 11; 539, 179 Hughes, G., 533, 223; 537,5 Ishitsuka, M., 539, 179 Hughes, V.W., 531,9;539, 197 Ishiwatari, T., 535,52 Hultqvist, K., 533, 243 Ishizuka, T., 539, 179 Hüntemeyer, P., 533, 207; 539,13 Isocrate, R., 535,93 Huovinen, P., 535, 109 I¸˙ssever, Ç., 539,25 Hurling, S., 539,25 Itahashi, K., 535,52 Hurvits, G., 539, 218 Ito, A.S., 531,52 344 Cumulative author index to volumes 531–540 (2002) 326–371

Ito, K., 536, 327 Jin, S., 533, 223; 537,5 Ito, M., 537, 211 Joffe, D., 533, 237 Itoh, K., 533,1;539,40 Johansson, E.K., 533, 243 Itoh, R., 538, 11; 540,33 Johansson, P.D., 533, 243 Itow, Y., 539, 179 Johns, K., 531,52 Ivanchenko, V.N., 537, 201 Johns, W.E., 535, 43; 537, 192; 540,25 Ivanova, V.A., 535, 371 Johnson, C., 539,25 Iwamoto, M., 538, 11; 540,33 Johnson, D.P., 539,25 Iwasaki, H., 538, 11; 540,33 Johnson, M., 531,52 Iwasaki, Y., 538, 11; 540,33 Jolos, R.V., 534,63 Iwashita, T., 539, 179 Jon, G.C., 539,40 Izawa, K.-I., 534,93 Jonckheere, A., 531,52 Izotov, A., 535,85 Jones, C.R., 533, 207 Jones, D.R.T., 535, 193, 377 Jacholkowska, A., 533, 223; 537,5 Jones, G., 534,39 Jack, I., 535, 193 Jones, L.T., 533, 223; 537,5 Jackiw, R., 534, 181 Jones, L.W., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Jackson, H.E., 535,85 540, 43, 185 Jackson, J.N., 533, 243 Jones, M.A.S., 539,25 Jacobsson, B., 532, 259 Jones, P.G., 538, 275 Jacquet, M., 539,25 Jones, R.W.L., 533, 223; 537,5 Jacquod, Ph., 537,62 Jones, T.W., 531,9;539, 197 Jadach, S., 533,75 Jonke, L., 531, 311 Jaffe, D.E., 537, 211 Jönsson, L., 539,25 Jaffré, M., 531,52 Jonsson, P., 533, 243 Jaffre, M., 539,25 Jonsson, T., 539, 277 Jain, S., 531,52 Joram, C., 533, 243 Jain, V., 537, 211 Josa-Mutuberría, I., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Jakob, H.-P., 531,9;539, 197 540, 43, 185 Jakobs, K., 533, 223; 537,5 Joshi, G.C., 535, 289 Jalocha, P., 533, 243; 538,11 Jost, B., 533, 223; 537,5 James, C., 539, 218 Jöstlein, H., 531,52 Jamin, M., 538,71 Jostlein, H., 537,41 Janas, Z., 532,29 Jousset, J., 533, 223; 537,5 Janauschek, L., 539,25 Jovanovic, P., 533, 207; 539,13 Jang, H.K., 540,33 Jung, C.K., 539, 179 Janik, R.A., 538, 189 Jung, H., 539,25 Janot, P., 533, 223; 537,5 Jung, P., 535,85 Janssen, X., 539,25 Jungermann, L., 533, 243 Jarlskog, Ch., 533, 243 Junk, T.R., 533, 207; 539,13 Jarlskog, G., 533, 243 Juste, A., 531,52 Jarry, P., 533, 243 Jeans, D., 533, 243 Kabe, S., 537, 211 Jeitler, M., 533, 196; 536, 229; 537,28 Kadija, K., 538, 275 Jelen,´ K., 531,9;539, 197 Kado, M., 533, 223; 537,5 Jemanov, V., 539,25 Käfer, D., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; 540, 43, 185 Jeoung, H.Y., 531,9 Kagan, A.L., 538, 327; 539, 227; 540, 278 Jeremie, H., 533, 207; 539,13 Kageyama, A., 538,96 Jesik, R., 531,52 Kahl, W., 531,52 Jézéquel, S., 533,75 Kahn, S., 531,52 Jezequel, S., 533, 223; 537,5 Kaiser, R., 535,85 Jgoun, A., 535,85 Kajfasz, E., 531,52 Ji, X., 538, 289 Kajita, T., 539, 179 Jia, Y., 536,67 Kajiyama, Y., 539, 179 Jin, B.N., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; 540, 43, 185 Kalinin, A.M., 531,52 Jin, G.-M., 538,39 Kalinin, S., 539, 188 Cumulative author index to volumes 531–540 (2002) 326–371 345

Kalmus, G.E., 533, 196; 536, 229; 537,28 Kawasaki, M., 533, 294 Kalmykov, Y., 532, 179 Kawasaki, T., 538, 11; 540,33 Kaloper, N., 535,33 Kaya, A., 538, 164 Kalter, A., 533, 196; 536, 229; 537,28 Kayis-Topaksu, A., 539, 188 Kalyana Rama, S., 539, 289 Kayser, B., 537, 227 Kameda, J., 539, 179 Kayser, F., 533, 223; 537,5 Kamon, T., 538, 121 Kazumori, M., 537, 211 Kananov, S., 531,9;539, 197 Kçira, D., 531,9;539, 197 Kanaya, N., 533, 207; 539,13 Kearns, E., 539, 179 Kane, G.L., 531, 263; 536, 263 Keeler, R.K., 533, 207; 539,13 Kaneko, J., 540,33 Kehoe, R., 531,52 Kaneko, S., 538,96 Keig, W.E., 539, 179 Kaneyuki, K., 539, 179 Keil, F., 539,25 Kang, J.H., 538, 11; 540,33 Kekelidze, V., 533, 196; 536, 229; 537,28 Kang, J.S., 535, 43; 537, 192; 538, 11; 540, 25, 33 Keller, N., 539,25 Kant, D., 539,25 Kellogg, R.G., 533, 207; 539,13 Kanti, P., 538, 146 Kennedy, B.W., 533, 207; 539,13 Kanzaki, J., 533, 207; 539,13 Kennedy, J., 533, 223; 537,5;539,25 Kapica, M., 532,29 Kenyon, I.R., 539,25 Kapichine, M., 539,25 Keranen, R., 533, 243 Kaplan, D.E., 531, 127; 534, 124 Kerger, R., 531,9;539, 197 Kappes, A., 531,9;539, 197 Kermiche, S., 539,25 Kapusta, F., 533, 243 Kernel, G., 533, 243 Kernreiter, T., 538, 59, 137 Kapusta, P., 538, 11; 540,33 Kersevan, B.P., 533, 243 Karapetian, G., 533, 207; 539,13 Kersten, J., 538,87 Karev, A., 538, 275 Kettell, S.H., 537, 211 Karlen, D., 533, 207; 539,13 Khanov, A., 531,52 Karliner, M., 533, 60; 538, 321 Kharchilava, A., 531,52 Karlsson, M., 539,25 Khein, L.A., 531,9;539, 197 Karmanov, D., 531,52 Khovansky, V., 539, 188 Karmgard, D., 531,52 Khriplovich, I.B., 537, 125 Karschnick, O., 539,25 Khuri, R.R, 535,1 Karshon, U., 531,9;539, 197 Kibayashi, A., 539, 179 Kartvelishvili, V., 533, 207; 539,13 Kichimi, H., 538, 11; 540,33 Kasemann, S., 535,93 Kiefer, C., 535,11 Kasper, J., 533, 237 Kielczewska, D., 539, 179 Kasper, P.A., 539, 218 Kienzle-Focacci, M.N., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Kasper, P.H., 535, 43; 537, 192; 540,25 540, 43, 185 Kataoka, S.U., 538, 11; 540,33 Kiesling, C., 539,25 Katayama, N., 538, 11; 540,33 Kiiskinen, A., 533, 243 Katkov, I.I., 531,9;539, 197 Kikuchi, Y., 539,40 Kato, H., 533,1 Kile, J., 533, 223; 537,5 Kato, I., 539, 179 Kim, C.K., 540, 309 Katsanevas, S., 533, 243 Kim, C.L., 531,9;539, 197 Katsouﬁs, E., 533, 243 Kim, C.S., 535, 249 Katz, S.D., 534,87 Kim, D.H., 533, 207; 539,13 Katz, U.F., 531,9;539, 197 Kim, D.W., 538, 11; 540,33 Kaur, M., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; 540, 43, 185 Kim, D.Y., 535, 43; 537, 192; 540,25 Kawagoe, K., 533, 207; 539,13 Kim, H., 535,5;536, 154; 538, 11, 11; 540, 33, 33 Kawai, H., 538, 11, 393; 540,33 Kim, H.I., 539, 179 Kawakami, Y., 538, 11; 540,33 Kim, H.J., 538, 11; 540,33 Kawamoto, T., 533, 207; 539,13 Kim, H.O., 538, 11; 540,33 Kawamura, H., 536, 344 Kim, J.E., 540, 289 Kawamura, N., 538, 11; 540,33 Kim, J.K., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; 540, 43, 185 Kawamura, T., 539, 188 Kim, J.Y., 531,9;539, 197 346 Cumulative author index to volumes 531–540 (2002) 326–371

Kim, S.B., 539, 179 Kobayashi, M., 537, 211 Kim, S.K., 531, 52; 540,33 Kobayashi, T., 533, 207; 539, 13, 179 Kim, T.H., 540,33 Kobel, M., 533, 207; 539,13 Kim, W., 537,21 Koblitz, B., 539,25 Kim, Y.K., 539, 197 Koch, H.R., 540, 207 Kimura, K., 537,86 Koch, U., 533, 196; 536, 229; 537,28 Kind, O., 531,9;539, 197 Kodaira, J., 536, 344 King, B.T., 533, 243 Kodama, K., 539, 188 King, S.F., 531, 263 Koffeman, E., 531,9;539, 197 Kinoshita, K., 538,11 Kohama, M., 539, 179 Kintz, N., 535,93 Kohli, J.M., 531,52 Kirchner, R., 532,29 Kohno, T., 531,9;539, 197 Kirillov, A.A., 532, 185; 535,22 Koike, M., 539, 179 Kirkby, J., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; 540, 43, 185 Kokkinias, P., 533, 243 Kirpichnikov, I.V., 532,8 Kokott, T.P., 533, 207; 539,13 Kiselev, A.V., 534,83 Kolesnikov, V.I., 538, 275 Kishimoto, T., 531, 190 Kolev, D., 539, 188 Kisielewska, D., 531,9;539, 197 Kolhinen, V.J., 532, 222 Kisselev, A., 535,85 Kollegger, T., 538, 275 Kitamura, S., 531,9;539, 197 Kolster, H., 535,85 Kitano, R., 539, 102 Kolya, S.D., 539,25 Kitching, P., 535, 85; 537, 211 Komamiya, S., 533, 207; 539,13 Kittel, W., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; 540, 43, 185 Komatsu, M., 539, 188 Kiyo, Y., 535, 145 Komatsubara, T.K., 537, 211 Kjaer, N.J., 533, 243 Konaka, A., 537, 211 Kjellberg, P., 539,25 Kondo, T., 531, 245; 535, 271 Klapdor-Kleingrothaus, H.V., 532,71 Kong, O.C.W., 537, 217 Kleber, V., 540, 207 König, A.C., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Klein, K., 533, 207; 539,13 540, 43, 185 Klein, M., 539,25 Königsmann, K., 535,85 Kleinknecht, K., 533, 196, 223; 536, 229; 537,5,28 Konijn, J., 539, 188 Kleinwort, C., 539,25 Konstantinidis, N., 533, 223; 537,5 Klemm, D., 536, 101 Kontros, K., 533,43 Klier, A., 533, 207; 539,13 Kooijman, P., 531,9;539, 197 Klima, B., 531,52 Koop, I.A., 537, 201 Klimek, K., 531,9;539, 197 Koop, T., 531,9;539, 197 Klimenko, A.A., 532,8;535,77 Kopal, M., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; 540, 43, 185 Klimentov, A., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Kopeikin, S., 532,1 540, 43, 185 Köpke, L., 533, 196; 536, 229; 537,28 KLOE Collaboration, 535, 37; 536, 209; 537,21;538,21 Koptev, V., 540, 207 Klug, T., 535,93 Korbel, V., 539,25 Kluge, E.E., 533, 223; 537,5 Kormos, L., 533, 207; 539,13 Kluge, T., 539,25 Korol, A.A., 537, 201 Kluge, W., 535, 37; 536, 209; 537, 21; 538,21 Korotkov, V., 535,85 Kluit, P., 533, 243 Korotkova, N.A., 531,9 Kluth, S., 533, 207; 539,13 Korpar, S., 538,11 Kmiecik, M., 540, 199 Körs, B., 532, 141 Knecht, M., 532,55 Korzhavina, I.A., 531,9;539, 197 Kneringer, E., 533, 223; 537,5 Koshelev, A.S., 536, 138 Knies, G., 539,25 Koshiba, M., 539, 179 Knowles, I., 536, 229; 537,28 Koshio, Y., 539, 179 Knuteson, B., 531,52 Koshuba, S.V., 537, 201 Ko, B.R., 535, 43; 537, 192; 540,25 Kostka, P., 539,25 Ko, C.M., 533, 259 Kostritskiy, A.V., 531,52 Ko, W., 531,52 Kostyuk, A.P, 531, 195 Kobayashi, K., 539, 179 Kotanski,´ A., 531,9;539, 197 Cumulative author index to volumes 531–540 (2002) 326–371 347

Kotcher, J., 531,52 Kubischta, W., 533, 196; 536, 229; 537,28 Kotelnikov, S.K., 539,25 Kubodera, K., 533,25 Kothari, B., 531,52 Kucharczyk, M., 533, 243 Kotik, E., 535,85 Kudomi, N., 531, 190 Kotwal, A.V., 531,52 Kuhl, T., 533, 207 Kötz, U., 531,9;539, 197 Kuhn, D., 533, 223; 537,5 Kourkoumelis, C., 533, 243 Kuhr, T., 539,25 Koutouev, R., 539,25 Kulasiri, R., 538, 11; 540,33 Koutov, A., 539,25 Kuleshov, S., 531,52 Koutsenko, V., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Kuliev, A.A., 532, 173 540, 43, 185 Kulik, B., 532, 357 Kouznetsov, O., 533, 243 Kulik, Y., 531,52 Kowal, A.M., 531,9;539, 197 Kulikov, V., 535, 37; 536, 209; 537, 21; 538,21 Kowal, M., 531,9;539, 197 Kumagai, K., 539,40 Kowalewski, R.V., 533, 207; 539,13 Kumar, S., 538, 11; 540,33 Kowalski, H., 531,9;539, 197 Kumar Ghosh, D., 537, 217 Kowalski, M., 538, 275 Kume, K., 531, 190 Kowalski, T., 531,9;539, 197 Kummer, W., 532, 373 Kowalski-Glikman, J., 539, 126 Kunin, A., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; 540, 43, 185 Kozelov, A.V., 531,52 Kunitomo, H., 536, 327 Kozlov, V., 535,85 Kuno, Y., 537, 211 Kozlovsky, E.A., 531,52 Kunori, S., 531,52 Kräber, M., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Kuo, C., 535, 37; 536, 209; 537,21;538,21 540, 43, 185 Kupco, A., 531,52 Kraemer, R.W., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Küpper, A., 539,25 540, 43, 185 Kupper, M., 533, 207; 539,13 Krakauer, D., 531,9;539, 197 Kuraev, E.A., 538,45 Krämer, T., 533, 207; 539,13 Kuramoto, H., 531, 190 Krane, J., 531,52 Kurca,ˇ T., 539,25 Kraus, I., 538, 275 Kuriki, M., 537, 211 Krauss, B., 535,85 Kurowska, J., 533, 243 Krehbiel, H., 539,25 Kusnezov, D., 537,62 Kreisel, A., 531,9;539, 197 Kutschke, R., 535, 43; 537, 192; 540,25 Krenz, W., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; 540, 43, 185 Kuze, M., 531,9;539, 197 Kreps, M., 538, 275 Kuzmin, A., 540,33 Kress, T., 533, 207; 539,13 Kuzmin, V.A., 531,9;539, 197 Kreyerhoff, G., 535, 201 Kuznetsov, V.E., 531,52 Kreymer, A.E., 535, 43; 537, 192; 540,25 Kwak, J.W., 535, 43; 537, 192; 540,25 Krieger, P., 533, 207; 539,13 Kwan, S., 539, 218 Krishnaswamy, M.R., 531,52 Kwee, H.J., 538, 115 Krivkova, P., 531,52 Kwon, Y.-J., 538, 11; 540,33 Krivokhijine, V.G., 535,85 Kyberd, P., 533, 207; 539,13 Križan, P., 538,11 Kycia, T.F., 537, 211 Krokovny, P., 538, 11; 540,33 Kyle, G., 535,85 Kroll, P., 532,99 Kyriakis, A., 533, 223; 537,5 Krop, D., 533, 207; 539,13 Kropp, W.R., 539, 179 L3 Collaboration, 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Kroseberg, J., 539,25 540, 43, 185 Krüger, A., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; 540,43 Labarga, L., 531,9;539, 197 Krüger, A., 540, 185 Labes, H., 531,9;539, 197 Krüger, K., 539,25 Labzowsky, L.N., 534,52 Krumnack, N., 539, 197 Lacava, F., 535, 37; 536, 209; 537, 21; 538,21 Krumstein, Z., 533, 243 Lachenmaier, T., 533, 191 Kryemadhi, A., 535, 43; 537, 192; 540,25 Lacourt, A., 533, 196; 536, 229; 537,28 Krzywdzinski, S., 531,52 Ladron de Guevara, P., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Kubantsev, M., 531,52 540, 43, 185 348 Cumulative author index to volumes 531–540 (2002) 326–371

Lafferty, G.D., 533, 207; 539,13 Lee, J.H., 531,9;539, 197 Laforge, B., 533, 243 Lee, K.B., 535, 43; 537, 192; 540,25 Lagamba, L., 535,85 Lee, S.B., 531,9 Lai, A., 533, 196; 536, 229; 537,28 Lee, S.H., 538, 11; 540,33 Lai, W.P., 536, 203 Lee, S.W., 531,9;539, 197 Laktineh, I., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Lee, W.M., 531,52 540, 43, 185 Lee, X.-G., 540, 213 Lalak, Z., 538, 426 Lee-Franzini, J., 535, 37; 536, 209; 537, 21; 538,21 Lalazissis, G.A., 532,29 Lees, J.-P., 533, 223; 537,5 Lamanna, G., 533, 196 Leﬂat, A., 531,52 Lamb, D., 539,25 Leggett, C., 531,52 Lambert, N.D., 540, 301 Le Goff, J.M., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Lammers, S., 531,9;539, 197 540, 43, 185 Lamsa, J., 533, 243 Lehner, F., 531,52 Lançon, E., 533, 223; 537,5 Lehto, M., 533, 223; 537,5 Landi, G., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; 540, 43, 185 Leibenguth, G., 533, 223; 537,5 Landon, M.P.J., 539,25 Leibovich, A.K., 539, 242 Landsberg, G., 531,52 Leinonen, L., 533, 243 Landshoff, P.V., 533, 277 Leins, A., 533, 207; 539,13 Landsman, H., 533, 207; 539,13 Leinson, L.B., 532, 267 Lane, J.B., 531,9;539, 197 Leißner, B., 539,25 Lanfranchi, G., 535, 37; 536, 209; 537,21;538,21 Leiste, R., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; 540, 43, 185 Langacker, P., 540, 247 Leitner, R., 533, 243 Lange, J.S., 538,11 Lelas, D., 531,9;539, 197 Lange, W., 539,25 Lellouch, D., 533, 207; 539,13 Langmann, E., 533, 168 Lemaire, M.-C., 533, 223; 537,5 Langs, D.C., 539, 218 Lemaitre, V., 533, 223; 537,5 Lanske, D., 533, 207; 539,13 Lemonne, J., 533, 243 Larios, F., 531, 231 Lemos, J.P.S., 538,1 Lasio, G., 533, 237 Lemrani, R., 539,25 Laštovicka,ˇ T., 539,25 Lendermann, V., 539,25 Lätt, J., 540,43 Lengyel, A., 533,43 Lattimer, J.M., 531, 167 Lenisa, P., 535,85 Laurelli, P., 533, 223; 537,5 Lenske, H., 532, 179 Lauss, B., 535,52 Lenti, M., 533, 196; 536, 229; 537,28 Laycock, P., 539,25 Lenzen, G., 533, 243 Layter, J.G., 533, 207; 539,13 Lenzi, S., 535,93 Laziev, A., 535,85 Leone, D., 535, 37; 536, 209; 537, 21; 538,21 Lazzeroni, C., 533, 196; 536, 229; 537,28 Leoni, S., 540, 199 Learned, J.G., 539, 179 Leonidopoulos, C., 531,52 Lebailly, E., 539,25 Lepeltier, V., 533, 243 Lebeau, M., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Lepora, N.F., 533, 131; 536, 338 540, 43, 185 Lerda, A., 540, 104 Lebedev, A., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; 539, 25; Lesiak, T., 533, 243 540, 43, 185 Leslie, J., 539, 218 Lebrun, P., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; 540, 43, 185 Lesniak,´ L., 538, 266 Lecomte, P., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Letts, J., 533, 207; 539,13 540, 43, 185 Lévai, P., 538, 275 Lecoq, P., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; 540, 43, 185 Levai, P., 538, 282 Le Coultre, P., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Levchenko, B.B., 531,9;539, 197 540, 43, 185 Leveraro, F., 535, 43; 537, 192; 540,25 Leder, G., 533, 243; 538, 11; 540,33 Levi, G., 531,9;539, 197 Ledroit, F., 533, 243 Levin, E., 532,37 Lee, C.-A., 531, 112 Levinson, L., 533, 207; 539,13 Lee, C.-Y., 536, 154 Levman, G.M., 531,9;539, 197 Lee, H.W., 537, 117 Levonian, S., 539,25 Cumulative author index to volumes 531–540 (2002) 326–371 349

Levtchenko, M., 534, 28; 535, 59; 536, 24, 217; 540, 43, 185 Litke, A.M., 533, 223; 537,5 Levtchenko, P., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Littenberg, L.S., 537, 211 540, 43, 185 Littlewood, C., 533, 207 Levy, A., 531,9;539, 197 Liu, D.W., 539, 179 Lewis, M., 540,1 Liu, J.-Y., 540, 213 Lewitowicz, M., 537,45 Liu, W., 533, 259 Lhuillier, D., 539,8 Liu, X., 531,9;539, 197 Li, C., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; 540, 43, 185 Liu, Y., 536, 203 Li, H., 536,67 Liu, Y.-Y., 538,39 Li, H.B., 536, 203 Liu, Z.A., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; 540, 43, 185 Li, J., 531, 52; 536, 203; 540,33 Liventsev, D., 538, 11; 540,33 Li, K.K., 537, 211 Lloyd, S.L., 533, 207; 539,13 Li, L., 539, 197 Lobanov, A., 535, 187 Li, Q.Z., 531,52 Lo Bianco, G., 540, 199 Li, X., 537, 306 Lobodzinska, E., 539,25 Li, Z.-H., 540, 213 Lobodzinski, B., 539,25 Li, Z.-Y., 538,39 Locci, E., 533, 223; 537,5 Liao, W., 538, 301 Loebinger, F.K., 533, 207; 539,13 Liebig, W., 533, 243 Loewy, A., 537, 147 Liebing, P., 535,85 Loginov, A., 539,25 Liesen, D., 534,52 Lohmann, W., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Lietti, S.M., 540, 252 540, 43, 185 Ligabue, F., 533, 223; 537,5 Löhner, H., 538,27 Ligeti, Z., 538, 327; 539, 242 Löhr, B., 531,9;539, 197 Lightwood, M.S., 531,9;539, 197 Lohrmann, E., 531,9;539, 197 Liguori, G., 535, 43; 537, 192; 540,25 Loizides, J.H., 539, 197 Liguori Neto, R., 534,45 Lokhtin, I.P., 537, 261 Likhoded, S., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Loktionova, N., 539,25 540, 43, 185 Lombardo, F.C., 539,1 Liko, D., 533, 243 Lombardo, U., 533,17 Lillich, J., 533, 207; 539,13 London, D., 533, 65; 538, 309 Lim, H., 531,9;539, 197 Long, G.-L., 533, 253 Lim, I.T., 531,9;539, 197 Long, K.R., 531,9;539, 197 Lima, J.G.R., 531,52 Longhin, A., 531,9;539, 197 Lima-Santos, A., 538, 435 Limentani, S., 531,9;539, 197 Longo, E., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; 540, 43, 185 Lin, C.H., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; 540, 43, 185 Loomis, C., 533, 223; 537,5 Lin, J., 533, 223; 537,5 Lopes, J.H., 533, 243 Lin, W.T., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; 540, 43, 185 Lopes da Silva, P., 533, 196; 536, 229; 537,28 Linch, W.D., 535, 280 Lopez, A.M., 535, 43; 537, 192; 540,25 Lincoln, D., 531,52 Lopez, J., 533, 223; 537,5 Linde, F.L., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Lopez, J.M., 533, 243 540, 43, 185 Lopez-Duran Viani, A., 531,9;539, 197 Lindemann, T., 535,85 Lorenzon, W., 535,85 Lindner, M., 538,87 Loukas, D., 533, 243 Lindstroem, M., 539,25 Loverre, P.F., 539, 188 Ling, T.Y., 531,9;539, 197 LoVetere, M., 533, 237 Link, J.M., 535, 43; 537, 192; 540,25 Lu, F., 535, 37; 536, 209; 537, 21; 538,21 Linn, S.L., 531,52 Lü, H., 534, 155, 172 Linnemann, J., 531,52 Lu, J., 533, 207; 539,13 Lipartia, E., 533, 285 Lu, Q., 532, 240 Lipkin, H.J., 533, 60; 540,97 Lu, R.-S., 540,33 Lipniacka, A., 533, 243 Lu, W.-F., 540, 309 Lipton, R., 531,52 Lu, Y.S., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; 540, 43, 185 List, B., 539,25 Lu, Z.-H., 538,39 Lista, L., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; 540, 43, 185 Lübcke, M., 534, 195 350 Cumulative author index to volumes 531–540 (2002) 326–371

Lübelsmeyer, K., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Madaras, R.J., 531,52 540, 43, 185 Maddox, E., 531,9;539, 197 Lubian, J., 534,45 Mader, W., 533, 207; 539,13 Lubimov, V., 539,25 Madigojine, D., 533, 196; 536, 229; 537,28 Lubrano, P., 533, 196; 536, 229; 537,28 Madouri, F., 536, 321 Lucherini, V., 535,52 Maeda, Y., 533,1;540, 207 Luci, C., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; 540, 43, 185 Maggi, G., 533, 223; 537,5 Lucotte, A., 531,52 Maggi, M., 533, 223; 537,5 Lüders, S., 539,25 Magill, S., 531,9;539, 197 Ludhova, L., 535,52 Magnin, J., 535, 43; 537, 192; 539, 218; 540,25 Ludovici, L., 539, 188 Maharana, J., 533, 146 Ludwig, J., 533, 207; 539,13 Mahlke-Krüger, H., 539,25 Lueking, L., 531,52 Mahmud, H., 532,29 Lugo, A.R., 539, 143 Maier, A., 533, 196; 536, 229; 537,28 Luisi, C., 536, 209 Maier, R., 540, 207 Luitz, S., 536, 229; 537,28 Maj, A., 540, 199 Lüke, D., 539,25 Majerotto, W., 538, 353 Lukierski, J., 538, 375; 539, 266 Majumder, G., 538, 11; 540,33 Lukina, O.Yu., 531,9;539, 197 Makins, N.C.R., 535,85 Luminari, L., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Maklioueva, I., 539, 188 540, 43, 185 Malakhov, A.I., 538, 275 Lunardi, S., 535,93 Malden, N., 539,25 Lundberg, B., 539, 218 Malek, A., 533, 243 Lundstedt, C., 531,52 Malek, M., 539, 179 Luo, C., 531,52 Malgeri, L., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Lupi, A., 531,9;539, 197 540, 43, 185 Luppi, E., 533, 196, 237 Malinin, A., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Lüst, D., 532, 141 540, 43, 185 Lustermann, W., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Malinovski, E., 539,25 540, 43, 185 Malinovski, I., 539,25 Lütjens, G., 533, 223; 537,5 Maltezos, S., 533, 243 Lutz, P., 533, 243 Malvezzi, S., 535, 43; 537, 192; 540,25 Lyakhovich, S.L., 534, 201 Malyshev, V.L., 531,52 Lyakhovsky, V., 538, 375 Maña, C., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; 540, 43, 185 Lykken, J., 534, 106 Manankov, V., 531,52 Lynch, J.G., 533, 223; 537,5 Mandelkern, M., 533, 237 Lyons, L., 533, 243 Mandl, F., 533, 243; 540,33 Lytkin, L., 539,25 Mangano, G., 534,8 Lyubovitskij, V.E., 533, 285 Mangeol, D., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; 540, 43, 185 Ma, E., 536, 259 Mankel, R., 531,9;539, 197 Ma, J.P., 537, 233 Mann, R.B., 535, 315; 537, 329 Ma, K.J., 531,9 Mannelli, I., 533, 196; 536, 229; 537,28 Ma, W.G., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; 540, 43, 185 Männer, W., 533, 223; 537,5 Maalampi, J., 532, 202 Mannocchi, G., 533, 223; 537,5 Maartens, R., 532, 153 Manohar, A.V., 539,59 Maas, A., 535,85 Mans, J., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; 540, 43, 185 Macchiolo, A., 533, 207 Mansouri, F., 538, 239 Macdonald, J.A., 537, 211 Manvelyan, R., 533, 138 Machado, M.V.T., 533,43 Mao, H.S., 531,52 Machavariani, A.I., 540,81 Maor, U., 532,37 Machefert, F., 533, 223; 537,5 Maracek,ˇ R., 539,25 Maciel, A.K.A., 531,52 Marage, P., 539,25 MacNaughton, J., 533, 243; 538, 11; 540,33 Marcellini, S., 533, 207; 539,13 Macpherson, A., 533, 207; 539,13 March-Russell, J., 539, 119 Macrì, M., 533, 237 Marchant, T.E., 533, 207; 539,13 Cumulative author index to volumes 531–540 (2002) 326–371 351

Marchetto, F., 533, 196, 237; 536, 229; 537,28 Martyniak, J., 539,25 Marco, J., 533, 243 Maru, N., 532,93 Marco, R., 533, 243 Maruelli, P., 536, 229 Marechal, B., 533, 243 Marukyan, H., 535,85 Marel, G., 533, 196 Maruyama, T., 539, 179 Marfatia, D., 532, 15, 19; 534, 120; 536, 79; 537, 179; 538, 346; Marzano, F., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; 540, 247 540, 43, 185 Margetis, S., 538, 275 Masaike, A., 534,39 Marginean, N., 535,93 Maselli, S., 531,9;539, 197 Margoni, M., 533, 243 Masetti, G., 533, 207; 539,13 Margotti, A., 531,9;539, 197 Masetti, L., 533, 196 Marie, F., 539,8 Mashimo, T., 533, 207; 539,13 Marin, J.-C., 533, 243 Masik, J., 533, 243 Marinelli, M., 533, 237 Maslennikov, A., 539, 188 Marinelli, N., 533, 223; 537,5 Masoli, F., 535,85 Marini, A., 540, 179 Massafferri, A., 535, 43; 537, 192; 539, 218; 540,25 Marini, G., 531,9;539, 197 Massam, T., 531,9 Mariotti, C., 533, 243 Mastroberardino, A., 531,9;539, 197 Maris, M., 534,17 Mastroyiannopoulos, N., 533, 243 Markert, C., 538, 275 Masuda, Y., 534,39 Markou, A., 533, 243 Matea, I., 537,45 Markou, C., 533, 223; 537,5 Mateos, D., 538, 366 Marks, J., 539,25 Matheys, J.P., 537,28 Markun, P., 531,9 Mato, P., 533, 223; 537,5 Markytan, M., 533, 196; 536, 229; 537,28 Matorras, F., 533, 243 Marlow, D.R., 537, 211 Matos, T., 538, 246 Marnelius, R., 534, 201 Matsuda, Y., 534,39 Marotta, R., 540, 104 Matsuishi, T., 538,11 Marouelli, P., 533, 196; 537,28 Matsumoto, S., 538, 11; 540,33 Marras, D., 533, 196; 536, 229; 537,28 Matsumoto, T., 538, 11; 540,33 Marshall, R., 539,25 Matsuno, S., 539, 179 Marshall, T., 531,52 Matsuoka, K., 531, 190 Martelli, F., 533, 196; 537,28 Matsushita, T., 531,9;539, 197 Martemianov, M., 535, 37; 536, 209; 537, 21; 538,21 Matsuyama, T., 536,49 Martens, J., 531,9 Matsuzawa, K., 531,9;539, 197 Martens, K., 539, 179 Matsyuk, M., 535, 37; 536, 209; 537,21;538,21 Martí, G.V., 534,45 Matteuzzi, C., 533, 243 Martin, A.D., 531, 216 Mättig, P., 533, 207; 539,13 Martin, A.J., 533, 207; 539,13 Mattingly, M.C.K., 531,9;539, 197 Martin, F., 533, 223; 537,5 Matute, E.A., 538,66 Martin, J.F., 531,9;539, 197 Mauger, C., 539, 179 Martin, J.P., 531, 28, 39; 533, 207; 534, 28; 535, 59; 536, 24, 217; Maxﬁeld, S.J., 539,25 539, 13; 540, 43, 185 Mayes, B.W., 538, 275 Martin, M.I., 531,52 Mayet, F., 538, 257 Martin, V., 536, 229; 537,28 Maynard, C.M., 532,63 Martin, V.J., 533, 196 Mayorov, A.A., 531,52 Martínez, G., 538,27 MayTal-Beck, S., 539, 218 Martínez, M., 531,9;539, 197 Mazumdar, K., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Martinez, M., 533, 223; 537,5 540, 43, 185 Martinez-Rivero, C., 533, 243 Mazzitelli, F.D., 539,1 Martini, M., 533, 196; 536, 229; 537,28 Mazzitelli, G., 540, 179 Martino, J., 539,8 Mazzocchi, C., 532,29 Martinovic,ˇ L., 536, 250 Mazzucato, E., 533, 196; 536, 229; 537,28 Marton, J., 535,52 Mazzucato, F., 533, 243 Martucci, L., 536, 101 Mazzucato, M., 533, 243 Martyn, H.-U., 539,25 McArthur, D.M.E., 535, 295 352 Cumulative author index to volumes 531–540 (2002) 326–371

McCance, G.J., 531,9;539, 197 Merzliakov, S., 540, 207 McCarthy, R., 531,52 Mes, H., 533, 207; 539,13 McCubbin, N.A., 531,9;539, 197 Meschini, M., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; McDonald, W.J., 533, 207; 539,13 540, 43, 185 McGrew, C., 539, 179 Messi, R., 535, 37; 536, 209; 537, 21; 538,21 McIlhany, K., 535,85 Messier, M.D., 539, 179 McIntosh, J.A.L., 538, 207 Messina, A., 538, 130 McKenna, J., 533, 207; 539,13 Messina, M., 539, 188 McMahon, T., 531,52 Messineo, A., 533, 223; 537,5 McMahon, T.J., 533, 207; 539,13 Mestvirishvili, A., 533, 196; 536, 229; 537,28 McNamara, P.A., 533, 223; 537,5 Mestvirishvili, I., 533, 196 McNeil, R.R., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Metlica, F., 531,9;539, 197 540, 43, 185 Metreveli, Z., 533, 237 Mc Nulty, R., 533, 243 Metsaev, R.R., 531, 152 McPherson, R.A., 533, 207; 537, 211; 539,13 Metzger, W.J., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Meadows, B., 539, 218 540, 43, 185 Medcalf, T., 533, 223; 537,5 Mexner, V., 535,85 Medvedev, P.B., 532, 291 Meyer, A., 531,9;539, 197 Meer, D., 539,25 Meyer, A.B., 539,25 Meggiolaro, E., 537, 173 Meyer, H., 539,25 Mehen, T., 539,59 Meyer, J., 539,25 Mehta, A., 539,25 Meyer, P.-O., 539,25 Mehta, P., 535, 219 Meyer, W.T., 533, 243 Mei, W., 535, 37; 536, 209; 537, 21; 538,21 Meyers, P.D., 537, 211 Meier, K., 539,25 Meyners, N., 535,85 Meijers, F., 533, 207; 539,13 Mezzadri, M., 535, 43; 537, 192; 540,25 Meinhard, H., 539, 188 Miao, C., 531,52 Meissner, F., 535,85 Michel, B., 533, 223; 537,5 Melanson, H.L., 531,52 Michelini, A., 533, 207; 539,13 Mele, S., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; 540, 43, 185 Miele, G., 534,8 Melikyan, A., 533, 146 Miettinen, H., 531,52 Meljanac, S., 531, 311 Migliore, E., 533, 243 Melkumov, G.L., 538, 275 Migliozzi, P., 539, 188 Mellado, B., 531,9;539, 197 Mihalcea, D., 531, 52; 539, 218 Melzer, O., 539, 188 Mihara, S., 533, 207; 539,13 Menary, S., 531,9;539, 197 Mihul, A., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; 540, 43, 185 Menasce, D., 535, 43; 537, 192; 540,25 Mikenberg, G., 533, 207; 539,13 Menden, F., 535,85 Mikirtychiants, S., 540, 207 Mendez, H., 535, 43; 537, 192; 540,25 Mikloukho, O., 535,85 Mendez, L., 535, 43; 537, 192; 540,25 Mikocki, S., 539,25 Mendez-Lorenzo, P., 533, 207; 539,13 Mikulec, I., 533, 196; 536, 229; 537,28 Menegazzo, R., 535,93 Milazzo, L., 535, 43; 537, 192 Menezes, R., 537, 321 Milburn, R.H., 539, 218 Meng, J., 532, 209 Milcent, H., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Menges, W., 533, 207; 539,13 540, 43, 185 Menichetti, E., 533, 196, 237; 536, 229; 537,28 Mildenberger, J., 537, 211 Menicucci, A., 535,37 Milekovic,´ M., 531, 311 Merino, G., 533, 223; 537,5 Miley, H.S., 532,8 Merkin, M., 531,52 Milite, M., 531,9;539, 197 Merle, E., 533, 223; 537,5 Miller, D.B., 531,9;539, 197 Merlo, M.M., 535, 43; 537, 192; 540,25 Miller, D.J., 533, 207; 539,13 Merola, L., 531, 28, 39; 534, 28; 535, 37, 59; 536, 24, 209, 217; Million, B., 540, 199 537, 21; 538, 21; 540, 43, 185 Milner, R., 535,85 Meroni, C., 533, 243 Milstead, D., 539,25 Merritt, F.S., 533, 207; 539,13 Mimura, Y., 538, 406 Merritt, K.W., 531,52 Minakata, H., 532, 275; 537, 249 Cumulative author index to volumes 531–540 (2002) 326–371 353

Minard, M.-N., 533, 223; 537,5 Monge, R., 533, 243 Mindur, B., 531,9;539, 197 Monteil, S., 533, 223; 537,5 Mine, S., 539, 179 Montenegro, J., 533, 243 Minenkov, Y., 540, 179 Montero, Á., 533, 322 Minic, D., 534, 147; 536, 305 Montgomery, H.E., 531,52 Minowa, M., 536,18 Montiel, E., 535, 43; 537, 192; 540,25 Miquel, R., 533, 223; 537,5 Montvay, I., 537,69 Mir, Ll.M., 533, 223; 537,5 Moore, R.W., 531,52 Mirabelli, G., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Moraes, D., 533, 243 540, 43, 185 Morales, A., 532,8 Mirea, A., 531,9;539, 197 Morales, J., 532,8 Mironov, A., 532, 350 Moreau, F., 539,25 Miscetti, S., 535, 37; 536, 209; 537, 21; 538,21 Moreno, E.F., 534, 185 Mischke, A., 538, 275 Moreno, S., 533, 243 Mishra, C.S., 531,52 Morettini, P., 533, 243 Misiejuk, A., 533, 223; 537,5 Mori, S., 538,11 Mitaroff, W., 533, 243; 538,11 Mori, T., 533, 207; 538, 11; 539, 13; 540,33 Mitchell, J., 539,8 Morii, M., 539, 179 Mitchell, R., 535, 43; 537, 192; 540,25 Morimoto, K., 534,39 Mitsuda, C., 539, 179 Moritz, M., 531,9;539, 197 Miura, M., 539, 179 Moriyama, S., 536, 18; 539, 179 Miyabayashi, K., 538, 11; 540,33 Moroi, T., 533, 294; 539, 303 Miyabayashi, Y., 538,11 Moroni, L., 535, 43; 537, 192; 540,25 Miyake, H., 538, 11; 540,33 Morozov, A., 532, 350; 539,25 Miyanishi, M., 539, 188 Morris, J.V., 539,25 Miyano, K., 539, 179 Moser, H.-G., 533, 223; 537,5 Miyata, H., 538, 11; 540,33 Mostafa, M., 531,52 Mizoguchi, S., 537, 130 Moulson, M., 535, 37; 536, 209; 537, 21; 538,21 Mjoernmark, U., 533, 243 Moutoussi, A., 533, 223; 537,5 Mkrtchyan, T., 539,25 Mozrzymas, M., 538, 375 Mnich, J., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; 540, 43, 185 Mü, S., 536, 209; 537,21 Moa, T., 533, 243 Muanza, G.S., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Moch, M., 533, 243 540, 43, 185 Mocioiu, I., 534, 114 Muccifora, V., 535,85 Modena, I., 540, 179 Muciaccia, M.T., 539, 188 Modestino, G., 540, 179 Mück, W., 531, 301 Moed, S., 533, 207; 539,13 Mueller, A.H., 539,46 Moenig, K., 533, 243 Mueller, U., 533, 243 Mohanta, R., 540, 241 Muenich, K., 533, 243 Mohanty, G.B., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Muijs, A.J.M., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; 540, 43, 185 540, 43, 185 Mohapatra, R.N., 532, 77; 536,94 Mulders, M., 533, 243 Mohr, R., 539,25 Mulders, P.J., 532,87 Mohr, W., 533, 207; 539,13 Mulhauser, F., 535,52 Mohrdieck, S., 539,25 Müller, A.-S., 533, 223; 537,5 Mokhov, N., 531,52 Müller, K., 539,25 Moleti, A., 540, 179 Müller, S., 535, 37; 538,21 Molnár, J., 538, 275 Mülsch, D., 535, 349 Molokanova, N., 533, 196; 536, 229; 537,28 Multamäki, T., 535, 170 Moloney, G.R., 538, 11; 540,33 Munday, D.J., 533, 196; 536, 229; 537,28 Momen, A., 534, 167 Mundim, L., 533, 243 Monaco, V., 531,9;539, 197 Muramatsu, N., 537, 211 Mondal, M., 540,52 Muri, C., 534,45 Mondal, N.K., 531,52 Murín, P., 539,25 Mondardini, M.R., 537,41 Murray, W., 533, 243 Mondragon, M.N., 539,25 Murtas, F., 533, 223; 535, 37; 536, 209; 537, 21; 538,21 354 Cumulative author index to volumes 531–540 (2002) 326–371

Murtas, G.P., 533, 223; 537,5 Naroska, B., 539,25 Muryn, B., 533, 243 Naryshkin, Y., 535,85 Musgrave, B., 531,9;539, 197 Nascimento, J.R.S., 537, 321 Musicar, B., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Nasir, S.M., 534, 195 540, 43, 185 Nass, A., 535,85 Mussa, R., 533, 237 Nassalski, J., 533, 196; 536, 229; 537,28 Mussardo, G., 536, 169 Nasseri, F., 538, 223 Musy, M., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; 540, 43, 185 Nassiakou, M., 533, 243 Mutaf, Y., 531,52 Natale, A.A., 537, 275 Muto, S., 534,39 Natale, S., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; 540, 43, 185 Mutter, A., 533, 207; 539,13 Natkaniec, Z., 538, 11; 540,33 Myatt, G., 533, 243 Naumann, J., 539,25 Myhrer, F., 533,25 Naumann, N.A., 531,52 Myklebust, T., 533, 243 Naumann, Th., 539,25 Myung, Y.S., 531,1;537, 117 Navarria, F., 533, 243 Nawrocki, K., 533, 243 NA48 Collaboration, 537,28 Neal, H.A., 531, 52; 533, 207; 539,13 Naftali, E., 532,37 Nedosekin, A., 535, 37; 536, 209; 537, 21; 538,21 Nagahiro, H., 536,49 Needham, M.D., 536, 229; 537,28 Nagai, K., 533, 207; 539,13 Neergård, K., 537, 287 Nagaitsev, A., 535,85 Neﬁodov, A.V., 534,52 Nagamine, T., 538, 11; 540,33 Negele, J.W., 533, 322 Nagano, K., 531,9;539, 197 Negodaeva, K., 535,85 Nagasaka, Y., 538, 11; 540,33 Negret, J.P., 531,52 Nagashima, Y., 539, 179 Negrini, M., 533, 237 Nagovizin, V., 539,25 Negus, P., 533, 223; 537,5 Nagy, E., 531,52 Nehme, A., 532,55 Nagy, S., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; 540, 43, 185 Nehring, M., 535, 43; 537, 192; 540,25 Nahm, K., 540, 309 Neichi, K., 540,33 Nakadaira, T., 538,11 Nekipelov, M., 540, 207 Nakahata, M., 539, 179 Nekrasov, M.L., 531, 225 Nakajima, H., 537, 165 Nellen, G., 539,25 Nakamura, I., 533, 207; 539,13 Nelson, J.M., 538, 275 Nakamura, K., 539, 179, 188 Nemecek, S., 533, 243 Nakamura, M., 539, 188 Nenoff, N., 538,33 Nakano, E., 538, 11; 540,33 Ness, S., 536, 315 Nakano, T., 537, 211; 539, 188 Nessi-Tedaldi, F., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Nakao, M., 538, 11; 540,33 540, 43, 185 Nakaya, T., 539, 179 Neubert, M., 535, 127; 539, 227; 540, 278 Nakayama, S., 539, 179 Neuhofer, G., 533, 196; 536, 229; 537,28 Nam, J.W., 538, 11; 540,33 Newman, H., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Namba, T., 536, 18; 539, 179 540, 43, 185 Nandi, S., 538, 406 Newman, P.R., 539,25 Nang, F., 531,52 Neyens, G., 537, 45; 538,33 Nania, R., 531,9;539, 197 Neyret, D., 539,8 Napier, A., 539, 218 Ng, C., 537, 211 Napoli, D.R., 535,93 Ng, J.N., 536,83 Napolitano, M., 531, 28, 39; 534, 28; 535, 37, 59; 536, 24, 209, Ng, S., 537, 211; 538, 366 217; 537,21;538, 21; 540, 43, 185 Ngac, A., 533, 223; 537,5 Nappi, A., 533, 196; 536, 229; 537,28 Nguyen, A., 539, 218 Nappi, E., 535,85 Nguyen, F., 535, 37; 536, 209; 537, 21; 538,21 Nara, Y., 531, 209 Nicolaidou, R., 533, 243 Narain, M., 531,52 Nicolo, D., 538,77 Narasimham, V.S., 531,52 Niebergall, F., 539,25 Narison, S., 540, 233 Niebuhr, C., 539,25 Narita, K., 539, 188 Nielsen, J., 533, 223; 537,5 Cumulative author index to volumes 531–540 (2002) 326–371 355

Niemi, A.J., 534, 195 Odintsov, S.D., 531, 143; 540, 167 Niessen, T., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; O’Donnell, P.J., 540,97 540, 43, 185 Oﬁerzynski, R., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Niezurawski, P., 533, 243 540, 43, 185 Nigro, A., 531,9;539, 197 Oganessyan, K., 535,85 Niizeki, T., 539,40 Oganessyan, K.A., 532,87 Nikolaev, N.N., 538,45 Ogawa, S., 538, 11; 539, 188; 540,33 Nikolenko, M., 533, 243 Oguri, V., 531,52 Nilles, H.P., 536, 270 Oh, A., 533, 207; 539,13 Nilsson, B.S., 533, 223; 537,5 Oh, B.Y., 531,9;539, 197 Nisati, A., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; 540, 43, 185 Oh, S., 535, 249 Nishida, S., 538, 11; 540,33 Ohlsson, T., 532, 259; 537,95 Nishijima, K., 539, 179 Ohm, H., 540, 207 Nishikawa, J., 533,1 Ohnishi, T., 533,1 Nishikawa, K., 539, 179 Ohno, F., 538, 11; 540,33 Nishimura, R., 539, 179 Ohnuma, H., 539,40 Nishimura, T., 531,9;539, 197 Ohshima, T., 538, 11; 540,33 Nishino, H., 532, 334; 535, 337; 540, 125 Ohsumi, H., 531, 190 Nisius, R., 533, 207; 539,13 Ohta, N., 539, 153 Nitoh, O., 538, 11; 540,33 Okabe, T., 538, 11; 540,33 Nitta, K., 539, 179 Okada, A., 539, 179 Niu, K., 539, 188 Okamura, H., 533,1 Niwa, K., 539, 188 Okazawa, H., 539, 179 Nix, O., 539,25 Okpara, A., 533, 207; 539,13 Noguchi, S., 540,33 Okuno, S., 538, 11; 540,33 Nojiri, S., 531, 143; 540, 167 Okusawa, T., 539, 188 Nomerotski, A., 531,52 Olaiya, E., 533, 196; 536, 229; 537,28 Nomura, D., 540, 233 Olaya, D., 535, 43; 537, 192; 540,25 Nomura, Y., 532, 111; 538, 359 Oldeman, R.G.C., 539, 188 Nonaka, N., 539, 188 Olechowski, M., 536, 270 Norton, A., 533, 196; 536, 229; 537,28 Olive, A., 538, 146 Norton, P.R., 533, 223; 537,5 Olive, K.A., 532, 318; 539, 107 Notz, D., 531,9;539, 197 Olivier, B., 531,52 Nowak, G., 539,25 Olkiewicz, K., 531,9;539, 197 Nowak, H., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Olsen, S.L., 538, 11; 540,33 540, 43, 185 Olshevski, A., 533, 243 Nowak, R.J., 531,9;539, 197 Olsson, J.E., 539,25 Nowak, S., 539, 126 O’Neale, S.W., 533, 207; 539,13 Nowak, W.-D., 535,85 O’Neil, D., 531,52 Nowell, J., 533, 223; 537,5 O’Neill, T.G., 535,85 Nozaki, T., 538, 11; 540,33 Onengüt, G., 539, 188 Numao, T., 537, 211 Onofre, A., 533, 243 Nunnemann, T., 531,52 Onuki, Y., 540,33 Nunokawa, H., 537, 249 OPAL Collaboration, 533, 207; 539,13 Nussinov, S., 538, 321 Orava, R., 533, 243 Nuzzo, S., 533, 223; 537,5 Orear, J., 537,41 Nyberg, J., 540, 199 Oreglia, M.J., 533, 207; 539,13 Nygren, A., 533, 243 O’Reilly, B., 535, 43; 537, 192; 540,25 Organtini, G., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Oakes, R.J., 539,67 540, 43, 185 Obayashi, Y., 539, 179 Orihara, H., 539,40 Obertino, M., 533, 237 Oriti, D., 532, 363 Oblakowska-Mucha, A., 533, 243 Orito, S., 533, 207; 539,13 Obraztsov, V., 533, 243 Oros-Peusquens, A.M., 538,33 Ocariz, J., 533, 196; 536, 229; 537,28 Ortega, R., 538,27 Ochs, A., 531,9;539, 197 Ortiz de Solórzano, A., 532,8 356 Cumulative author index to volumes 531–540 (2002) 326–371

Osetrov, S.B., 532,8;535,77 Pantea, D., 535, 43; 537, 192; 540,25 O’Shaughnessy, K., 539, 218 Panzer-Steindel, B., 533, 196; 536, 229; 537,28 O’Shea, V., 533, 223; 537,5 Paolucci, P., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Oshima, N., 531,52 540, 43, 185 Osipov, A.A., 539,76 Paoluzi, L., 535, 37; 536, 209; 537,21;538,21 Ostendorf, R., 538,27 Papadopoulos, I.M., 539, 188 Osterberg, K., 533, 243 Papadopoulou, Th.D., 533, 243 Ostrowicz, W., 538, 11; 540,33 Papageorgiou, K., 531,52 Otboev, A.V., 537, 201 Pape, L., 533, 243 Ouraou, A., 533, 243 Paramatti, R., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Ouyang, Q., 533, 223; 537,5 540, 43, 185 Owen, B.R., 535,85 Parashar, N., 531,52 Oxman, L.E., 531, 305 Parenti, A., 531,9;539, 197 Oyama, Y., 539, 179 Paris, A., 535, 43; 537, 192; 540,25 Oyanguren, A., 533, 243 Park, C.W., 538, 11; 540,33 Oz, Y., 537, 147 Park, D.K., 532, 305; 535,5 Ozaki, H., 538, 11; 540,33 Park, H., 535, 43; 537, 192; 538, 11; 540, 25, 33 Ozerov, D., 539,25 Park, I.H., 531,9;539, 197 Park, K.S., 538, 11; 540, 25, 33 Pac, M.Y., 531,9;539, 197 Park, S.K., 531,9 Pacheco, A., 533, 223; 537,5 Park, T.-S., 533,25 Pacheco, A.J., 534,45 Parke, S., 537, 249 Padhi, S., 531,9;539, 197 Parker, M.A., 533, 196; 536, 229; 537,28 Padley, P., 531,52 Parkes, C., 533, 243 Padron, I., 534,45 Parodi, F., 533, 243 Paganis, S., 531,9;539, 197 Parrini, G., 533, 223; 537,5 Paganoni, M., 533, 243 Pahl, C., 533, 207; 539,13 Parsons, H., 537,28 Paiano, S., 533, 243 Partridge, R., 531,52 Pak, N.K., 531, 119 Parua, N., 531,52 Pakhlov, P., 538, 11; 540,33 Parzefall, U., 533, 243 Pakhtusova, E.V., 537, 201 Pascaud, C., 539,25 Pakvasa, S., 535, 181 Pascolo, J.M., 533, 223; 537,5 Palacios, J.P., 533, 243 Pasqualucci, E., 535, 37; 536, 209; 537, 21; 538,21 Palese, M., 532, 129 Passalacqua, L., 533, 223; 535, 37; 536, 209; 537, 5, 21; 538,21 Palestini, S., 533, 196; 536, 229; 537,28 Passaleva, G., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Palka, H., 533, 243; 538, 11; 540,33 540, 43, 185 Palla, F., 533, 223; 537,5 Passeri, A., 533, 243; 535, 37; 536, 209; 537,21;538,21 Pálla, G., 538, 275 Passon, O., 533, 243 Palladino, V., 539, 188 Pastor, S., 534,8 Pallavicini, M., 533, 237 Pastrone, N., 533, 196, 237; 536, 229; 537,28 Pallin, D., 533, 223; 537,5 Pásztor, G., 533, 207; 539,13 Pallottino, G.V., 540, 179 Pate, S.F., 535,85 Palmonari, F., 531,9;539, 197 Patel, G.D., 539,25 Palomares, C., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Pater, J.R., 533, 207; 539,13 540, 43, 185 Patera, V., 535, 37; 536, 209; 537, 21; 538,21 Palomares-Ruiz, S., 531,90 Paterno, M., 531,52 Palutan, M., 535, 37; 536, 209; 537,21;538,21 Patkós, A., 537,77 Pan, L.J., 531,52 Patrascioiu, A., 532, 135 Pan, Y.B., 533, 223; 537,5 Patricelli, S., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Panagiotou, A.D., 538, 275 540, 43, 185 Panassik, V., 539,25 Patrick, G.N., 533, 207; 539,13 Pandoulas, D., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Patrignani, C., 533, 237 540, 43, 185 Patwa, A., 531,52 Paneque, D., 533, 223; 537,5 Paul, E., 531,9;539, 197 Panman, J., 539, 188 Paul, T., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; 540, 43, 185 Cumulative author index to volumes 531–540 (2002) 326–371 357

Pauluzzi, M., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Petersen, B., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; 540, 43, 185 540, 43, 185 Paus, C., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; 540, 43, 185 Petkov, P., 535,93 Pauss, F., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; 540, 43, 185 Petridis, A., 538, 275 Pavan, P., 535,93 Petrini, M., 536, 161 Pavel, N., 531,9;539, 197 Pétroff, P., 531,52 Pavlovski˘ı, O.V., 538, 202 Petrolini, A., 533, 243 Pavšic,ˇ M., 539, 133 Petrolo, E., 535, 37; 536, 209; 537,21;538,21 Pawlak, J.M., 531,9;539, 197 Petrucci, F., 533, 196; 536, 229; 537,28 Pawlik, B., 531,52 Petrucci, M.C., 531,9;539, 197 Pawlowski, J.M., 540, 153 Petrus, A., 540, 207 Payre, P., 533, 223; 537,5 Peyaud, B., 533, 196; 536, 229; 537,28 Peak, L.S., 540,33 Pham, T.N., 539,67 Pearce, P.A., 534, 216 Philipsen, O., 535, 138 Pearson, M.R., 533, 223; 537,5 Phillips, J., 535, 280 Pedace, M., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Phillips, J.P., 539,25 540, 43, 185 Pi, S.-Y., 534, 181 Pedrini, D., 535, 43; 537, 192; 540,25 Piacitelli, G., 533, 178 Pedroza, J.L., 535,93 Piai, M., 533,94 Peez, M., 539,25 Picca, D., 535,37 Pelfer, P.G., 531,9;539, 197 Piccini, M., 533, 196; 536, 229; 537,28 Pellegrino, A., 531,9;539, 197 Piccolo, D., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Pellmann, I., 533, 196; 536, 229; 537,28 540, 43, 185 Pellmann, I.-A., 539, 197 Pickering, A.G.M., 535, 377 Peloso, M., 534,8 Piedra, J., 533, 243 Peng, K.C., 539, 218 Piegaia, R., 531,52 Penin, A.A., 538, 335 Pierazzini, G., 533, 196; 536, 229; 537,28 Penionzhkevich, Yu.E., 537,45 Pierce, A., 538, 359 Pensotti, S., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Pierella, F., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; 540, 43, 185 540, 43, 185 Penttilä, S.I., 534,39 Pieri, L., 533, 243 Pepe, I.M., 535, 43; 537, 192; 540,25 Pierre, F., 533, 243 Pepe, M., 533, 196; 536, 229; 537,28 Pietrzyk, B., 533, 223; 537,5 Peralta, L., 533, 243 Pignanelli, M., 540, 199 Perepelitsa, V., 533, 243 Piilonen, L.E., 538, 11; 540,33 Perera, L.P., 539, 218 Pikna, M., 538, 275 Pérez, A., 531,90 Pilcher, J.E., 533, 207; 539,13 Perez, E., 539,25 Pimenta, M., 533, 243 Pérez, M.A., 531, 231 Pinfold, J., 533, 207; 539,13 Perez, P., 533, 223; 537,5 Pinsky, L., 538, 275 Perez, S., 531, 289 Pioppi, M., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Perl, K., 538, 275 540, 43, 185 Pernicka, M., 533, 196; 536, 229; 537,28 Piotto, E., 533, 243 Peroni, C., 531,9;539, 197 Pire, B., 535, 117; 540, 324 Perret, P., 533, 223; 537,5 Pirjol, D., 533,8;539,59 Perret-Gallix, D., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Piroué, P.A., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; 540, 43, 185 540, 43, 185 Perrotta, A., 533, 243 Pirozzi, G., 535, 37; 536, 209; 537, 21; 538,21 Perroud, J.-P., 540,33 Pisano, C., 540,75 Pesando, I., 536, 121 Pistolesi, E., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Pesci, A., 531,9;539, 197 540, 43, 185 Pesen, E., 539, 188 Pitzl, D., 539,25 Petcov, S.T., 531, 90; 533, 94; 534, 17; 538,77 Pizzella, G., 540, 179 Peters, A., 533, 196; 536, 229; 537,28 Płaczek, W., 533,75 Peters, M., 540,33 Plane, D.E., 533, 207; 539,13 Peters, O., 531,52 Pleiter, D., 532,63 358 Cumulative author index to volumes 531–540 (2002) 326–371

Plucinski, P., 531,9;539, 197 Privitera, P., 533, 243 Plunien, G., 534,52 Prokoﬁev, D., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Plyaskin, V., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; 540, 43, 185 540, 43, 185 Prokoﬁev, D.O., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Podiatchev, S., 535,85 540, 43, 185 Podobnik, T., 533, 243 Proskuryakov, A.S., 531,9;539, 197 Pogosov, V.S., 532,8 Prosper, H.B., 531,52 Pohl, M., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; 540, 43, 185 Protopopescu, S., 531,52 Poireau, V., 533, 243 Pruss, S.M., 537,41 Pojidaev, V., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Przybycien,´ M., 531,9;539, 13, 197 540, 43, 185 Przybycien,´ M.B., 531,9;539, 197 Pokorski, S., 535, 258; 538, 426 Przybycien, M.B., 531,52 Pokotilov, A., 535,1 Pucknell, V., 535,93 Pokrovskiy, N.S., 531,9;539, 197 Pühlhofer, F., 538, 275 Pol, M.E., 533, 243 Puimedón, J., 532,8 Polarski, D., 535,11 Pukhaeva, N., 533, 243 Poli, B., 533, 207; 539,13 Pullia, A., 533, 243 Polikarpov, M.I., 537, 291 Purohit, M.V., 539, 218 Polini, A., 531,9;539, 197 Pussieux, T., 539,8 Polok, G., 533, 243 Putz, J., 533, 223; 537,5 Polok, J., 533, 207; 539,13 Putzer, A., 533, 223; 537,5 Polonyi, J., 531, 316 Polyakov, D., 535, 321 Qi, B.J., 536, 203 Pomarol, A., 536, 277 Qian, J., 531,52 Pomeroy, V.R., 534,39 Qiao, C.-F., 536, 344 Ponomarev, V.Yu., 532, 179 Qiu, W.G., 538, 435 Ponta, T., 535,52 Quadt, A., 533, 207; 539,13 Pontecorvo, L., 535, 37; 536, 209; 537, 21; 538,21 Quartieri, J., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Pontoglio, C., 535, 43; 537, 192; 540,25 540, 43, 185 Pooth, O., 533, 207; 539,13 Quast, G., 533, 223; 537,5 Pope, B.G., 531,52 Quinn, B., 539, 218 Pope, C.N., 534, 172 Quinones, J., 535, 43; 537, 192; 540,25 Popkov, E., 531,52 Quintieri, L., 540, 179 Pordes, S., 533, 237 Porod, W., 538, 59, 137 Raach, H., 531,9;539, 197 Poropat, P., 533, 243 Rabbertz, K., 533, 207; 539,13 Porrati, M., 532, 48; 534, 209 Rädel, G., 539,25 Pöschl, R., 539,25 Radeztsky, S., 539, 218 Posocco, M., 531,9;539, 197 Radicioni, E., 539, 188 Pospelov, M., 534, 114; 538, 146 Radu, E., 536, 107 Potachnikova, I., 539,25 Rafatian, A., 539, 218 Potashov, S., 535,85 Ragusa, F., 533, 223; 537,5 Pothier, J., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; 540, 43, 185 Rahal-Callot, G., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Potrebenikov, Yu., 533, 196; 536, 229; 537,28 540, 43, 185 Potterveld, D.H., 535,85 Rahaman, M.A., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Potzel, W., 533, 191 540, 43, 185 Poulose, P., 534, 131 Rahimi, A., 535, 43; 537, 192; 540,25 Poutissou, J.-M., 537, 211 Raics, P., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; 540, 43, 185 Poutissou, R., 537, 211 Raithel, M., 535,85 Povh, B., 539,25 Raja, N., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; 540, 43, 185 Pozdniakov, V., 533, 243 Raja, R., 531,52 Prakash, M., 531, 167 Rajagopalan, S., 531,52 Prange, G., 533, 223; 537,5 Rajan, P., 533, 307 Prasuhn, D., 540, 207 Rajpoot, S., 532, 334; 535, 337; 540, 125 Prelz, F., 535, 43; 537, 192; 540,25 Rakers, S., 532, 179 Primavera, M., 535, 37; 536, 209; 537, 21; 538,21 Rakness, G., 535,85 Cumulative author index to volumes 531–540 (2002) 326–371 359

Rakow, P.E.L., 532,63 Renton, P., 533, 243 Ramallo, A.V., 533, 313 Reolon, A.R., 535,85 Ramberg, E., 531,52 Repond, J., 531,9;539, 197 Ramelli, R., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Rescigno, M., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; 540, 43, 185 540, 43, 185 Rames, J., 533, 243 Retyk, W., 538, 275 Ramírez, J., 532,1 Reucroft, S., 531, 28, 39, 52; 534, 28; 535, 59; 536, 24, 217; Ramirez, J.E., 535, 43; 537, 192; 540,25 540, 43, 185 Ramírez, M., 534,45 Reya, E., 540,75 Ramler, L., 533, 243 Reyes, M., 535, 43; 537, 192; 540,25 Rancoita, P.G., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Reyna, D., 539,25 540, 43, 185 Rhodes, C.S., 531, 135 Rander, J., 533, 223; 537,5 Ribeiro, R.F., 537, 321 Ranieri, A., 533, 223; 537,5 Riccardi, C., 535, 43; 537, 192; 540,25 Ranieri, R., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Ricciardi, S., 539, 188 540, 43, 185 Richard, F., 533, 243 Ranjard, F., 533, 223; 537,5 Richards, D.G., 532,63 Rapidis, P.A., 531,52 Richert, J., 531,71 Rappoport, V., 535,85 Richter, A., 532, 173, 179 Räsänen, S.S., 535, 109 Rick, H., 533, 207; 539,13 Raso, G., 533, 223; 537,5 Ridel, M., 531,52 Raspereza, A., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Ridky, J., 533, 243 540, 43, 185 Riemann, S., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Rathmann, F., 540, 207 540, 43, 185 Ratti, S.P., 535, 43; 537, 192; 540,25 Rigby, M., 531,9;539, 197 Ratz, M., 538,87 Righini, P., 539, 188 Rauschenberger, J., 539,25 Rigolin, S., 531, 263 Rautenberg, J., 531,9;539, 197 Rijssenbeek, M., 531,52 Raval, A., 539, 197 Riles, K., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; 540, 43, 185 Ravanini, F., 534, 216 Ring, C., 535,93 Ray, A., 531, 187 Ring, P., 532,29 Razis, P., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; 540, 43, 185 Ripp-Baudot, I., 533, 243 Read, A., 533, 243 Risler, C., 539,25 Reay, N.W., 531, 52; 539, 218 Rith, K., 535,85 Rebecchi, P., 533, 243 Rivera, C., 535, 43; 537, 192; 540,25 Redlinger, G., 537, 211 Rivers, R.J., 539,1 Redondo, I., 531,9;539, 197 Rizatdinova, F., 531,52 Redwine, R., 535,85 Rizvi, E., 539,25 Reeder, D.D., 531,9;539, 197 Roberts, R.G., 531, 216 Reeves, J.H., 532,8 Robertson, N.A., 533, 223; 537,5 Reggiani, D., 535,85 Robins, S., 531,9;539, 197 Rehn, J., 533, 243 Robinson, D., 535,85 Reid, D., 533, 243 Robmann, P., 539,25 Reid, J.G., 538, 275 Robutti, E., 533, 237 Reidy, J.J., 539, 218 Rocchi, A., 540, 179 Reimer, P., 535, 85; 539,25 Rockwell, T., 531,52 Reinhardt, R., 533, 243 Roco, M., 531,52 Reisert, B., 539,25 Rodrigues, E., 531,9;539, 197 Reitz, B., 532, 179 Rodriguez, D., 533, 243 Rembser, C., 533, 207; 539,13 Roe, B.P., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; 540, 43, 185 Ren, D., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; 540, 43, 185 Roeckl, E., 532,29 Renardy, J.-F., 533, 223 Roethel, W., 533, 237 Renfordt, R., 538, 275 Roland, C., 538, 275 Renk, B., 533, 223; 536, 229; 537,5 Roland, G., 538, 275 Renkel, P., 533, 207; 539,13 Rolandi, L., 533, 223; 537,5 Renner, R., 531,9;539, 197 Roldão, C.G., 540, 252 360 Cumulative author index to volumes 531–540 (2002) 326–371

Romanenko, N., 532, 202 Ruhlmann-Kleider, V., 533, 243 Romano, G., 539, 188 Ruiz, H., 533, 223; 537,5 Romero, A., 533, 243 Ruiz Morales, E., 536, 193 Romero, L., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Rulikowska-Zar¸ebska, E., 531,9;539, 197 540, 43, 185 Rumerio, P., 533, 237 Ronchese, P., 533, 243 Runge, K., 533, 207; 539,13 Rondeshagen, D., 539, 188 Rusack, R., 533, 237 Rondio, E., 533, 196; 536, 229; 537,28 Rusakov, S., 539,25 Roney, J.M., 533, 207; 539,13 Rusetsky, A., 533, 285 Ronga, F., 540, 179 Ruske, O., 531,9;539, 197 Roosen, R., 539,25 Ruspa, M., 531,9;539, 197 Root, N., 540,33 Russakovic, N., 535,37 Rosa, G., 539, 188 Rust, D.R., 533, 207; 539,13 Rosati, S., 533, 207; 539,13 Rutherfoord, J., 531,52 Rosca, A., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; 540, 43, 185 Rutherford, S.A., 533, 223; 537,5 Rosen, J., 533, 237 Ruuskanen, P.V., 535, 109 Rosenberg, E., 533, 243 Ryabtchikov, D., 533, 243 Rosenbleck, C., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Rybicki, A., 538, 275 540, 43, 185 Rybicki, K., 538, 11; 539, 25; 540,33 Rosier-Lees, S., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Ryckbosch, D., 535,85 540, 43, 185 Rykaczewski, H., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Rosowsky, A., 533, 223 540, 43, 185 Rossbach, D., 538,33 Rossi, A., 535, 207 Sa, B.-H., 537, 268 Rostomyan, A., 535,85 Sabetfakhri, A., 531,9;539, 197 Rostovtsev, A., 539,25 Sabirov, B.M., 531,52 Roth, M., 533,75 Sabra, W.A., 537, 383 Roth, S., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; 540, 43, 185 Sacchi, R., 531,9;539, 197 Rothberg, J., 533, 223; 537,5 Sacco, R., 533, 196; 536, 229; 537,28 Rothe, H.J., 539, 296 Sachs, K., 533, 207; 539,13 Roudeau, P., 533, 243 Sadovsky, A., 533, 243 Rougé, A., 533, 223; 537,5 Saeki, T., 533, 207; 539,13 Roux, B., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; 540, 43, 185 Sagawa, H., 538, 11; 539, 40; 540,33 Rovelli, T., 533, 243 Saha, S.K., 531, 187 Rovere, M., 535, 43; 537, 192; 540,25 Sahr, O., 533, 207; 539,13 Roy, D.P., 535, 243; 540,14 Sailer, K., 531, 316 Roy, P., 535, 181 Saito, T., 533,1 Roy, S., 531, 281 Saitoh, S., 538, 11; 540,33 Royon, C., 531,52 Saitta, B., 539, 188 Rozanov, A., 539, 188 Saji, C., 539, 179 Rozanska, M., 538,11 Sajot, G., 531,52 Rozen, Y., 533, 207; 539,13 Sakai, H., 533,1 Ruan, X.C., 536, 203 Sakai, K., 534,39 Rubin, H.A., 539, 218 Sakai, Y., 538, 11; 540,33 Rubin, P., 533, 196 Sakamoto, N., 533,1 Rubinov, P., 531,52 Sakemi, Y., 535,85 Rubinstein, R., 537,41 Sakharov, A., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Rubio, J.A., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; 540, 43, 185 540, 43, 185 Sakoda, S., 533,1 Ruchti, R., 531,52 Sakuda, M., 539, 179 Rudolph, G., 533, 223; 537,5 Sakurai, N., 539, 179 Rudy, Z., 540, 207 Sala, S., 535, 43; 537, 192; 540,25 Ruelle, P., 539, 172 Salehi, H., 531,9;539, 197 Ruggieri, F., 533, 223; 535, 37; 536, 209; 537, 5, 21; 538,21 Salgado, C.A., 532, 222 Ruggiero, G., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Salicio, J., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; 540, 43, 185 540, 43, 185 Salmi, L., 533, 243 Cumulative author index to volumes 531–540 (2002) 326–371 361

Salnikov, A.A., 537, 201 Savrié, M., 533, 196; 536, 229; 537,28 Salt, J., 533, 243 Savvidy, G.K., 533, 138 Sammer, T., 538, 275 Sawyer, L., 531,52 Sampson, J., 535,93 Saxon, D.H., 531,9;539, 197 Sanchez, E., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Scarlett, C., 535,85 540, 43, 185 Scarpa, M., 533, 196; 537,28 Sánchez-Hernández, A., 535, 43; 537, 192; 540,25 Schaefer, S., 535, 358 Sander, H.-G., 533, 223; 537,5 Schael, S., 533, 223; 537,5 Sanders, D.A., 539, 218 Schäfer, A., 535,85 Sanders, M.P., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Schäfer, C., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; 540, 43, 185 540, 43, 185 Sanderson, A.C., 535,52 Schagen, S., 531,9;539, 197 Sandoval, A., 538, 275 Schaile, A.D., 533, 207; 539,13 Sandulescu, N., 535, 103 Schaile, O., 533, 207; 539,13 Sanguinetti, G., 533, 223; 537,5 Schaller, L.A., 535,52 Sanjiev, I., 535,85 Schamberger, R.D., 531, 52; 535, 37; 536, 209; 537,21;538,21 Sankey, D.P.C., 539,25 Scharff-Hansen, P., 533, 207; 539,13 Sann, H., 538, 275 Schätzel, S., 539,25 Santacesaria, R., 539, 188 Schegelsky, V., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Santangelo, P., 535, 37; 536, 209; 537, 21; 538,21 540, 43, 185 Santha, A.K.S., 539, 218 Scheid, W., 533, 265 Santoro, A., 531,52 Scheins, J., 539,25 Santoro, A.F.S., 539, 218 Schellman, H., 531,52 Santos, D., 538, 257 Schieck, J., 539,13 Santos, L.F., 537,62 Schierholz, G., 532,63 Santoso, Y., 539, 107 Schiff, D., 539,46 Santovetti, E., 535, 37; 536, 209; 537,21;538,21 Schill, C., 535,85 Santroni, A., 533, 237 Schilling, F.-P., 539,25 Saperstein, E.E., 533,17 Schinzel, D., 533, 196; 537,28 Saracino, G., 535, 37; 536, 209; 537,21;538,21 Schioppa, M., 531,9;539, 197 Sarangi, S., 536, 185 Schlatter, D., 533, 223; 537,5 Saremi, S., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; 540, 43, 185 Schleichert, R., 540, 207 Sarkar, S., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; 540, 43, 185 Schlenstedt, S., 531,9;539, 197 Sarkar, T., 534, 167 Schleper, P., 539,25 Sarkar, U., 532,71 Schmeling, S., 533, 223; 537,5 Sarkisyan, E.K.G., 533, 207; 539,13 Schmidke, W.B., 531,9;539, 197 Sarsa, M.L., 532,8 Schmidt, D., 539, 25, 25 Sartorelli, G., 531,9;539, 197 Schmidt, F., 535,85 Sarycheva, L.I., 537, 261 Schmidt, K., 532,29 Sasaki, C., 537, 280 Schmidt, S., 539,25 Sasaki, T., 537, 211 Schmidt, S.A., 533, 196; 536, 229; 537,28 Satapathy, M., 538, 11; 540,33 Schmidt-Kaerst, S., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Sato, F., 535,85 540, 43, 185 Sato, O., 539, 188 Schmidt-Ott, W.-D., 537,45 Sato, T., 537, 211 Schmitt, S., 539,25 Sato, Y., 539, 188 Schmitz, D., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Satou, Y., 533,1 540, 43, 185 Satpathy, A., 538,11 Schmitz, N., 538, 275 Satta, A., 539, 188 Schnakenburg, I., 540, 137 Satuła, W., 531,61 Schneekloth, U., 531,9;539, 197 Saull, P.R.B., 531,9;539, 197 Schneider, H., 540, 207 Savage, M.J., 535, 177 Schneider, M., 539,25 Savin, A.A., 531,9;539, 197 Schneider, O., 538, 11; 540,33 Savin, I., 535,85 Schnell, G., 535,85 Savklı,¸ Ç., 531, 161 Schnurbusch, H., 531,9;539, 197 Savoy-Navarro, A., 533, 243 Schoeffel, L., 539,25 362 Cumulative author index to volumes 531–540 (2002) 326–371

Schoerner-Sadenius, T., 539,13 Semenov, S., 538, 11; 540,33 Scholberg, K., 539, 179 Sen, A.A., 532, 159 Schönharting, V., 533, 196; 536, 229; 537,28 Sen, N., 531,52 Schöning, A., 539,25 Senyo, K., 538, 11; 540,33 Schopper, H., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Seo, S.H., 533, 237 540, 43, 185 Serednyakov, S.I., 537, 201 Schörner, T., 539,25 Serin, L., 533, 223; 537,5 Schotanus, D.J., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Serin-Zeyrek, M., 539, 188 540, 43, 185 Servoli, L., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Schrenk, S., 538, 11; 540,33 540, 43, 185 Schrieder, G., 532, 179 Seth, K.K., 533, 237 Schröder, M., 533, 207; 539,13 Sethi, S., 532, 159 Schröder, V., 539,25 Settles, R., 533, 223; 537,5 Schroer, B., 538, 415 Seuster, R., 533, 207; 538, 11; 539, 13; 540,33 Schuck, P., 531,61 Sever, R., 539, 188 Schué, Y., 533, 196; 536, 229; 537,28 Sevior, M.E., 538, 11; 540,33 Schüler, K.P., 535,85 Seyboth, P., 538, 275 Schuller, F.P., 540, 119 Sfetsos, K., 536, 294 Schultz, J., 533, 237 Sﬁligoi, I., 535, 37; 536, 209; 537,21;538,21 Schultz-Coulon, H.-C., 539,25 Sguazzoni, G., 533, 223; 537,5 Schumacher, M., 533, 207; 539,13 Shabalina, E., 531,52 Schutz, Y., 538,27 Shaﬁ, Q., 531, 112 Schwanda, C., 533, 243; 540,33 Shamanov, V., 539, 188 Schwanenberger, C., 539,25 Sharapov, A.A., 531, 255 Schwartz, A.J., 539, 218 Sharapov, E.I., 534,39 Schwartzman, A., 531,52 Sharkey, E., 539, 179 Schweda, K., 532, 179 S h ar y, V. V. , 537, 201 Schwering, B., 533, 243 Shatunov, Yu.M., 537, 201 Schwering, G., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Shcheglova, L.M., 531,9;539, 197 540, 43, 185 Sheaff, M., 535, 43; 537, 192; 539, 218; 540,25 Schwick, C., 533, 207; 539,13 Shears, T.G., 533, 207; 539,13 Schwickerath, U., 533, 243 Sheikh, J.A., 533, 253 Schwind, A., 535,85 Sheikh-Jabbari, M.M., 535, 328; 538, 180 Sciabà, A., 533, 223; 537,5 Shekelyan, V., 539,25 Sciacca, C., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Sheldon, P.D., 535, 43; 537, 192; 540,25 540, 43, 185 Shen, B.C., 533, 207; 539,13 Sciascia, B., 535, 37; 536, 209; 537, 21; 538,21 Shen, Y.-G., 537, 187 Sciubba, A., 535, 37; 536, 209; 537, 21; 538,21 Shepherd-Themistocleous, C.H., 533, 207; 539,13 Sciulli, F., 531,9;539, 197 Sherstnev, A.V., 534,97 Scott, J., 531,9;539, 197 Sherwood, P., 533, 207; 539,13 Scott, W.G., 533, 207; 535, 163, 229; 539,13 Shevchenko, S., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Scuri, F., 535, 37; 536, 209; 537, 21; 538,21 540, 43, 185 Seager, P., 533, 223 Sheviakov, I., 539,25 Sedgbeer, J.K., 533, 223; 537,5 Shibata, T., 539, 179 Sedlák, K., 539,25 Shibata, T.-A., 535,85 Sefkow, F., 539,25 Shibuya, H., 538, 11; 539, 188; 540,33 Segar, A., 533, 243 Shimizu, Y., 531, 245; 535, 271 Segoni, I., 535, 43; 537, 192; 540,25 Shimoyama, N., 538,96 Seibert, J., 535,85 Shinkawa, T., 537, 211 Seidl, G., 537,95 Shiozawa, M., 539, 179 Seiler, E., 532, 135 Shirai, J., 539, 179 Seitz, B., 535,85 Shiromizu, T., 535, 315 Sekiguchi, K., 533,1 Shiu, G., 536,1 Sekulin, R., 533, 243 Shivarov, N., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Selonke, F., 531,9;539, 197 540, 43, 185 Selvaggi, G., 533, 223; 537,5 Shivpuri, R.K., 531,52 Cumulative author index to volumes 531–540 (2002) 326–371 363

Shoemaker, F.C., 537, 211 Smirnova, N.A., 537,45 Shoutko, V., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Smirnova, O., 533, 243 540, 43, 185 Smith, A.J.S., 537, 211 Shpakov, D., 531,52 Smith, A.M., 533, 207; 539,13 Shtarkov, L.N., 539,25 Smith, D.A., 534,39 Shukla, S., 537,41 Smith, E., 531,52 Shumilov, E., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Smith, R.P., 531,52 540, 43, 185 Smith, W.H., 531,9;539, 197 Shupe, M., 531,52 Smizanska, M., 533, 223; 537,5 Shutov, V., 535,85 Smolnikov, A.A., 532,8;535,77 Shvorob, A., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Smy, M.B., 539, 179 540, 43, 185 Snellman, H., 532, 259 Shwartz, B., 538, 11; 540,33 Snigirev, A.M., 537, 261 Si, Z.G., 539, 235 Snihur, R., 531,52 Sidorov, V., 538, 11; 540,33 Snow, G.R., 531,52 Sidorov, V.A., 537, 201 Snow, J., 531,52 Sidwell, R.A., 531, 52; 539, 218 Snyder, S., 531,52 Siebel, M., 533, 243 So, H., 535, 363 Siedenburg, T., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Soares, M., 531,9;539, 197 540, 43, 185 Sobel, H.W., 539, 179 Sieler, U., 533, 223; 537,5 Sobie, R., 533, 207; 539,13 Siklér, F., 538, 275 Sobolev, Yu., 537,45 Silagadze, Z.K., 537, 201 Soff, G., 534,52 Silvestris, L., 533, 223; 537,5 Sokoloff, M.D., 539, 218 Simak, V., 531,52 Sokolov, A., 533, 243 Simani, M.C., 535,85 Solano, A., 531,9;539, 197 Simon, A., 535,85 Solano Salinas, C.J., 539, 218 Simone, S., 539, 188 Söldner-Rembold, S., 533, 207; 539,13 Simopoulou, E., 533, 223; 537,5 Solodukhin, S.N., 533, 153 Singh, H., 531,52 Solomin, A.N., 531,9 Singh, J.B., 538, 11; 540,33 Solomon, J., 531,52 Sinram, K., 535,85 Soloviev, Y., 539,25 Siopsis, G., 536, 315 Soluk, R., 537, 211 Sirghi, D.L., 535,52 Son, D., 531, 9, 28, 39; 534, 28; 535, 59; 536, 24, 217; 539, 197; Sirghi, F., 535,52 540, 43, 185 Sirois, Y., 539,25 Son, D.T., 539,46 Siroli, G., 533, 207; 539,13 Song, J.S., 539, 188 Sirotenko, V., 531,52 Song, Y., 531,52 Sisakian, A., 533, 243 Soni, A., 533,37 Sistemich, K., 540, 207 Sopczak, A., 533, 243 Sitar, B., 538, 275 Sorella, S.P., 531, 305 Skalozub, V., 533, 182 Sorín, V., 531,52 Skillicorn, I.O., 531,9;539, 197 Sorrentino, S., 539, 188 Skrinsky, A.N., 537, 201 Sosebee, M., 531,52 Skripkin, A.G., 537, 201 Sosnovtsev, V., 531,9;539, 197 Skrzypczak, E., 538, 275 Sosnowski, R., 533, 243 Skrzypek, M., 533,75 Sotnikova, N., 531,52 Skuja, A., 533, 207; 539,13 Souga, C., 540, 43, 185 Slattery, P., 531,52 Soustruznik, K., 531,52 Slaughter, A.J., 539, 218 South, D., 539,25 Sloan, T., 539,25 Souza, M., 531,52 Słominski,´ W., 531,9;539, 197 Sozzi, M., 533, 196; 536, 229; 537,28 Smadja, G., 533, 243 Spada, F.R., 539, 188 Smalska, B., 531,9;539, 197 Spadaro, T., 535, 37; 536, 209; 537, 21; 538,21 Smirnov, A.D., 531, 237 Spagnolo, S., 533, 207; 539,13 Smirnov, P., 539,25 Spano, F., 533, 207; 539,13 364 Cumulative author index to volumes 531–540 (2002) 326–371

Spanos, V.C., 538, 353 Stopa, P., 531,9;539, 197 Spaskov, V., 539,25 Stösslein, U., 535,85 Spassov, T., 533, 243 Stoyanov, B., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Specka, A., 539,25 540, 43, 185 Spillantini, P., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Stoyanova, D.A., 531,52 540, 43, 185 Straessner, A., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Spiriti, E., 535, 37; 536, 209; 537, 21; 538,21 540, 43, 185 Spitzer, H., 539,25 Strand, R.C., 537, 211 Squier, G.T.A., 538, 275 Strang, M.A., 531,52 St-Laurent, M., 531,9;539, 197 Straub, P.B., 531,9;539, 197 Stachyra, A.L., 539, 179 Straumann, U., 539,25 Stahl, A., 533, 207; 539,13 Strauss, J., 533, 243 Staiano, A., 531,9;539, 197 Strauss, M., 531,52 Stairs, D.G., 531,9;539, 197 Striet, J., 540, 319 Stamen, R., 539,25 Strikman, M., 537, 51; 540, 220 Stancari, G., 533, 237 Ströbele, H., 538, 275 Stancari, M., 533, 237 Ströher, H., 540, 207 Stanco, L., 531,9;539, 197 Ströhmer, R., 533, 207; 539,13 Standage, J., 531,9;539, 197 Strolin, P., 539, 188 Stanev, T., 538, 251 Strom, D., 533, 207; 539,13 Stanic,ˇ S., 538, 11; 540,33 Strong, J.A., 533, 223; 537,5 Stanitzki, M., 533, 243 Strovink, M., 531,52 Stanoiu, M., 537,45 Strumia, A., 539,91 Stanton, N.R., 531, 52; 539, 218 Studenikin, A., 535, 187 Staric,ˇ M., 540,33 Stugu, B., 533, 243 Starinets, A., 532,48 Stutte, L., 531,52 Stefanski, R.J., 539, 218 Styczen, J., 540, 199 Steffens, E., 535,85 Suchkov, S., 531,9;539, 197 Steijger, J.J.M., 535,85 Suda, K., 533,1 Stein, H.J., 540, 207 Sudhakar, K., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Steinbrück, G., 531,52 540, 43, 185 Steinhauser, M., 538, 335 Suetsugu, K., 535,85 Stella, B., 539,25 Sugi, A., 538,11 Stenson, K., 535, 43; 537, 192; 539, 218; 540,25 Sugimoto, S., 537, 211 Stenzel, H., 533, 223; 537,5 Sugiyama, A., 538, 11; 540,33 Stephens, K., 533, 207; 539,13 Sugiyama, H., 532, 275 Stephens, R.W., 531,52 Sugiyama, K., 538, 173 Steuer, M., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Sulak, L.R., 539, 179 540, 43, 185 Sullivan, G.W., 539, 179 Stewart, E.D., 538, 213 Sultanov, G., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Stewart, I.W., 539,59 540, 43, 185 Stewart, J., 535,85 Sumino, Y., 535, 145 Stickland, D.P., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Sumisawa, K., 538, 11; 540,33 540, 43, 185 Sumiyoshi, T., 538, 11; 540,33 Stiewe, J., 539,25 Summers, D.J., 539, 218 Stifutkin, A., 531,9;539, 197 Sun, L.Z., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; 540, 43, 185 Stirling, W.J., 531, 216 Sun, Y., 533, 253 Stocchi, A., 533, 243 Surrow, B., 531,9;539, 197 Stock, R., 538, 275 Susa, T., 538, 275 Stöcker, H., 531, 195 Sushkov, S., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Stoker, D., 531,52 540, 43, 185 Stolin, V., 531,52 Susinno, G., 531,9;539, 197 Stone, A., 531,52 Suszycki, L., 531,9;539, 197 Stone, J.L., 539, 179 Sutcliffe, P.M., 540, 146 Stone, J.R., 537, 211 Suter, H., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; 540, 43, 185 Stonjek, S., 531,9;539, 197 Sutton, M.R., 531,9;539, 197 Cumulative author index to volumes 531–540 (2002) 326–371 365

Suzuki, A., 539, 179 Tang, H.Q., 536, 203 Suzuki, A.T., 539, 179 Tang, X.W., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Suzuki, H., 538, 197; 539,40 540, 43, 185 Suzuki, K., 538,11 Tanihata, I., 532, 209 Suzuki, S., 538, 11; 540,33 Tanimoto, M., 538,96 Suzuki, S.Y., 538,11 Tapper, A.D., 531,9;539, 197 Suzuki, T., 537, 291 Tapper, R.J., 531,9;539, 197 Suzuki, Y., 539, 179 Tarem, S., 533, 207; 539,13 Svoboda, R., 539, 179 Tarjan, P., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; 540, 43, 185 Swain, J.D., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Taroian, S., 535,85 540, 43, 185 Tasaka, S., 539, 179 Swain, S.K., 540,33 Tasevsky, M., 533, 207; 539,13 Swart, M., 539,25 Taševský, M., 539,25 Swynghedauw, M., 533, 223; 537,5 Tassi, E., 531,9;539, 197 Szabo, R.J., 533, 168 Tatishvili, G., 533, 196; 536, 229; 537,28 Szczekowski, M., 533, 243 Taureg, H., 533, 196; 536, 229; 537,28 Szentpétery, I., 538, 275 Taurok, A., 533, 196; 536, 229; 537,28 Szép, Zs., 537,77 Tauscher, L., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Szépfalusy, P., 537,77 540, 43, 185 Szeptycka, M., 533, 243 Tavares-Velasco, G., 531, 231 Sziklai, J., 538, 275 Taylor, G., 533, 223; 537,5 Szillasi, Z., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Taylor, G.N., 538, 11; 540,33 540, 43, 185 Taylor, L., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; 540, 43, 185 Szleper, M., 533, 196; 536, 229; 537,28 Taylor, R.J., 533, 207; 539,13 Sznajder, A., 531,52 Taylor, W., 531,52 Sztuk, J., 531,9;539, 197 Tchetchelnitski, S., 539,25 Szuba, D., 531,9;539, 197 Tchrakian, D.H., 540, 146 Szuba, J., 531,9;539, 197 Teaney, D., 539,53 Szumlak, T., 533, 243 Tegenfeldt, F., 533, 243 Szymanowski, L., 535, 117; 540, 324 Teixeira-Dias, P., 533, 223; 537,5 Tejessy, W., 533, 223; 537,5 Tabarelli, T., 533, 243 Tellili, B., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; 540, 43, 185 Taffard, A.C., 533, 243 Tempesta, P., 533, 223; 537,5 Tai, A., 537, 268 Tenchini, R., 533, 223; 537,5 Takach, S., 539, 218 Tentindo-Repond, S., 531,52 Takahashi, T., 533, 294; 538, 11; 539, 303; 540,33 Terakawa, A., 539,40 Takahisa, K., 531, 190 Teramoto, Y., 538, 11; 540,33 Takaishi, T., 540, 159 Terenzi, R., 540, 179 Takamura, A., 537,86 Terkulov, A., 535,85 Takasaki, F., 538, 11; 540,33 Terning, J., 535,33 Takasu, Y., 536,18 Terranova, F., 533, 243 Takemori, D., 539, 179 Terrón, J., 531,9;539, 197 Takeuchi, H., 539, 179 Teryaev, O.V., 535, 117; 540, 324 Takeuchi, Y., 539, 179 Tessarin, S., 535,85 Takita, M., 539, 179 Testoni, J.E., 534,45 Talby, M., 531,52 Teubert, F., 533, 223; 537,5 Tamai, K., 540,33 Teughels, S., 537, 45; 538,33 Tamanyan, A.G., 532,8 Teuscher, R., 533, 207; 539,13 Tamaryan, S., 532, 305; 535,5 Teyssier, D., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Tamii, A., 533,1 540, 43, 185 Tamura, N., 538, 11; 540,33 Tezuka, I., 539, 188 Tanaka, K., 536, 344 Thomas, A.W., 531,77 Tanaka, M., 538, 11, 121; 540,33 Thomas, E., 535,85 Tanaka, R., 533, 223; 537,5 Thompson, A.S., 533, 223; 537,5 Tandler, J., 531,9;539, 197 Thompson, G., 539,25 Tang, C.H., 536, 203 Thompson, I.J., 533, 265 366 Cumulative author index to volumes 531–540 (2002) 326–371

Thompson, J.C., 533, 223; 537,5 Treille, D., 533, 243 Thompson, L.F., 533, 223; 537,5 Tretyak, V.I., 535,77 Thompson, P.D., 539,25 Tricoli, A., 533, 207; 539,13 Thomson, M.A., 533, 207; 539,13 Tricomi, A., 533, 223; 537,5 Thorne, K., 539, 218 Trigger, I., 533, 207; 539,13 Thorne, R.S., 531, 216 Tripathi, A.K., 539, 218 Tiecke, H., 531,9;539, 197 Tripathi, S.M., 531,52 Tilquin, A., 533, 223; 537,5 Trippe, T.G., 531,52 Timmermans, C., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Tristram, G., 533, 243 540, 43, 185 Trochimczuk, M., 533, 243 Timmermans, J., 533, 243 Trocmé, B., 533, 223; 537,5 Ting, S.C.C., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Trócsányi, Z., 533, 207; 539,13 540, 43, 185 Troitskaya, N.I., 535, 371 Ting, S.M., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Troncon, C., 533, 243 540, 43, 185 Truöl, P., 539,25 Tinti, N., 533, 243 Tsenov, R., 539, 188 Tioukov, V., 539, 188 Tsipolitis, G., 539,25 Tipton, B., 535,85 Tsuboyama, T., 538, 11; 540,33 Tittel, K., 533, 223; 537,5 Tsujikawa, S., 536,9 Tjon, J., 531, 161 Tsukamoto, T., 538, 11; 540,33 Tkatchev, A., 537,28 Tsukerman, I., 539, 188 Tkatchev, L., 533, 243 Tsur, E., 533, 207; 539,13 Tobien, N., 539,25 Tsurin, I., 539,25 Tobin, M., 533, 243 Tsurugai, T., 531,9;539, 197 Todorovova, S., 533, 243 Tuchming, B., 533, 223; 537,5 Tokuda, S., 538, 11; 540,33 Tully, C., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; 540, 43, 185 Tokushuku, K., 531,9;539, 197 Tung, K.L., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Tolun, P., 539, 188 540, 43, 185 Tomalin, I.R., 533, 223; 537,5 Tuning, N., 531,9;539, 197 Tomaradze, A., 533, 237, 243 Tupper, G.B., 535,17 Tomasz, F., 539,25 Turaev, D., 532, 185 Tome, B., 533, 243 Turc,ˇ D., 539, 179 Tomoto, M., 538,11 Turcato, M., 531,9;539, 197 Tomura, T., 538, 11; 540,33 Turcot, A.S., 531,52 Tonazzo, A., 533, 243 Turkot, F., 537,41 Tong, G.L., 535, 37; 536, 209; 537,21;538,21 Turlay, R., 533, 196; 536, 229; 537,28 Tonwar, S.C., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Turnau, J., 539,25 540, 43, 185 Turner-Watson, M.F., 533, 207; 539,13 Toppan, F., 539, 266 Turney, J.E., 539,25 Torokoff, K., 534, 195 Tuts, P.M., 531,52 Torrence, E., 533, 207; 539,13 Tyapkin, I.A., 533, 243 Tortora, L., 535, 37; 536, 209; 537, 21; 538,21 Tyapkin, P., 533, 243 Tortosa, P., 533, 243 Tye, S.-H.H., 536, 185 Toshito, T., 539, 179, 188 Tymieniecka, T., 531,9;539, 197 Tóth, J., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; 540, 43, 185 Tytgat, M., 535,85 Totsuka, Y., 539, 179 Tzamarias, S., 533, 243 Tovey, S.N., 538, 11; 540,33 Tzamariudaki, E., 539,25 Townsend, P.K., 538, 366 Tze, C.-H., 536, 305 Toya, D., 533, 207; 539,13 Trabelsi, A., 533, 223 Uchigashima, N., 533,1 Trabelsi, K., 538,11 Udluft, S., 539,25 Trainor, T.A., 538, 275 Ueda, I., 533, 207; 539,13 Tran, P., 533, 207; 539,13 Uehara, S., 538, 11; 540,33 Travnicek, P., 533, 243 Uehara, Y., 537, 256 Traynor, D., 539,25 Ueno, K., 540,33 Trefzger, T., 533, 207; 539,13 Uesaka, T., 533,1 Cumulative author index to volumes 531–540 (2002) 326–371 367

Uiterwijk, J.W.E., 539, 188 Van Gulik, R., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Ujvári, B., 533, 207; 539,13 540, 43, 185 Ukita, N., 535, 363 Vaniev, V., 531,52 Ukleja, A., 531,9;539, 197 Van Kooten, R., 531,52 Ukleja, J., 531,9;539, 197 Vankov, H., 538, 251 Ulbricht, J., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Van Leeuwen, M., 538, 275 540, 43, 185 Van Lysebetten, A., 533, 243 Ullaland, O., 533, 243 Van Mechelen, P., 539,25 Umemori, K., 531,9;539, 197 Vannerem, P., 533, 207; 539,13 Unal, G., 533, 196; 536, 229; 537,28 Van Remortel, N., 533, 243 Uno, S., 538, 11; 540,33 Van Vulpen, I., 533, 243 Urban, M., 539,25 Varelas, N., 531,52 Urbanowski, K., 540,89 Varga, D., 538, 275 Urciuoli, G.M., 535,85 Varner, G., 538, 11; 540,33 Ureña-López, L.A., 538, 246 Varvell, K.E., 538, 11; 540,33 Uribe, C., 535, 43; 537, 192; 540,25 Vary, J.P., 536, 250 Ushida, N., 539, 188 Vasenko, A.A., 532,8 Usik, A., 539,25 Vasiliev, S.I., 532,8;535,77 Usov, Yu.V., 537, 201 Vasiljev, A.V., 537, 201 Uvarov, V., 533, 243 Vasquez, R., 540, 43, 185 Uwer, P., 539, 235 Vassilevich, D.V., 532, 373 Vassiliev, S., 539,25 Vaandering, E.W., 535, 43; 537, 192; 540,25 Vassiliou, M., 538, 275 Vachon, B., 533, 207; 539,13 Vayaki, A., 533, 223; 537,5 Vagenas, E.C., 533, 302 Vazdik, Y., 539,25 Vagins, M.R., 539, 179 Vázquez, F., 535, 43; 537, 192; 540,25 Vahsen, S.E., 538, 11; 540,33 Vázquez, M., 531,9;539, 197 Valassi, A., 533, 223; 537,5 Vázquez-Poritz, J.F., 534, 155 Valente, E., 531, 28, 39; 534, 28; 535, 37, 59; 536, 24, 209, 217; Vegni, G., 533, 243 537, 21; 538, 21; 540, 43, 185 Veillet, J.-J., 533, 223; 537,5 Valente, P., 535, 37; 536, 209; 537, 21; 538,21 Velasco, M., 533, 196; 536, 229; 537,28 Valenti, G., 533, 243 Veloso, F., 533, 243 Valeriani, B., 535, 37; 536, 209; 537,21;538,21 Velthuis, J.J., 531,9;539, 197 Valkár, S., 539,25 Veltri, M., 533, 196; 536, 229; 537,28 Valkárová, A., 539,25 Venanzoni, G., 535, 37; 536, 209; 537, 21; 538,21 Vallage, B., 533, 196, 223; 536, 229; 537,5,28 Veneziano, S., 535, 37; 536, 209; 537,21;538,21 Vallée, C., 539,25 Ventura, A., 535, 37; 536, 209; 537,21;538,21 Vance, S.E., 531, 209 Venturi, A., 533, 223; 537,5 Vancea, I.V., 536, 114 Venugopalan, R., 539,53 Van Dalen, J.A., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Venus, W., 533, 243 540, 43, 185 Verbeure, F., 533, 243 Van Dam, P., 533, 243 Verdier, P., 533, 243 Van Dantzig, R., 539, 188 Verdini, P.G., 533, 223; 537,5 Van de Bruck, C., 531, 135 Veres, G.I., 538, 275 Van den Berg, A.M., 532, 179 Vertogradov, L.S., 531,52 Van den Brand, J.F.J., 535,85 Verzi, V., 533, 243 Van der Aa, O., 533, 223; 537,5 Verzocchi, M., 533, 207; 539,13 Van der Bij, J.J., 536, 107 Veselov, A.I., 537, 291 Van der Steenhoven, G., 535,85 Veszpremi, V., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Van de Vyver, B., 539, 188 540, 43, 185 Van de Vyver, R., 535,85 Vesztergombi, G., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Van de Walle, R.T., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; 538, 275; 540, 43, 185 540, 43, 185 Vetlitsky, I., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Van Dierendonck, D., 531,39 540, 43, 185 Van Eldik, J., 533, 243 Vetterli, M.C., 535,85 Van Giai, N., 535, 103 Vichnevski, A., 539,25 368 Cumulative author index to volumes 531–540 (2002) 326–371

Vicinanza, D., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Wachsmuth, H., 533, 223; 537,5 540, 43, 185 Wacker, K., 539,25 Vicini, A., 531,83 Wackeroth, D., 533,75 Videau, H., 533, 223; 537,5 Wadhwa, M., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Videau, I., 533, 223; 537,5 540, 43, 185 Vidnovic, T., 533, 237 Wagner, J., 539,25 Viertel, G., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Wagner, P., 531,71 540, 43, 185 Wahl, H., 533, 196; 536, 229; 537,28 Vikhrov, V., 535,85 Wahl, H.D., 531,52 Vilain, P., 539, 188 Wahlen, H., 533, 243 Vilanova, D., 533, 243 Wakasa, T., 533,1 Vilja, I., 535, 170 Wakui, T., 533,1 Villa, S., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; 540, 43, 185 Walczak, R., 531,9;539, 197 Villar, J.A., 532,8 Walker, A., 533, 196; 536, 229; 537,28 Villeneuve-Seguier, F., 531,52 Walker, R., 531,9;539, 197 Vincter, M.G., 535,85 Waller, D., 533, 207; 539,13 Wallny, R., 539,25 Viollier, R.D., 535,17 Wallraff, W., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Viren, B., 539, 179 540, 43, 185 Visco, M., 540, 179 Walter, C.W., 539, 179 Visschers, J.L., 539, 188 Wambach, J., 532, 179 Visser, J., 535,85 Wan, S.-L., 536, 241 Vitale, L., 533, 243 Wang, B., 538, 435 Vitev, I., 538, 282 Wang, C.C., 540,33 Vitulo, P., 535, 43; 537, 192; 540,25 Wang, C.H., 538, 11; 540,33 Vivargent, M., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Wang, H., 531,52 540, 43, 185 Wang, H.-W., 538,39 Vlachos, S., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Wang, J.G., 538, 11; 540,33 540, 43, 185 Wang, L.-T., 531, 263; 536, 263 Vlasov, N.N., 531,9;539, 197 Wang, M.-Z., 538, 11; 540,33 Vodopianov, I., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Wang, M.Z., 536, 203 540, 43, 185 Wang, Q., 532, 240; 537, 233 Vogel, H., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; 540, 43, 185 Wang, S.-F., 538,39 Vogt, C., 532,99 Wang, T., 533, 223; 537,5 Vogt, H., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; 540, 43, 185 Wang, T.T., 536, 263 Volkov, A.A., 531,52 Wang, X.-N., 540,62 Vollmer, C.F., 533, 207; 539,13 Wang, X.L., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Vo lm er, J . , 535,85 540, 43, 185 Von Brentano, P., 534,63 Wang, Z.-G., 536, 241 Von Feilitzsch, F., 533, 191 Wang, Z.-M., 531,52 Von Krogh, J., 533, 207; 539,13 Wang, Z.M., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Von Neumann-Cosel, P., 532, 173, 179 540, 43, 185 Von Wimmersperg-Toeller, J.H., 533, 223; 537,5 Wanke, R., 533, 196; 536, 229; 537,28 Vorobiev, A.P., 531,52 Warchol, J., 531,52 Vorobiev, I., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Ward, B.F.L., 533,75 540, 43, 185 Ward, C.P., 533, 207; 539,13 Vorobiev, M., 539,25 Ward, D.R., 533, 207; 539,13 Vorobyov, A.A., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Ward, J.J., 533, 223; 537,5 540, 43, 185 W¸as, Z., 533,75 Voss, H., 533, 207; 539,13 Washbrook, A.J., 533, 243 Voss, K.C., 531,9;539, 197 Wasserbaech, S., 533, 223; 537,5 Vossebeld, J., 533, 207; 539,13 Wasserman, I., 536,1 Vossnack, O., 537,28 Watanabe, S., 539, 218 Vranic,´ D., 538, 275 Watanabe, Y., 538, 11; 539, 179; 540,33 Vrba, V., 533, 243 Watari, T., 532, 252; 534,93 Vyvey, K., 537, 45; 538,33 Watkins, P.M., 533, 207; 539,13 Cumulative author index to volumes 531–540 (2002) 326–371 369

Watson, A.T., 533, 207; 539,13 Wijngaarden, D.A., 531,52 Watson, N.K., 533, 207; 539,13 Wilbert, J., 535,85 Watts, G., 531,52 Wild, R., 535, 193 Watzlawik, K.-H., 540, 207 Wilkens, H., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Waugh, B., 539,25 540, 43, 185 Wayne, M., 531,52 Wilkes, R.J., 539, 179 Weber, A., 531,9;539, 197 Wilkin, C., 540, 207 Weber, G., 539,25 Wilkinson, G., 533, 243 Weber, M., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; 539, 25; Wilksen, T., 539,25 540, 43, 185 Willis, S., 531,52 Webster, M., 535, 43; 537, 192; 540,25 Wilquet, G., 539, 188 Weerts, H., 531,52 Wilschut, H.W., 538,27 Wegener, D., 539,25 Wilson, G.W., 533, 207; 539,13 Wei, Z.-Y., 538,39 Wilson, J.A., 533, 207; 539,13 Weidenmüller, H.A., 534,63 Wilson, J.R., 535, 43; 537, 192; 540,25 Weiner, N., 534, 124 Wimpenny, S.J., 531,52 Weiser, C., 533, 243 Winde, M., 539,25 Weiskopf, C., 535,85 Wing, M., 531,9;539, 197 Weiss-Babai, R., 539, 218 Wingerter-Seez, I., 533, 196; 536, 229; 537,28 Wells, P.S., 533, 207; 539,13 Winhart, A., 533, 196; 536, 229; 537,28 Wendland, J., 535,85 Winter, G.-G., 539,25 Wengler, T., 533, 207; 539,13 Winter, K., 539, 188 Wenig, S., 538, 275 Winter, M., 533, 243 Werkema, S., 533, 237 Winter, W., 532, 259 Wermes, N., 533, 207; 539,13 Winterroth, E., 532, 129 Werner, C., 539,25 Wise, M.B., 539, 242 Werner, M., 539,25 Wise, T., 535,85 Werner, N., 539,25 Wislicki, W., 533, 196; 536, 229; 537,28 Wessels, M., 539,25 Wiss, J., 535, 43; 537, 192; 540,25 Wessoleck, H., 531,9;539, 197 Wissing, Ch., 539,25 Wesson, P.S., 538, 159 Witchey, N., 539, 218 West, B.J., 531,9;539, 197 Witek, M., 533, 243 West, P., 540, 137 Wittgen, M., 533, 196; 536, 229; 537,28 Wetterling, D., 533, 207; 539,13 Witzig, C., 537, 211 Wetzler, A., 538, 275 Wobisch, M., 539,25 Whisnant, K., 532, 15; 537, 179 Woehrling, E.-E., 539,25 White, A., 531,52 Wolf, G., 531,9;533, 223; 537,5;539, 13, 197 White, G., 539,25 Wolff, T., 539, 188 White, J.T., 531,52 Wölﬂe, S., 531,9 White, R., 533, 223; 537,5 Wolin, E., 539, 218 White, T.O., 533, 196; 536, 229; 537,28 Womersley, J., 531,52 Whiteson, D., 531,52 Won, E., 538, 11; 540,33 Whitmore, J.J., 531,9;539, 197 Wong, H.T., 536, 203 Whitten, C., 538, 275 Wood, B.P., 532, 19; 537, 179 Wichmann, R., 531,9;539, 197 Wood, D.R., 531,52 Wick, K., 531,9;539, 197 Woods, P.J., 532,29 Wicke, D., 533, 243 Wörtche, H.J., 532, 179 Wickens, J., 533, 243 Wotton, S.A., 533, 196; 536, 229; 537,28 Widhalm, L., 533, 196; 536, 229; 537,28 Wotzasek, C., 537, 321 Wiedenmann, W., 533, 223; 537,5 Wright, S.V., 531,77 Wieland, O., 540, 199 Wronka, S., 533, 196; 536, 229; 537,28 Wienemann, P., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Wu, H.-Y., 538,39 540, 43, 185 Wu, J., 533, 223; 537,5 Wiener, J., 539, 218 Wu, S.C., 536, 203 Wiesand, S., 539,25 Wu, S.L., 533, 223; 537,5 Wiggers, L., 531,9;539, 197 Wu, X., 533, 223; 537,5 370 Cumulative author index to volumes 531–540 (2002) 326–371

Wu, Y., 537,21 Yang, M.Z., 536,34 Wünsch, E., 539,25 Yang, S.M., 539, 218 Wunsch, M., 533, 223; 537,5 Yashima, J., 540,33 Wyatt, A.C., 539,25 Yasuda, T., 531,52 Wyatt, T.R., 533, 207; 539,13 Yatsunenko, Y.A., 531,52 Wynhoff, S., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Yeh, S.C., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; 540, 43, 185 540, 43, 185 Yen, S., 535,85 Wyss, R., 535,93 Yi, D., 539, 218 Yip, K., 531,52 Xia, L., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; 540, 43, 185 Yokomakura, H., 537,86 Xiang, L., 540,9 Yokoyama, M., 538,11 Xiao, Z.-G., 538,39 Yoneyama, S., 535,85 Xiao, Z.G., 536,59 Yoo, I.K., 538, 275 Xie, Y., 533, 223; 537,5 Yoo, J., 539, 179 Xin, B., 536, 203 Yoon, C.S., 539, 188 Xing, Y.-Z., 540, 213 Yoshida, K., 538,11 Xing, Z.-Z., 533, 85; 539,85 Yoshida, M., 539, 179 Xiong, W., 535, 43; 537, 192; 540,25 Yoshida, R., 531,9;539, 197 Xu, Q., 531,52 Yoshida, S., 531, 190; 539, 218 Xu, R., 533, 223; 537,5 Yoshimura, Y., 537, 211 Xu, Y., 535, 37; 536, 209; 537, 21; 538,21 Youm, D., 531, 276 Xu, Z.Z., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; 540, 43, 185 Youngman, C., 531,9;539, 197 Xue, S., 533, 223; 537,5 Youssef, S., 531,52 Yu, J., 531,52 Yabsley, B.D., 538, 11; 540,33 Yu, Y., 535, 37; 536, 209; 537, 21; 538,21 Yager, P.M., 535, 43; 537, 192; 540,25 Yuan, C., 533, 223; 537,5 Yako, K., 533,1 Yuan, F., 540,62 Yamada, R., 531,52 Yuan, V.W., 534,39 Yamada, S., 531,9;539, 179, 197 Yuan, Y., 538, 11; 540,33 Yamada, Y., 537, 130; 538, 11; 540,33 Yue, C., 536,67 Yamaguchi, A., 538, 11; 540,33 Yue, Q., 536, 203 Yamaguchi, S., 538, 173 Yusa, Y., 538, 11; 540,33 Yamamoto, A., 536,18 Yushchenko, O., 533, 243 Yamamoto, J., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; 540, 43, 185 Zabawa, R., 538,52 Yamamoto, M., 537, 165 Žácek,ˇ J., 539,25 Yamashita, S., 533, 207; 539,13 Zacek, V., 533, 207; 539,13 Yamashita, T., 531,9;539, 197 Zachariadou, K., 533, 223; 537,5 Yamashita, Y., 538, 11; 540,33 Zaffaroni, A., 536, 161 Yamauchi, M., 538, 11; 540,33 Zakharov, V.I., 537, 291 Yamazaki, T., 535,70 Zakrzewski, J.A., 531,9;539, 197 Yamazaki, Y., 531,9;539, 197 Zálešák, J., 539,25 Yamin, P., 531,52 Zalewska, A., 533, 243 Yanagida, T., 532, 252; 534, 93; 538, 107 Zalewski, P., 533, 243 Yanagisawa, C., 539, 179 Zalite, An., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Yanai, H., 540,33 540, 43, 185 Yang, B.Z., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Zalite, Yu., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; 540, 43, 185 540, 43, 185 Yang, C.B., 534,69 Zaliznyak, R., 539, 218 Yang, C.G., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Zallo, A., 535, 43; 537, 192; 540,25 540, 43, 185 Zanabria, M., 531,52 Yang, H., 532, 240 Zanon, D., 536, 101 Yang, H.J., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Zaranek, J., 538, 275 540, 43, 185 Zarnecki,˙ A.F., 531,9;539, 197 Yang, K.-C., 533, 271 Zavrtanik, D., 533, 243 Yang, M., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; 540, 43, 185 Zawiejski, L., 531,9;539, 197 Cumulative author index to volumes 531–540 (2002) 326–371 371

Zdesenko, Yu.G., 535,77 Zieblinski, M., 540, 199 Zeitnitz, C., 533, 223; 537,5 Ziegler, A., 531,9;539, 197 Zema, P.F., 535,37 Ziegler, Ar., 531,9;539, 197 Zemba, G.R., 537, 141 Ziegler, T., 533, 223; 537,5 Zer-Zion, D., 533, 207; 539,13 Zielinski, M., 531,52 Zeuner, W., 531,9;539, 197 Zieminska, D., 531,52 ZEUS Collaboration, 531,9;539, 197 Zieminski, A., 531,52 Zeyrek, M.T., 539, 188 Zilizi, G., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217 Zhalov, M., 537, 51; 540, 220 Zimányi, J., 538, 275 Zhang, B.-G., 538,39 Zimin, N.I., 533, 243 Zhang, C., 539, 218 Zimmermann, B., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Zhang, D.-X., 531,67 540, 43, 185 Zhang, J., 533, 223; 537,5;538, 11; 540,33 Zhang, J.-Z., 532, 215; 539, 162 Zinchenko, A., 533, 196; 536, 229; 537,28 Zhang, L., 533, 223; 537,5 Zintchenko, A., 533, 243 Zhang, X., 531,52 Ziolkowski, M., 533, 196 Zhang, Y., 535, 43; 536, 67; 537, 192; 540,25 Ziolkowski , M., 536, 229 Zhang, Z., 539,25 Ziolkowski, M., 537,28 Zhang, Z.P., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; 538, 11; Zioutas, K., 531, 175 540, 33, 43, 185 Zito, G., 533, 223; 537,5 Zhao, J., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; 540, 43, 185 Zivkovic, L., 539,13 Zhao, L., 532, 345 Zmeskal, J., 535,52 Zhao, W., 533, 223; 537,5 Zobernig, G., 533, 223; 537,5 Zhautykov, B.O., 531,9;539, 197 Zohrabian, H., 535,85 Zheng, H., 531,52 Zöller, M., 535, 59; 536, 24, 217; 540, 43, 185 Zheng, H.Q., 536,59 Zöller, M.Z., 531, 28, 39; 534,28 Zhilich, V., 538, 11; 540,33 Zhokin, A., 539,25 Zoller, Ph., 533, 243 Zhou, B., 531,52 Zomer, F., 539,25 Zhou, D.-M., 537, 268 Žontar, D., 538, 11; 540,33 Zhou, S.-G., 532, 209 Zotkin, S.A., 531,9;539, 197 Zhou, Z., 531,52 Zucchelli, P., 532, 166; 539, 188 Zhou, Z.Y., 536, 203 Zucchiatti, A., 540, 199 Zhu, G.Y., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; 540, 43, 185 Zuo, W., 540, 213 Zhu, H.-D., 538,39 Zupan, M., 533, 243 Zhu, R.Y., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; 540, 43, 185 Zur Nedden, M., 539,25 Zhu, S., 537, 351 Zutshi, V., 531,52 Zhu, Y.-T., 538,39 Zverev, E.G., 531,52 Zhu, Z.M., 538,11 Zverev, M.V., 533,17 Zhuang, H.L., 531, 28, 39; 534, 28; 535, 59; 536, 24, 217; Zweber, P., 533, 237 540, 43, 185 Zichichi, A., 531, 9, 28, 39; 534, 28; 535, 59; 536, 24, 217; Zylberstejn, A., 531,52 539, 197; 540, 43, 185 Zylicz,˙ J., 532,29