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0370-2693/2005 Published by Elsevier B.V. doi:10.1016/S0370-2693(05)00566-6 vi Instructions to authors

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L. ALVAREZ-GAUMÉ J.-P. BLAIZOT M. CVETICˇ GENEVA VILLAZZANO (TRENTO) PHILADELPHIA, PA

M. DOSER D.F. GEESAMAN H. GEORGI GENEVA ARGONNE, IL CAMBRIDGE, MA

G.F. GIUDICE N. GLOVER W. HAXTON GENEVA DURHAM SEATTLE, WA

V. METAG L. ROLANDI W.-D. SCHLATTER GIESSEN GENEVA GENEVA

H. WEERTS T. YANAGIDA EAST LANSING, MI TOKYO

VOLUME 615, 2005

Amsterdam – Boston – Jena – London – New York – Oxford Paris – Philadelphia – San Diego – St. Louis Physics Letters B 615 (2005) 1–13 www.elsevier.com/locate/physletb

Nutty dyons

Yves Brihaye a, Eugen Radu b

a Physique-Mathématique, Universite de Mons-Hainaut, Mons, Belgium b Department of Mathematical Physics, National University of Ireland Maynooth, Maynooth, Ireland Received 25 February 2005; received in revised form 6 April 2005; accepted 7 April 2005 Available online 15 April 2005 Editor: N. Glover

Abstract We argue that the Einstein–Yang–Mills–Higgs theory presents nontrivial solutions with a NUT charge. These solutions approach asymptotically the Taub–NUT spacetime and generalize the known dyon black hole configurations. The main properties of the solutions and the differences with respect to the asymptotically flat case are discussed. We find that a non-Abelian magnetic monopole placed in the field of gravitational dyon necessarily acquires an electric field, while the magnetic charge may take arbitrary values.  2005 Elsevier B.V. All rights reserved.

1. Introduction

A feature of certain gauge theories is that they admit classical solutions which are interpreted as representing magnetic monopoles. For non-Abelian gauge fields interacting with a Higgs scalar, there exist even regular configurations with a finite mass, as proven by the famous ’t Hooft–Polyakov solution [1]. Typically, the magnetic monopoles admit also electrically charged generalizations—so-called dyons, the Julia–Zee solution [2] of the SU(2)-Higgs theory possibly being the best known case. These solutions admits also gravitating generalizations, both regular and black hole solutions being considered in the literature (see [3] for a general review of this top- ics). In SU(2)-Einstein–Yang–Mills–Higgs (EYMH) theory, a branch of globally regular gravitating dyons emerges smoothly from the corresponding flat space solutions. The non-Abelian black hole solutions emerge from the globally regular configurations, when a finite regular event horizon radius is imposed [4,5]. These solutions cease to exist beyond some maximal value of the coupling constant α (which is proportional to the ratio of the vector meson mass and Planck mass).

E-mail address: [email protected] (Y. Brihaye).

It has been speculated that such configurations might have played an important role in the early stages of the evolution of the Universe. Also, various analyses indicate that the monopole and dyon solutions are important in quantum theories. Since general relativity shares many similarities with gauge theories, one may ask whether Einstein’s equations present solutions that would be the gravitational analogous of the magnetic monopoles and dyons. The first example of such a solution was found in 1963 by Newman, Unti and Tamburino (NUT) [6,7]. This metric has become renowned for being “a counterexample to almost anything” [8] and represents a generalization of the Schwarzschild vacuum solution [9] (see [10] for a simple derivation of this metric and historical review). It is usually interpreted as describing a gravitational dyon with both ordinary and magnetic mass.1 The NUT charge which plays a dual role to ordinary mass, in the same way that electric and magnetic charges are dual within Maxwell theory [11].By continuing the NUT solution through its horizon one arrives in the Taub universe [7], which may be interpreted as a homogeneous, nonisotropic cosmology with the spatial topology S3. As discussed by many authors (see, e.g., [13,14]), the presence of magnetic-type mass (the NUT parameter n) introduces a “Dirac-string singularity” in the metric (but no curvature singularity). This can be removed by appropriate identifications and changes in the topology of the spacetime manifold, which imply a periodic time coordinate. Moreover, the metric is not asymptotically flat in the usual sense although it does obey the required fall-off conditions. A large number of papers have been written investigating the properties of the gravitational analogs of magnetic monopoles [15,16], the vacuum Taub–NUT solution being generalized in different directions. The corresponding configuration in the Einstein–Maxwell theory has been found in 1964 by Brill [17]. This Abelian solution has been generalized for the matter content of the low-energy string theory, a number of NUT-charged configurations being exhibited in the literature (see, e.g., [18] for a recent example and a large set of references). A discussion of the non-Abelian counterparts of the Brill solution is presented in [19]. These configurations generalize the well-known SU(2)-Einstein–Yang–Mills hairy black hole solutions [20], presenting, as a new feature, a nontrivial electric potential. However, the “no global non-Abelian charges” results found for asymptotically flat EYM static configurations [21] are still valid in this case, too. Here we present arguments for the existence of NUT-charged generalizations of the known EYMH black hole solutions [4,5]. Apart from the interesting question of finding the properties of a Yang–Mills–Higgs dyon in the field of a gravitational dyon, there are a number of other reasons to consider this type of solutions. In some supersymmetric theories, closure under duality forces us to consider NUT-charged solutions. Furthermore, dual mass solutions play an important role in Euclidean quantum gravity [22] and therefore cannot be discarded in spite of their causal pathologies. Also, by considering this type of asymptotics, one may hope to attain more general features of gravitating non-Abelian dyons. The Letter is structured as follows: in the next section we present the general framework and analyse the field equations and boundary conditions. In Section 3 we present our numerical results. We conclude with Section 4, where our results are summarized.

2. General framework and equations of motion

2.1. Action principle

The action for a gravitating non-Abelian SU(2) gauge ﬁeld coupled to a triplet Higgs ﬁeld with vanishing Higgs self-coupling is

1 Note that the Taub–NUT spacetime plays also an important role outside general relativity. For example, the asymptotic motion of monopoles in (super-)Yang–Mills theories corresponds to the geodesic motion in a Euclideanized Taub-NUT background [12].However,these developments are outside the interest of this work. Y. Brihaye, E. Radu / Physics Letters B 615 (2005) 1–13 3 √ R 1 1 S = −gd4x − Tr F F µν − Tr D ΦDµΦ , (1) 16πG 2 µν 4 µ with Newton’s constant G. The ﬁeld strength tensor is given by Fµν = ∂µAν − ∂νAµ − ie[Aµ,Aν], with Dµ = ∂µ − ie[Aµ, ] being the covariant derivative and e the Yang–Mills coupling constant. µν Varying the action (1) with respect to g , Aµ and Φ we have the ﬁeld equations

1 1 √ 1 1 √ R − g R = 8πGT , √ D −gF µν = ie Φ,DνΦ , √ D −gDµΦ = 0, µν 2 µν µν −g µ 4 −g µ (2) where the stress-energy tensor is 1 1 1 T = 2Tr F F gαβ − g F F αβ + Tr D ΦD Φ − g D ΦDαΦ . (3) µν µα νβ 4 µν αβ 2 µ ν 4 µν α

2.2. Metric ansatz and symmetries

We consider NUT-charged spacetimes whose metric can be written locally in the form dr2 θ 2 ds2 = + P 2(r) dθ2 + sin2 θdϕ2 − N(r)σ2(r) dt + 4n sin2 dϕ , (4) N(r) 2 the NUT parameter n being defined as usually in terms of the coefficient appearing in the differential dt + 4n sin2(θ/2)dϕ.Hereθ and ϕ are the standard angles parametrizing an S2 with ranges 0 θ π,0 ϕ 2π. Apart from the Killing vector K0 = ∂t , this line element possesses three more Killing vectors characterizing the NUT symmetries θ K = sin ϕ∂ + cos ϕ cot θ∂ + 2n cos ϕ tan ∂ , 1 θ ϕ 2 t θ K = cos ϕ∂ − sin ϕ cot θ∂ − 2n sin ϕ tan ∂ , 2 θ ϕ 2 t K3 = ∂ϕ − 2n∂t . (5) These Killing vectors form a subgroup with the same structure constants that are obeyed by spherically symmetric solutions [Ka,Kb]= abcKc. The n sin2(θ/2) term in the metric means that a small loop around the z-axis does not shrink to zero at θ = π. This singularity can be regarded as the analogue of a Dirac string in electrodynamics and is not related to the usual degeneracies of spherical coordinates on the two-sphere. This problem was first encountered in the vacuum NUT metric. One way to deal with this singularity has been proposed by Misner [8]. His argument holds also independently of the precise functional form of N and σ . In this construction, one considers one coordinate patch in which the string runs off to infinity along the north axis. A new coordinate system can then be found with the string running off to infinity along the south axis with t = t + 4nϕ, the string becoming an artifact resulting from a poor choice of coordinates. It is clear that the t coordinate is also periodic with period 8πnand essentially becomes an Euler angle coordinate on S3. Thus an observer with (r,θ,ϕ)= const follows a closed timelike curve. These lines cannot be removed by going to a covering space and there are no reasonable spacelike surface. One finds also that surfaces of constant radius have the topology of a three-sphere, in which there is a Hopf fibration of the S1 of time over the spatial S2 [8]. Therefore for n different from zero, the metric structure (4) generically shares the same troubles exhibited by the vacuum Taub–NUT gravitational field [23], and the solutions cannot be interpreted properly as black holes. 4 Y. Brihaye, E. Radu / Physics Letters B 615 (2005) 1–13

2.3. Matter ﬁelds ansatz

While the Higgs ﬁeld is given by the usual form

Φ = φτ3, (6) the computation of the appropriate SU(2) connection compatible with the Killing symmetries (5) isamorein- volved task. This can be done by applying the standard rule for calculating the gauge potentials for any spacetime group [24,25]. According to Forgacs and Manton, a gauge ﬁeld admit a spacetime symmetry if the spacetime trans- L = L formation of the potential can be compensated by a gauge transformation [24] Ki Aµ DµWi , where stands for the Lie derivative. Taking into account the symmetries of the line element (4) we ﬁnd the general form 1 2 θ A = dt + 4n sin dϕ u(r)τ3 + ν(r)τ3 dr + ω(r)τ1 +˜ω(r)τ2 dθ 2e 2 + cos θτ3 + ω(r)τ2 −˜ω(r)τ1 sin θ dϕ . (7)

This gauge connection remains invariant under a residual U(1) gauge symmetry which can be used to set ν = 0. Also, because the variables ω and ω˜ appear completely symmetrically in the EYMH system, the two amplitudes must be proportional and we can always set ω˜ = 0 (after a suitable gauge transformation). Thus, similar to the n = 0 case, the gauge potential is described by two functions ω(r) and u(r) which we shall refer to as magnetic and electric potential, respectively.

2.4. Field equations and known solutions

Within the above ansatz, the classical equations of motion can be derived from the following reduced action 2 3 1 n σ N S = dr dt σ 1 − NP 2 − PP N + 2P (σ NP ) + 8πG P 2 2 2 2 2 2 2 1 σ(ω − 1 + 2nu) P u ω u 1 − Nσω 2 + − − + σNP2φ 2 + σω2φ2 , (8) e2 2P 2 2σ 2 σN 2 where the prime denotes the derivative with respect to the radial√ coordinate r. At this point, we fix the metric gauge by choosing P(r)= r2 + n2, which allows a straightforward analysis of the relation with the Abelian configurations. Dimensionless quantities are obtained by considering the rescalings r → r/(ηe), φ → φη, n → n/(ηe), u → ηeu (where η is the asymptotic√ magnitude of the Higgs field). As a result, the field equations depend only on the coupling constant α = 4πGη. The EYMH equations reduce to the following system of five nonlinear differential equations 2 n N rN = − N + σ 2 − 1 2 3 1 P 2 2 2 2 1 P u ω u 1 − 2α2 Nω 2 + ω2 − 1 + 2nu 2 + + + NP2φ 2 + ω2φ2 , 2P 2 2σ 2 σ 2N 2 2 2 2 2 2 n σ(1 − σ ) α σ 2ω u σ = + P 2φ 2 + 2ω 2 + , rP2 r σ 2N 2 2 2 (ω − 1 + 2nu) u (Nσ ω ) = σω + φ2 − , P 2 σ 2N Y. Brihaye, E. Radu / Physics Letters B 615 (2005) 1–13 5 NσP2φ = 2σω2φ, P 2u 2ω2u 2nσ = − ω2 − 1 + 2nu . (9) σ σN P 2 Two explicit solution of the above equations are well known. The vacuum Taub–NUT one corresponds to 2(Mr + n2) ω(r) =±1,u(r)= 0,σ(r)= 1,φ(r)= 1,N(r)= 1 − . (10) r2 + n2 The U(1) Brill solution [17] has the form nQ − Q r ω(r) = 0,u(r)= u + m e ,σ(r)= 1,φ(r)= 1, 0 r2 + n2 2(Mr + n2) α2(Q2 + Q2 ) N(r)= 1 − + e m , (11) r2 + n2 4(r2 + n2) and describes a gravitating Abelian dyon with a mass M, electric charge Qe and magnetic charge Qm ≡ 1 − 2u0n, u0 being an arbitrary constant, corresponding to the asymptotic value of the electric potential. It can be stressed that the Brill solution possesses two, one or zero horizons, according to the values of the free parameters Qe, M, u(∞). In the same way as in the case of Reissner–Nordström solutions, the extremal Brill solution can be defined as the solutions with a degenerate horizon at r = r0. This gives the following conditions, fixing M and r0 α2 r = M, M2 + n2 − Q2 + Q2 = 0. (12) 0 4 e m As we will see later, it is convenient to further specify the arbitrary constant u(∞) in such a way the u(r0) = 0, this implying 1 − Q nQ − MQ m + m e = 0, (13) 2n M2 + n2 which fixes Qm and leaves Qe as the only remaining free parameter. In the following we will refer to this solution as to the extremal Brill solution. As far as we could see, it is not possible to express M and Qm in a closed form depending on (α,n,Qe), but the solution can be constructed numerically.

2.5. Boundary conditions

We want the metric (4) to describe a nonsingular, asymptotically NUT spacetime outside an horizon located at r = rh.HereN(rh) = 0 is only a coordinate singularity where all curvature invariants are finite. A nonsingular extension across this null surface can be found just as at the event horizon of a black hole. If the time is chosen to be periodic, as discussed above, this surface would not be a global event horizon, although it would still be an apparent horizon. The regularity assumption implies that all curvature invariants at r = rh are finite. The corresponding expansion as r → rh is 2 2 N(r)= N1(r − rh) + O(r − rh) ,σ(r)= σh + σ1(r − rh) + O(r − rh) , 2 2 3 ω(r) = ωh + ω1(r − rh) + O(r − rh) ,u(r)= u1(r − rh) + u2(r − rh) + O(r − rh) , 2 φ(r)= φh + φ1(r − rh) + O(r − rh) , (14) 2 = 2 + 2 where Ph rh n and 1 (ω2 − 1)2 1 u2P 2 N = − α2 h + 1 h + ω φ2 , 1 1 2 2 2 h h rh 2Ph 2 σh 6 Y. Brihaye, E. Radu / Physics Letters B 615 (2005) 1–13 2 2 2 2 2 n σh(1 − σ ) α σ 2ω u σ = h + h P 2φ2 + ω2 + 1 1 , 1 2 h 1 2 1 2 2 rhP rh σ N h h 1 2 2 2 2 2 ω ω − 1 σ u nσ (ω − 1) u1ω u r 2ω φh ω = h h + φ2 ,u= 1 1 − h h + h − 1 h ,φ= h , 1 2 h 2 4 2 2 1 2 (15) N1 Ph 2σh Ph N1Ph Ph N1Ph σh, u1, ωh, φh being arbitrary parameters. The analysis of the field equations as r →∞gives the following expression in terms of the constants c, u0, Qe, ˜ φ1, M 2M 2n2 − α2(φ˜2 + (1 − 2nu2)2 + Q2) M(2n2 + α2φ˜2) N(r)∼ 1 − − 1 0 e + 1 +···, r r2 r3

2 ˜2 2 ˜2 α φ 4α φ M − 1−u2r σ ∼ 1 − 1 − 1 +···,ω(r)∼ ce 0 +···, 2r2 3r3 ˜ ˜ 2 2 ˜2 φ φ M Q n(1 − 2nu ) Qe(6n + α φ ) φ ∼ 1 − 1 + 1 +···,u(r)∼ u − e + 0 − 1 +···. (16) r r2 0 r r2 6r3 Note that similar to the n = 0 asymptotically flat case, the magnitude of the electric potential at infinity cannot 2 exceed that of the Higgs field, |u0| < 1. The constant M appearing in the asymptotic expansion of the metric function N(r) can be interpreted as the total mass of solutions (this can be proven rigorously by applying the general formalism proposed in [28]). Note that M and n are unrelated on a classical level. Also, no purely monopole solution can exist for a nonvanishing NUT charge (i.e., one cannot consistently set u = 0 unless ω =±1, in which case the vacuum Taub–NUT solution is recovered). Thus, a non-Abelian magnetic monopole placed in the field of gravitational dyon necessarily acquires an electric field. We close this section by remarking that the definition of the non-Abelian charges is less clear for n = 0. Although we may still define a ’t Hooft field strength tensor, in the absence of a nontrivial two-sphere at infinity on which to integrate, the only reasonable definition the non-Abelian magnetic and electric charges is in terms of the asymptotic (3) 2 behavior of the gauge field. By analogy to the asymptotically flat case, Qe and Qm are defined from Ftr Qe/r (3) = − and Fθφ Qm sin θ (a similar problem occurs for an U(1) field [27]). Thus, since Qm 1 2nu0, the usual quantization relation for the magnetic charge is lost for n = 0, which is a consequence of the pathological large scale structure of a NUT-charged spacetime.

3. Numerical results

Although an analytic or approximate solution appears to be intractable, we present in this section numerical arguments that the known EYMH black hole solutions can be extended to include a NUT parameter. The equations of motion (9) have been solved for a large set of the parameters (α, n, Qe,rh), looking for solutions interpolating between the asymptotics (14) and (16). NUT-charged solutions are found for any n = 0EYMH dyonic black hole configuration by slowly increasing the parameter n (since the transformation n →−n leaves the field equations unchanged except for the sign of the electric potential, we consider here only positive values of n). As expected, these configurations have many features in common with the n = 0 solutions discussed in [4]; they also present new features that we will pointed out in the discussion. Typical profiles for the metric functions N(r) and σ(r) and for the electric potential u(r) are presented in Fig. 1, for a dyonic black hole solution as well as for two NUT-charged solutions. The gauge function ω(r) and the Higgs scalar φ(r) interpolates monotonically between some constant values on the event horizon and zero respectively one at infinity, without presenting any local extremum (see Fig. 3).

2 This depends on the asymptotic structure of the spacetime. For example, in an anti-de Sitter spacetime, u0 may take arbitrary values [26]. Y. Brihaye, E. Radu / Physics Letters B 615 (2005) 1–13 7

Fig. 1. The functions N(r), σ(r) and u(r) are plotted for three typical solutions for the same values of (rh,Qe,α).

The domain of existence of the non-Abelian nutty dyons can be determined in the space of parameters. If we ﬁx the electric charge Qe of the solution, then there likely exist a volume VQ in the parameter space of (α,n,rh) inside which non-Abelian solutions exist and on the side of which they become singular and/or bifurcate into Abelian solution of the type of the Brill solution. For n = 0 the domain of the (α, rh) plane where non-Abelian solutions exist was determined in [5] for Qe = 0 and in [4] for Qe = 0. The determination of VQ is of course a huge task. In this Letter, we will not attempt to determine the shape of VQ accurately but rather attempt to sketch it by analyzing the pattern of solutions on some generic lines in the space of parameters. For deﬁniteness we set Qe = 0.2 in our numerical analysis, although nontrivial solutions have been found also for other values of the electric charge.

3.1. n varying

First, we have integrated the system of equations (9) with ﬁxed values for α, rh and Qe and increased the NUT charge n. Our values here are α = 1.0, rh = 0.2 and Qe = 0.2 corresponding to a generic values for the parameters (the corresponding n = 0 gravitating dyon was constructed in [4]). As far as the function u(r) is concerned, there exists a main difference between the case n = 0 and n = 0. Indeed in the case n = 0 this function behave 4 asymptotically like u(r) ∼ u0 + Qe/r + O(1/r ) while in the presence of a NUT charge the behaviour is instead 2 u(r) ∼ u0 + Qe/r + K/r , where, as seen from (16), the constant K increases with n. Thus, when n becomes large, it becomes more difﬁcult to construct numerical solutions with a good enough accuracy,3 for a given value of the electric charge Qe. The effect of increasing n apparently depend strongly of the value α.Forα small (typically α 1) the pattern can be summarized by the following points: (i) No local extrema of N(r) are found for small enough values of n. For larger n, the function N(r) develops a local maximum and also a local minimum, say NM and Nm at some intermediate, n-depending values of r.Forn large enough, we have Nmax > 1. No local minimum of N persist for large enough n, the minimum of N(r) (Nm = 1) being attained as r →∞. (ii) The second metric function σ(r) still remains monotonically increasing but the value σ(rh) diminishes when n increases. (iii) The asymptotic

3 To integrate the equations, we used the differential equation solver COLSYS which involves a Newton–Raphson method [29]. 8 Y. Brihaye, E. Radu / Physics Letters B 615 (2005) 1–13

Fig. 2. The values of the parameters M, σ(rh), Nm, NM , φ(rh), ω(rh) and u(∞) = u0 are shown as a function of n for solutions with rh = 0.2, Qe = 0.2 and two different values of the coupling constant α. value u(∞) also decreases for increasing n. With the values choosen, we have u(∞) ≈ 0.189 for n = 0; we ﬁnd. u(∞) = 0forn ≈ 0.25 and negative values for larger n. These effects are illustrated on Fig. 2a. On this ﬁgure we have set 0

Fig. 3. The metric functions N(r), σ(r)and the matter functions ω(r), φ(r) and u(r) are shown as a function of r for ﬁxed values of (rh,Qe,n) and three different values of α. The functions N(r) and u(r) of the corresponding extremal Brill solution (with ω(r) = 0,h(r)= 1) are also exhibited.

Nevertheless, it seems that there are two possible patterns for n →∞: for values of α smaller than a critical value αˆ , solutions with large values of n seem to occur, while for α>αˆ , the solutions bifurcate into a Brill solution (for Qe = 0.2 we ﬁnd αˆ ∼ 1.5). The occurence of these two patterns is reminiscent to the case of n = 0 gravitating dyons. Note also that, as shown in these plots, the mass parameter M takes negative values for large enough values of n. This is not a surprise, since something similar happens already for the U(1) Brill solution (11).

3.2. α varying

We now discuss the behaviour of the solutions for a varying α and the other parameters fixed. In absence of ∈[ + 2 ] a NUT charge it is know [4,5] that non-Abelian dyonic black hole exist for rh 0, 3 4Qe/2 . For fixed Qe and rh and increasing α they bifurcates into an extremal Reissner–Nordström solution at α ∼ αc.Thevalueαc depends of course on rh and Qe.For rh 1thevalueαc ≈ 1.4 is found numerically and depends weakly on Qe. ∼ + 2 ≈ + 2 + 2 For rh 3 4Qe/2wehaveαc (3 4Qe)/(1 Qe)/2. For n>0 we see (e.g., on Fig. 3) that the local maximum characterizing the function N(r) of a nutty solution (at least for large enough values of the NUT charge n) progressively disappears in favor of a local minimum when α increases. This minimum appears far outside the event horizon r = rh and becomes deeper. In fact, the minimal value Nm approaches zero when α tends to a critical value, say αc(n, Qe,rh). If we denote by rm the value of the radial variable where N(rm) = 0 (with rm >rh) our numerical results strongly indicate that the non-Abelian solution converges into an extremal Brill solution on the interval r ∈[rm, ∞) for α → αc. Indeed, the matter functions’ profiles u, w, φ and the metric functions σ,N all approaches the profiles of the corresponding extremal Brill solution with the same αc,Qe,n. This result is illustrated on Fig. 3 for n = 1, rh = 0.2 and Qe = 0.2; in this case, we find αc ≈ 2.35 but we believe that the result holds for generic values of (n, rh,Qe). The determination of the critical value αc(n, rh,Qe) is not aimed in this Letter. However, it seems that the value αc depends weakly of n, for example, we find αc ≈ 2.22 for n ∈[3, 4]. Nevertheless we can conclude that nutty dyons exist on a finite interval of α and bifurcate into extremal Brill solutions for α = αc. 10 Y. Brihaye, E. Radu / Physics Letters B 615 (2005) 1–13

Fig. 4. The proﬁles of the functions N(r), σ(r) and u(r) are represented for Qe = 0.2, n = 0.5, α = 1 and three different values of rh.

3.3. rh varying

In the case of gravitating dyonic black holes with event horizon rh, the solutions approach the corresponding regular gravitating solution on the interval ]rh, ∞[ when the limit rh → 0 is considered. It is therefore a natural question to investigate how nutty-dyons behave in the same limit. Considering this problem for a few generic values of (α, n) we reach the conclusion that, in the limit rh → 0, the nutty dyon becomes singular at r = 0 because the value σ(rh) tends to zero. This situation is illustrated on Fig. 4 where the functions N(r), σ(r) and u(r) are plotted for three different values of rh and α = 1, n = 0.5. Remarkably, this figure reveals that the functions σ(r) and u(r) are rather independant of rh (it is also true for w(r), φ(r) which are not represented)√ while the function N(r) indeed involves nontrivially with rh.Notealso that for the metric gauge choice P(r)= r2 + n2, the area of two-sphere dΩ2 = P 2(r)(dθ 2 + sin2 θdϕ2) does not vanish at r = 0. However, by choosing a Schwarzschild gauge choice P(r)= r, a straightforward analysis of the corresponding field equations (which can easily be derived from (8)) implies that it is not possible to take a consistent set of boundary conditions at r = 0 without introducing a curvature singularity at that point. Therefore, no globally regular EYMH solutions are found for n = 0. The determination of the domain of nutty dyons for fixed (α,n,Qe) and increasing the horizon radius rh is very likely an involved problem. Already in the case n = 0, discussed in [4] the numerical analysis reveals several (up to three) branches of solutions on some definite intervals of the parameter rh. We believe that similar patterns could occur for n>0 but their analysis is out of the scope of this Letter.

4. Further remarks

In this work we have analysed the basic properties of gravitating YMH system in the presence of a NUT charge. We have found that despite the existence of a number of similarities to the n = 0 case (for example, the presence of a maximal value of the coupling constant α), the NUT-charged solutions exhibits some new qualitative features. The static nature of a n = 0 spherically symmetric gravitating non-Abelian solution implies that it can only produce a “gravitoelectric” field. There both non-Abelian monopole and dyon black hole solutions are possible to exist, with a well defined zero event horizon radius limit. For a nozero NUT charge, the existence of the cross Y. Brihaye, E. Radu / Physics Letters B 615 (2005) 1–13 11 metric term gϕt shows that the solutions have also a “gravitomagnetic” field. The term gϕt does not produce an ergoregion but it will induce an effect similar to the dragging of inertial frames [30]. In this case we have found that only non-Abelian dyons are possible to exist and the usual magnetic charge quantization relation is lost. The total mass of these solutions may be negative and the configurations do not survive in the limit of zero event horizon radius. A discussion of possible generalizations of this work should start with the radially excited nutty dyons, for which the gauge function ω(r) possesses nodes. These configurations are very likely to exist, continuing for n>0 the excited configurations discussed in [4]. Also, in our analysis, to simplify the general picture, we set the Higgs potential V(φ)to zero. We expect to find the same qualitative results for a nonvanishing scalar potential (at least if the parameters are not to large). It would be a challenge to construct axially symmetric NUT-charged dyons (the corresponding n = 0 monopoles are discussed in [31]). Such dyon solutions would present a nonvanishing angular momentum, generalizing the Abelian Kerr-Newman–NUT configurations (a set of asymptotically flat rotating solutions have been considered recently in [32]). Similar to the case n = 0, the solutions discussed in this work can also be generalized by including a more general matter content. However, we expect that these more general configurations will present the same generic properties discussed in this work. This may be important, since there are many indications that the NUT charge is an important ingredient in low energy string theory [27], conclusion enhanced by the discovery of “duality” transformations which relate superficially very different configurations. In many situations, if the NUT charge is not included in the study, some symmetries of the system remain unnoticed (see, e.g., [33] for such an example). Therefore, we may expect the NUT charge to play a crucial role in the duality properties of a (supersymmetric-) theory presenting gravitating non-Abelian dyons. Unfortunately, the pathology of closed timelike curves is not special to the vacuum Taub–NUT solution but afflicts all solutions of Einstein equations solutions with “dual” mass in general [28]. This condition emerges only from the asymptotic form of the fields, and is completely insensitive to the precise details of the nature of the source, or the precise nature of the theory of gravity at short distances where general relativity may be expected to break down [23]. This a causal behavior precludes the nutty dyons solutions discussed in this Letter having a role classically and implies a number of pathological properties of these configurations. Nevertheless, there are various features suggesting that the Euclidean version of NUT-charged solutions play an important role in quantum gravity [22]. For example, the entropy of such solutions generically do not obey the simple “quarter-area law”. As usual, a positive-definite metric is found by considering in (4) the analytical continuation t → it, n → in, which gives P 2(r) = r2 − n2. In this case, the absence of conical singularities at the root rh of the function N(r) imposes a periodicity in the Euclidean time coordinate

= 4π β , (17) N (rh)σ (rh) which should be equal with the one to remove the Dirac string β = 8πn. In the usual approach, the solution’s parameters must be restricted such that the fixed point set of the Killing vector ∂t is regular at the radial position r = rh. We find in this way two types of regular solutions, “bolts” (with arbitrary rh = rb >n) or “nuts” (rh = n), depending on whether the fixed point set is of dimension two or zero (see [13] for a discussion of these solutions in the vacuum case and [34] for a recent generalization with anti-de Sitter asymptotics). We expect that the Euclidean nutty dyons will present some new features as compared to the Lorentzian counterparts. For example, globally regular solutions may exist in this case, since r = rh corresponds to the origin of the coordinate system (note also that the SU(2) Yang–Mills system is known to present self-dual solutions in the background of a vacuum Taub–NUT instanton [35]). In the absence of closed form solutions, the properties of these non-self dual EYMH solutions cannot be predicted directly from those of the Lorentzian configurations. However, similar to the Lorentzian case, they can be studied in a systematic way, by using both analytical and numerical arguments. For example, the magnitude of the electric potential at infinity of the Euclidean solutions, is 12 Y. Brihaye, E. Radu / Physics Letters B 615 (2005) 1–13

not restricted. Also, the condition β = 8πn implies N (rh)σ (rh) = 2n and introduces a supplementary constraint on the matter functions as r → rh. A study of such solutions may be important in a quantum gravity context.

Acknowledgements

E.R. thanks D.H. Tchrakian for useful discussions. Y.B. is grateful to the Belgian FNRS for ﬁnancial support. The work of E.R. is carried out in the framework of Enterprise-Ireland Basic Science Research Project SC/2003/390 of Enterprise-Ireland.

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Search for a Lorentz invariance violation contribution in atmospheric neutrino oscillations using MACRO data

G. Battistoni a, Y. Becherini b, S. Cecchini b,c, M. Cozzi b,H.Dekhissib,d, L.S. Esposito b,G.Giacomellib, M. Giorgini b,G.Mandriolib,S.Manzoorb,e, A. Margiotta b, L. Patrizii b,V.Popab,f, M. Sioli b,∗,G.Sirrib, M. Spurio b, V. Togo b

a INFN Sezione di Milano, I-20133 Milano, Italy b Dipartimento di Fisica dell’Università di Bologna and INFN, I-40127 Bologna, Italy c INAF-IASF Sezione di Bologna, I-40129 Bologna, Italy d Facultè des Sciences, Universitè Mohamed Ier, Oujda, Morocco e RPD, PINSTECH, P.O. Nilore, Islamabad, Pakistan f Institute for Space Sciences, R77125 Bucharest, Romania Received 7 March 2005; received in revised form 5 April 2005; accepted 6 April 2005 Available online 13 April 2005 Editor: L. Rolandi

Abstract The energy spectrum of neutrino-induced upward-going muons in MACRO has been analysed in terms of relativity principles violating effects, keeping standard mass-induced atmospheric neutrino oscillations as the dominant source of νµ → ντ transitions. The data disfavor these exotic possibilities even at a subdominant level, and stringent 90% C.L. limits are placed on −24 −26 the Lorentz invariance violation parameter |v| < 6 × 10 at sin 2θv = 0and|v| < 2.5–5 × 10 at sin 2θv =±1. These limits can also be re-interpreted as upper bounds on the parameters describing violation of the equivalence principle.  2005 Elsevier B.V. All rights reserved.

PACS: 14.60.Pq; 14.60.Lm; 11.30.Cp

Keywords: Neutrino mass and mixing; Lorentz and Poincaré invariance

Neutrino mass-induced oscillations are the best ex- Two ﬂavor νµ → ντ oscillations are strongly favored planation of the atmospheric neutrino problem [1–4]. over a wide range of alternative solutions such as νµ → νsterile oscillations [5,6], νµ → νe oscillations [3,4] or other exotic possibilities [7,8]. * Corresponding author. E-mail addresses: [email protected], In this Letter, we assume standard mass-induced [email protected] (M. Sioli). neutrino oscillations as the leading mechanism for ﬂa-

0370-2693/$ – see front matter  2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physletb.2005.04.010 G. Battistoni et al. / Physics Letters B 615 (2005) 14–18 15 vor transitions and we treat Lorentz invariance flavor In this case, the νµ survival probability is transitions as a subdominant effect [9]. More specifi- P(ν → ν ) cally, we constrain mass-induced neutrino oscillation µ µ 2 2 18 parameters to the ones obtained with a global fit of = 1 − sin 2θv sin 2.54 × 10 vLEν , (4) all MACRO neutrino data. Then we study the en- where v = (vνv − vνv ) is the neutrino MAV differ- ergy spectrum of a limited sample of neutrino-induced 3 2 ence in units of c. Notice that neutrino flavor oscil- upward-going muons and its compatibility with the lations induced by VLI are characterized by an LE inclusion of competitive oscillation scenarios. In the ν dependence of the oscillation probability (Eq. (4)), to literature, neutrino oscillations induced by violation of be compared with the L/E behavior of mass-induced (CPT-conserving) Lorentz invariance (VLI) and viola- ν oscillations (Eq. (2)). tion of the equivalence principle (VEP) are described When both mass-induced transitions and VLI- within the same formalism. In the following we will induced transitions are considered simultaneously, the mention only VLI for simplicity. muon neutrino survival probability can be expressed In this scenario, neutrinos can be described in terms as [9–11] of three distinct bases: flavor eigenstates, mass eigen- 2 2 states and velocity eigenstates, the latter being char- P(νµ → νµ) = 1 − sin 2Θ sin Ω, (5) acterized by different maximum attainable velocities (MAVs) in the limit of infinite momentum. where the global mixing angle Θ and the term Ω are Both mass-induced oscillations and VLI transitions given by: are treated in the two-family approximation and we 2Θ = atan(a1/a2), assume that mass and velocity mixings occur inside | | = 2 + 2 the same families (e.g., ν2 and ν3 ). Ω a1 a2 (6) The usual interpretation of the atmospheric neu- → with trino oscillations is νµ ντ induced by the mixing | m | m 2 of the two mass eigenstates ν2 and ν3 , and two a1 = 1.27 m sin 2θmL/Eν | | weak eigenstates νµ and ντ , i.e., + × 18 iη 2 10 v sin 2θvLEνe , m m m m |ν = ν cos θ + ν sin θ , = 2 µ 2 23 3 23 a2 1.27 m cos 2θmL/Eν m m m m |ν =−ν sin θ + ν cos θ , (1) 18 τ 2 23 3 23 + 2 × 10 v cos 2θvLEν . (7) m where θ (≡ θm) is the flavor-mass mixing angle. The 2 23 Here m , L and Eν are expressed, as in Eqs. (2) survival probability of muon neutrinos at a distance L and (4),ineV2, km and GeV, respectively. The addi- from production is tional factor eiη connects the mass and velocity eigenstates, and for the moment it is assumed to be real P(νµ → νµ) (η = 0orπ). Note that formulae (2) and (5) do not = − 2 2 2 1 sin 2θm sin 1.27m L/Eν , (2) depend on the sign of the mixing angle and/or on 2 2 = 2 − 2 2 the sign of the v and m parameters; this is not where m (mνm mνm ) is expressed in eV , L in 3 2 so in the case of mixed oscillations, where the rela- km and the neutrino energy Eν in GeV. Notice the de- 2 tive sign between the mass-induced and VLI-induced pendence on L/Eν in the argument of the second sin oscillation terms is important. The whole domain of term. variability of the parameters can be accessed with the In the VLI case, the two flavor eigenstates |νµ, |ντ 2 requirements m 0, 0 θm π/2, v 0 and and the two velocity eigenstates |νv, |νv are con- 2 3 −π/4 θv π/4. θ v ≡ θ nected through the mixing angle 23 ( v) in analogy The same formalism also applies to violation of the with mass-induced oscillations: equivalence principle, after substituting v/2 with the v v v v adimensional product |φ|γ ; γ is the difference of |νµ= ν cos θ + ν sin θ , 2 23 3 23 the coupling constants for neutrinos of different types | =− v v + v v ντ ν2 r sin θ23 ν3 cos θ23. (3) to the gravitational potential φ [12]. 16 G. Battistoni et al. / Physics Letters B 615 (2005) 14–18

Fig. 1. Energy dependence of the νµ → νµ survival probability for mass-induced oscillations alone (continuous line) and mass-induced + VLI −25 oscillations for v = 2 × 10 and sin 2θv = 0, ±0.3, ±0.7and±1 (dashed lines for positive values, dotted lines for negative values). 2 2 The neutrino pathlength was ﬁxed at L = 10000 km and we assumed m = 0.0023 eV , θm = π/4.

As shown in [10], in a footnoted comment of [13] current interactions νµ + N → µ + X; upgoing muons and more recently in [11], the most sensitive tests of were identified with the streamer tube system (for VLI can be made by analysing the high energy tail tracking) and the scintillator system (for time-of-flight of atmospheric neutrinos at large pathlength values. measurement). Early results concerning atmospheric As an example, Fig. 1 shows the energy dependence neutrinos were published in [15] and in [1] for the up- of the νµ → νµ survival probability as a function of throughgoing muon sample and in [16] for the low the neutrino energy, for neutrino mass-induced oscil- energy semi-contained and upgoing-stopping muon lations alone and for both mass and VLI-induced os- events. Matter effects in the νµ → νsterile channel were cillations for v = 2 × 10−25 and different values of presented in [5] and a global analysis of all MACRO sin 2θv parameter. Note the large sensitivity for large neutrino data in [2]. neutrino energies and large mixing angles. Given the In order to analyse the MACRO data in terms of very small neutrino mass (mν 1 eV), neutrinos with VLI, we used a subsample of 300 upthroughgoing energies larger than 100 GeV are extremely relativis- muons whose energies were estimated via multiple tic, with Lorentz γ factors larger than 1011. Coulomb scattering in the 7 horizontal rock absorbers MACRO [14] was a multipurpose large area detec- in the lower apparatus [17,18]. The energy estimate tor (∼10000 m2 sr acceptance for an isotropic flux) was obtained using the streamer tubes in drift mode, located in the Gran Sasso underground Lab, shielded which allowed to considerably improve the spatial res- by a minimum rock overburden of 3150 hg/cm2.The olution of the detector (∼3 mm). The overall neutrino detector had global dimensions of 76.6 × 12 × 9.3m3 energy resolution was of the order of 100%, mainly and used limited streamer tubes and scintillation coun- dominated by muon energy losses in the rock be- ters to detect muons. νµ’s were detected via charged low the detector (note that Eµ0.4Eν). Upgoing G. Battistoni et al. / Physics Letters B 615 (2005) 14–18 17 muon neutrinos of this sample have large zenith angles (> 120◦) and the median value of neutrino pathlengths is slightly larger than 10000 km. Following the analysis in Ref. [18], we selected a low and a high energy sample by requiring that the rec rec reconstructed neutrino energy Eν should be Eν < rec 30 GeV and Eν > 130 GeV. The number of events surviving these cuts is Nlow = 49 and Nhigh = 58, respectively; their median energies, estimated via Monte Carlo, are 13 GeV and 204 GeV (assuming mass- induced oscillations). The analysis then proceeds by fixing the neutrino mass oscillation parameters at the values obtained with the global analysis of all MACRO neutrino data [2]: Fig. 2. 90% C.L. upper limits on the Lorentz invariance viola- 2 2 2 ¯ tion parameter v versus sin 2θv. Standard mass induced oscilla- m = 0.0023 eV ,sin 2θm = 1. Then, we scanned tions are assumed in the two-flavor νµ → ντ approximation, with 2 2 the plane of the two free parameters (v, θv)using m = 0.0023 eV and θm = π/4. The dashed line shows the limit the function obtained with the same selection criteria of Ref. [18] to define the low and high energy samples; the continuous line is the final result obtained with the selection criteria optimized for the present analy- high MC 2 ¯ 2 Ni − αN (v, θv; m , θm) sis (see text). χ2 = i , (8) σi i=low The energy cuts described above (the same used in Ref. [18]), were optimized for mass-induced neutrino where N MC is the number of events predicted by i oscillations. In order to maximize the sensitivity of the Monte Carlo, α is a constant which normalizes the analysis for VLI induced oscillations, we performed number of Monte Carlo events to the number of ob- a blind analysis, based only on Monte Carlo events, served events and σ is the overall error comprehensive i to determine the energy cuts which yield the best per- of statistical and systematic uncertainties. formances. The results of this study suggest the cuts We used the Monte Carlo simulation described Erec < 28 GeV and Erec > 142 GeV; with these cuts in [18] with different neutrino fluxes in input [19–22]. ν ν the number of events in the real data are N = 44 The largest relative difference of the extreme values low events and N = 35 events. The limits obtained with of the MC expected ratio N /N is 13%. How- high low high this selection are shown in Fig. 2 by the continuous ever, in the evaluation of the systematic error, the main line. As expected, the limits are now more stringent sources of uncertainties for this ratio (namely the pri- than for the previous choice. mary cosmic ray spectral index and neutrino cross In order to understand the dependence of this re- sections) have been separately estimated and their ef- 2 fects added in quadrature (see [18] for details): in this sult with respect to the choice of the m parameter, 2 work, we use a conservative 16% theoretical system- we varied the m values around the best-fit point. 2 ± atic error on the ratio Nlow/Nhigh. The experimental We found that a variation of m of 30% moves systematic error on the ratio was estimated to be 6%. up/down the upper limit of VLI parameters by at most In the following, we show the results obtained with the a factor 2. computation in [22]. Finally, we computed the limit on v marginal- The inclusion of the VLI effect does not improve ized with respect to all the other parameters left free 2 2 2 the χ in any point of the (v, θv) plane, compared to to variate inside the intervals: m = m ± 30%, ¯ mass-induced oscillations stand-alone, and proper up- θm = θm ± 20%, −π/4 θv π/4 and any value of per limits on VLI parameters were obtained. The 90% the phase η. We obtained the 90% C.L. upper limit −25 C.L. limits on v and θv, computed with the Feld- |v| < 3 × 10 . man and Cousins prescription [23],areshownbythe An independent and complementary analysis was dashed line in Fig. 2. performed on a sample of events with a reconstructed 18 G. Battistoni et al. / Physics Letters B 615 (2005) 14–18

rec neutrino energy 25 GeV

Measurement of exclusive ρ+ρ− production in mid-virtuality two-photon interactions and study of the γγ∗ → ρρ process at LEP

L3 Collaboration P. Achard t, O. Adriani q, M. Aguilar-Benitez y, J. Alcaraz y, G. Alemanni w, J. Allaby r, A. Aloisio ac, M.G. Alviggi ac, H. Anderhub aw, V.P. Andreev f,ah,F.Anselmoh, A. Areﬁev ab, T. Azemoon c, T. Aziz i,P.Bagnaiaam,A.Bajoy,G.Baksayz,L.Baksayz, S.V. Baldew b,S.Banerjeei,Sw.Banerjeed, A. Barczyk aw,au, R. Barillère r, P. Bartalini w, M. Basile h,N.Batalovaat, R. Battiston ag,A.Bayw, F. Becattini q, U. Becker m,F.Behneraw, L. Bellucci q, R. Berbeco c, J. Berdugo y,P.Bergesm, B. Bertucci ag,B.L.Betevaw,M.Biasiniag, M. Biglietti ac,A.Bilandaw, J.J. Blaising d, S.C. Blyth ai, G.J. Bobbink b,A.Böhma, L. Boldizsar l,B.Borgiaam,S.Bottaiq, D. Bourilkov aw, M. Bourquin t, S. Braccini t,J.G.Bransonao,F.Brochud,J.D.Burgerm, W.J. Burger ag,X.D.Caim, M. Capell m, G. Cara Romeo h, G. Carlino ac, A. Cartacci q, J. Casaus y, F. Cavallari am, N. Cavallo aj, C. Cecchi ag, M. Cerrada y,M.Chamizot, Y.H. Chang ar,M.Chemarinx,A.Chenar,G.Cheng,G.M.Cheng,H.F.Chenv, H.S. Chen g, G. Chiefari ac, L. Cifarelli an, F. Cindolo h,I.Clarem,R.Clareal, G. Coignet d,N.Colinoy, S. Costantini am,B.delaCruzy, S. Cucciarelli ag, R. de Asmundis ac, P. Déglon t, J. Debreczeni l,A.Degréd,K.Dehmeltz, K. Deiters au, D. della Volpe ac, E. Delmeire t, P. Denes ak, F. DeNotaristefani am,A.DeSalvoaw, M. Diemoz am, M. Dierckxsens b, C. Dionisi am, M. Dittmar aw,A.Doriaac,M.T.Dovaj,5, D. Duchesneau d, M. Duda a, B. Echenard t,A.Eliner,A.ElHagea, H. El Mamouni x, A. Engler ai,F.J.Epplingm, P. Extermann t, M.A. Falagan y, S. Falciano am,A.Favaraaf, J. Fay x,O.Fedinah, M. Felcini aw, T. Ferguson ai, H. Fesefeldt a, E. Fiandrini ag, J.H. Field t, F. Filthaut ae,P.H.Fisherm, W. Fisher ak, I. Fisk ao, G. Forconi m, K. Freudenreich aw,C.Furettaaa, Yu. Galaktionov ab,m, S.N. Ganguli i, P. Garcia-Abia y, M. Gataullin af, S. Gentile am,S.Giaguam, Z.F. Gong v,G.Grenierx,O.Grimmaw, M.W. Gruenewald p, M. Guida an, V.K. Gupta ak,A.Gurtui,L.J.Gutayat, D. Haas e, D. Hatzifotiadou h, T. Hebbeker a,A.Hervér,J.Hirschfelderai,H.Hoferaw,

0370-2693/$ – see front matter  2005 Published by Elsevier B.V. doi:10.1016/j.physletb.2005.04.011 20 L3 Collaboration / Physics Letters B 615 (2005) 19–30

M. Hohlmann z,G.Holzneraw,S.R.Houar,B.N.Jing, P. Jindal n, L.W. Jones c, P. de Jong b, I. Josa-Mutuberría y, M. Kaur n, M.N. Kienzle-Focacci t,J.K.Kimaq, J. Kirkby r, W. Kittel ae, A. Klimentov m,ab, A.C. König ae, M. Kopal at, V. Koutsenko m,ab, M. Kräber aw, R.W. Kraemer ai,A.Krügerav, A. Kunin m, P. Ladron de Guevara y, I. Laktineh x, G. Landi q, M. Lebeau r,A.Lebedevm,P.Lebrunx,P.Lecomteaw, P. Lecoq r,P.LeCoultreaw,J.M.LeGoffr,R.Leisteav,M.Levtchenkoaa, P. Levtchenko ah,C.Liv, S. Likhoded av,C.H.Linar,W.T.Linar,F.L.Lindeb, L. Lista ac, Z.A. Liu g, W. Lohmann av, E. Longo am,Y.S.Lug,C.Luciam, L. Luminari am, W. Lustermann aw,W.G.Mav,L.Malgerir, A. Malinin ab,C.Mañay,J.Mansak, J.P. Martin x, F. Marzano am, K. Mazumdar i, R.R. McNeil f,S.Meler,ac,L.Merolaac, M. Meschini q, W.J. Metzger ae,A.Mihulk,H.Milcentr, G. Mirabelli am,J.Mnicha, G.B. Mohanty i,G.S.Muanzax, A.J.M. Muijs b,B.Musicarao,M.Musyam, S. Nagy o, S. Natale t, M. Napolitano ac, F. Nessi-Tedaldi aw,H.Newmanaf,A.Nisatiam, T. Novak ae,H.Nowakav, R. Oﬁerzynski aw,G.Organtiniam,I.Palat,C.Palomaresy, P. Paolucci ac, R. Paramatti am,G.Passalevaq,S.Patricelliac,T.Paulj, M. Pauluzzi ag, C. Paus m,F.Paussaw, M. Pedace am,S.Pensottiaa, D. Perret-Gallix d,D.Piccoloac, F. Pierella h, M. Pioppi ag,P.A.Pirouéak, E. Pistolesi aa, V. Plyaskin ab, M. Pohl t, V. Pojidaev q, J. Pothier r,D.Prokoﬁevah, G. Rahal-Callot aw, M.A. Rahaman i, P. Raics o,N.Rajai, R. Ramelli aw,P.G.Rancoitaaa,R.Ranieriq, A. Raspereza av, P. Razis ad,D.Renaw, M. Rescigno am, S. Reucroft j, S. Riemann av,K.Rilesc,B.P.Roec, L. Romero y,A.Roscaav, C. Rosemann a, C. Rosenbleck a,S.Rosier-Leesd,S.Rotha, J.A. Rubio r, G. Ruggiero q, H. Rykaczewski aw,A.Sakharovaw, S. Saremi f,S.Sarkaram, J. Salicio r, E. Sanchez y, C. Schäfer r, V. Schegelsky ah, H. Schopper u, D.J. Schotanus ae, C. Sciacca ac,L.Servoliag, S. Shevchenko af,N.Shivarovap, V. Shoutko m, E. Shumilov ab, A. Shvorob af,D.Sonaq,C.Sougax, P. Spillantini q, M. Steuer m, D.P. Stickland ak,B.Stoyanovap,A.Straessnert, K. Sudhakar i, G. Sultanov ap, L.Z. Sun v,S.Sushkova,H.Suteraw,J.D.Swainj, Z. Szillasi z,3,X.W.Tangg,P.Tarjano, L. Tauscher e, L. Taylor j, B. Tellili x, D. Teyssier x,C.Timmermansae, Samuel C.C. Ting m, S.M. Ting m, S.C. Tonwar i,J.Tóthl,C.Tullyak,K.L.Tungg, J. Ulbricht aw, E. Valente am,R.T.VandeWalleae, R. Vasquez at, V. Veszpremi z, G. Vesztergombi l, I. Vetlitsky ab,G.Viertelaw, S. Villa al, M. Vivargent d,S.Vlachose, I. Vodopianov z,H.Vogelai,H.Vogtav, I. Vorobiev ai,ab, A.A. Vorobyov ah, M. Wadhwa e, Q. Wang ae,X.L.Wangv,Z.M.Wangv, M. Weber r, S. Wynhoff ak,L.Xiaaf,Z.Z.Xuv, J. Yamamoto c,B.Z.Yangv,C.G.Yangg,H.J.Yangc,M.Yangg,S.C.Yehas, An. Zalite ah, Yu. Zalite ah, Z.P. Zhang v,J.Zhaov,G.Y.Zhug,R.Y.Zhuaf, H.L. Zhuang g,A.Zichichih,r,s,B.Zimmermannaw, M. Zöller a

a 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 L3 Collaboration / Physics Letters B 615 (2005) 19–30 21

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, China 6 h University of Bologna, and INFN, Sezione di Bologna, I-40126 Bologna, Italy i Tata Institute of Fundamental Research, Mumbai (Bombay) 400 005, India j Northeastern University, Boston, MA 02115, USA k Institute of Atomic Physics and University of Bucharest, R-76900 Bucharest, Romania l Central Research Institute for Physics of the Hungarian Academy of Sciences, H-1525 Budapest 114, Hungary 2 m Massachusetts Institute of Technology, Cambridge, MA 02139, USA n Panjab University, Chandigarh 160 014, India o KLTE-ATOMKI, H-4010 Debrecen, Hungary 3 p Department of Experimental Physics, University College Dublin, Belﬁeld, Dublin 4, Ireland q INFN, Sezione di Firenze, and University of Florence, I-50125 Florence, Italy r European Laboratory for Particle Physics, CERN, CH-1211 Geneva 23, Switzerland s World Laboratory, FBLJA Project, CH-1211 Geneva 23, Switzerland t University of Geneva, CH-1211 Geneva 4, Switzerland u University of Hamburg, D-22761 Hamburg, Germany v Chinese University of Science and Technology, USTC, Hefei, Anhui 230 029, 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 Radboud University 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 National Central University, Chung-Li, Taiwan as Department of Physics, National Tsing Hua University, Taiwan at Purdue University, West Lafayette, IN 47907, USA au Paul Scherrer Institut, PSI, CH-5232 Villigen, Switzerland av DESY, D-15738 Zeuthen, Germany aw Eidgenössische Technische Hochschule, ETH Zürich, CH-8093 Zürich, Switzerland

Received 18 March 2005; received in revised form 5 April 2005; accepted 6 April 2005 Available online 14 April 2005 Editor: L. Rolandi 22 L3 Collaboration / Physics Letters B 615 (2005) 19–30

Abstract + − ∗ Exclusive ρ ρ production in two-photon collisions between a quasi-real√ photon, γ , and a mid-virtuality photon, γ ,is studied with data collected at LEP at centre-of-mass energies 183 GeV s 209 GeV with a total integrated luminosity of − ∗ + − 684.8pb 1. The cross section of the γγ → ρ ρ process is determined as a function of the photon virtuality, Q2,andthe 2 2 2 two-photon centre-of-mass energy, Wγγ, in the kinematic region: 0.2GeV Q 0.85 GeV and 1.1GeV Wγγ 3GeV. + − ∗ These results, together with previous L3 measurements of ρ0ρ0 and ρ ρ production, allow a study of the γγ → ρρ process over the Q2-region 0.2GeV2 Q2 30 GeV2.  2005 Published by Elsevier B.V.

1. Introduction duction were found to have a similar dependence on Wγγ and to be of similar magnitude. However, the The L3 Collaboration has recently measured the ex- ρ+ρ− cross section is systematically higher than the clusive production of ρ0ρ0 [1,2] and ρ+ρ− [3] pairs ρ0ρ0 one. This is in contrast with the suppression + − in the two-photon fusion process: and different Wγγ dependence of ρ ρ production 0 0 + − + − ∗ + − [6] with respect to ρ ρ [7] observed in data with e e → e e γγ → e e ρρ, (1) 2 Q ≈ 0 and Wγγ 2 GeV. We note that despite the where the beam electrons7 radiate virtual photons wide range of theoretical models [8,9], ρ-pair produc- which interact and produce a hadronic final state. One tion at Q2 ≈ 0 is still not well understood. Therefore of the photons, γ , is quasi-real, characterised by a the experimental study of the Q2-evolution of ρ-pair 2 = small value of its squared four momentum, Pγ production is important to understand vector meson m2 ≈ 0, whereas the other one, γ ∗, has a significant pair-production in two-photon interactions. γ 0 0 2 =− 2 =− 2 − 2 Previously, we performed a measurement of ρ ρ virtuality, Q Pγ ∗ mγ ∗ mγ . Our measurements cover the two-photon centre-of-mass en- production [2] for intermediate virtualities: ergy 0.2GeV2 Q2 0.85 GeV2. (3) 1.1GeV W 3GeV. (2) γγ In this Letter, we complement that study with the first The two measurements [1,3] done at large virtual- measurement of the process ities, 1.2GeV2 Q2 30 GeV2, provide a testing + − + − ∗ + − + − ground for a recently-developed QCD-based model e e → e e γγ → e e ρ ρ (4) [4]. This model describes well the Q2-dependence of in the kinematic region (2) and (3). These data allow the ρ0ρ0 production at large momentum transfer [5]. + − to follow the Q2-evolution of the ρρ-production over The measured cross sections for ρ0ρ0 and ρ ρ pro- two orders of magnitude in this variable. The analysis techniques employed in this study are 1 Supported by the German Bundesministerium für Bildung, similar to those of our previous measurements [2,3]. Wissenschaft, Forschung und Technologie. The data used, corresponding to an integrated lumi- 2 Supported by the Hungarian OTKA fund under contract num- nosity of 684.8 pb−1,arethesameasinRef.[2] bers T019181, F023259 and T037350. 3 Also supported by the Hungarian OTKA fund under contract and were collected by the L3 detector√[10] at LEP at number T026178. centre-of-mass energies 183 GeV s 209 GeV. 4 Supported also by the Comisión Interministerial de Ciencia y Scattered beam electrons which have radiated photons Tecnología. with virtualities in the range (3) can be “tagged” by 5 Also supported by CONICET and Universidad Nacional de La the Very Small Angle Tagger (VSAT) [11].TheVSAT Plata, CC 67, 1900 La Plata, Argentina. is an electromagnetic calorimeter, constructed with 6 Supported by the National Natural Science Foundation of China. BGO crystals, with a geometrical acceptance covering 7 Throughout this Letter, the term “electron” denotes both elec- the polar angle range 5 mrad θ 10 mrad, for az- tron and positron. imuthal angles in the ranges −1.25 rad φ 1.25 rad L3 Collaboration / Physics Letters B 615 (2005) 19–30 23 and π − 1.25 rad φ π + 1.25 rad. When the elec- the fits is taken. To make the selection robust against tron with the largest scattering angle is detected in instrumental noise and backgrounds and to reduce the the VSAT, the virtuality of the photon it radiated is, sensitivity to the Monte Carlo simulation of fake pho- within 1% precision, equal to the transverse momen- tons, we retain events with one additional photon, not 2 0 0 tum squared, pt , of the final state hadron system: used in the π π pair, if the photon energy is less than 300 MeV and does not exceed 10% of the energy of 2 = − ≈ 2 ≈ 2 Q 2EbEs(1 cos θs) EbEsθs pt , (5) the π 0π 0 pair. 2 where Eb is the beam energy, and Es and θs are the The transverse momentum squared, pt , of the four- energy and the scattering angle of the tagged electron. pion system is used to measure the Q2 of the event Therefore the VSAT is not used to directly measure and is required to be in the range 0.2–0.85 GeV2.For Q2, but rather to select exclusive final states by corre- selection of an exclusive final state, the acoplanarity lating the direction of the transverse momentum vector angle, φaco, calculated from the difference between the of the tagged electron with the detected hadron sys- azimuthal angle of the tagged electron, φtag,shown tem. in Fig. 1(b), and the azimuthal angle of the four-pion system, is required to be less than 0.4 rad, as shown in Fig. 1(c). The data contain a contribution from η 2. Event selection production, as visible in the π +π −π 0 mass spectrum, shown in Fig. 1(d). This background is removed by re- The reaction e+e− → e+e−ρ+ρ− contributing to quiring M(π+π −π 0)>0.65 GeV. the process After all cuts, 414 events are retained. Their four- + − → + − + − 0 0 pion mass spectrum is shown in Fig. 2(a). The region e e e etagπ π π π (6) 1.1GeV Wγγ 3 GeV is populated by 387 events, is identified by one and only one scattered electron, which are used for the cross section determination. A ± etag, detected in the VSAT, two charged pions mea- strong signal from ρ production is observed in the sured in the tracking chamber, and energy clusters M(π±π 0) spectrum, shown in Fig. 2(b). The cluster- from the two-photon decays of the π 0’s, deposited ing of entries at the crossing of the ρ± mass bands in in the BGO electromagnetic calorimeter. These events the correlation plot of the masses of the π ±π 0 com- are collected by two independent track-triggers [12]. binations, shown in Fig. 2(c), gives evidence for a The trigger efficiency, as determined from the data it- signal from ρ+ρ− intermediate states. No structure is self, is (60 ± 3)%. observed in the correlation plot of the masses of the Single-tagged events are selected by requiring just π +π − and π 0π 0 combinations, shown in Fig. 2(d). one electromagnetic cluster with energy greater then We also inspected the two- and three-pion mass distri- 50% of the beam energy reconstructed in the VSAT. butions, shown in Fig. 3, for production of higher-mass The event candidates must have exactly two tracks resonances. The only statistically-significant signal is ± ± 0 0 with zero total charge. The tracks must come from the from the a2 (1320) state in the π π π mass spec- interaction vertex, have transverse momentum greater trum, as seen in Fig. 3(f). than 100 MeV and an energy loss in the tracking chamber compatible with the pion hypothesis. The selected events should contain a π 0π 0 pair, therefore we 3. Data analysis consider event candidates that have four or five photons, identified as isolated clusters in the electromag- 3.1. Monte Carlo modelling netic calorimeter. Photons having energies greater than 60 MeV are paired to reconstruct neutral pions, which To estimate the number of ρ+ρ− events in the are required to be in the mass window 100 MeV selected four-pion data sample, we consider non- M(γγ) 170 MeV, as shown in Fig. 1(a). The mass interfering contributions from the processes: of a π 0 candidate is constrained to the nominal value ∗ → + − by a 1-C kinematic fit. If more than one π 0π 0 com- γγ ρ ρ , ∗ ± ∓ bination exists, the one with the smallest χ2 sum of γγ → ρ π π 0, 24 L3 Collaboration / Physics Letters B 615 (2005) 19–30

+ − Fig. 1. Distributions for π π π0π0 candidates. (a) Two-photon invariant mass for the selected π0’s (two entries per event); (b) azimuthal angle, φtag, of the tagged electron for tags in the inner side of the LEP ring (in) and, folded over it, for tags in the outer side of the LEP ring + − 0 0 + − 0 (out); (c) acoplanarity angle, φaco, between the tagged electron and the π π π π system and (d) mass of the π π π system (two entries per event). The data are compared to the four-pion Monte Carlo. The estimated background is indicated by the hatched histograms. The arrows indicate the selection cuts.

∗ → ± ∓ γγ a2 (1320)π , 3.2. Background estimation ∗ + − γγ → π π π 0π 0, non-resonant. (7) The contribution to the selected events from e+e− About 40 million Monte Carlo events of the processes annihilation and from the process e+e− → e+e−τ +τ − (7) are generated with the EGPC [13] program, which is negligible. Random coincidences with off-momen- uses the luminosity function from Ref. [14]. Particle tum beam electrons, which give signals in the VSAT, production and decay is uniform in phase-space. The are a source of background. The flux of these par- generated events are passed through the full L3 detec- ticles is dominantly on the outer side of the LEP tor simulation using the GEANT [15] and GHEISHA ring. Therefore, this background would cause an ex- [16] programs and processed in the same way as the cess in the number of events having a tag on the data, reproducing the detector behaviour as monitored outer side of the accelerator ring, Nout, with respect in the different data-taking periods. to the inner side, Nin. In the selected data, the ratio For acceptance calculations, Monte Carlo events Nout/Nin = 1.04 ± 0.10 is close to unity, indicating are assigned a Q2-dependent weight, evaluated us- that this background is small. This conclusion is cor- ing the GVDM form-factor [17] for both interacting roborated by the good agreement observed between photons. The detection efficiencies of the process (4) the φtag distribution of the selected data and Monte 2 are listed in Tables 1 and 2 for bins in Q and Wγγ. Carlo event samples, shown in Fig. 1(b). The efficiencies for the four-pion final states of all the Two sources of background remain. The first is processes (7) are of similar magnitude. partially-reconstructed events from two-photon inter- L3 Collaboration / Physics Letters B 615 (2005) 19–30 25

± 0 Fig. 2. Mass distributions for the selected events: (a) the four-pion system, Wγγ;(b)theπ π combinations (four entries per event); (c) corre- − + + − lation between the π π0 and π π0 pairs (two entries per event) and (d) correlation between the π π and π0π0 pairs. The two-dimensional distributions have a bin width of 55 × 55 MeV2, the size of the boxes is proportional to the number of entries and both plots have the same vertical scale. actions with higher particle multiplicities, when tracks passing the selection, are combined with the distribu- or photons escape detection. The second is signal tion of selected π +π −π 0π 0 Monte Carlo events so as events with one or more photons substituted by pho- to reproduce the φaco distribution observed in the data, ton candidates due to noise. To estimate the accepted as shown in Fig. 1(c). The estimated background lev- background we use background-like event samples ex- els are listed in Tables 1 and 2. As data samples are tracted from the experimental data. The ﬁrst back- used in the background estimation, they contain also ground is modelled with selected π ±π ±π 0π 0 events, a fraction of events with fake tags and thus take into in which at least two charged particles have not been account the effect of this background. detected and by π +π −π 0π 0π 0 events in which one π 0 is excluded from consideration. An event-mixing 3.3. Fit method technique is employed in order to reproduce events from the second background: one or two photons In order to determine the differential ρ+ρ− pro- forming a π 0 are excluded from a selected event and duction rate, a maximum likelihood ﬁt of the data replaced by photons from another data event. The to a sum of Monte Carlo samples of the processes 2 φaco distributions of the background-like data samples, (7) is performed in intervals of Q and Wγγ using 26 L3 Collaboration / Physics Letters B 615 (2005) 19–30

± Fig. 3. (a), (c), (e), (g) Mass distributions of the π π0 combinations (four entries per event) in four Q2-intervals. Distributions for the entire 2 2 2 + − 0 0 kinematic region 1.1GeV Wγγ 3 GeV and 0.2GeV Q 0.85 GeV of: (b) The sum of the π π and π π mass spectra (two entries per event). (d) The neutral three-pion combinations (two entries per event). (f) The charged three-pion combinations (two entries per + − ± event). (h) The sum of the π π π0 and π π0π0 mass spectra (four entries per event). The points represent the data, the hatched areas + − show the ρ ρ component and the open areas show the sum of the other contributing processes. The fraction of the different components are determined by the fit and the total normalisation is to the number of the events. a box method [1–3,18]. The inputs to the fit are the six The analysis procedure is optimised for deriving two-pion masses in an event, namely, the four combi- the ρ+ρ− contribution and only the ρ+ρ− content and nations π ±π 0 and the two combinations π +π − and the sum of the rest of the contributing processes, de- π 0π 0. They provide a complete description of a four- noted as “other 4π”, are considered for cross section pion event in our model of isotropic production and measurements. To check the quality of the fit, the two- phase space decay. and three-pion mass distributions of the data are com- L3 Collaboration / Physics Letters B 615 (2005) 19–30 27

Table 1 + − + − + − ∗ + − Detection efﬁciencies, ε, background fractions, Bg, and cross sections of the reactions e e → e e ρ ρ , γγ → ρ ρ and of the sum 2 of the rest of the contributing processes, “other 4π”, as a function of Q for 1.1GeV Wγγ 3 GeV. The values of the differential cross sections are corrected to the centre of each bin. The ﬁrst uncertainties are statistical, the second systematic. An overall normalization uncertainty of 5% for the trigger is not included 2 2 2 Q range ε Bg σee [pb] dσee/dQ [pb/GeV ] σγγ [nb] σγγ [nb] + − + − + − [GeV2] [%] [%] ρ ρ ρ ρ ρ ρ other 4π 0.20–0.28 0.8 14 7.4 ± 2.4 ± 1.992± 29 ± 23 5.7 ± 1.8 ± 1.410.9 ± 2.2 ± 1.5 0.28–0.40 1.2 14 5.7 ± 1.8 ± 1.347± 15 ± 10 4.3 ± 1.4 ± 1.012.2 ± 1.8 ± 1.4 0.40–0.55 1.1 15 5.6 ± 1.6 ± 1.137± 11 ± 7.34.9 ± 1.4 ± 1.013.3 ± 2.0 ± 1.8 0.55–0.85 0.7 18 7.7 ± 2.5 ± 2.025± 8.2 ± 6.55.3 ± 1.7 ± 1.412.1 ± 2.2 ± 1.9

Table 2 + − + − + − ∗ + − Detection efﬁciencies, ε, background fractions, Bg, and cross sections of the reactions e e → e e ρ ρ , γγ → ρ ρ andofthesumof 2 2 2 the rest of the contributing processes, “other 4π”, as a function of Wγγ for 0.2GeV Q 0.85 GeV . The ﬁrst uncertainties are statistical, the second systematic. An overall normalization uncertainty of 5% for the trigger is not included

Wγγ-range ε Bg σee [pb] σγγ [nb] σγγ [nb] + − + − [GeV] [%] [%] ρ ρ ρ ρ other 4π 1.10–1.40 0.6 25 4.9 ± 1.8 ± 1.33.9 ± 1.5 ± 1.19.0±2.4±1.7 1.40–1.65 0.9 18 6.7 ± 1.6 ± 1.37.6 ± 1.9 ± 1.514.8±2.7±2.5 1.65–1.85 1.1 15 5.1 ± 1.5 ± 0.98.4 ± 2.4 ± 1.615.8±3.1±2.3 1.85–2.10 1.1 13 3.9 ± 1.4 ± 0.85.9 ± 2.0 ± 1.218.3±3.0±2.7 2.10–2.40 1.2 10 2.2 ± 1.0 ± 0.53.2 ± 1.4 ± 0.811.5±2.1±1.8 2.40–3.00 1.2 10 2.2 ± 1.0 ± 0.51.9 ± 0.9 ± 0.58.5±1.5±1.5

pared in Fig. 3 with those of a mixture of Monte Carlo and Wγγ bin using the program GALUGA [20], which event samples from the processes (7), in proportions performs O(α4) QED calculations. The same proce- determined by the fit. The observed experimental dis- dure was used in our previous studies [1–3]. The cross tributions are reasonably well described by the Monte section σγγ is derived from the measured cross sec- Carlo model. tion using the relation σγγ = σee/LTT. Thus, σγγ represents an effective cross section containing contributions from both transverse and longitudinal photon 4. Results polarisations. The cross section of the process γγ∗ → ρ+ρ− is listed in Table 1 as a function of Q2 and in The cross sections of the process Table 2 as a function of Wγγ. The sum of the cross sections of the other contributing processes is also given + − → + − + − e e e e ρ ρ in Tables 1 and 2. 2 in bins of Q and Wγγ, σee, are listed in Tables 1 Several sources of systematic uncertainty are con- and 2. The statistical uncertainties, also listed in Ta- sidered. The contribution of the selection procedure bles 1 and 2, are those of the fit. The differential cross is in the range 12–18%; Monte Carlo statistics in the 2 section, dσee/dQ , derived from σee, is listed in Ta- range 1.3–2.1%; the fit procedure in the range 11– ble 1. When evaluating the differential cross section, a 20%. Half of the changes of the acceptance when no correction based on the Q2-dependence of the ρ+ρ− form factor re-weighting of the Monte Carlo events is Monte Carlo sample is applied, so as to assign the performed is considered as model uncertainty. It is in cross section value to the centre of the corresponding the range 0.5–5%. The background correction proce- Q2-bin [19]. dure introduces systematic uncertainties in the range To evaluate the cross section, σγγ, of the process 2–6%. All contributions are added in quadrature to ob- γγ∗ → ρ+ρ−, the integral of the transverse photon tain the systematic uncertainties, quoted in Tables 1 2 luminosity function, LTT, is computed for each Q and 2. Finally, a normalization uncertainty of 5% ac- 28 L3 Collaboration / Physics Letters B 615 (2005) 19–30 counts for the uncertainty of the trigger efficiency de- 0.85 GeV2 there is a clear enhancement of ρ0ρ0 pro- termination. duction relative to ρ+ρ−.Thisissimilartowhatwas observed at Q2 ≈ 0 [6,7], but in contrast with the high Q2-region, where both cross sections have similar de- ∗ + − 5. Study of γγ → ρρ process pendence on Wγγ and the ρ ρ is systematically higher than the ρ0ρ0. These differences are clearly Combining the present results with the L3 data seen in the ratio on ρρ production from Refs. [1–3], we compare the + − + − 0 0 σ (ρ ρ ) ρ ρ to the ρ ρ channels and their evolution as R = ee , 0 0 afunctionofQ2. The cross section of the process σee(ρ ρ ) γγ∗ → ρρ is plotted in Fig. 4 as a function of where the sum is for the region 1.1GeV Wγγ W .ForW 2.1 GeV and 0.2GeV2 Q2 γγ γγ 2.1 GeV. In the domain 0.20 GeV2 Q2 0.85 GeV2 we measure R = 0.62 ± 0.10(stat.) ± 0.09(syst.),a value that can only be explained by the presence of an isospin I = 2 intermediate state or by a mixture of different states [8,9]. The value of this ratio for 1.2GeV2 Q2 8.5GeV2 is R = 1.81 ± 0.47(stat.) ± 0.22(syst.) [3], close to the factor 2, expected for an isospin I = 0 state. Such variation suggests different ρ-pair production mechanisms at low and high Q2. 2 The differential cross section dσee/dQ of the reaction e+e− → e+e−ρρ is shown in Fig. 5(a). The L3 measurements span a Q2-region of two orders of magnitude, over which the differential cross sections show a monotonic fall of more than four orders of magnitude. The ρρ data are fitted to a form [21] expected from QCD-based calculations [22]:

dσ 1 ee ∼ , 2 n 2 2 2 (8) dQ Q (Q +Wγγ )

where n is a constant and Wγγ is the average Wγγ value, 1.9 GeV for this measurement. Although this 2 formula is expected to be valid only for Q Wγγ, we find it provides a good parametrisation of the Q2- evolution of the ρρ data. A fit to the ρ+ρ− data finds an exponent n = 2.3 ± 0.2 with χ2/d.o.f. = 1.4/7. A value n = 2.9 ± 0.1 was found for ρ0ρ0 with χ2/d.o.f. = 6.9/10 [2]. Only the statistical uncertainties are considered in the fits. The results of the fits are shown in Fig. 5(a). The fits indicate a cross-over of the differential cross sections of ρ+ρ− and ρ0ρ0 produc- ∗ Fig. 4. Cross section of the process γγ → ρρ as a function of tion in the vicinity of Q2 ≈ 1GeV2. 2 Wγγ in three Q regions. The results from this measurement, full The measured cross section of the process γγ∗ → points in (a), are compared to previous L3 measurements of the ρρ 2 production [1–3]. The bars show the statistical uncertainties. Some ρρ as a function of Q is shown in Fig. 5(a). The + − 0 0 points from the previous measurements are displaced horizontally change of the relative magnitude of ρ ρ and ρ ρ for better readability. production is clearly visible when comparing the low- L3 Collaboration / Physics Letters B 615 (2005) 19–30 29

2 Fig. 5. The ρρ production cross section as a function of Q ,for1.1GeV Wγγ 3 GeV: (a) differential cross section of the process + − + − ∗ e e → e e ρρ and (b) cross section of the process γγ → ρρ. The results from this measurement, full points in the region Q2 < 1GeV2, are presented together with previous L3 measurements of the ρρ production [1–3]. The bars indicate the statistical uncertainties. Some points from the previous measurements are displaced horizontally for better readability. The lines in (a) represent the results of fits using the QCD-inspired form of Eq. (9). The lines in (b) represent the results of a fit to the ρ0ρ0 data based on the GVDM model [17] and of a fit basedonaρ-pole parametrisation. and the high-Q2 regions. A parametrisation, based on tion and allow to follow the evolution of ρρ cross the GVDM model [17]: sections over a Q2-region of two orders of magnitude. 2 2 A QCD-based form, derived for the description of σγγ Wγγ,Q = σγγ(Wγγ)F Q , 2 the differential cross section dσee/dQ of the process + 2 2 + − → + − 2 2 1 Q /4mV 0.22 e e e e ρρ at high Q , is found to provide a F Q = rV + , 2 (1 + Q2/m2 )2 1 + Q2/m2 good parametrisation of the Q -evolution of the ρρ V =ρ,ω,φ V 0 2 2 2 (9) data in the entire interval 0.2GeV Q 30 GeV , over which the differential cross sections show a with r = 0.65, r = 0.08, r = 0.05 and m = ρ ω φ 0 monotonic decrease of more than four orders of mag- 1.4 GeV reproduces well the Q2-dependence of the nitude, for 1.1GeV W 3GeV. ρ0ρ0 data, as shown in Ref. [2] and indicated by γγ The Q2-dependence of the cross section of the the line in Fig. 5(b). The fit finds a cross section of ∗ process γγ → ρ0ρ0 is well reproduced by a para- 13.6 ± 0.7nbfortheWγγ region 1.1GeV Wγγ 2 2 + − metrisation based on the GVDM model over the entire 3 GeV at Q = 0. The Q -evolution of ρ ρ data + − Q2-region. On the other hand, the ρ ρ data cannot cannot be satisfactorily described by this form. In ad- 0 0 be satisfactorily described by such a parametrisation. dition, as shown in Fig. 5(b), the ρ ρ data cannot 2 2 A ρ-pole description of the Q -dependence for both be described by the much steeper Q -fall of a ρ-pole 0 0 + − + − ρ ρ and ρ ρ data is excluded. parametrisation [2]; the same is true for the ρ ρ + − The relative magnitude of ρ ρ and ρ0ρ0 produc- cross section since it is decreasing with Q2 less steeply tion changes in the vicinity of Q2 ≈ 1GeV2, suggest- than the ρ0ρ0 one. ing different ρ-pair production mechanisms at low and high Q2. 6. Conclusions

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+ − + − Measurement of the KL → e e e e decay rate A. Lai, D. Marras

Dipartimento di Fisica dell’Università e Sezione dell’INFN di Cagliari, I-09100 Cagliari, Italy

A. Bevan, R.S. Dosanjh, T.J. Gershon, B. Hay 1, G.E. Kalmus, C. Lazzeroni, D.J. Munday, M.D. Needham 2, E. Olaiya, M.A. Parker, T.O. White, S.A. Wotton

Cavendish Laboratory, University of Cambridge, Cambridge CB3 0HE, UK 3

G. Barr, G. Bocquet, A. Ceccucci, T. Cuhadar, D. Cundy, G. D’Agostini, N. Doble, V. Falaleev, L. Gatignon, A. Gonidec, B. Gorini, G. Govi, P. Grafström, W. Kubischta, A. Lacourt, A. Norton, S. Palestini, B. Panzer-Steindel, G. Tatishvili 4,H.Taureg, M. Velasco, H. Wahl 5

CERN, CH-1211 Genève 23, Switzerland

C. Cheshkov, P. Hristov, V. Kekelidze, D. Madigojine, N. Molokanova, Yu. Potrebenikov, A. Zinchenko

Joint Institute for Nuclear Research, Dubna, Russian Federation

I. Knowles, V. Martin, R. Sacco, A. Walker

Department of Physics and Astronomy, University of Edinburgh, JCMB King’s Buildings, Mayﬁeld Road, Edinburgh EH9 3JZ, UK

M. Contalbrigo, P. Dalpiaz, J. Duclos, P.L. Frabetti, A. Gianoli, M. Martini, F. Petrucci, M. Savrié

Dipartimento di Fisica dell’Università e Sezione dell’INFN di Ferrara, I-44100 Ferrara, Italy

A. Bizzeti 6, M. Calvetti, G. Collazuol 7, G. Graziani, E. Iacopini, M. Lenti, G. Ruggiero

Dipartimento di Fisica dell’Università e Sezione dell’INFN di Firenze, I-50125 Firenze, Italy

H.G. Becker, M. Eppard, H. Fox 8, K. Eppard, A. Kalter, K. Kleinknecht, U. Koch, L. Köpke, P. Lopes da Silva, P. Marouelli, I. Melzer-Pellmann, A. Peters, B. Renk, S.A. Schmidt, V. Schönharting, Y. Schué, R. Wanke, A. Winhart, M. Wittgen

Institut für Physik, Universität Mainz, D-55099 Mainz, Germany 9

J.C. Chollet, L. Fayard, L. Iconomidou-Fayard, J. Ocariz, G. Unal, I. Wingerter-Seez

Laboratoire de l’Accélératur Linéaire, IN2P3-CNRS, Université de Paris-Sud, 91406 Orsay, France 10

G. Anzivino, P. Cenci, E. Imbergamo, G. Lamanna 11, P. Lubrano, A. Mestvirishvili, A. Nappi, M. Pepe, M. Piccini

Dipartimento di Fisica dell’Università e Sezione dell’INFN di Perugia, I-06100 Perugia, Italy

R. Casali, C. Cerri, M. Cirilli 12, F. Costantini, R. Fantechi, L. Fiorini, S. Giudici, I. Mannelli, G. Pierazzini, M. Sozzi

Dipartimento di Fisica, Scuola Normale Superiore e Sezione INFN di Pisa, I-56100 Pisa, Italy

J.B. Cheze, J. Cogan 13, M. De Beer, P. Debu, A. Formica, R. Granier de Cassagnac 14, E. Mazzucato, B. Peyaud, R. Turlay, B. Vallage

DSM/DAPNIA, CEA Saclay, F-91191 Gif-sur-Yvette, France

M. Holder, A. Maier, M. Ziolkowski

Fachbereich Physik, Universität Siegen, D-57068 Siegen, Germany 15

R. Arcidiacono, C. Biino, N. Cartiglia, F. Marchetto, E. Menichetti, N. Pastrone

Dipartimento di Fisica Sperimentale dell’Università e Sezione dell’INFN di Torino, I-10125 Torino, Italy

J. Nassalski, E. Rondio, M. Szleper, W. Wislicki, S. Wronka

Soltan Institute for Nuclear Studies, Laboratory for High Energy Physics, PL-00-681 Warsaw, Poland 16

H. Dibon, G. Fischer, M. Jeitler, M. Markytan, I. Mikulec, G. Neuhofer, M. Pernicka, A. Taurok, L. Widhalm

Österreichische Akademie der Wissenschaften, Institut für Hochenergiephysik, A-1050 Wien, Austria 17 Received 16 March 2005; accepted 31 March 2005 Available online 9 April 2005 Editor: W.-D. Schlatter A. Lai et al. / Physics Letters B 615 (2005) 31–38 33

Abstract + − + − The decay rate of the long-lived neutral K mesonintothee e e e ﬁnal state has been measured with the NA48 detector at the CERN SPS. Using data collected in 1998 and 1999, a total of 200 events has been observed with negligible background. + − + − −8 This observation corresponds to a branching ratio of Br(KL → e e e e ) = (3.30 ± 0.24stat ± 0.23syst ± 0.10norm) × 10 .  2005 Elsevier B.V. All rights reserved.

1. Introduction [1,2] and thus depends on the structure of the KL → γ ∗γ ∗ vertex. Phenomenological models include vector meson dominance of the photon propagator [3], + − + − The KL → e e e e decay is expected to pro- QCD inspired models [4], intermediate pseudo-scalar ∗ ∗ ceed mainly via the intermediate state KL → γ γ and vector mesons [5] and models based on chiral perturbation theory [6]. The probability for both virtual photons to convert into e+e− pairs is calculated to be in the range (5.89–6.50) × 10−5 [2,7]. E-mail address: [email protected] (M. Lenti). 1 The chiral model prediction of [7] corresponds to Present address: EP Division, CERN, 1211 Genève 23, → + − + − = × −8 Switzerland. Br(KL e e e e ) 3.85 10 , including the 2 Present address: NIKHEF, PO Box 41882, 1009 DB Amster- effect of a form factor, which increases the width by dam, The Netherlands. 4%. The interference term due to the identity of parti- 3 Funded by the UK Particle Physics and Astronomy Research cles has been calculated to change the branching ratio Council. by only 0.5%. 4 On leave from Joint Institute for Nuclear Research, Dubna The decay was first observed by the CERN NA31 141980, Russian Federation. 5 Also at Dipartimento di Fisica dell’Università e Sezione experiment [8] based on 2 observed events and has dell’INFN di Ferrara, I-44100 Ferrara, Italy. been confirmed by later measurements [9].Herewe 6 Dipartimento di Fisica dell’Università di Modena e Reggio report the result obtained from the 1998 and 1999 data Emilia, via G. Campi 213/A, I-41100 Modena, Italy. taking periods by the NA48 experiment at the CERN 7 Present address: Scuola Normale Superiore e Sezione INFN di SPS. Pisa, I-56100 Pisa, Italy. 8 Present address: Physikalisches Institut, Universität Freiburg, D-79104 Freiburg, Germany. 9 Funded by the German Federal Minister for Research and 2. Experimental setup and data taking Technology (BMBF) under contract 7MZ18P(4)-TP2. 10 Funded by Institut National de Physique des Particules et de The NA48 experiment is designed specifically to Physique Nucléaire (IN2P3), France. measure the direct CP violation parameter Re(/) 11 Present address: Dipartimento di Fisica dell’Università di Pisa e Sezione INFN di Pisa. using simultaneous beams of KL and KS. To produce 12 Present address: CERN, CH-1211 Genève 23, Switzerland. the KL beam, 450 GeV/c protons are extracted from 13 Present address: Centre de Physique des Particules de Mar- the accelerator during 2.4 s every 14.4 s and 1.1×1012 seille, Université de la Méditerranée, IN2P3-CNRS, F-13288 Mar- of these are delivered to a beryllium target. Using di- seille, France. 14 pole magnets to sweep away charged particles and Present address: Laboratoire Leprince-Ringuet, Ecole Poly- collimators to define a narrow beam, a neutral beam technique, IN2P3-CNRS, F-91128 Palaiseau, France. × 7 ± 15 Funded by the German Federal Minister for Research and of 2 10 KL per burst and divergence 0.15 mrad Technology (BMBF) under contract 056SI74. enters the decay region. The fiducial volume begins 16 Supported by the Committee for Scientific Research grant 126 m downstream of the target and is contained in 2P03B07615 and using computing resources of the Interdisciplinary an evacuated cylindrical steel vessel 89 m long and Center for Mathematical and Computational Modelling of the Uni- 2.4 m in maximum diameter. The vessel is terminated versity of Warsaw. 17 Funded by the Austrian Ministery for Traffic and Research un- at the downstream end by a Kevlar-fiber composite −3 der the contract GZ 616.360/2-IV GZ 616.363/2-VIII, and by the window of a thickness corresponding to 3 × 10 Fonds für Wissenschaft und Forschung FWF No. P08929-PHY. radiation length and is followed immediately by the 34 A. Lai et al. / Physics Letters B 615 (2005) 31–38 main NA48 detector. The sub-detectors which are used More details on the 4-track trigger can be found in + − + − in the KL → e e e e analysis are described be- [12]. low. A magnetic spectrometer consisting of a dipole magnet is preceded and followed by two sets of drift 3. Data analysis chambers. The drift chambers are each comprised of eight planes of sense wires, two horizontal, two The decay vertex of candidate events was recon- vertical and two along each of the 45◦ directions. structed from the 4-track barycenter position in the Only the vertical and horizontal planes are instru- transverse direction, calculated as a function of the mented in the third chamber. The volume between the vertex longitudinal position; each track is weighted by chambers is filled with helium at atmospheric pres- its momentum to take into account the multiple scat- sure. The momentum resolution is p/p = 0.65% at tering effect. The 4-track vertex longitudinal position 45 GeV/c. was calculated by minimizing the sum of the squared Two segmented plastic scintillator hodoscope planes transverse distance of each track from the transverse are placed after the helium tank and provide signals for vertex position; the closest distance of approach of the the trigger. 4-track vertex is defined as the square root of this sum A liquid krypton filled calorimeter (LKr) is used at its minimum. for measuring the energy, position and time of elec- Events were preselected by requiring two positive tromagnetic showers. Space and time resolutions of and two negative tracks; each couple of tracks must better than 1.3 mm and 300 ps, respectively, have been form a 2-track vertex with distance of closest approach achieved for energies above 20 GeV. The energy res- smaller than 10 cm and an axial position of the ver- olution was determined to be σ(E) = 0√.032 ⊕ 0.090 ⊕ tex less than 210 m downstream of the target; each E E E 0.0042, with E measured in GeV. track must be compatible in time with any other within A hadron calorimeter composed of 48 steel plates, 8ns. each 24 mm thick, interleaved with scintillator is The tracks extrapolated at the LKr were required to used in trigger formation and for particle identifica- be in a fiducial area given by an octagon about 5 cm tion. smaller than the outside perimeter of the calorimeter A detailed description of the detector can be found and an inner radius of 15 cm; the distance to any dead in [10]. calorimeter cell (about 80 out of 13500) had to exceed The data used in this analysis were recorded in the 2 cm. The separation between each track extrapolated 1998–1999 data taking period. Candidate events were at LKr entry face was required to be greater than 5 cm. selected by a two-stage trigger. At the first level, a trig- The momentum of each track, measured by the mag- ger requiring adjacent hits in the hodoscope is put in netic spectrometer, was required to exceed 2 GeV, well coincidence with a total energy condition ( 35 GeV), above the detector noise of 100 MeV per cluster in the defined by adding the energy deposited in the hadronic LKr. calorimeter with that seen by the trigger in the LKr Electron candidates were identified by requiring calorimeter. The second level trigger uses information that cluster centers in the LKr be within 1.5 cm of the from the drift chambers to reconstruct tracks and in- extrapolation of each track (the rms width of electro- variant masses. For the 4-track part of the trigger, the magnetic showers in LKr is 2.2 cm). To reject pion number of clustered hits in each of the first, second, showers, the ratio of cluster energy to track momen- and fourth drift chamber had to be between 3 and 7. tum E/p was required to be greater than 0.9. Track- All possible 2-track vertices were calculated online. associated clusters with E/p < 0.8 were classified as At least two vertices within 6 m of each other in the pions. axial direction had to be found. In order to determine The fiducial volume was defined by the axial po- the efficiency of the 4-track trigger, downscaled events sition of the vertex being between 127.5 and 210 m that passed a trigger based just on the total energy con- downstream of the target. Within this volume, 4-track dition were recorded (the downscaling factor changed vertices were determined with a typical longitudinal from 100 to 60 depending on the data taking period). resolution of 0.5 m, as estimated by the Monte Carlo A. Lai et al. / Physics Letters B 615 (2005) 31–38 35 simulation. The total energy had to be within 50 and 200 GeV.

+ − + − 3.1. Selection of KL → e e e e candidates and background rejection

+ − + − Candidate events for the decay KL → e e e e with all tracks identiﬁed as electrons were selected. The following four classes of background sources were relevant:

• Events with two decays KL → πeν occurring at the same time and for which the pions were misidentified as electrons. Being due to two coincident kaon decays the invariant mass of the system could be around and above the nominal KL mass. These events were largely rejected by requiring a good vertex quality: the 4-track ver- + − + − tex closest distance of approach (defined above) Fig. 1. Correlation of e e e e invariant mass with the squared 2 had to be smaller than 5 cm. Because of miss- transverse momentum pt of the reconstructed kaon. The box is the ing transverse momentum in this and most other signal region. background decays, we required the square of 2 + − the transverse momentum pt of the reconstructed ton conversion to a e e pair. Each pair of op- kaon with respect to the line joining the de- positely charged tracks was therefore required to cay vertex and the KL target to be less than be separated by more than 2 cm in the first drift 0.0005 (GeV/c)2. We chose not to cut harder chamber. According to the Monte Carlo simu- in order to include most signal events with final lation, there was no remaining background with state radiation. The position of the cut is indicated converted photons. + − + − in Fig. 1. The Monte Carlo simulation indicates • Events KL → π π e e [11,12], with the pions that 8.3% of the signal events were lost by the misidentified as electrons. Due to the misidenti- 2 requirement on pt . In addition, as already men- fication probability of 0.5% [13] this background tioned in the preselection, it was required that each was found to be negligible. track be compatible in time with any other within + − + − 8 ns. A study of sidebands in this time distribution The invariant e e e e mass distribution result- shows that the background from this source was ing from this selection is shown in Fig. 2. Note the negligible. slightly asymmetric shape of the KL mass peak, which 0 0 0 0 0 0 • Events KL → π π ,π π π , where the π s un- is due to photons radiated off the electrons in the final dergo single or double Dalitz decays or photons state. convert in the material of the detector, so that 2 Finally, a mass window of 475 MeV/c2 < + − + − positive and 2 negative electrons are detected. Due m(e e e e )<515 MeV/c2 was set to define the to the missing photons, the invariant mass of the final sample. In total, 200 candidate events were se- + − + − e e e e system is below the nominal KL mass. lected, 62 from the 1998 data period and 138 from the + − • Events KL → γγ and KL → e e γ , with con- 1999 one. version of the photons in the material upstream The Monte Carlo simulation of the background → + − 0 of the spectrometer also yield invariant masses shows that the contribution from KL π π πDalitz, → 0 0 0 → 0 0 around the nominal KL mass. The conversion KL π πDalitzπDalitz and KL πDalitzπDalitz decays probability in the material of the NA48 detector in the signal region was negligible (less than 0.1% at is of similar magnitude as that for internal pho- 90% C.L.). As a cross-check in the data, we defined 36 A. Lai et al. / Physics Letters B 615 (2005) 31–38

In addition, at least one extra cluster in the calorimeter with energy larger than 2 GeV, separated by more than 15 cm from each extrapolated track was required, with a time compatible within 3 ns with the average track time. The invariant mass of the e+e−γ system was required to be in the range of 120–140 MeV/c2 and the + − 0 invariant mass of the π π πDalitz system had to be in the range of 475–515 MeV/c2. Monte Carlo studies + − 0 showed that background from KL → π π π with one of the external photons converting in the material of the detector was completely eliminated by the requirement on the minimum distance of the electron tracks in the ﬁrst drift chamber being larger than 2 cm. All other backgrounds have been estimated to be negligible. After applying all selection criteria, a total of × 6 → + − 0 2.988 10 KL π π πDalitz decays were found (0.822 × 106 in the 1998 data sample and 2.166 × 106 + − + − Fig. 2. Invariant mass of the e e e e system. The data are shown in the 1999 one). as dots with error bars, the Monte Carlo prediction for the signal and backgrounds, normalized to the data, is shown as histogram. The position of the mass cut is also indicated. 3.3. Acceptance determination and systematic uncertainty 2 2 a control region 0.0010

The difference between signal and normalization Table 1 → + − + − trigger efficiency should cancel when computing the Systematic uncertainty contributions to Br(KL e e e e ) decay rate. We cross-checked this assumption by com- Source paring the 4-track trigger efficiency as a function of Trigger efficiency ±2.0% 4-track total energy (instead of the kaon energy) with Background estimation ±1.0% the signal spectrum. We also used a partially biased not E/p cut efficiency ±2.0% ± downscaled trigger based only on LKr information to Detector acceptance 3.0% Radiative corrections ±5.6% cross-check directly the signal trigger efficiency, ob- ± taining results in agreement with the previous method. Total 7.0% We assigned a systematic uncertainty of 2% to the 4-track trigger efficiency. the momentum dependence of the E/p cut efficiency In order to be insensitive to the real kaon energy were taken into account. The correction induced by the spectrum, the acceptance correction was applied to E/p cut decreases the branching ratio by a factor of the data in bins of kaon energy (5 GeV wide) and 0.944 ± 0.020 where the error is systematic. + − + − the corresponding average KL → e e e e /KL → In Table 1 the different contributions to the system- + − π π π 0 acceptance ratio was evaluated to be atic uncertainty are listed. Dalitz + − + − 2.89. The average acceptance for KL → e e e e → + − 0 was 5.67% and 2.03% for KL π π πDalitz decays, for events generated in the range 50 GeV

0.03syst and γCP = 0.13 ± 0.10stat ± 0.03syst. Clearly, the obtained precision on the parameters βCP and γCP is limited by the event statistics. The main sources for the systematic uncertainties are the φ dependence of the detector acceptance and the effect of wrong pairings. By imposing γCP = 0, the ﬁtted value of βCP is −0.13 ± 0.10stat. Our result is consistent with the hypothesis of a CP =−1 amplitude as expected from a KL decay.

Acknowledgements

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Measurement of the γγ → π +π − and γγ → K+K− processes at energies of 2.4–4.1 GeV

Belle Collaboration H. Nakazawa j, S. Uehara j,K.Abej,K.Abeat,H.Aiharaav,M.Akatsux,Y.Asanoba, V. Aulchenko b,T.Aushevn, S. Bahinipati f,A.M.Bakichaq,Y.Banaj, S. Banerjee ar, I. Bedny b,U.Bitenco, I. Bizjak o, S. Blyth ac, A. Bondar b, A. Bozek ad,M.Brackoˇ j,v,o, J. Brodzicka ad,A.Chenz,B.G.Cheond,R.Chistovn, Y. Choi ap, Y.K. Choi ap, A. Chuvikov ak,J.Dalsenow, M. Danilov n, M. Dash bc,A.Drutskoyf, S. Eidelman b, Y. E n ar i x,D.Epifanovb,S.Fratinao, N. Gabyshev b,A.Garmashak, T. Gershon j, G. Gokhroo ar, K. Hayasaka x, H. Hayashii y, Y. Hoshi at,S.Houz,W.-S.Houac, T. Iijima x, A. Imoto y, K. Inami x,A.Ishikawaj,R.Itohj,M.Iwasakiav,Y.Iwasakij, J.H. Kang bd, J.S. Kang q, S.U. Kataoka y, N. Katayama j,H.Kawaic, T. Kawasaki af, H.R. Khan aw,H.Kichimij,H.J.Kims,J.H.Kimap,S.K.Kimao,S.M.Kimap, S. Korpar v,o, P. Križan u,o,P.Krokovnyb, R. Kulasiri f,C.C.Kuoz,A.Kuzminb, Y. - J . Kwo n bd,G.Lederm,S.E.Leeao,T.Lesiakad,J.Lian,S.-W.Linac,D.Liventsevn, G. Majumder ar,F.Mandlm, T. Matsumoto ax,A.Matyjaad,W.Mitaroffm,H.Miyakeah, H. Miyata af,R.Mizukn, D. Mohapatra bc,T.Moriaw, T. Nagamine au, Y. Nagasaka k, E. Nakano ag, M. Nakao j, Z. Natkaniec ad,S.Nishidaj,O.Nitohay, T. Nozaki j, S. Ogawa as, T. Ohshima x, T. Okabe x, S. Okuno p,S.L.Olseni, W. Ostrowicz ad, P. Pakhlov n,H.Palkaad,C.W.Parkap,H.Parks,N.Parslowaq, L.S. Peak aq, R. Pestotnik o, L.E. Piilonen bc,H.Sagawaj,Y.Sakaij,N.Satox, T. Schietinger t, O. Schneider t, J. Schümann ac,K.Senyox, H. Shibuya as,B.Shwartzb, J.B. Singh ai, A. Somov f,N.Soniai, R. Stamen j,S.Stanicˇ ba,1,M.Staricˇ o,K.Sumisawaah, T. Sumiyoshi ax, S.Y. Suzuki j, O. Tajima j, F. Takasaki j,N.Tamuraaf, M. Tanaka j, Y. Teramoto ag, X.C. Tian aj, T. Tsuboyama j, T. Tsukamoto j,T.Uglovn,S.Unoj, G. Varner i, S. Villa t,C.C.Wangac,C.H.Wangab, M. Watanabe af, Y. Watanabe aw, B.D. Yabsley bc, A. Yamaguchi au,Y.Yamashitaae, M. Yamauchi j, J. Ying aj,Y.Yusaau, L.M. Zhang an,Z.P.Zhangan, V. Zhilich b,D.Žontaru,o

a Aomori University, Aomori, Japan b Budker Institute of Nuclear Physics, Novosibirsk, Russia c Chiba University, Chiba, Japan d Chonnam National University, Kwangju, South Korea e Chuo University, Tokyo, Japan f University of Cincinnati, Cincinnati, OH, USA g University of Frankfurt, Frankfurt, Germany h Gyeongsang National University, Chinju, South Korea i University of Hawaii, Honolulu, HI, USA j High Energy Accelerator Research Organization (KEK), Tsukuba, Japan k Hiroshima Institute of Technology, Hiroshima, Japan l Institute of High Energy Physics, Chinese Academy of Sciences, Beijing, PR China m Institute of High Energy Physics, Vienna, Austria n Institute for Theoretical and Experimental Physics, Moscow, Russia o J. Stefan Institute, Ljubljana, Slovenia p Kanagawa University, Yokohama, Japan q Korea University, Seoul, South Korea r Kyoto University, Kyoto, Japan s Kyungpook National University, Taegu, South Korea t Swiss Federal Institute of Technology of Lausanne, EPFL, Lausanne, Switzerland u University of Ljubljana, Ljubljana, Slovenia v University of Maribor, Maribor, Slovenia w University of Melbourne, Victoria, Australia x Nagoya University, Nagoya, Japan y Nara Women’s University, Nara, Japan z National Central University, Chung-li, Taiwan aa National Kaohsiung Normal University, Kaohsiung, Taiwan ab National United University, Miao Li, Taiwan ac Department of Physics, National Taiwan University, Taipei, Taiwan ad H. Niewodniczanski Institute of Nuclear Physics, Krakow, Poland ae Nihon Dental College, Niigata, Japan af Niigata University, Niigata, Japan ag Osaka City University, Osaka, Japan ah Osaka University, Osaka, Japan ai Panjab University, Chandigarh, India aj Peking University, Beijing, PR China ak Princeton University, Princeton, NJ, USA al RIKEN BNL Research Center, Brookhaven, NY, USA am Saga University, Saga, Japan an University of Science and Technology of China, Hefei, PR China ao Seoul National University, Seoul, South Korea ap Sungkyunkwan University, Suwon, South Korea aq University of Sydney, Sydney, NSW, Australia ar Tata Institute of Fundamental Research, Bombay, India as Toho University, Funabashi, Japan at Tohoku Gakuin University, Tagajo, Japan au Tohoku University, Sendai, Japan av Department of Physics, University of Tokyo, Tokyo, Japan aw Tokyo Institute of Technology, Tokyo, Japan ax Tokyo Metropolitan University, Tokyo, Japan ay Tokyo University of Agriculture and Technology, Tokyo, Japan az Toyama National College of Maritime Technology, Toyama, Japan ba University of Tsukuba, Tsukuba, Japan bb Utkal University, Bhubaneswer, India bc Virginia Polytechnic Institute and State University, Blacksburg, VA, USA bd Yonsei University, Seoul, South Korea Belle Collaboration / Physics Letters B 615 (2005) 39–49 41

Received 22 December 2004; accepted 29 March 2005 Available online 7 April 2005 Editor: M. Doser

Abstract + − + − − We have measured π π and K K production in two-photon collisions using 87.7fb 1 of data collected with the Belle + − detector at the asymmetric energy e e collider KEKB. The cross sections are measured to high precision in the two-photon ∗ center-of-mass energy (W) range between 2.4GeV

PACS: 12.38.Qk; 13.25.Gv; 13.66.Bc; 13.85.Lg

Keywords: Two-photon collisions; Mesons; QCD; Charmonium

1. Introduction pion or kaon pairs at large energy and momentum transfers, in which the amplitude is expressed by a Exclusive processes with hadronic final states test hard γγ → qq¯ subprocess and a form factor describ- various model calculations motivated by perturbative ing the soft transition from qq¯ to the meson pair. DKV, and non-perturbative QCD. Two-photon production of as well as BL, predict the sin−4 θ ∗ dependence of the exclusive hadronic final states is particularly attractive angular differential cross section, which is an impor- due to the absence of strong interactions in the initial tant test of these approaches. It is interesting to in- state and the possibility of calculating γγ → qq¯ am- vestigate experimentally an energy scale where those plitudes. The perturbative QCD calculation by Brod- theoretical predictions become valid. The recent mea- sky and Lepage (BL) [1] is based on factorization surements of γγ → π +π − and K+K− performed by of the amplitude into a hard scattering amplitude for ALEPH [5] are consistent, within their errors, with the γγ → qqq¯ q¯ and a single-meson distribution ampli- BL’s prediction of the energy dependence, but not the tude. Their prediction gives the dependence√ on the normalization. However, their dataset is not sufficient center-of-mass (c.m.) energy W(≡ s) and scatter- to conclusively test the W and sin−4 θ ∗ dependences. ing angle θ ∗ for γγ → M+M− processes In this report, we measure γγ → π +π − and → + − 2 2 γγ K K processes with high precision, and dσ + − 16πα |F (s)| γγ → M M ≈ M , (1) make quantitative comparisons with QCD predictions. | ∗| 4 ∗ − d cos θ s sin θ This analysis is based on an 87.7fb 1 data sample col- where M represents a meson and FM denotes its elec- lected at or near the Υ(4S) resonance energy, accumu- tromagnetic form factor. Vogt [2], based on the pertur- lated with the Belle detector [6] located at KEKB [7]. bative approach, claimed a need for soft contributions, as his result for the hard contribution was well below the experimental cross section obtained by CLEO [3]. 2. KEKB accelerator and Belle detector Diehl, Kroll and Vogt (DKV) proposed [4] the soft handbag contribution to two-photon annihilation into KEKB is a colliding beam accelerator of 8 GeV electrons and 3.5 GeV positrons designed to produce ¯ E-mail address: [email protected] (H. Nakazawa). copious BB meson pairs to observe CP violation. 1 On leave from Nova Gorica Polytechnic, Nova Gorica, Slove- The Belle detector, with a 1.5 T solenoidal mag- nia. netic field, surrounds the interaction point and sub- 42 Belle Collaboration / Physics Letters B 615 (2005) 39–49

◦ ◦ tends the polar angle range 17 <θlab < 150 , mea- ened in the next section), the invariant mass of these sured from the z axis, which is aligned opposite the two tracks be below 4.5GeV/c2, and that the squared positron beam. It is described in detail in Ref. [6]. missing mass of the event be above 2 GeV2/c4. Here, Briefly, charged track momenta and their decay points the two tracks are assumed to be massless parti- are measured by the central drift chamber (CDC) and cles. The latter two requirements eliminate radiative silicon vertex detector (SVD). The hadron identity of Bhabha and initial state radiation events. The remain- these charged tracks is determined using information ing events consist of two-photon production of e+e−, from the time-of-flight counters (TOF), the aerogel µ+µ−, π +π −, K+K−, and pp¯ final states as well as threshold Cerenkovˇ counters (ACC), and the specific unvetoed e+e− → τ +τ − events according to a Monte ionization in the CDC. Hadron/electron discrimina- Carlo (MC) study [8]. tion is performed using the above information as well The predicted versus measured range and trans- as the energy deposition and shower profile in the verse deviation of hits in the KLM are used to con- segmented CsI(Tl)-crystal electromagnetic calorime- struct a normalized likelihood Rµ that a track extrap- ter (ECL). Hadron/muon discrimination is achieved olated from the CDC is a muon rather than a pion using information from the neutral-kaon and muon de- or kaon. Here, we classify an event as arising from + − tector (KLM), which consists of glass resistive plate γγ → µ µ if either track has Rµ > 0.66. Simi- counters embedded in the solenoid’s iron flux return. larly, the TOF, ACC, CDC, and ECL information is used to construct a normalized likelihood Re that a re- 3. Event selection constructed track is an electron rather than a hadron. We classify an event as arising from γγ → e+e− if The signal events are collected predominantly by either track has Re > 0.66. For above two separations a trigger that requires two charged tracks penetrating 93% of signal events survive for both modes accord- through the CDC and TOF, with an opening angle in ing to the MC study described later. The TOF, ACC the rϕ plane (perpendicular to the z axis) of at least and CDC information is used to construct another nor- ◦ 135 . malized likelihood Rp that a reconstructed track is We select signal candidates according to the fol- a proton rather than a kaon, with a high value corre- lowing criteria. There must be exactly two oppositely- sponding to a proton-like track. The scatterplot of this charged reconstructed tracks satisfying the following quantity for the negative track vs. that for the positive conditions: −0.47 cos θlab 0.82 for the polar an- track in each event is shown in Fig. 1(a). Note that the + − + − gle θlab of each track; pt > 0.8GeV/c for the mo- peak near the origin contains both K K and π π mentum component in the rϕ plane of each track; candidates. Events above the hyperbolic curve dr 1 cm and |dz| < 2 cm for the origin of each − + + − R − . R − . = . , track relative to the nominal e e collision point; and p 1 01 p 1 01 0 0101 |dz1 − dz2| 1 cm for the two tracks’ origin differ- shown in the inset of Fig. 1(a), are deemed to arise ence along the z axis, where the origin is defined by the from γγ → pp¯. closest approach of the track to the nominal collision After removing events that appear to arise from point in the rϕ plane. The event is vetoed if it contains two-photon production of µ+µ−, e+e−, and pp¯ ac- any other reconstructed charged track with transverse cording to the above criteria, the remaining sample momentum above 0.1GeV/c. consists of two-photon production of K+K−, π +π −, Cosmic rays are suppressed by demanding that and residual µ+µ−,aswellase+e− → τ +τ − pro- the opening angle α between the two tracks satisfy duction where each τ lepton decays to a single pion cos α −0.997. The signal is enriched relative to or muon. Information from the TOF, ACC, and CDC other backgrounds by requiring that the scalar sum of is used to form a normalized likelihood ratio RK that the momentum of the two tracks be below 6 GeV/c, a reconstructed track is a kaon rather than a pion (or the total energy deposited in the ECL be below 6 GeV, muon), with a high value corresponding to a kaon- the magnitude of the net transverse momentum of the like track. The scatter plot of this quantity for the above-selected two charged tracks in the e+e− c.m. negative track vs. that for the positive track is shown frame be below 0.2GeV/c (this condition is tight- in Fig. 1(b). Note that the peak near the origin con- Belle Collaboration / Physics Letters B 615 (2005) 39–49 43

(a) (b)

Fig. 1. Two-dimensional plots of likelihood ratios for hadron identification: (a) Rp and (b) RK . The cut boundaries are shown in the top-view insets. tains π +π −, τ +τ −, and residual µ+µ− events. Events 4. Background rejection − + above the diagonal line R + R = 1.2, shown in K K + − the inset of Fig. 1(b), are classified as K K candi- The spectrum of the residual γγ → µ+µ− back- − + dates, while events below the line R + R = 0.8are ground within the π +π − sample is obtained from a + − K K + − classified as π π candidates (including τ τ and MC simulation program AAFH [9], based on a full + − residual µ µ backgrounds). Events in the diagonal O(α4) QED calculation, with a data sample corre- band between these two lines are discarded. sponding to an integrated luminosity of 174.2fb−1 + − The π π sample is somewhat contaminated by that is processed by the full detector simulation pro- + − non-exclusive two-photon background γγ → π π X gram and then subjected to trigger simulation and the + − + − as well as the e e → τ τ process, in roughly equal above event selection criteria. After calibration of the proportion. We note that these backgrounds appear at muon identification efficiency to match that in the high values of the magnitude of the net transverse mo- data using identified γγ → µ+µ− events, the resid- | + + −| + − + − mentum pt pt in the e e c.m. frame, and are ual µ µ background is scaled by the integrated lu- often accompanied by photons from the prompt decay minosity ratio and then subtracted from the π +π − of a neutral pion in the final state. Therefore, we reject sample. (The contamination amounts to 19–42%, de- + − events in the π π sample that contain a photon with pending on W .) energy above 400 MeV (Eγ -veto). The distributions The excess in the Eγ -vetoed histogram of Fig. 2(a) | + + −| + − of pt pt for the π π candidates before and af- above the smooth curve from the signal MC, deter application of this veto are shown as the histograms scribed in more detail below, is attributed to non- in Fig. 2(a). exclusive γγ → π +π −X events that are not rejected + − + − + − + − The yields of the π π and K K events are by the Eγ -veto; most of the e e → τ τ events expressed as functions of three variables: W derived are rejected by this veto. A similar excess appears ∗ from the invariant mass of the two mesons, | cos θ | in Fig. 2(b) for the γγ → K+K− process. Assum- | + + −| and pt pt . Eighty-five 20 MeV wide bins in W ing that this remaining background is proportional to times six bins in the cosine of the γγ c.m. scattering net transverse momentum, we determine the slope us- ∗ angle θ times twenty bins in net transverse momen- ing the difference between data and MC in the range | + + −| tum are used in the ranges 2.4GeV

+ − + − + − + − Fig. 2. |p + p | distribution for π π (a) and K K (b) candidates. The dashed and solid histograms in π π indicate the distribution t t + − of events before and after Eγ -veto (which is not applied to the K K candidates), respectively. The arrows indicate the upper boundaries of | + + −| + − pt pt for the signal. The residual muon background has been subtracted from the π π distribution. The curves show the signal MC distribution which is normalized to the signal candidates at the leftmost bin.

∗ + − + − Fig. 3. Number of events (| cos θ | < 0.6) obtained for the γγ → π π (solid) and γγ → K K (dashed) samples after the background subtraction. there is no dependence on the scattering angle θ ∗.Us- γγ → π +π − (γ γ → K+K−). Thus, we obtain 6919 ing the smoothed slope, the estimated non-exclusive and 6234 signal events for π +π − and K+K−, respec- background, which amounts to at most 4% (9%) for tively. The background-subtracted yields, integrated π +π − (K+K−) below 3.1 (3.3) GeV but as much as over net transverse momentum and scattering angle, 28% (26%) in the highest W bin, is subtracted from are shown as a function of W in Fig. 3. each bin. Finally, we restrict our signal region to net The signatures of the χc0(1P) and χc2(1P) reso- transverse momentum below 0.05 (0.10) GeV/c for nances are observed in both π +π − and K+K− chan- Belle Collaboration / Physics Letters B 615 (2005) 39–49 45 nels. By fitting each W distribution outside the range value of the trigger efficiency is ∼ 93% for events in 3.3–3.7 GeV to a cubic polynomial, we see an ex- the acceptance. cess of 129 (153) events in the π +π − (K+K−) chan- The efficiency-corrected measured differential + − + − nel in the χc0 range of 3.34–3.44 GeV and a corre- cross sections for γγ → π π and γγ → K K , ∗ sponding excess of 54 (33) events in the χc2 range normalized to the partial cross section σ0 for | cos θ | < of 3.54–3.58 GeV. We obtain consistent results from 0.6, are shown in Fig. 4 for each 100 MeV wide W a fit of each distribution to a cubic polynomial plus bin. The partial cross sections σ0 for both processes, a Breit–Wigner (χc0) or a Gaussian (χc2) peak. The integrated over the above scattering angle range, are + − χc0 statistical significance is 6.2σ(8.2σ)in the π π shown in Fig. 5 (along with their ratio) and itemized (K+K−) channel, where σ is the standard deviation. in Table 1 as a function of W . The χc2 statistical significance is 4.8σ (3.7σ )inthe π +π − (K+K−) channel. The significances are taken from the square root of the difference of the goodness- 6. Systematic errors of-fit values from the two fits where the peak term in the above fit function is included or excluded. As- 4 ∗ suming a flat (sin θ ) shape for the χc0 (χc2) reso- The dominant systematic errors are listed in Ta- nance [10], we subtract the above excesses bin by bin ble 2. The error in track finding efficiency is estimated from each angular distribution in the above W ranges. by comparing η → π +π −π 0 and η → γγ samples in data and MC. The uncertainty due to trigger efficiency is estimated by comparing the yields of γγ → µ+µ− 5. Derivation of the cross section in real and simulated data [9] after accounting for the background from e+e− → µ+µ− nγ events (varying The differential cross section for a two-photon with W from 0.5–4.6%), which have the same topol- process to a two-body final state arising from an ogy [13]. To estimate the systematic error in the K/π + − + − electron–positron collision is given by separation for the π π (K K ) mode, which is dominated by the contribution from the fake rate for + − + − dσ | ∗|; → K K (π π ) events, we have compared the two- ∗ W, cos θ γγ X d| cos θ | dimensional likelihood ratio distribution of the experi- | ∗|; + − → + − mental data in Fig. 1(b) to that from the corresponding = N(W, cos θ e e e e X) ∗ ∗ , signal MC, normalized appropriately. We assign the Lγγ(W) W | cos θ | (W,| cos θ |) L dt (2) so-estimated background contribution to the systematic error from this source. The uncertainty in the rel- where N and denote the number of the signal events ative muon identification efficiency between real and and a product of detection and trigger efficiencies, simulated data is used to determine the error asso- respectively; L dt is the integrated luminosity, and ciated with the residual µ+µ− subtraction from the L is the luminosity function, defined as γγ π +π − sample. We use an error of 100% of the sub- dσ ; + − → + − tracted value for the non-exclusive background sub- dW (W e e e e X) Lγγ(W) = . traction. We allow the number of χ events to fluctu- σ(W; γγ → X) cJ ate by up to 20% of the measured excess to estimate ∗ + − The efficiencies (W,| cos θ |) for γγ → π π the error due to the χc subtraction that is applied for and γγ → K+K− are obtained from a full Monte the energy bins in the range 3.3GeV

−1 | ∗| + − + − Fig. 4. Angular dependence of the cross section, σ0 dσ/d cos θ ,fortheπ π (closed circles) and K K (open circles) processes. The − ∗ curves are 1.227 sin 4 θ . The errors are statistical only.

+ − + − ∗ − Fig. 5. Cross section for (a) γγ → π π ,(b)γγ → K K in the c.m. angular region | cos θ | < 0.6 together with a W 6 dependence line derived from the ﬁt of s|RM |. (c) Cross section ratio. The solid line is the result of the ﬁt for the data above 3 GeV. The errors indicated by short ticks are statistical only. Belle Collaboration / Physics Letters B 615 (2005) 39–49 47

Table 1 + − + − ∗ Cross sections and errors for the γγ → π π and γγ → K K processes, in the angular range | cos θ | < 0.6 γγ → π+π− γγ → K+K−

W, GeV σ0, nb stat. err., nb syst. err., nb σ0, nb stat. err., nb syst. err., nb 2.4–2.5 0.832 0.026 0.083 0.595 0.019 0.062 2.5–2.6 0.625 0.024 0.070 0.549 0.018 0.059 2.6–2.7 0.636 0.025 0.067 0.497 0.018 0.054 2.7–2.8 0.530 0.023 0.059 0.377 0.016 0.043 2.8–2.9 0.448 0.022 0.048 0.364 0.016 0.043 2.9–3.0 0.407 0.021 0.042 0.313 0.015 0.038 3.0–3.1 0.302 0.019 0.032 0.266 0.014 0.034 3.1–3.2 0.247 0.017 0.026 0.205 0.013 0.027 3.2–3.3 0.202 0.016 0.022 0.199 0.013 0.027 3.3–3.4 0.153 0.016 0.023 0.130 0.012 0.022 3.4–3.5 0.103 0.014 0.018 0.093 0.011 0.018 3.5–3.6 0.093 0.014 0.017 0.093 0.011 0.016 3.6–3.7 0.079 0.011 0.013 0.070 0.009 0.012 3.7–3.8 0.068 0.011 0.014 0.062 0.008 0.011 3.8–3.9 0.043 0.009 0.010 0.048 0.007 0.009 3.9–4.0 0.035 0.008 0.010 0.039 0.007 0.008 4.0–4.1 0.041 0.009 0.013 0.039 0.007 0.008

Table 2 where M denotes either a pion or a kaon. Here, they in- Contributions to the systematic errors. A range is shown when the troduce the “annihilation form factor” R2M (s), which error has a W dependence can be determined experimentally. + − + − Source π π K K Fig. 4 shows normalized angular distributions as + − + − Tracking efﬁciency 4% 4% a function of W ,forπ π and K K modes. The Trigger efﬁciency 4% 4% solid curves indicate the expectations from a sin−4 θ ∗ K/π separation 0–1% 2–4% behavior predicted by BL and DKV models µµ background subtraction 5–17% 0% Non-exclusive background subtraction 4–27% 7–20% 1 dσ 1.227 = . (4) Luminosity function 5% 5% σ d| cos θ ∗| 4 ∗ Integrated luminosity 1% 1% 0 sin θ χc subtraction (3.3GeV

Table 3 + − Results for the product of the two-photon decay width and the branching fraction, Γγγ(χcJ )B(χcJ → M M ). The second column gives the observed χcJ yields in the W region of 3.34–3.44 GeV (3.54–3.58 GeV) for χc0 (χc2). The ﬁrst and second errors for ΓγγB are statistical and systematic, respectively + − Number of events Γγγ(χcJ )B(χcJ → M M ) [eV] Signiﬁcance + − γγ → χc0 → π π 129 ± 18 15.1 ± 2.1 ± 2.36.2σ + − γγ → χc0 → K K 153 ± 17 14.3 ± 1.6 ± 2.38.2σ + − γγ → χc2 → π π 54 ± 10 0.76±0.14±0.11 4.8σ + − γγ → χc2 → K K 33 ± 80.44±0.11±0.07 3.7σ

2 2 0.05 GeV and s|R2K |=0.68 ± 0.01 ± 0.05 GeV rules. This compensates for the partial account of the by ﬁts to the data in this W range. Fig. 5 shows the SU(3) breaking by BL who used the same wave func- observed cross sections in comparison with the recent tions for pions and kaons so that the cross section ratio ALEPH data [5]. The lines indicate expectations with is equal to the fourth power of the ratio of the kaon the s|R2M (s)| values from the ﬁt above. Our data are and pion decay constants. consistent with W −6 behavior predicted by BL [1] and BF [14] models.

We can also directly obtain the power n of the W - 8. Two-photon decay width of χ cJ resonances n dependence (σ0 ∝ W ) from the data. We ﬁnd n = −7.9 ± 0.4 ± 1.5forπ +π − and −7.3 ± 0.3 ± 1.5 + − The measured yields of χc0 and χc2 events can be for K K ,for3.0GeV

9. Conclusion ence and Technology of India; the BK21 program of the Ministry of Education of Korea and the CHEP − Using 87.7fb 1 of data collected with the Belle SRC program of the Korea Science and Engineering detector at KEKB, we have measured with high pre- Foundation; the Polish State Committee for Scientiﬁc + − cision the cross sections for the γγ → π π and Research under contract No. 2P03B 01324; the Min- + − γγ → K K processes in the two-photon c.m. en- istry of Science and Technology of the Russian Fed- ergy range 2.4GeV

Transition from baryonic to mesonic freeze-out

J. Cleymans a, H. Oeschler b,K.Redlichc, S. Wheaton a

a UCT-CERN Research Centre and Department of Physics, University of Cape Town, Rondebosch 7701, South Africa b Institut für Kernphysik, Technische Universität Darmstadt, D-64289 Darmstadt, Germany c Institute of Theoretical Physics, University of Wrocław, Pl-45204 Wrocław, Poland Received 6 December 2004; received in revised form 24 February 2005; accepted 22 March 2005 Available online 9 April 2005 Editor: V. Metag

Abstract + + The recently discovered sharp peak in the K /π ratio in relativistic heavy-ion collisions is discussed in the framework of the statistical model. In this model a rapid change is expected as the hadronic gas undergoes a transition from a baryon- = dominated to a meson-dominated gas. The transition occurs√ at a temperature T 140 MeV and baryon chemical potential µB = 410 MeV corresponding to an incident energy of sNN = 8.2 GeV. The maximum in the Λ/π ratio is well reproduced + + by the statistical model, but the change in the K /π ratio is much less pronounced than the one observed by the NA49 − − − + − + Collaboration. The calculated smooth increase of the K /π ratio and the shape of the Ξ /π and Ω /π ratios exhibiting maxima at different incident energies is consistent with the presently available experimental data. We conclude that the measured particle ratios with 20–30% deviations agree with a hadronic freeze-out scenario. These deviations seem to occur just in the transition from baryon-dominated to meson-dominated freeze-out.  2005 Elsevier B.V. All rights reserved.

The NA49 Collaboration has recently performed difference in heavy-ion collisions. This transition has a series of measurements of Pb–Pb collisions at 20, been referred to in Ref. [1] as the “horn”. A strong 30, 40, 80 and 158 A GeV beam energies [1–3]. variation with energy of the Λ/π ratio has been pre- When these results are combined with measurements dicted on the basis of arguments put forward in [10]. at lower beam energies from the AGS [4–9] they re- It has also been suggested recently in Ref. [11] that veal an unusually sharp variation with beam energy in this is a signal of the special critical point of the QCD the Λ/π, with π≡3/2(π + + π −), and K+/π + phase diagram [12–14] at high baryon density. In this ratios. Such a strong variation with energy does not Letter we explore another, less spectacular, possibility occur in pp collisions and therefore indicates a major for the origin of the sharp maximum, namely as being due to the transition from a baryon-dominated to a meson-dominated hadronic gas. The distinction be- E-mail address: [email protected] (J. Cleymans). ing based on whether the entropy of the hadronic gas

√ Fig. 1. The net baryon density as a function of sNN calcu- Fig. 2. The entropy density normalised to T 3 as a function of the lated along the chemical freeze-out curve [15] corresponding to beam energy as calculated in the statistical model using THERMUS E/N = 1GeV. [21]. is dominated by baryons or by mesons. For this purpose we study various quantities√ along the freeze-out Above this value the entropy is carried mainly by curve [15] as a function of sNN. mesonic degrees of freedom. It is remarkable that the In the statistical model a steep rise at low ener- entropy density divided by T 3 is constant over the en- gies and a subsequent ﬂattening off leading to a mild tire freeze-out curve, except for the low-energy region maximum in the K+/π + ratio, was predicted many corresponding to the SIS energy region. The line de- years ago [16–18]. The sharpness of the observed peak noting the transition from a baryon-dominated to a therefore comes as a surprise. On the other hand, a meson-dominated hadron gas is shown in Fig. 3.This sharp peak in the Λ/π ratio was predicted by the line crosses the freeze-out curve at a temperature of statistical model [18] and is in good agreement with T = 140 MeV, when the baryon chemical potential the data. While the statistical model cannot explain equals µB = 410 MeV. The corresponding invariant + + √ the sharpness of the peak in the K /π ratio, there energy is sNN = 8.2GeV. are nevertheless several phenomena, giving rise to The strong decrease in the net baryon density seen the rapid change, which warrant a closer look at the in Fig. 1 is due to the fact that low energies are char- model. In Fig. 1 we show the net baryon density cal- acterized by a very low multiplicity of mesons and, culated along√ the chemical freeze-out curve [15] as a correspondingly, a very large baryon-to-meson ratio. function of sNN. This curve shows a clear maximum As a consequence, the baryon chemical potential is with the net baryon density decreasing rapidly towards also very large. As the beam energy is increased, me- higher energies. son production increases and the baryon chemical po- To get a better estimate of the statistical parameters tential decreases. The number of strange baryons pro- in the transition region we show in Fig. 2 the entropy duced in heavy-ion collisions at different collision en- density as a function of beam energy following the ergies will follow the net baryon density since a large freeze-out curve given in [15]. The separate contri- baryon chemical potential will also enhance the num- bution of mesons and of baryons to the total entropy ber of hyperons. + + is also shown in this ﬁgure. There√ is a clear change The corresponding K /π ratio is shown in Fig. 4. of baryon to meson dominance at sNN = 8.2GeV. As is well known [19,20], the statistical model de- 52 J. Cleymans et al. / Physics Letters B 615 (2005) 50–54

Fig. 3. The chemical freeze-out curve [15] together with the regions where baryonic (or mesonic) contributions to the entropy density dominate, separated by the dashed line. Calculated in the statistical model using THERMUS [21]. scription leads to a mild maximum in this ratio which does not reproduce the so-called “horn” observed by the NA49 Collaboration [1]. The observed deviations at the highest SPS energy have been interpreted as a lack of full chemical equilibrium in the strangeness Fig. 4. (a) The Λ/π ratio as a function of beam energy. (b) The + + − − sector, leading to a strangeness suppression factor, γs , K /π and K /π ratios as a function of energy. The solid and deviating from its equilibrium value by about thirty dashed lines are the predictions of the statistical model calculated percent. Detailed fits using the statistical model in using THERMUS [21]. The data points are from Refs. [1–4,7,9]. the region of the “horn” show rapid variations in γs [20] which do not lend themselves to any interpretation. There is no corresponding peak in the K−/π − combination of the facts that strangeness has to be bal- ratio because the production of K− is not tied to anced, the baryon chemical potential decreases rapidly that of baryons. As the relative number of baryons with energy and the multi-strange baryons have suc- decreases with increasing energy, there is no corre- cessively higher thresholds. The values are listed in sponding decrease in the number of K− as is the Table 1. case with K+ as these must be balanced by strange It is to be expected that if these maxima do not all baryons. occur at the same temperature, i.e., at the same beam It is worth noting that the maxima in the ratios energy, then the case for a phase transition is not very for multi-strange baryons occur at ever higher beam strong. The observed behavior seems to be governed energies. This can be seen clearly in Fig. 5 for the by properties of the hadron gas. More detailed experi- Ξ −/π + ratio which peaks at a higher value of the mental studies of multi-strange hadrons will allow the beam energy. The ratio Ω−/π + also shows a (very verification or disproval of the trends shown in this weak) maximum, as can be seen in Fig. 5. The higher Letter. It should be clear that the Ω−/π + ratio is very the strangeness content of the baryon, the higher in broad and shallow and it will be difficult to find a max- energy is the maximum. This behavior is due to a imum experimentally. J. Cleymans et al. / Physics Letters B 615 (2005) 50–54 53

If the change in properties of the above excitation functions were associated with a genuine deconﬁne- ment phase transition one would expect these changes to occur at the same beam energy. It is clear that more data are needed to clarify the precise nature of the sharp variation observed by the NA49 Collabora- tion.

Acknowledgements

We thank C. Blume for his help with the NA49 data. We acknowledge the support of the German Bun- desministerium für Bildung und Forschung (BMBF), the Polish State Committee for Scientiﬁc Research (KBN) grant 2P03 (06925), the National Research Foundation (NRF, Pretoria) and the URC of the Uni- versity of Cape Town. − + − + Fig. 5. The Ξ /π (full line) and Ω /π (dashed line) ratios as a function of beam energy calculated using THERMUS [21].The data points are from Refs. [22–24]. The square points correspond to + − + References the (Ω + Ω)/π¯ ratio while the round points are for the Ξ /π ratio. [1] M. Ga´zdzicki, NA49 Collaboration, J. Phys. G: Nucl. Part. Phys. 30 (2004) S701. [2] S.V. Afanasiev, et al., NA49 Collaboration, Phys. Rev. C 66 Table 1 (2002) 054902. Maxima in particle ratios [3] T. Anticic, et al., Phys. Rev. Lett. 93 (2004) 022302; √ Ratio Maximum at sNN (GeV) Maximum value C. Blume, et al., SQM04 Proceedings, J. Phys. G, in press. [4] L. Ahle, et al., E802 Collaboration, Phys. Rev. C 57 (1998) Λ/π 5.10.052 − + 466. Ξ /π 10.20.011 + + [5] L. Ahle, et al., E802 Collaboration, Phys. Rev. C 60 (1999) K /π 10.80.22 − + 044904; Ω /π 27.00.0012 L. Ahle, et al., E802 Collaboration, Phys. Rev. C 60 (1999) 064901. [6] L. Ahle, et al., E866/E917 Collaboration, Phys. Lett. B 490 In conclusion, while the statistical model cannot (2000) 53. + + [7] S. Albergo, et al., Phys. Rev. Lett. 88 (2002) 062301. explain the sharpness of the peak in the K /π [8] S. Ahmad, et al., Phys. Lett. B 381 (1996) 3. ratio, its position corresponds precisely to a transi- [9] J. Klay, et al., E895 Collaboration, Phys. Rev. C 68 (2003) tion from a baryon-dominated to a meson-dominated 054905. hadronic gas. This transition occurs at a temperature [10] M. Ga´zdzicki, M.I. Gorenstein, Acta Phys. Pol. B 30 (1999) = = 2705. T 140 MeV, a baryon√ chemical potential µB = [11] R. Stock, J. Phys. G: Nucl. Part. Phys. 30 (2004) S633. 410 MeV and an energy sNN 8.2 GeV. In the [12] M.G. Alford, K. Rajagopal, F. Wilczek, Phys. Lett. B 442 statistical model this transition leads to a sharp peak (1998) 247; in the Λ/π ratio, and to moderate peaks in the Y.H. Atta, T. Ikeda, Phys. Rev. D 67 (2003) 014028. K+/π +, Ξ −/π + and Ω−/π + ratios. Furthermore, [13] Z. Fodor, S.D. Katz, J. High Energy Phys. 0203 (2002) 014; these peaks are at different energies in the statistical Z. Fodor, S.D. Katz, hep-lat/0402006. [14] F. Karsch, K. Redlich, A. Tawﬁk, Phys. Lett. B 571 (2003) 67. model. The statistical model predicts that the maxima − + − + [15] J. Cleymans, K. Redlich, Phys. Rev. Lett. 81 (1998) 5284. in the Λ/π, Ξ /π and Ω /π occur at increas- [16] J. Cleymans, H. Oeschler, K. Redlich, Phys. Rev. C 59 (1999) ing beam energies. 1663. 54 J. Cleymans et al. / Physics Letters B 615 (2005) 50–54

[17] F. Becattini, J. Cleymans, A. Keränen, E. Suhonen, K. Redlich, [20] F. Becattini, M. Ga´zdzicki, A. Keränen, J. Manninen, R. Stock, Phys. Rev. C 64 (2001) 024901. Phys. Rev. C 69 (2004) 024905. [18] P. Braun-Munzinger, J. Cleymans, H. Oeschler, K. Redlich, [21] S. Wheaton, J. Cleymans, hep-ph/0407174. Nucl. Phys. A 697 (2002) 902. [22] C. Meurer, et al., J. Phys. G 30 (2004) S14; [19] P. Braun-Munzinger, K. Redlich, J. Stachel, in: R. Hwa, X.N. S.V. Afanasiev, et al., Phys. Lett. B 358 (2002) 275. Wang (Eds.), Quark–Gluon Plasma 3, World Scientiﬁc, Singa- [23] P. Chung, et al., Phys. Rev. Lett. 91 (2003) 202301. pore, 2003, nucl-th/0304013. [24] C. Alt, et al., nucl-ex/0409004. Physics Letters B 615 (2005) 55–60 www.elsevier.com/locate/physletb

Absence of structure in the 20,22Ne + 118Sn quasi-elastic barrier distribution

E. Piasecki a,Ł.Swiderski´ a,P.Czosnykaa, M. Kowalczyk a,K.Piaseckia, M. Witecki a, T. Czosnyka b,J.Jastrz˛ebski b, A. Kordyasz b, M. Kisielinski´ b, T. Krogulski c, M. Mutterer d, S. Khlebnikov e, W.H. Trzaska f, K. Hagino g,N.Rowleyh

a Institute of Experimental Physics, Warsaw University, Poland b Heavy Ion Laboratory, Warsaw University, Poland c Institute of Experimental Physics, University in Bialystok, Poland d Institut für Kernphysik, Technische Universität, Darmstadt, Germany e Radium Institute, St. Petersburg, Russia f University of Jyväskylä, Finland g Tohoku University, Sendai, Japan h Institut de Recherches Subatomiques/Universite Louis Pasteur (UMR 7500), Strasbourg, France Received 15 October 2004; received in revised form 4 February 2005; accepted 24 March 2005 Available online 7 April 2005 Editor: V. Metag

Abstract Motivated by the extreme deformation parameters of the projectile, we have measured quasi-elastic scattering for 20Ne + 118Sn. In contrast to calculations based on known collective states, the experimental barrier distribution is structureless. A comparison with the system 22Ne + 118Sn shows that this smoothing is unlikely to be due to nucleon- or α-transfer channels, and is more likely to be due to coupling to many other weak channels.  2005 Elsevier B.V. All rights reserved.

PACS: 25.70.Bc; 25.70.Hi; 25.70.Jj

Keywords: Coupled channels; Coulomb barrier distribution; 20Ne; 22Ne; Quasi-elastic scattering

Our understanding of the interplay of nuclear reac- of barrier distributions, which show the remarkable ef- tion channels has been greatly enhanced by the study fects of strong coupling to collective excitations. Such work has mainly taken place in the context of fusion E-mail address: [email protected] (E. Piasecki). [1], where the structures are observed in the “experi-

0370-2693/$ – see front matter  2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physletb.2005.03.070 56 E. Piasecki et al. / Physics Letters B 615 (2005) 55–60 mental barrier distribution” [2] coupling to collective states gives rise to a struc- 2 ture in the barriers that must be crossed if the d (E σfus) Dfus(E) = . (1) target and projectile nuclei are to come into close dE2 contact. However, after crossing the barriers, the Here σfus is the fusion cross section and E is the inci- system may quasi-fission (binary separation be- dent centre-of-mass energy. In many cases, this func- fore forming an equilibrated compact CN) or form tion gives a clear fingerprint of the couplings involved a compound nucleus that may then yield evapora- (rotational or vibrational states in the target and/or pro- tion residues or may itself fission. This is partic- jectile in varied degrees of complexity). In some cases ularly true of very heavy systems leading to the multi-nucleon transfer channels may also play an im- creation of superheavy elements. Thus three cross portant role [3–5]. sections must be measured, two of which (fusion– One can also define barrier distributions in terms of fission and quasi-fission) yield angular distribu- the quasi-elastic cross section, that is the sum of elas- tions which must be integrated. Thus a complex tic, inelastic and transfer channels [6]. This has several experiment is necessary if one wishes to exploit advantages: Eq. (1). However, quasi-elastic scattering is the complementary part of the barrier-crossing flux • The quasi-elastic barrier distribution is defined in (and related to it by unitarity) and in principle terms of a first derivative rather than a second there are no additional complications in the quasi- derivative: elastic distribution for such systems, though there is of course the practical consideration of discrim- d(σqe/σR) D =− . (2) inating against fission fragments in the detectors. qe dE This leads to a smaller statistical error in Dqe com- Because of the above potentially interesting appli- pared with Dfus for a given statistical error in the cations it is important to understand as well as possible corresponding cross section. This could make the the relationship between Dqe and Dfus. use of Dqe interesting for low-intensity, for exam- Despite the above-mentioned advantages, there is ple radioactive, beams. a possible disadvantage to Dqe. While theoretical cal- • Eq. (2) is most readily derived and understood for culations generally give very similar results for Dqe = a scattering angle θ π, though we can apply the and Dfus [6,7], it has been noted that the experimental formula for any large scattering angle by intro- quasielastic distribution, while having the same over- ducing an effective energy into the cross section: all shape as the fusion distribution, may possess less σqe(Eeff) ≈ σqe(E, θ) with [6] detailed structure. This disagreement was observed in 16 + 144 2 Ref. [6] for the system O Sm. Here, a small sec- E = E. (3) ondary peak (arising essentially from coupling to the eff 1 + cosec(θ/2) lowest octupole phonon state in the target) is present This corrects for centrifugal effects and means in Dfus but not in Dqe. On the other hand, for the sys- 40 90 that σqe can be measured simultaneously at several tem Ca + Zr, similar phonon structures are present effective energies with a single beam energy by in both distributions [6]. The reason for this difference using detectors disposed at different scattering an- is not clear. gles. Along with the use of degraders, this greatly The systems 16O + 116,119Sn have already been facilitates studies with a cyclotron accelerator [8]. measured at the Warsaw cyclotron [8]. For those sys- • The measurement of the total quasi-elastic cross tems, σfus has also been measured [9] and neither Dqe section means that relatively simple detectors can nor Dfus display any significant structure, either theo- be exploited since there is no need for a great res- retically or experimentally. olution either in mass, charge or energy. Our aim in the present work is to try to elucidate • For systems leading to a heavy compound nu- the above problem by taking a system where the pro- cleus (CN), Eq. (1) should be written for the total jectile has a known large deformation that leads to a “barrier-passing” cross section. In other words the prediction of very marked structure in both Dqe and E. Piasecki et al. / Physics Letters B 615 (2005) 55–60 57

Dfus and to measure its quasi-elastic scattering. We have chosen the system 20Ne+ 118Sn for the very large deformation of the projectile: β2 = 0.46, β4 = 0.27. These values seem to be representative of results of many scattering experiments with projectiles like electrons, protons, pions and α-particles [10]. The experimental method was very similar to that described in Ref. [8]. That is, we measured quasielastic large-angle scattering excitation functions using 30 10×10 mm PIN diodes placed at 130◦, 140◦ and 150◦ in the laboratory system and two “Rutherford” semiconductor detectors (of 6 mm diameter) placed at 35◦ with respect to the beam. The experiment consisted of measuring the backward/forward counting ratio as a function of the projectile energy. The geometry of the Fig. 1. Examples of the Q-value spectra measured for 20,22Ne scat- ◦ ◦ PIN diode set-up (including the detector–target sepa- tered at 140 and 35 on 118Sn for energy corresponding to max- rations) was the same as in Ref. [8]. In addition, 4 tele- imum of the barriers. The calculated distribution is presented by a scopes at backward angles (two each at 120◦ and 160◦) dashed line. For easier comparison the spectra have been normalized to a common peak value. were used to measure the relative intensity of light charged particle transfers. The light charged particle transfer was not measured at forward angles, however, tion, the stability of the gain, the offset and resolution the CC calculations indicate that at these energies it is of the electronics and detectors as well as the beam- strongly backward peaked. energy were continuously monitored using a precise The 20Ne beam with intensity of 1–40 pnA (de- pulse generator and an α-particle source. pending on the energy) from the Warsaw Cyclotron Data analysis was performed as in Ref. [8]: from impinged on the 130 µg/cm2 target of 118Sn (enriched the kinetic energy spectra, assuming two-body kine- 2 matics, we calculate the Q-value spectrum (see Fig. 1). to 96.6%) on a 65 µg/cm Al2O3 backing, which was taken into account in data analysis. Then the number of counts in the integration limits − The use of nickel foils as energy degraders and the ( 3, 15) MeV for the forward detectors and − different detectors angles enable measurements at ef- ( 3, 22) MeV for the backward ones are obtained, E is calculated and the σ /σ ratios constructed. fective energies Eeff with sufficiently small steps. En- eff qe R ergy calibration was performed using a precise pulse Data were binned over 0.75 MeV intervals and a generator and Bi–Po α-particle sources (the estimated Savitzky–Golay filter (cubic, 5 points) was applied. pulse height defect of the semiconductor detectors was Then, using Eq. (2), the barrier distributions were de- negligibly small). termined. The energy resolution of the experiment is of crit- The telescopes enabled us to check that the projec- ical importance for assessing the meaning of the re- tile-like fragments following elastic, inelastic, p- and sults. It was continuously monitored during the exper- α-transfer reactions were contained within the integra- iment and turned out to vary in time within the limits tion limits, while protons and α-particles (for example, 0.8–1.6 MeV (FWHM). Apart from the contributions evaporated from fusion products) were rejected. Nei- coming from straggling in the degrader and the energy ther fission nor quasi-fission fragments were observed. loss in the target (≈ 0.3 MeV each), the main contri- The possible products of n-transfer reaction, in our bution to the width comes from characteristics of the measurement not experimentally distinguishable from beam. Precise information on the beam-energy spread the inelastic excitations, are in the same energy range was determined from the energy spectra registered in and are also necessarily included in the quasi-elastic the forward detectors. Other effects, for example, the events. geometry of detection, had very little influence on the The experimental results for σqe/σR and Dqe are energy resolution. In addition to checking the resolu- shown in Fig. 2(a) and (b) (filled circles). Also shown 58 E. Piasecki et al. / Physics Letters B 615 (2005) 55–60

20 118 Fig. 3. (a) Experimental Dqe for Ne + Sn compared with the theoretical distribution (solid line). The dotted line shows the no-coupling result. See text. Both calculated distributions are folded with the experimental resolution 1.6 MeV. The dashed line shows 22 118 the results without folding. (b) Experimental Dqe for Ne + Sn compared with the theoretical calculation folded with a 2.0 MeV resolution (solid line), and ignoring the resolution (dashed line). Fig. 2. (a) Experimental values of σqe/σR as a function of Eeff (see text) for the systems 20Ne + 118Sn (filled circles) and 22Ne + 118Sn (open circles). (b) The derived barrier distributions Dqe. essentially shifts the entire distribution to lower energies (effectively a renormalisation of the central po- are the results for a 22Ne beam (open circles) which tential). Here the shift is 0.9 MeV. In addition, we will be discussed below. We note that there is no sig- included couplings to the lowest vibrational state in ∗ = = nificant structure in either of the Dqe distributions. the target (E 1.41 MeV; β2 0.11). Also shown in This is in evident disagreement with theoretical cal- Fig. 3(a) is the no-coupling calculation (dotted curve). culations which we shall present below. We see that the coupled channels calculations result in In Fig. 3(a), the solid line is the result of a coupled- a Dqe distribution which possesses two distinct peaks channels (CC) calculation performed using the pro- in contrast to the experimental curve which is rela- gram CCQUEL [11]. (Unless otherwise stated, the tively smooth and centered around 62 MeV. theoretical curves take into account the experimental One should emphasize that this structure cannot be + resolution.) This calculation includes the states 0 , removed by changing β2 and β4 values within the ob- 2+,4+,6+ in the 20Ne ground state rotational band. served scatter quoted in Ref. [10]. Calculations were The results converge rapidly as the number of states performed with inclusion of reorientation terms but is increased. It was verified that truncation of calcula- their omission affects only the details of the distrib- tions at the 6+ level is entirely sufficient for our pur- ution and not the presence of the structure. The two- poses. In addition, this calculation takes account of the peaked structure arising from the 20Ne rotational band strong octupole phonon state in the projectile (E∗ = differs from the smooth barrier distribution obtained 154 5.62 MeV; β3 = 0.39). As observed in other cases for heavy deformed nuclei (for example, Sm [13]) [12], however, the presence of a high-lying phonon due to the large excitation energies in the present case E. Piasecki et al. / Physics Letters B 615 (2005) 55–60 59

(E(2+) = 1634 keV as opposed to 82 keV). These tems 40Ca + 90,96Zr are very different from each other reduce considerably the influence of the higher-lying and it has been suggested [5,6,15] that this is due to the members of the rotational band and force a rapid presence of strong multi-nucleon transfer channels [4] convergence to a barrier distribution possessing pro- for the 96Zr case (6 neutrons outside the N = 50 closed nounced structure. We note, however, that our calcu- shell with favorable Q values). (Though note that this lations seem to give the correct overall width of the explanation is not universally accepted [16].) Unfor- barrier distribution. tunately it is very difficult to include transfer channels It is important to mention that the correctness of in the CC calculations and we chose, therefore, to do the experimental procedure and of the resolution deter- an additional experiment using a 22Ne beam. Here mination was confirmed a posteriori through the bar- we have much smaller effective Q values [17] than rier distribution measurements for 20Ne+ natNi, which for 20Ne (see Table 1) and this should give rise to a were performed simultaneously with the present mea- reduced probability of α transfer. Indeed, our experi- surements. Despite the fact that the Ni target was a nat- ment confirmed that while in the case of 20Ne beam ural mixture of isotopes and despite its smaller atomic at the barrier energy the α-transfer contributes 3% of number (which would normally be expected to give the total scattering cross section at backward angles rise to a weaker barrier structure), a clear structure, in (120◦–160◦) and much less below the barrier, in the agreement with calculations, was found [14]. case of 22Ne it is weaker by a factor of about 6. The Discussing the possible reasons of disagreement contribution of proton transfer is still smaller. between experimental and calculated barrier distribu- The calculated curve in Fig. 3(b) takes the values tions, we would like to draw attention to Fig. 1.We β2 = 0.47 and β4 = 0.10 [10]. Although the defor- note there that while the measured large-angle nega- mation parameter β4 is considerably smaller than that tive Q-value spectrum extends to quite high energies, for 20Ne, the calculation predicts a similar though less the calculated spectrum fails to reproduce anything pronounced structure in the Dqe distribution. However, like the observed intensities in that region. This indi- the experimental results for Dqe (Fig. 1(b)) show that cates the importance of high-lying excitations in ei- for 22Ne this function is still very smooth, suggesting ther the target or projectile, which may in some way that the absence of structure in the 20Ne case is not wash out the predicted structures. However, cutting out due to strong α transfers. This is suggested also by the these high Q-value contributions to the quasi-elastic similarity of the Q spectra measured with both neon cross section has rather little effects on the distribution isotopes (Fig. 1). of barriers. It seems, therefore, that it is not simply Can the barrier distribution be smoothed out by the presence of extra contributions to the cross section neutron transfer? Indeed, in the case of 16O + 144Sm, which destroy the structure in the barrier distribution, the disappearance of the high-energy peak [6] was at- but rather the way they interfere with (damp) the cou- tributed to the dominance of the contribution of neu- plings to the collective states. tron transfer channels to σqe at that energy (where Can these high-lying excitations be caused by σqe/σR is small). In our case, however, the structures 20 transfer reactions? Since Ne is a good α-cluster lie in an energy range where σqe/σR is still of the order nucleus, it could be that α-transfers affect the above of 0.5–0.8. We have no measure of the neutron-transfer results by providing more channels, more barriers and cross sections, but judging from the negative Q values thus a greater overlap and smoothing of the structures. (see Table 1) they are not expected to be important ei- In fact it has been observed that the Dfus for the sys- ther.

Table 1 118 * Effective Q values for stripping in the Ne + Sn collisions (in parentheses the Qgg values are given) 1n 2n 1p 2p α 20Ne −9.6 (−10.4) −11.2 (−12.9) −2.1 (−7.7) +2.8 (−8.5) +9.5 (−3.6) 22Ne −3.2 (−3.9) 0.0 (−1.5) −4.4 (−10.2) −2.8 (−14.1) +4.2 (−8.6) * For pick-up reactions the effective Q values are all negative, with the exception of 2n-transfer, where Q =+0.8 MeV. In this experiment pick-up of charged particles was never observed. 60 E. Piasecki et al. / Physics Letters B 615 (2005) 55–60

Thus, for the moment we do not consider the pos- fact that present results confront us with clear limi- sible influence of transfer reactions on the Dqe distri- tations of the standard CC approach. Experimentally, bution as very likely. investigations of this could consist from measurements It is appropriate to mention that since the high en- of Dfus for present systems and looking for other ones. ergy excitations were observed also for the 16O + ASn Theoretically, one should look for possibilities of re- systems [8], while it was much weaker for the 20Ne + laxing the present limitations on the model space. natNi system, where the predicted structure was observed [14], it is likely that this high-lying intensity corresponds mainly to excitations of the 118Sn tar- Acknowledgements get rather than to excitations of 20Ne projectile. We have, however, already noted that high-lying collec- We are grateful to Marie-Antoinette Saettel for her tive states simply shift the entire barrier distribution to help in providing the targets for this experiment. The lower energies, and so it might be expected that this work was funded in part by Grant No. 2 P03B 02624 of intensity corresponds to weak coupling to many non- the KBN and also supported by the cooperation agree- collective states. ment between the IN2P3 (France) and Polish Labora- tories. In this context it would be valuable to measure σfus 20,22 118 for Ne + Sn and to find out if Dfus is similar to Dqe or whether it still possess the predicted structure. If the predicted structures were observed, this could References again point to the influence of many weak channels, [1] M. Dasgupta, D.J. Hinde, N. Rowley, A.M. Stefanini, Annu. which may contribute differently in fusion and scatter- Rev. Nucl. Part. Sci. 48 (1998) 401. ing. This would mean that principal assumption of the [2] N. Rowley, G.R. Satchler, P.H. Stelson, Phys. Lett. B 254 standard CC model about truncation of available chan- (1991) 25. nels to just a few, strongly coupled collective ones, is [3] H. Timmers, et al., Phys. Lett. B 399 (1997) 35. too radical for the description of scattering of 20,22Ne [4] G. Montagnoli, et al., J. Phys. G: Nucl. Part. Phys. 23 (1997) A 1439. on Sn nuclei. If they were not, this would mean that [5] N. Rowley, in: Yu.Ts. Oganessian, V. Zagrebaev (Eds.), Proc. the model space is too limited even for description of of International Conference on Fusion Dynamics at the Ex- fusion. tremes, Dubna, 25–27 May, 2000, World Scientific, Singapore, This rises a number of interesting questions. Can 2001, p. 296. [6] H. Timmers, et al., Nucl. Phys. A 584 (1995) 190. we simulate the smoothing action of the weak chan- [7] K. Hagino, N. Rowley, Phys. Rev. C 69 (2004) 054610. nels by schematically including some extra couplings, [8] E. Piasecki, et al., Phys. Rev. C 65 (2002) 054611. recovering the missing strength in high Q-value bins? [9] V. Tripathi, et al., Phys. Rev. C 65 (2001) 014614. Is it possible to do it in more realistic way? Is the im- [10] G.S. Blanpied, et al., Phys. Rev. C 38 (1988) 2180. portance of weak channels a reason for similar effects [11] K. Hagino, N. Rowley, unpublished. [12] K. Hagino, et al., Phys. Rev. Lett. 79 (1997) 2014. observed with other systems [6]? How to recognize [13] J.X. Wei, et al., Phys. Rev. Lett. 67 (1991) 3368. systems not posing such problems? [14] L. Swiderski,´ et al., Int. J. Mod. Phys., in press. Concluding, the usefulness of the quasi-elastic scat- [15] V.I. Zagrebaev, Phys. Rev. C 67 (2003) 061601. tering for the barrier distribution investigations de- [16] G. Montagnoli, et al., Eur. Phys. J. A 15 (2002) 351. pends on a deeper understanding of some of these [17] R. Bass, Nuclear Reactions with Heavy Ions, Springer-Verlag, Berlin, 1980, 152. points. However, perhaps even more important is the Physics Letters B 615 (2005) 61–67 www.elsevier.com/locate/physletb

Inﬂuence of NN-rescattering effect on the photon asymmetry of d(γ,π −)pp reaction

Eed M. Darwish

Physics Department, Faculty of Science, South Valley University, Sohag 82524, Egypt Received 14 December 2004; received in revised form 28 January 2005; accepted 6 April 2005 Available online 15 April 2005 Editor: J.-P. Blaizot

Abstract − The inﬂuence of ﬁnal-state NN-rescattering on the beam asymmetry Σ for linearly polarized photons in π photoproduction on the deuteron in the energy range from π-threshold through the ∆(1232)-resonance has been investigated. Numerical results for this spin observable are predicted and compared with recent experimental data from the LEGS Spin Collaboration. Final-state NN-rescattering is found to be quite important and leads to a better agreement with existing experimental data. Furthermore, the differences with other theoretical models have been discussed.  2005 Elsevier B.V. All rights reserved.

PACS: 24.70.+s; 14.20.-c; 29.27.Hj; 25.30.Fj

Keywords: Polarization phenomena in reactions; Spin observables; Polarized beams; Final-state interactions

1. Introduction fact that in contrast to the differential cross section, which is a sum of the absolute squares of the t-matrix It is a well-known fact that polarization observables elements, these polarization observables contain inter- allow a further and much more detailed analysis of ference terms of the various reaction amplitudes in the process under study compared to the differential different combinations and, therefore, may be more cross section alone. Because polarization observables sensitive to small amplitudes and to small contribu- contain a much richer information on the dynamics of tions of interesting dynamical effects. the system than attainable without beam and/or target In recent years a great effort, both from theoret- polarization and without polarization analysis of the ical [1–7] and experimental [8–11] points of view, particles in the ﬁnal state. The reason for this is the has been devoted to the analysis of single-pion photoproduction with polarized beams and/or polarized targets. In [1], π − photoproduction on the deuteron has E-mail address: [email protected] (E.M. Darwish). been studied within a diagrammatic approach includ-

0370-2693/$ – see front matter  2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physletb.2005.04.018 62 E.M. Darwish / Physics Letters B 615 (2005) 61–67 ing nucleon–nucleon (NN) and pion–nucleon (πN) can only be considered as a first step towards a more rescattering in the final state. Special emphasize was realistic description of spin observables. given for the analyzing powers connected to beam In this Letter we investigate, therefore, the in- and target polarization, and to polarization of one fluence of final-state NN interaction on the photon of the final protons. First preliminary model calcula- asymmetry for the reaction d(γ,π −)pp in the en- tions for the photon asymmetry Σ of the γd → π −pp ergy region from π-threshold through the ∆(1232)- reaction have been given in the pure impulse ap- resonance. To our knowledge, the influence of NN- proximation (IA) [2]. The comparison between these FSI effect on this spin observable has never been predictions for Σ and the preliminary experimen- studied before. The πN-rescattering contribution has tal data from LEGS Spin Collaboration [12] gives a been considered as negligible in the region of the clear indication that the effects of final-state inter- ∆(1232)-resonance [14,15] and thus it is not consid- action (FSI) may be important. The deuteron tensor ered in the present work. Our main goal is to analyze analyzing powers of the reaction d(γ,π −)pp have the recent experimental data from LEGS [12]. Further- been studied in the IA [3] without inclusion of any more, it was an open question whether the inclusion FSI or two-body exchange current contributions. In of rescattering contributions would lead to a good de- our previous papers [4–6], various polarization ob- scription of the available data. servables in inclusive single-pion photoproduction on In the next section we will define the photon asym- the deuteron using a polarized photon beam and/or metry Σ in terms of the transition matrix amplitude. an oriented deuteron target have been investigated in In Section 3 we will present and discuss the numeri- the pure IA only, i.e., by neglecting any FSI effects cal results of our calculations and compare them with and possible two-body contributions to the produc- the experimental data and other predictions. Finally, tion operator. In particular, a complete survey on all we summarize our conclusions in Section 4. single- and double-polarization observables like beam and target asymmetries was given. In [7] the influence of final-state NN-rescattering on the helicity struc- 2. Linear photon asymmetry ture of the inclusive reaction γd → π −pp has been investigated. The differential polarized cross-section difference for the parallel and antiparallel helicity The beam asymmetry Σ for linearly polarized pho- states has been predicted and compared with recent tons is defined in analogy to deuteron photodisinte- experimental data from MAMI (Mainz/Pavia) [13].It gration [16] writing the differential cross section for has been shown that the effect of NN-rescattering linearly polarized photons and unpolarized deuterons is much less important in the polarized differen- in the form tial cross-section difference than in the unpolarized dσ dσ0 one. (θ ,φ ) = (θ ) dΩ π π dΩ π The photon asymmetry Σ is very sensitive to the in- π π × + γ ternal mechanisms of the reaction and, therefore, can 1 P Σ(θπ ) cos 2φπ , (1) be a very useful test to impose constraints on the theoretical models. The work has been partly motivated by where dσ0/dΩπ denotes the semi-inclusive unpolar- preliminary experimental results, for the γd → π −pp ized differential cross section of incoherent pion photoproduction on the deuteron, where only the final channel, with the LEGS Brookhaven National Labo- γ ratory [12] which shows strong and not trivial angu- pion is detected without analyzing its energy [14], P lar dependences of this observable. In agreement with is the degree of linearly polarized photons [16], θπ and these preliminary data, one can see in [2,6] that the φπ represent the polar and azimuthal pion angles and predictions in the pure IA can hardly provide a reason- Σ is the photon asymmetry for linearly polarized pho- able description of the data since major discrepancies tons. Then one has [4,16] are found. As already noted in [2,6], the effect of NN- rescattering is quite important. This means in particu- dσ0 Σ =−W00, (2) lar that the calculation in the spectator nucleon model dΩπ E.M. Darwish / Physics Letters B 615 (2005) 61–67 63 with for NN-rescattering in the final state. Further details with respect to the matrix elements are not discussed 1 1−m 110 W00 = √ (−) d C md −m 0 here and can be found in [14]. 2 3 d smt,mγ md md qmax × dq dΩ ρ M(tµ) 3. Results and discussion pNN s sm,mγ md 0 Here we present and discuss our results for the pho- (tµ) × M (3) ton asymmetry Σ for linearly polarized photons of s−m,mγ −m d π − photoproduction on the deuteron with inclusion j1j2j denoting with Cm1m2m a Clebsch–Gordan coefficient, of NN-rescattering in the final state. These results mγ the photon polarization, md the spin projection are also compared to the preliminary experimental of the deuteron, s and m total spin and its projection data of LEGS [12] and the preliminary IA calculations of the two outgoing nucleons, respectively, t their to- of Lee [2]. The results presented here are calculated tal isospin, µ the isospin projection of the pion, qmax by using the elementary photoproduction operator of the maximum value of pion momentum, ΩpNN the Schmidt et al. [17] and the deuteron wave function solid angle of the relative momentum pNN of the final of Paris potential [18]. For the half-off-shell NN- NN system and ρs the phase space factor. For further scattering amplitude, the separable representation [19] details with respect to the kinematical variables and of the realistic Paris potential has been used. All par- quantum numbers we refer to our previous work [14]. tial waves with total angular momentum J 3are For the transition M-matrix we include, in this included. work, besides the pure IA, the driving term from NN- We start with presenting our results for the linear rescattering, so that the total transition matrix reads photon asymmetry Σ at different photon lab-energies = M(tµ) = M(tµ) IA + M(tµ) NN ωγ 200, 270, 330, 370, 420 and 500 MeV in Fig. 1 smm m smm m smm m , (4) γ d γ d γ d as a function of emission pion angle θπ in the lab- where the first term represents the transition amplitude oratory frame. The solid curves show the results of in the pure IA and the second is the corresponding one the full calculation, i.e., when NN-rescattering is in-

− Fig. 1. Linear photon asymmetry of the differential cross section for linearly polarized photons for π photoproduction on the deuteron as a function of emission pion angle in the laboratory frame at ﬁxed values of photon lab-energies. Notation of curves: dashed—IA; solid—IA + NN-rescattering. 64 E.M. Darwish / Physics Letters B 615 (2005) 61–67

+ Fig. 2. The ratio of the linear photon asymmetry with NN-rescattering ΣIA NN to the one in the IA ΣIA as a function of emission pion angle at the same photon lab-energies as in Fig. 1. cluded, while the dashed curves show the contribu- Fig. 3 shows the sensitivity of our results for the tion of the IA alone in order to clarify the importance linear photon asymmetry Σ to the photon lab-energy ◦ ◦ of NN-FSI effect. In order to show in greater detail ωγ at three fixed values of pion angle θπ = 0 ,90 the relative influence of NN-rescattering effect on the and 180◦ for photon lab-energies between 200 and linear photon asymmetry, we show in Fig. 2 the ef- 500 MeV. In order to show the relative influence of fect of NN-rescattering relative to the IA by the ratio NN-rescattering effect on the linear photon asymme- ΣIA+NN/ΣIA, where ΣIA denotes the photon asym- try, we show in the bottom panels of Fig. 2 the ef- metryintheIA and ΣIA+NN the one including the fect of NN-rescattering relative to the IA by the ratio contribution of NN-rescattering. ΣIA+NN/ΣIA, where ΣIA denotes the photon asym- In the photon energy domain of this work, the mag- metry in the IA and ΣIA+NN the one including the netic multipoles dominate over the electric ones, due contribution of NN-rescattering. In view of these re- to the excitation of the ∆-resonance. This is clear sults, one notes that NN-rescattering—the difference from the dominantly negative values of Σ as shown in between the solid and the dashed curves—is quite Fig. 1. On the contrary, the left-top and right-bottom small, almost completely negligible at extreme for- panels in Fig. 1 show that small positive values are ward and backward angles. found at ωγ = 200 and 500 MeV. We see also that the Fig. 4 shows a comparison of our numerical re- asymmetry Σ is sensitive to the energy of the incom- sults for the linear photon asymmetry Σ in the pure ing photon. It is noticeable, that the photon asymmetry IA (dashed curves) and with NN-rescattering (solid ◦ ◦ Σ vanish at θπ = 0 which is not the case at 180 .At curves) with experimental data. In view of the fact extreme forward and backward emission pion angles that experimental data for this spin observable are one sees, that the photon asymmetry is relatively small not available in a final form, we compare our predic- in comparison to the results when θπ changes from tions with the preliminary experimental data from the about 30◦ to 120◦. One notices also, that the contribu- LEGS Spin Collaboration [12] as depicted in Fig. 4. tion from NN-rescattering is much important in this We see that the general feature of the data is repro- region, in particular in the peak position. For lower and duced. However, the discrepancy is rather significant higher photon energies, one finds the strongest effect in the region where the photon energy close to the by NN-rescattering. ∆-resonance. This could be due to the higher-order E.M. Darwish / Physics Letters B 615 (2005) 61–67 65

− Fig. 3. Linear photon asymmetry Σ for d(γ,π )pp as a function of the photon lab-energy ωγ at ﬁxed values of pion angle θπ in the laboratory frame with NN-rescattering (top panels) and their ratios with respect to the pure IA (bottom panels). Notation as in Fig. 1.

− Fig. 4. Photon asymmetry Σ for the reaction d(γ,π )pp at ωγ = 270 and 330 MeV photon lab-energy in comparison with the preliminary data from LEGS [12]. Notation as in Fig. 1. rescattering mechanisms which are neglected in this ferences between theory and experiment are still ev- work. In the same ﬁgure, we also show the results from ident. It is appear that our model is still not capable the IA only (dashed curves). It is seen that the NN- of describing the measured photon asymmetry, even rescattering yields an about 10% effect in the region if NN-FSI is included. Future efforts must be made of the peak position. We found that this is mainly due to remove the remaining discrepancies such as a com- to the interference between the IA amplitude and the plete three-body treatment of the ﬁnal πNN system. NN-FSI amplitude. In agreement with our previous Now we compare our results for the linear pho- results [6], one notes that the pure IA (dashed curves ton asymmetry with the preliminary model predic- in Fig. 4) cannot describe the experimental data. The tions of Lee [2] as shown in Fig. 5. The solid curves inclusion of NN-FSI leads at ωγ = 270 MeV to a show the results of the present calculations when NN- quite satisfactory description of the data, whereas at rescattering is included and the dashed ones show our 330 MeV NN-FSI effect is small and therefore dif- results in the IA alone. The preliminary IA results 66 E.M. Darwish / Physics Letters B 615 (2005) 61–67

− Fig. 5. Photon polarization asymmetry Σ for d(γ,π )pp reaction at two photon lab-energies. Dashed curves: IA of present calculations; solid curves: IA plus NN-FSI of present calculations; dotted curves: preliminary IA calculations of Lee [2]. of Lee [2] are represented in this figure by the dot- well as with the ‘totally neglected’ rescattering mech- ted curves. It is very clear that the results for the IA anisms in [2] would be very interesting to understand of the present work showed certain significant differ- the origin of this discrepancy. ences to the preliminary IA results [2] which cannot be attributed to the use of different elementary pion photoproduction operators and/or from different real- 4. Summary istic NN potential models used for the deuteron wave function. It is clear from the right panel of Fig. 5 that In this Letter we have investigated the influence of the discrepancy is rather significant in the region of the final-state NN-rescattering effect on the linear pho- ∆(1232)-resonance, in particular, in the peak position. ton asymmetry Σ for the γd → π −pp reaction in the As already mentioned in the beginning of this sec- photon energy range from π-threshold to 500 MeV. tion, our results are calculated using the effective La- We have found that the effect due to the final two- grangian model developed by Schmidt et al. [17]. nucleon interaction to be small, but it can has signif- The main advantage of this model is that it has been icant contribution to the photon asymmetry through constructed to give a realistic description of the ∆- its interference with the dominant term from the im- resonance region. It is also given in an arbitrary frame pulse approximation. Furthermore, the linear photon of reference and allows a well defined off-shell contin- asymmetry is found to be sensitive to the energy of the uation as required for studying pion production on nu- incident photon. In comparison with the preliminary clei. As shown in Figs. 1–3 in [14], the results for this experimental data from LEGS [12], the inclusion of model are in good agreement with recent experimental NN-FSI effect leads to a better agreement with exper- data as well as with other theoretical predictions. On imental data. With respect to the comparison with the the other hand, the well-known dynamical model of preliminary results of [2] in the IA, we found a large Sato and Lee [20] has been used in [2]. This model has difference between both calculations in the peak posi- given also a successful description of the pion photo- tion. The origin of this difference is still not clear. production data. Therefore, the big difference between We would like to conclude that the results presented both predictions in the IA results cannot be attributed here for linear photon asymmetry of d(γ,π −)pp can to the use of different elementary operators. This can be used as a basis for the simulation of the behav- be interpreted as lack of understanding of the nonres- iour of this asymmetry and for an optimal planning of onant background, which in dynamical models is re- new experiments of this reaction with polarized pho- lated to the pion cloud. It seems that pion cloud effects ton beams. An experimental check of these predictions are not yet consistently included in dynamical mod- for the linear photon asymmetry covering a large range els. An independent evaluation for Σ in the pure IA as for the pion angle would provide an additional sig- E.M. Darwish / Physics Letters B 615 (2005) 61–67 67 nificant test of our present understanding of this spin [4] E.M. Darwish, Nucl. Phys. A 735 (2004) 200. observable. Furthermore, an independent evaluation in [5] E.M. Darwish, Int. J. Mod. Phys. E 13 (2004) 1191. the framework of effective field theory would be very [6] E.M. Darwish, J. Phys. G: Nucl. Part. Phys. 31 (2005) 105. [7] E.M. Darwish, Nucl. Phys. A 748 (2005) 596. interesting. Future improvements should include fur- [8] St. Goertz, W. Meyer, G. Reicherz, Prog. Part. Nucl. Phys. 49 ther investigations including FSI as well as two-body (2002) 403; effects. St. Goertz, W. Meyer, G. Reicherz, Prog. Part. Nucl. Phys. 51 (2003) 309, Erratum. [9] See, for example, in: M. Anghinolfi, M. Battaglieri, R. De Acknowledgements Vita (Eds.), Proceedings of the 2nd International Symposium on the Gerasimov–Drell–Hearn 2002, Sum Rule and the Spin Structure of the Nucleon, Genova, Italy, 3–6 July 2002, World This work is supported in part by the Biblio- Scientific, Singapore, 2003. theca Alexandrina—Center for Special Studies and [10] B. Krusche, S. Schadmand, Prog. Part. Nucl. Phys. 51 (2003) Programs—under grant No. 2602314 Sohag 2nd- 399. Sohag. I am indebted to Prof. H. Arenhövel as well [11] V. Burkert, T.-S.H. Lee, Int. J. Mod. Phys. E 13 (2004) 1035. [12] A. Sandorfi, M. Lucas, private communication; as the members of his work group for fruitful discus- M. Lucas, in: LOWq Workshop on Electromagnetic Nuclear sions and valuable comments. I would like to thank Reactions at Low Momentum Transfer, 23–25 August 2001, Profs. T.-S. Harry Lee, T. Sato for useful discussions Halifax, Nova Scotia, Canada, 2001; and Prof. S.A.E. Khallaf for a careful reading of this A. Sandorfi in [9]. Letter. [13] P. Pedroni, private communication; C.A. Rovelli, Diploma Thesis, University of Pavia, Italy, 2002. [14] E.M. Darwish, H. Arenhövel, M. Schwamb, Eur. Phys. J. A 16 (2003) 111. References [15] M.I. Levchuk, M. Schumacher, F. Wissmann, nucl-th/0011041. [16] H. Arenhövel, Few-Body Syst. 4 (1988) 55. [1] A.Yu. Loginov, A.A. Sidorov, V.N. Stibunov, Phys. At. [17] R. Schmidt, H. Arenhövel, P. Wilhelm, Z. Phys. A 355 (1996) Nucl. 63 (2000) 391. 421. [2] T.-S.H. Lee, private communication; [18] M. Lacombe, et al., Phys. Lett. B 101 (1981) 139. T.-S.H. Lee, in: LOWq Workshop on Electromagnetic Nuclear [19] J. Haidenbauer, W. Plessas, Phys. Rev. C 30 (1984) 1822; Reactions at Low Momentum Transfer, 23–25 August 2001, J. Haidenbauer, W. Plessas, Phys. Rev. C 32 (1985) 1424. Halifax, Nova Scotia, Canada, 2001. [20] T. Sato, T.-S.H. Lee, Phys. Rev. C 54 (1996) 2660. [3] A.Yu. Loginov, A.V. Osipov, A.A. Sidorov, V.N. Stibunov, nucl-th/0407045. Physics Letters B 615 (2005) 68–78 www.elsevier.com/locate/physletb

Top quark pair production and decay at a polarized photon collider

A. Brandenburg a,1,Z.G.Sib,2

a DESY-Theorie, 22603 Hamburg, Germany b Department of Physics, Shandong University, Jinan, Shandong 250100, China Received 17 March 2005; received in revised form 2 April 2005; accepted 2 April 2005 Available online 12 April 2005 Editor: N. Glover

Abstract Top quark pair production by (polarized) γγ collisions offers an interesting testing ground of the Standard Model and its extensions. In this Letter we present results for differential cross sections of top quark pair production and decay including QCD radiative corrections. We take into account the full dependence on the top quark spins. We give analytic and numerical results for single and double differential angular distributions of tt¯ decay products which are due to top quark polarizations and spin correlations in the intermediate state.  2005 Elsevier B.V. All rights reserved.

Keywords: Photon collider; Top quarks; QCD corrections; Polarization; Spin correlations

1. Introduction

At a future linear lepton collider, backscattered laser light may provide very high-energy photons [1], which would allow for a very interesting physics program [2,3]. In particular, top quark pair production is possible with large rates in (un)polarized photon–photon fusion. The measurement of the process γγ → ttX¯ is an important test of the Standard Model (SM). The ﬁrst order QCD corrections to this process have already been calculated and found to be large [4–7]. The electroweak virtual plus soft-photonic O(α) corrections are also known [8].This process will also provide information on possible anomalous γtt¯ couplings [9,10] without contributions from Ztt¯ couplings present in e+e− collisions. Once the Higgs boson is discovered, it will be of primary importance to determine whether its properties are as predicted within the SM. In this respect, the process γγ → ttX¯ may play an important role. For example,

E-mail address: [email protected] (Z.G. Si). 1 Work supported by a Heisenberg fellowship of DFG. 2 Work supported in part by NSFC and NCET.

0370-2693/$ – see front matter  2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physletb.2005.04.003 A. Brandenburg, Z.G. Si / Physics Letters B 615 (2005) 68–78 69 heavy quark production in polarized γγ collisions will help to determine the parity of the Higgs boson produced as a resonance and decaying into top quark pairs [6,7]. In particular, if a Higgs boson is no CP eigenstate, spin correlations of the top quark pairs will help to probe the scalar and pseudoscalar couplings of the Higgs boson to the top quark [11,12]. For this kind of studies, predictions for top quark pair production and decay at a photon collider must be as precise as possible within the SM. In particular, the spin state of the intermediate tt¯ pair must be taken into account. (The role of the top quark polarization in probing the tt¯ threshold dynamics in γγ collisions was discussed in [13].) The purpose of this Letter is therefore to study the processes γγ → ttX¯ → + + X, + jets + X, all jets, (1.1) where stands for a charged lepton, with polarized photons from backscattered laser beams. We include QCD radiative corrections and take into account polarization and spin correlation effects of the intermediate tt¯ pairs. Leading order results and QCD corrections for the cross section and for top quark spin observables in the process ¯ 2 γγ → ttX are summarized in Sections 2 and 3. Numerical results to order α αs for the effective lepton collider cross section and for several decay distributions are given in Section 4.

2. Kinematics and leading order results

2 0 The production of top quark pairs by photon scattering at leading order α αs is described by the reaction ¯ γ(p1,λ1) + γ(p2,λ2) → t(k1,st ) + t(k2,st¯). (2.1)

Here, p1,p2,k1 and k2 denote the momenta of the particles, λ1 and λ2 are the helicities of the photons, and the vectors st and st¯ describe the spins of top quark and antiquark. These fulﬁl the relations 2 = 2 =− · = · = st st¯ 1 and k1 st kt¯ st¯ 0. (2.2)

In the (anti)top rest frame the spin of the (anti)top is described by a unit vector sˆt (sˆt¯). We choose the specific rest frames that are obtained by a rotation-free Lorentz boost from the zero momentum frame of the tt¯ quarks (tt¯-ZMF). Both the tt¯-ZMF and the t and t¯ rest frames will be used to construct spin observables from the final state momenta of the tt¯ decay products. We use the tt¯-ZMF rather than the c.m. frame of the colliding high-energy photons, since the latter system is probably more difficult to reconstruct experimentally. For the 2 → 2 process of Eq. (2.1), the two frames coincide. The differential cross section for the process of Eq. (2.1) can be written as follows:

N 2 dσ(λ1,λ2,st ,st¯) = |M0| dΓ2, (2.3) 2sγγ 2 where the two-particle phase space measure is denoted by dΓ2, sγγ = (p1 + p2) and N = 3 is the number of colours. A simple calculation gives: 2 4 2 2 16α Qt π |M | = A + B p · (s + s¯) + B | ↔ p · (s + s¯) + C (s · s¯) + D (p · s )(p · s¯) 0 − 2 2 2 0 0 1 t t 0 λ1 λ2 2 t t 0 t t 0 1 t 2 t (1 β z ) + D0|z→−z(p2 · st )(p1 · st¯) , (2.4) with 2 2 4 2 2 2 2 4 2 2 A0 = 1 + 2β 1 − z − β 1 + 1 − z + λ1λ2 1 − 2β 1 − z − β z 2 − z , (2.5) 4m 2 2 2 2 2 B0 = λ1 1 − 2β + β z + λ2 1 − β z , (2.6) sγγ 70 A. Brandenburg, Z.G. Si / Physics Letters B 615 (2005) 68–78 2 4 2 2 2 4 2 2 C0 = 1 − 2β + β 1 + 1 − z + λ1λ2 1 − 2β + β z 2 − z , (2.7) 2 2 4(1 + βz)(1 − z )(1 − λ1λ2)β D0 =− . (2.8) sγγ

Here, Qt = 2/3, m is the top quark mass, 4m2 β = 1 − , (2.9) s ¯ ˆ ˆ and z is the cosine of the scattering angle in the tt-ZMF, i.e., z = pˆ γ · k, where pˆ γ (k) is the direction of one of the photons (of the top quark) in that frame.

3. NLO results for γγ → ttX¯

In this section we present results for the inclusive reaction γγ → ttX¯ (3.1) 2 to order α αs . Apart from the cross section we study observables that depend on the spins of the top quark and antiquark. For polarized photons, observables of the form s O = 2St · aˆ (3.2) can have non-zero expectation values. Here, aˆ is an arbitrary reference direction and St is the top quark spin operator. The expectation value of Os is related to a single spin asymmetry: σ(↑) − σ(↓) Os = , (3.3) σ(↑) + σ(↓) where the arrows on the right-hand side refer to the spin state of the top quark with respect to the quantization axis aˆ. We will consider here two choices for aˆ, aˆ = kˆ (helicity basis), aˆ = pˆ (beam basis), (3.4) where kˆ denotes the direction of the top quark in the tt¯-ZMF and pˆ is the direction of the lepton beam coming from the left in that frame, which coincides to good approximation with the direction of one of the high-energy photons. Top quark polarization perpendicular to the plane spanned by pˆ and kˆ is induced by absorptive parts in the one-loop amplitude. This effect is, however, quite small (∼ a few percent) [14]. Analogous observables may of course be deﬁned for the top antiquark. Apart from the above single spin observables, we also consider observables of the form d ˆ O = 4(St · aˆ)(St¯ · b). (3.5) ˆ Here, aˆ and b are arbitrary reference directions and St¯ is the top antiquark spin operator. The expectation value of Od is related to a double spin asymmetry: σ(↑↑) + σ(↓↓) − σ(↑↓) − σ(↓↑) Od = . (3.6) σ(↑↑) + σ(↓↓) + σ(↑↓) + σ(↓↑) For the reference directions we will consider here aˆ =−bˆ = kˆ, aˆ = bˆ = pˆ. (3.7) A. Brandenburg, Z.G. Si / Physics Letters B 615 (2005) 68–78 71

Finally, we also present results for the observable

˜ d 4 O = S · S¯. (3.8) 3 t t The above double spin asymmetries Od and O˜ d have also proved useful for an analysis of spin correlations of top quark pairs in hadronic collisions [15]. The NLO cross section for the reaction γγ → ttX¯ may be written in terms of two scaling functions:

α2Q4 σ(s,m,λˆ ,λ ) = t c(0)(ρ, λ ,λ ) + 4πα c(1)(ρ, λ ,λ ) . (3.9) 1 2 m2 1 2 s 1 2 Likewise, the unnormalized expectation values of the above spin observables are of the form

α2Q4 σ O = t d(0)(ρ, λ ,λ ) + 4πα d(1)(ρ, λ ,λ ) , (3.10) a m2 a 1 2 s a 1 2 where a = 1 corresponds to the observable O˜ d defined in Eq. (3.8), a = 2(3) to the observable Od defined in Eq. (3.5) in the helicity (beam) basis, and a = 4(5) corresponds to the single spin observable Os defined in Eq. (3.2) in the helicity (beam) basis. The variable ρ is defined as

4m2 ρ = . (3.11) sγγ

(0) (0) The lowest order scaling functions c √and da can be computed analytically. We use the following auxiliary functions, which vanish in the limit β = 1 − ρ → 0:

1 1 2 1 2 2 = ln(x) + 2β ,= ln(x) + 2β + β3 ,= ln(x) + 2β + β3 + β5 , 1 β 2 β3 3 3 β5 3 5 (3.12) where x = (1 − β)/(1 + β). We then obtain:

ρ2 c(0)(ρ, λ ,λ ) = Nπβρ 1 + ρ − ρ2 + λ λ − 1 + ρ − − λ λ , 1 2 1 2 2 1 2 1

Nπβρ ρ2 d(0)(ρ, λ ,λ ) =− 1 + ρ + ρ2 + (1 + 2ρ)λ λ + 1 − − (1 + ρ)λ λ , 1 1 2 3 1 2 2 1 2 1

1 + 8ρ − 7ρ2 + ρ3 + (5 − 3ρ + ρ2)λ λ d(0)(ρ, λ ,λ ) = Nπβρ 1 2 2 1 2 3 ρ3 ρ2 − −1 + ρ − 2ρ2 + + 1 + λ λ , 2 2 1 2 2 √ 9 − 20 ρ − 6ρ + 14ρ3/2 + 29ρ2 + 6ρ5/2 − 20ρ3 + 3ρ4 d(0)(ρ, λ ,λ ) = Nπβρ − 3 1 2 15 √ −21 − 20 ρ + 25ρ + 14ρ3/2 − 16ρ2 + 6ρ5/2 − 3ρ3 + λ1λ2 15 √ ρ3 + −1 − 4ρ − ρ2 − 2 ρ + − 2ρ3/2 + ρ5/2 2 √ ρ2 √ + 1 + 2 ρ + 3ρ + λ λ (1 − ρ)2 , 2 1 2 3 72 A. Brandenburg, Z.G. Si / Physics Letters B 615 (2005) 68–78

Fig. 1. Left: scaling functions c(0)(ρ, 1, 1) (dotted), c(0)(ρ, 1, −1) (dash-dotted), c(1)(ρ, 1, 1) (full), and c(1)(ρ, 1, −1) (dashed). Right: scaling (0) (0) − (1) (1) − functions d1 (ρ, 1, 1) (dotted), d1 (ρ, 1, 1) (dash-dotted), d1 (ρ, 1, 1) (full), and d1 (ρ, 1, 1) (dashed).

(0) (0) − (1) (1) − Fig. 2. Left: scaling functions d2 (ρ, 1, 1) (dotted), d2 (ρ, 1, 1) (dash-dotted), d2 (ρ, 1, 1) (full), and d2 (ρ, 1, 1) (dashed). Right: (0) (0) − (1) (1) − scaling functions d3 (ρ, 1, 1) (dotted), d3 (ρ, 1, 1) (dash-dotted), d3 (ρ, 1, 1) (full), and d3 (ρ, 1, 1) (dashed).

ρ d(0)(ρ, λ ,λ ) = Nπβρ 1 − ρ − ln(x) (λ + λ ), 4 1 2 2 1 2 √ 1 − ρ √ 1 − ρ √ d(0)(ρ, λ ,λ ) =−Nπβρ 1 − 6 ρ + ρ3/2 + 2 + 2 ρ + 3ρ − ρ3/2 − ρ2 (λ − λ ). 5 1 2 3 2 2 1 2 (3.13) (1) (1) The functions c and da are obtained by a numerical integration. The scaling functions for the spin-averaged cross section and all spin observables are plotted in Figs. 1–3 for different choices of the photon helicities as a function of 1 η = − 1. (3.14) ρ The result for unpolarized photons can be inferred from 1 c(0),(1)(ρ, 0, 0) = c(0),(1)(ρ, 1, 1) + c(0),(1)(ρ, 1, −1) , (3.15) 2 A. Brandenburg, Z.G. Si / Physics Letters B 615 (2005) 68–78 73

(0) (0) − (1) (1) − Fig. 3. Left: scaling functions d4 (ρ, 1, 1) (dotted), d4 (ρ, 1, 1) (dash-dotted), d4 (ρ, 1, 1) (full), and d4 (ρ, 1, 1) (dashed). Right: (0) (0) − (1) (1) − scaling functions d5 (ρ, 1, 1) (dotted), d5 (ρ, 1, 1) (dash-dotted), d5 (ρ, 1, 1) (full), and d5 (ρ, 1, 1) (dashed). 1 d(0),(1)(ρ, 0, 0) = d(0),(1)(ρ, 1, 1) + d(0),(1)(ρ, 1, −1) , (3.16) 1,2,3 2 1,2,3 1,2,3 (0),(1) = d4,5 (ρ, 0, 0) 0. (3.17) As a check we compared our result for the functions c(0),(1)(ρ, 0, 0) with the results given in Fig. 2 of Ref. [4] and found perfect agreement. We further compared the functions c(0),(1)(ρ, 1, ±1) to the results given in Table 1 of Ref. [6]. After a trivial rescaling to account for the different conventions used in the deﬁnition of the scaling functions, we also found agreement.

4. Effective cross sections and spin observables

4.1. The effective cross section for γγ → ttX¯

The total tt¯ cross section at a photon collider may be written at NLO QCD as (cf., e.g., [5])

ymax ymax 2 4 α Qt e e (0) (1) σ ¯ = dy dy f (y ,P ,P )f (y ,P ,P ) c + 4πα c . (4.1) tt m2 1 2 γ 1 e L γ 2 e L s 0 0 e The function fγ (y1,Pe,PL) is the normalized energy spectrum of the photons resulting from Compton backscat- tering of laser light off the high energy electron beam. It is explicitly given by:

− 1 f e(y, P ,P ) = N 1 − y + (2r − 1)2 − P P xr(2r − 1)(2 − y) . (4.2) γ e L 1 − y e L

Here, Pe (PL) is the polarization of the electron (laser) beam, and y is the fraction of the electron energy in the c.m. frame transferred to the photon. It takes values in the range x 0 y ≡ y , (4.3) x + 1 max with 4E E x = L e , 2 (4.4) me 74 A. Brandenburg, Z.G. Si / Physics Letters B 615 (2005) 68–78

Table 1 √ Results for the effective cross section at see = 500 GeV (P ,P ; P ,P )σLO (fb) σ NLO (fb) K = σ NLO/σ LO e1 e2 L1 L2 tt¯ tt¯ tt¯ tt¯ (0, 0; 0, 0) 49.81 76.44 1.53 (0.85, 0.85;−1, −1) 175.86 260.77 1.48 (0.85, 0.85;+1, +1) 15.96 26.89 1.68 (0.85, −0.85;−1, +1) 48.99 71.93 1.47

where EL (Ee) is the energy of the laser (electron) beam and me is the electron mass. In order to avoid the creation of an e+e− pair from the backscattered laser beam and the low energy laser beam, the maximal value for x is √ xmax = 2 1 + 2 . (4.5)

For a beam energy Ee = 250 GeV, this leads to an optimal laser energy

EL ≈ 1.26 eV, (4.6) which will be used in the following numerical results. Finally, y r = . (4.7) x(1 − y) The normalization factor N in Eq. (4.2) is determined by

ymax e = fγ (y, Pe,PL)dy 1. (4.8) 0 (0),(1) 2 The scaling functions c have to be evaluated at ρ = 4m /(y1y2see) and for polarizations = (i) (i) = λi Pγ yi,Pe ,PL ,i1, 2. (4.9)

The function Pγ (y, Pe,PL) describes the degree of polarization of photons scattered with energy fraction y, which is given by

1 1 P (y, P ,P ) = xrP 1 + (1 − y)(2r − 1)2 − (2r − 1)P + 1 − y . (4.10) γ e L N e e L − fγ (y, Pe,PL) 1 y √ Numerical results for σtt¯ are given in Table 1 for see = 500 GeV and different polarizations of the laser and = = = = electron beam. We use the values m√t 178 GeV, α 1/128 and αs(µ mt ) 0.1. The QCD corrections to σtt¯ are quite large. This is because for see = 500 GeV most of the top quark pairs are produced close to threshold where the Coulombic β−1 singularity from soft gluons is important.

4.2. Spin observables

The spin observables for γγ → ttX¯ discussed in Section 3 translate into observables built from the momenta of the tt¯ decay products. Thesinglespinasymmetries(3.2) cause a non-trivial one-particle inclusive decay distribution of the form

1 dσ(γγ → a1 + X) 1 = (1 + Bi cos θ1). (4.11) σ d cos θ1 2

Here, θ1 is the angle between the direction of a top quark decay product a1 measured in the top quark rest frame and one of the reference directions aˆ defined in Eq. (3.4). The coefficient Bi , with i = heli, beam for the helicity A. Brandenburg, Z.G. Si / Physics Letters B 615 (2005) 68–78 75 and beam bases, is determined by the top quark spin asymmetry (3.2) and by the so-called spin analysing power of the decay product a1, which will be discussed below. The double spin asymmetries (3.5) lead to a two-particle inclusive decay distribution of the following form: → + 1 dσ(γγ a1a2 X) 1 ¯ = (1 + Bi cos θ1 + Bi cos θ2 − Ci cos θ1 cos θ2), (4.12) σ d cos θ1d cos θ2 4 where θ1 is defined as above and θ2 is analogously the angle between one of the top antiquark decay products and ˆ one of the reference directions b defined in Eq. (3.7). The coefficients Ci are determined by the double spin asymmetries (3.5) and the spin analysing powers of the two decay products a1 and a2. Finally, a non-zero expectation value of the observable defined in Eq. (3.8) leads to a distribution of the form

1 dσ(γγ → a a + X) 1 1 2 = (1 − D cos ϕ), (4.13) σ d cos ϕ 2 where ϕ is the angle between the direction of ﬂight of the top decay product a1 and the antitop decay product a2 deﬁned in the t and t¯ rest frames, respectively. We recall that these rest frames have to be obtained by a rotation-free boost from the tt¯-ZMF. The spin analysing power of the t and t¯ decay products is encoded in the one-particle inclusive angular distributions dΓ/d cos θ for the decays

¯ t(st ) → a1(q1) + X1, t(st¯) → a2(q2) + X2. (4.14)

Here q1 and q2 are the momenta of a1 and a2, respectively, deﬁned in the rest frame of the (anti)top quark and θ is the angle between the polarization vector of the (anti)top quark and the direction of ﬂight of a1(a2). For a fully polarized ensemble of top quarks (antiquarks) these distributions are of the form

dΓ (1,2) Γ (1,2) = 1 ± κ(1,2) cos θ , (4.15) d cos θ 2 where Γ (1,2) is the partial width of the respective decay channel. The quantity κ(1,2) is the (anti)top-spin analysing power of a1,2. For the case of the standard (V − A) charged current interactions these distributions were computed to order αs for the semileptonic and non-leptonic channels in Refs. [16,17], respectively. (2) (1) As we work to lowest order in the electroweak couplings, Γ = Γ and κ2 = κ1 to all orders in αs ,ifthe channel a2 + X2 is the charge-conjugate of a1 + X1. + For semileptonic top decays t → b ν(g), the charged lepton is the most efficient analyser of the spin of the top quark. In the case of non-leptonic decays t → bqq¯(g) a good top spin analyser that can be identified easily is the least-energetic light quark jet. In Ref. [17] the coefficients κ(f ) were given to NLO accuracy for different choices of the spin analyser. To compute the coefficients Bi ,Ci and D we need the partial widths

(sl,h) = (sl,h) + (sl,h) Γ a0 4παsa1 , (4.16) where the indices sl and h stand for semileptonic and hadronic decay modes. Further, we need the dimensionful coefﬁcients

(sl,h) (,j) = (,j) + (,j) Γ κ b0 4παsb1 , (4.17) where (j) refers to using the charged lepton (least-energetic light quark jet) as spin analyser. For the determination −5 −2 of these coefﬁcients we use the Fermi constant GF = 1.16639 × 10 GeV , m = 178 GeV, mW = 80.42 GeV, 76 A. Brandenburg, Z.G. Si / Physics Letters B 615 (2005) 68–78

ΓW = 2.12 GeV, mb = 4.75 GeV, and all other quark and lepton masses are put to zero. (We do not use the narrow width approximation for the intermediate W boson.) We obtain, putting the CKM matrix elements |Vtb|= |Vqq |=1:

ah ah = 0.52221 GeV,asl = 0 ,ah =−0.01968(15) GeV,asl =−0.01097(5) GeV. (4.18) 0 0 N 1 1

For the relevant coefﬁcients b0,1 we obtain:

= sl j = =− j =− b0 a0 ,b0 0.26950 GeV,b1 0.01118(8) GeV,b1 0.02375(26) GeV. (4.19)

j The Durham algorithm was used as jet clustering scheme to obtain the four parton contribution to b1 . Within the leading pole approximation for the intermediate top quarks and antiquarks, the coefﬁcients of the single and double differential distributions (4.11)–(4.13) are obtained in terms of the following quantities:

ymax ymax α2Q4 1 σ = t dy dy f e(y ,P ,P )f e(y ,P ,P ) c(0)a(1) + πα c(1)a(1) + c(0)a(1) , s 2 1 2 γ 1 e L γ 2 e L 0 4 s 0 1 (4.20) m Γt 0 0

ymax ymax α2Q4 1 σ = t dy dy f e(y ,P ,P )f e(y ,P ,P ) d 2 2 1 2 γ 1 e L γ 2 e L m Γt 0 0 × (0) (1) (2) + (1) (1) (2) + (0) (1) (2) + (0) (1) (2) c a0 a0 4παs c a0 a0 c a1 a0 c a0 a1 , (4.21) ymax ymax α2Q4 1 N s = t dy dy f e(y ,P ,P )f e(y ,P ,P ) d(0)b(1) + πα d(1)b(1) + d(0)b(1) , r 2 1 2 γ 1 e L γ 2 e L r 0 4 s r 0 r 1 m Γt 0 0 (4.22) ymax ymax α2Q4 1 N d = t dy dy f e(y ,P ,P )f e(y ,P ,P ) r 2 2 1 2 γ 1 e L γ 2 e L m Γt 0 0 × (0) (1) (2) + (1) (1) (2) + (0) (1) (2) + (0) (1) (2) dr b0 b0 4παs dr b0 b0 dr b1 b0 dr b0 b1 . (4.23)

We then get to NLO in αs :

d d d s s N1 N2 N3 N4 N5 D = , Cheli = , Cbeam = , Bheli = , Bbeam = . (4.24) σd σd σd σs σs The LO and NLO results for these quantities are shown in Table 2 for favorable and realistic choices of electron and laser polarizations, using the same parameters as in Table 1. In most cases the QCD corrections are of the order of a few percent and thus much smaller than the corrections to the total tt¯ cross section. This was to be expected, since the bulk of the corrections is due to soft gluons which do not affect the tt¯ spin state. The biggest correction (∼ 11%) occurs for the coefﬁcient Bbeam if the least energetic light jet is used as spin analyser. Photon polarization is an important asset: it is necessary to obtain polarized top quarks and thus non-zero coefﬁcients Bheli and Bbeam. Further, the choice (Pe1,Pe2; PL1,PL2) = (0.85, 0.85;−1, −1), which increases the total yield of tt¯ pairs by more than a factor of 3 (see Table 1), in addition leads to larger tt¯ spin correlations. In particular, in the helicity basis the correlation is then almost 100% in the dilepton channel. A. Brandenburg, Z.G. Si / Physics Letters B 615 (2005) 68–78 77

Table 2 √ Results for double and single spin asymmetries at see = 500 GeV

(Pe1,Pe2; PL1,PL2) Dilepton Lepton–jet Jet–jet LO NLO LO NLO LO NLO D (0, 0; 0, 0) −0.670 −0.686 −0.346 −0.338 −0.178 −0.167 (0.85, 0.85;−1, −1) −0.806 −0.801 −0.416 −0.394 −0.215 −0.194 Cheli (0, 0; 0, 0) 0.811 0.826 0.418 0.408 0.216 0.201 (0.85, 0.85;−1, −1) 0.985 0.981 0.508 0.483 0.262 0.238 Cbeam (0, 0; 0, 0) −0.580 −0.606 −0.299 −0.299 −0.154 −0.148 (0.85, 0.85;−1, −1) −0.808 −0.804 −0.417 −0.396 −0.215 −0.195

(Pe1,Pe2; PL1,PL2) Lepton + XJet+ X LO NLO LO NLO

Bheli (0.85, 0.85;−1, −1) 0.658 0.655 0.340 0.323 Bbeam (0.85, −0.85;−1, 1) −0.684 −0.637 −0.353 −0.314

So-called non-factorizable corrections do neither contribute at NLO QCD to σtt¯ nor to the angular correlations considered above. A proof of this statement is given in [15].

5. Conclusions

¯ 2 We have computed a variety of spin observables for the process γγ → ttX up to order α αs . Together with the differential rates of polarized top and antitop quark decays at order αs , we have obtained the NLO QCD contributions to the fully differential cross section with intermediate top quark pair production at a photon collider.√ We have applied the above results to tt¯ production and decay at a future linear collider operating at s = 500 GeV. We have shown that for an appropriate choice of the polarizations of the laser and electron beam, the cross section and the double/single spin asymmetries can be quite large. While the QCD corrections to the cross section can be very large, most of the double/single spin asymmetries are affected at the level of only a few percent. The observables considered here will provide useful tools to analyse in detail the top quark pair production and decay dynamics. In particular, their precise measurement will test whether the top quark truly behaves as a quasi- free fermion as predicted in the Standard Model.

Acknowledgements

We wish to thank W. Bernreuther and P.M. Zerwas for discussions. A.B. was supported by a Heisenberg grant of the Deutsche Forschungsgemeinschaft. Z.G. Si wishes to thank DFG and MoE of China for ﬁnancial support of his short visit in Germany, and also thanks DESY Theory Group for its hospitality.

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Decay constants of the pseudoscalar charmonium and bottomonium

Zhi-Gang Wang a, Wei-Min Yang b, Shao-Long Wan b

a Department of Physics, North China Electric Power University, Baoding 071003, PR China b Department of Modern Physics, University of Science and Technology of China, Hefei 230026, PR China Received 15 November 2004; received in revised form 22 February 2005; accepted 23 March 2005 Available online 14 April 2005 Editor: W. Haxton

Abstract In this Letter, we investigate the structures of the pseudoscalar charmonium and bottomonium in the framework of the coupled rainbow Schwinger–Dyson equation and ladder Bethe–Salpeter equation with the confining effective potential (infrared modified flat bottom potential). As the current masses are very large, the dressing or renormalization for the c and b quarks are tender, however, mass poles in the timelike region are absent. The Euclidean time Fourier transformed quark propagator has no mass poles in the timelike region which naturally implements confinement. The Bethe–Salpeter wavefunctions for those mesons have the same type (Gaussian type) momentum dependence and center around zero momentum with spatial extension to about q2 = 1GeV2 which happen to be the energy scale for chiral symmetry breaking, the strong interactions in the infrared region result in bound states. The decay constants for those pseudoscalar heavy quarkonia are compatible with the values of experimental extractions and theoretical calculations.  2005 Elsevier B.V. All rights reserved.

PACS: 14.40.-n; 11.10.Gh; 11.10.St; 12.40.Qq

Keywords: Schwinger–Dyson equation; Bethe–Salpeter equation; Decay constant; Conﬁnement

1. Introduction the heavy quarks), the soft scale (the relative momentum of the heavy quark–antiquark |p|) and the ultrasoft Heavy quarkonium, bound state of the heavy quark scale (the typical kinetic energy of the heavy quark– and antiquark, characterized by at least three widely antiquark E), plays a special role in probing the strong separated energy scales: the hard scale (the mass m of interactions in both the perturbative and nonperturbative regions. By deﬁnition of the heavy quark, m is large in comparison with the typical hadronic scale E-mail address: [email protected] (Z.-G. Wang). ΛQCD, the corresponding processes can be success-

0370-2693/$ – see front matter  2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physletb.2005.03.080 80 Z.-G. Wang et al. / Physics Letters B 615 (2005) 79–86 fully described in perturbative quantum chromody- well as the meson structures, such as electromagnetic namics (QCD) due to the asymptotic freedom. How- form factors, radius, decay constants [18,22,23]. ever, the lower scales |p| and E, which are responsible During the past two years, the experiments have for the binding, cannot be accessible by perturbation discovered a number of new states, for example, the ηc − + theory. The appearance of multiscales in the dynamics in exclusive B → KKSK π decays by Belle [24], of the heavy quarkonium makes its quantitative study the narrow DsJ states by BaBar, CLEO and Belle [25], extremely difficult, the properties of the bound states evidence for the Θ+(1540) with quantum numbers and their decays can provide powerful test for QCD in of K+n [26], and the X(3872) through decay to both the perturbative and nonperturbative regions. π +π −J/ψ by Belle [27]. New experimental results The physicists propose many original approaches call for interpretations, offer opportunities to extend to deal with the long distance properties of QCD, such our knowledge about hadron spectrum and challenge as chiral perturbation theory [1], heavy quark effective our understanding of the strong interaction; further- theory [2], QCD sum rules [3], lattice QCD [4], pertur- more, they revitalize the study of heavy quarkonia and bative QCD [5], coupled Schwinger–Dyson equation stimulate a lot of theoretical analysis through the char- (SDE) and Bethe–Salpeter equation (BSE) method monia and bottomonia have been thoroughly investi- [6], nonrelativistic QCD [7], potential nonrelativistic gated. QCD [8], etc. All of those approaches have both out- The decay constants of the pseudoscalar charmo- standing advantages and obvious shortcomings in one nium and bottomonium (ηc and ηb) mesons play an or other ways. The coupled rainbow SDE and ladder important role in modern physics with the assumption BSE have given a lot of successful descriptions of of current-meson duality. The precise knowledge of the long distance properties of the low energy QCD the those values fηc and fηb will provide great im- and the QCD vacuum (for example, Refs. [9–12],for provements in our understanding of various processes recent reviews one can see Refs. [13,14]). The SDE convolving the ηc and ηb mesons, for example, the can naturally embody the dynamical symmetry break- process B → ηcK, where the mismatches between ing and confinement which are two crucial features of the theoretical and experimental values are large [28]. QCD, although they correspond to two very different The ηc meson is already observed experimentally, the energy scales [15,16]. On the other hand, the BSE is current experimental situation with the ηb meson is a conventional approach in dealing with the two body rather uncertain, yet the discovery of the ηb meson relativistic bound state problems [17].Fromthesolu- is one of the primary goals of the CLEO-c research tions of the BSE, we can obtain useful information program [29]; furthermore, the ηb meson may be ob- about the under-structure of the mesons and obtain served in run II at the Tevatron through the decay powerful tests for the quark theory. However, the obvi- modes into charmed states D∗D(∗) [30]. It is inter- ously drawback may be the model dependent kernels esting to combine those successful potential mod- for the gluon two-point Green’s function and the trun- els within the framework of coupled SDE and BSE cations for the coupled divergent SDE and BSE series to calculate the decay constants of the pseudoscalar in one or the other ways [18]. Many analytical and nu- heavy quarkonia such as ηc and ηb. For previous stud- merical calculations indicate that the coupled rainbow ies about the electroweak decays of the pseudoscalar SDE and ladder BSE with phenomenological potential mesons with the SDE and BSE, one can consult models can give model independent results and satis- Refs. [6,9–14]. In this Letter, we use an infrared mod- factory values [6,9–14]. The usually used effective po- ified flat-bottom potential (IMFBP) which takes the tential models are confining Dirac δ function potential, advantages of both the Gaussian distribution potential Gaussian distribution potential and flat bottom poten- and the FBP to calculate the decay constants of those tial (FBP) [13,14,19–21]. The FBP is a sum of Yukawa pseudoscalar heavy quarkonia. potentials, which not only satisfies chiral invariance The Letter is arranged as follows: we introduce the and fully relativistic covariance, but also suppresses IMFBP in Section 2; in Section 3–5,wesolvethe the singular point that the Yukawa potential has. It rainbow SDE and ladder BSE, explore the analytic- works well in understanding the dynamical chiral sym- ity of the quark propagators, investigate the dynam- metry breaking, confinement and the QCD vacuum as ical dressing and confinement, finally obtain the de- Z.-G. Wang et al. / Physics Letters B 615 (2005) 79–86 81

dV(0) d2V(0) dnV(0) cay constants for those pseudoscalar heavy quarkonia; = =···= = 0. (4) Section 6 is reserved for conclusion. dr dr2 drn The aj can be determined by solve the equations inferred from the flat bottom condition Eq. (4).Asin 2. Infrared modified flat bottom potential previous literature [18,21–23], n issettobe9.The phenomenological effective potential (IMFBP) can be The present techniques in QCD calculation cannot taken as give satisfactory large r behavior for the gluon two- 2 = 2 + 2 point Green’s function to implement the linear poten- G k G1 k G2 k . (5) tial confinement mechanism, in practical calculation, the phenomenological effective potential models always do the work. As in our previous work [18],we 3. SchwingerÐDyson equation use a Gaussian distribution function to represent the infrared behavior of the gluon two-point Green’s func- The SDE can provide a natural framework for in- tion, vestigating the nonperturbative properties of the quark and gluon Green’s functions. By studying the evo- 2 2 2 2 − k lution behavior and analytic structure of the dressed 4πG1 k = 3π e ∆ , (1) ∆2 quark propagators, we can obtain valuable information which determines the quark–antiquark interaction about the dynamical dressing phenomenon and con- through a strength parameter and a range parame- finement. In the following, we write down the rainbow ter ∆. This form is inspired by the δ function poten- SDE for the quark propagator, tial (in other words the infrared dominated potential) − S 1(p) = iγ · p +ˆm used in Refs. [19,20], which it approaches in the limit c,b ∆ → 0. For the intermediate momentum, we take the d4k λa λa + 4π γµ S(k)γν Gµν(k − p), FBP as the best approximation and neglect the large (2π)4 2 2 momentum contributions from the perturbative QCD (6) calculations as the coupling constant at high energy where is very small. The FBP is a sum of Yukawa poten- − S 1(p) = iA p2 γ · p + B p2 tials which is an analogy to the exchange of a series of particles and ghosts with different masses (Euclidean ≡ A p2 iγ · p + m p2 , (7) form), k k G (k) = δ − µ ν G k2 , (8) n µν µν 2 2 aj k G2 k = , (2) k2 + (N + jρ)2 ˆ j=0 and mc,b stands for the current quark mass that explicitly breaks chiral symmetry. where N stands for the minimum value of the masses, The full SDE for the quark propagator is a divergent ρ is their mass difference, and aj is their relative cou- series of coupled nonlinear integral equations for the pling constant. Due to the particular condition we take propagators and vertexes, we have to make truncations for the FBP, there is no divergence in solving the SDE. in one or other ways. The rainbow SDE has given a lot In its three-dimensional form, the FBP takes the fol- of successful descriptions of the QCD vacuum and low lowing form: energy hadron phenomena [6,13–16], in this Letter, we n e−(N+jρ)r take the rainbow SDE. If we go beyond the rainbow V(r)=− a . (3) λa j approximation, the bare vertex γµ 2 has to be substi- = r a j 0 tuted by the full quark–gluon vertex Γµ (qqg), which In order to suppress the singular point at r = 0, we satisfies the Slavnov–Tayler identity. In the weak cou- 2 → take the following conditions: pling limit, g 0, two Feynman diagrams contribute a to the vertex Γµ (qqg) at one-loop level due to the V(0) = const, non-Abelian nature of QCD, i.e., the self-interaction 82 Z.-G. Wang et al. / Physics Letters B 615 (2005) 79–86 of gluons [31]. If we neglect the contributions from the BSE for the pseudoscalar quarkonia, a three-gluon vertex Γµ (ggg) and retain an Abelian ver- a −1 P −1 P sion, the vertex Γ a(qqg) can be taken as λ Γ (qqp), S+ q + χ(q,P)S− q − µ 2 µ 2 2 where the vertex Γµ(qqp) is the quark–photon vertex 16π d4k which satisfies the Ward–Takahashi identity. In practi- = γ χ(k,P)γ G (q − k), 4 µ ν µν (9) cal calculation, we can take the vertex Γµ(qqp) to be 3 (2π) the Ball–Chiu and Curtis–Pennington vertex [32,33] where S(q) is the quark propagator, Gµν(k) is the so as to avoid solving the coupled SDE for the vertex gluon propagator, Pµ is the four-momentum of the Γµ(qqp). However, the nonperturbative properties of center of mass of the pseudoscalar quarkonia, qµ is QCD at the low energy region suggest that the SDEs the relative four-momentum between the quark and are strongly coupled nonlinear integral equations, no antiquark, γµ is the bare quark–gluon vertex, and theoretical work has ever proven that the contributions χ(q,P)is the Bethe–Salpeter wavefunction (BSW) of a from the vertex Γµ (ggg) can be safely neglected due the bound state. to the complex Dirac and tensor structures. The one We can perform the Wick rotation analytically and Feynman diagram contributions version of the ver- continue q and k into the Euclidean region.1 In the a tex Γµ (qqg), i.e., neglecting the contributions from lowest order approximation, the BSW χ(q,P) can be a a the vertex Γµ (ggg) in dressing the vertex Γµ (qqg) written as is inconsistent with the Slavnov–Tayler identity [31]. = 0 + · 0 If we take the assumption that the contributions from χ(q,P) γ5 iF1 (q, P ) γ PF2 (q, P ) the vertex Γ a(ggg) are not different greatly from the + · · 1 µ γ qq PF3 (q, P ) vertex Γ a(qqg), we can multiply the contributions µ + [ · · ] 0 a i γ q,γ P F4 (q, P ) . (10) from the vertex Γµ (qqg) by some parameters which effectively embody the contributions from the vertex The ladder BSE can be projected into the following a Γµ (ggg) [34]. four coupled integral equations, In this Letter, we assume that a Wick rotation to ∞ π Euclidean variables is allowed, and perform a rotation 0,1 = 3 2 analytically continuing p and k into the Euclidean re- H(i,j)Fj (q, P ) k dk sin θK(i, j), gion. The Euclidean rainbow SDE can be projected j j 0 0 (11) into two coupled integral equations for A(p2) and B(p2). Alternatively, one can derive the SDE from the expressions of the H(i,j)and K(i,j) are cumber- the Euclidean path-integral formulation of the theory, some and neglected here. thus avoiding possible difficulties in performing the Here we will give some explanations for the expres- Wick rotation [35]. As far as only numerical results sions of H(i,j).TheH(i,j)’s are functions of the 2 + are concerned, the two procedures are equal. In fact, quark’s Schwinger–Dyson functions (SDF) A(q 2 + · 2 + 2 + · 2 + 2 − the analytical structures of quark propagators have in- P /4 q P), B(q P /4 q P), A(q P /4 · 2 + 2 − · teresting information about confinement, we will make q P) and B(q P /4 q P). The relative four- detailed discussion about the c and b quarks propaga- momentum q is a quantity in the Euclidean space- tors respectively in Section 5. time while the center of mass four-momentum P must be continued to the Minkowski spacetime, i.e., P 2 = −m2 q · P ηc,ηb , this results in the varying throughout a complex domain. It is inconvenient to solve the SDE 4. BetheÐSalpeter equation at the resulting complex values of the quark momentum, especially for the heavy quarks. As the dressing The BSE is a conventional approach in dealing with effect is minor, we can expand A and B in terms of the two body relativistic bound state problems [17]. The precise knowledge about the quark structures of 1 To avoid possible difficulties in performing the Wick rotation, the mesons will result in better understanding of their one can derive the BSE from the Euclidean path-integral formula- properties. In the following, we write down the ladder tion of the theory. Z.-G. Wang et al. / Physics Letters B 615 (2005) 79–86 83

Taylor series of q · P , for example, 5. Coupled rainbow SDE and ladder BSE and the decay constants A q2 + P 2/4 + q · P In this section, we explore the coupled equations of = A q2 + P 2/4 + A q2 + P 2/4 q · P +···. the rainbow SDE and ladder BSE for the pseudoscalar heavy quarkonia numerically, the final results for the The other problem is that we cannot solve the SDE SDFs and BSWs can be plotted as functions of the in the timelike region as the two point gluon Green’s square momentum q2. function cannot be exactly inferred from the SU(3) In order to demonstrate the confinement of quarks, color gauge theory even in the low energy spacelike re- we have to study the analyticity of SDFs for the c and gion. In practical calculations, we can extrapolate the b quarks, and prove that there no mass poles on the real values of A and B from the spacelike region smoothly timelike q2 axial. In the following, we take the Fourier to the timelike region with suitable polynomial func- transform with respect to the Euclidean time T for the tions. To avoid possible violation with confinement in scalar part (Ss ) of the quark propagator [6,13,37], sense of the appearance of pole masses q2 =−m2(q2) +∞ in the timelike region, we must be care in choos- 2 ∗ dq4 B(q ) S (T ) = eiq4T , ing the polynomial functions [20].Fortheηc meson, s 2 2 2 2 2 2π q A (q ) + B (q ) = the mass is about 3.0 GeV, the extrapolation to the −∞ q 0 − 2 (14) timelike region with the quantity mηc /4 can be performed easily, however, the large mass of the ηb meson where the 3-vector part of q is set to zero. If S(q) has a makes the extrapolation into the deep timelike region mass pole at q2 =−m2(q2) in the real timelike region, troublesome. Although the η meson has not been ∗ −mT b the Fourier transformed Ss (T ) would fall off as e observed experimentally yet, the theoretical calcula- ∗ =− for large T or log Ss mT . tions indicate that its mass is about 9.4GeV[36].As In our numerical calculations, for small T ,theval- the dressed quark propagators comprise the notation ues of S∗ are positive and decrease rapidly to zero and 2 s of constituent quarks by providing a mass m(q ) = beyond with the increase of T , which are compatible 2 2 B(q )/A(q ), which corresponding to the dynamical with the result (curve tendency with respect to T ) from symmetry breaking phenomena for the light quarks. ∗ lattice simulations [38];forlargeT , the values of Ss We can simplify the calculation greatly and avoid the are negative, except occasionally a very small fraction problems concerning the extrapolations in solving the ∗ of positive values. The negative values for Ss indicate BSE by take the following propagator for the c and b an explicit violation of the axiom of reflection posi- quarks, tivity [39], in other words, the quarks are not physical observable, i.e., confinement. −1 2 S q = iγ · q + Mc,b, (12) For the c and b quarks, the current masses are very large, the dressing or renormalization is tender and the where the Mc,b is the Euclidean constituent quark curves are not steep which in contrast to the dynam- 2 = 2 2 = 2 mass with Mc,b mc,b(q ) q obtained from the ical chiral symmetry breaking phenomenon for the solution of the SDE (6). light quarks, mc(0)/mˆ c 1.5 and mb(0)/mˆ b 1.1, Finally we write down the normalization condition however, mass poles in the timelike region are ab- for the BSW, sent. At zero momentum, mc(0) = 1937 MeV and m (0) = 5105 MeV, while the Euclidean constituent b 4 −1 quark masses M = 1908 MeV and M = 5096 MeV, d q ∂S+ −1 c b N χ¯ χ(q,P)S 2 2 2 2 c 4 Tr − which defined by M = m (q ) = q , are compati- (2π) ∂Pµ ble with the constituent quark masses in the litera- −1 −1 ∂S− ture. From the plotted BSWs (see Fig. 1 as an ex- +¯χS+ χ(q,P) = 2Pµ, (13) ∂Pµ ample), we can see that the BSWs for pseudoscalar mesons have the same type (Gaussian type) momen- + where χ¯ = γ4χ γ4. tum dependence while the quantitative values are dif- 84 Z.-G. Wang et al. / Physics Letters B 615 (2005) 79–86

b quarks are taken in solving the BSE as the dressing is tender. We borrow some idea from the fact that the simple phenomenological model of Cornell potential (Coulomb potential plus linear potential) with constituent quark masses can give satisfactory mass spectrum for the heavy quarkonia3 and take larger values for the strength parameter and range parameter ∆, i.e., = 2.2 GeV and ∆ = 2.9GeV2,in the infrared region comparing with the corresponding ones used in Ref. [18]. Furthermore the masses of the pseudoscalar mesons are taken as input parameters. If we take the Euclidean constituent quark masses Mc = mc(0) and Mb = mb(0), the decay constants for = the ηc and ηb mesons change slightly, fηc 357 MeV Fig. 1. BSWs for charmonium. = and fηb 289 MeV. ferent from each other. Just like the lighter qq¯ and qQ¯ pseudoscalar mesons [18], the Gaussian type BSWs 6. Conclusion center around zero momentum with spatial extension to about q2 = 1GeV2 which happen to be the energy In this Letter, we investigate the under-structures scale for chiral symmetry breaking, the strong inter- of the pseudoscalar heavy quarkonia η and η in the actions in the infrared region result in bound states. c b framework of the coupled rainbow SDE and ladder Finally we obtain the values for the decay constants of BSE with the conﬁning effective potential (IMFBP). those pseudoscalar mesons which are deﬁned by After we solve the coupled rainbow SDE and ladder ifπ Pµ =0|¯qγµγ5q π(P) , BSE numerically, we obtain the SDFs and BSWs for the pseudoscalar heavy quarkonia η and η .Asthe d4k c b = N γ γ χ(k,P) , current masses of the c and b quarks are very large, c Tr µ 5 4 (15) (2π) the dressing or renormalization for the SDFs is ten- hereweuseπ to represent the pseudoscalar mesons,2 der and the curves are not steep which in contrast to the explicitly dynamical chiral symmetry break- = = fηc 349 MeV,fηb 287 MeV, (16) ing phenomenon for the light quarks, however, mass which are compatible with the results from the ex- poles in the timelike region are absent. We can sim- perimental extractions and theoretical calculations, plify the calculation greatly and avoid the problems = ± ≈ concerning the extrapolations in solving the BSE by fηc 335 75 MeV (exp) [40]; fηc 400 MeV = ± = ± making the substitution B(q2) → M and A(q2) → 1. (exp) [41]; fηc 420 52 MeV, fηb 705 27 MeV = ± The BSWs for the pseudoscalar heavy quarkonia have (theor) [42]; fηc 292 25 MeV (theor) [43]; ≈ = ± the same type (Gaussian type) momentum depen- fηc 350 MeV (theor) [44]; fηc 300 50 MeV (theor) [45]. In calculation, the values of mˆ c and mˆ b dence while the quantitative values are different from are taken as the current quark masses, mˆ c = 1250 MeV each other. The Gaussian type BSWs center around 2 = and mˆ b = 4700 MeV; the input parameters for the zero momentum with spatial extension to about q 2 FBP are N = 1.0Λ, V(0) =−11.0Λ, ρ = 5.0Λ and 1GeV which happen to be the energy scale for chi- Λ = 200 MeV, which are determined in study of the ral symmetry breaking, the strong interactions in the qq¯ and qQ¯ pseudoscalar mesons [18]. In this Letter, infrared region result in bound states. Our numerical the Euclidean constituent quark masses for the c and results for the values of the decay constants of the

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Enlarged Galilean symmetry of anyons and the Hall effect

P.A. Horváthy a, L. Martina b,P.C.Stichelc

a Laboratoire de Mathématiques et de Physique Théorique, Université de Tours, Parc de Grandmont, F-37 200 Tours, France b Dipartimento di Fisica dell’Università, and Sezione INFN di Lecce, Via Arnesano, CP 193, I-73 100 Lecce, Italy c An der Krebskuhle 21, D-33 619 Bielefeld, Germany Received 23 March 2005; accepted 2 April 2005 Available online 12 April 2005 Editor: N. Glover

Abstract Enlarged planar Galilean symmetry, built of both space–time and field variables and also incorporating the “exotic” central extension is introduced. It is used to describe non-relativistic anyons coupled to an electromagnetic field. Our theory exhibits an anomalous velocity relation of the type used to explain the anomalous Hall effect. The Hall motions, characterized by a Casimir of the enlarged algebra, become mandatory for some critical value(s) of the magnetic field. The extension of our scheme yields the semiclassical effective model of the Bloch electron.  2005 Elsevier B.V. All rights reserved.

1. Introduction ﬁeld yields the ﬁrst-order phase space Lagrangian

2 ˙ P ˙ L = Pi · Xi − + e(AiXi + A0) The planar Galilei group admits a two-fold “exotic” 2m central extension, labeled with m (the mass) and a sec- θ + P P˙ , (1.1) ond, “exotic” parameter κ [1]. Models which provide 2 ij i j a physical realization of this “exotic” symmetry have where θ =−κ/m2 is the non-commutative parameter. been presented in [2,3]. Below, we focus our atten- The Euler–Lagrange equations are tion on the theory of [3], since that of [2] is in fact an ∗ ˙ extended version of the latter. Minimal (symplectic) m Xi = Pi − emθij Ej , (1.2) coupling of the particle to an external electromagnetic ˙ ˙ Pi = eBij Xj + eEi, (1.3)

where Ei and B are the electric and magnetic ﬁeld, re- ∗ = − E-mail addresses: [email protected] (P.A. Horváthy), spectively, and m m(1 eθB) is an effective mass. [email protected] (L. Martina), These equations can also be obtained in a Hamil- [email protected] (P.C. Stichel). tonian framework, using the usual Hamiltonian H =

2 P /2m − eA0 and the modified Poisson brackets Now we consider the enlarged Hamiltonian structure. Then the equations of motion (1.2) and (1.3), aug- { }= m { }= m ˙ = ˙ ={ } Xi,Xj ∗ θij , Xi,Pj ∗ δij , mented with πi eXi are Hamiltonian, Y Y,H , m m with the usual Hamiltonian H = P2/2m − eE X , m i i {P ,P }= eB . (1.4) and the fundamental Poisson brackets (1.4), supple- i j m∗ ij mented with {Ei,πj }=δij . Then conserved quantities The most dramatic prediction of the model is that are readily constructed. Integration of the equation of when m∗ = 0, i.e., when the magnetic field takes the motion (1.3) shows that critical value Pi = Pi − eBij Xj − eEit (2.2) 1 B = B = , (1.5) is a constant of the motion. Using the commutation crit eθ relations the system becomes singular, and the only allowed { P }= { P }= motions follow the Hall law [3]. The Poisson brack- Xi, j δij , Pi, j 0, ets (1.4) are changed for new ones in a reduced phase {Pi,Ej }=0, {πi, Pj }=etδij , (2.3) space (see [3]). we find furthermore that (2.2) generates enlarged Requiring B = B amounts to a restriction to the crit translations (2.1). Similarly, lowest Landau level, and quantization allowed us to recover the “Laughlin” wave functions [3]. θ 2 eB 2 J = ij XiPj + P + X + ij Eiπj + s0, (2.4) In this Letter we generalize this model, and indicate 2 2 its relation to models used in solid state physics [4,5]. E t K = mX − P + e i t + mθ P − B π , i i i 2 ij j ij j (2.5) 2. Enlarged Galilean symmetry are conserved and generate enlarged rotations and boosts, respectively. Note that anyonic spin, repre- Adapting the idea of [6] to planar physics, we con- sented by the real number s0, has also been included. sider a homogeneous electric field Ei(t) and a constant The generators satisfy the enlarged Poisson relations magnetic field B, and view Ei and its canonical con- {Pi,H}=eEi, {Ki,H}=Pi, {J ,H}=0, jugate momentum, πi , as additional variables on an enlarged phase space. This latter is endowed with the {Pi, J }=−ij Pj , {Ki, J }=−ij Kj , enl = + ˙ enlarged Lagrangian L L πiEi. The πi are La- {Pi, Kj }=−mδij , {Pi, Pj }=−eBij , grange multipliers and one of the equations of motion 2 ˙ {Ki, Kj }=−m θij . (2.6) is Ei = 0, i.e., the electric field should actually be constant. A closed algebra is obtained, therefore, if the electro- The Galilean symmetry of combined particle + ho- magnetic fields Ei and B are considered as additional mogeneous field system is readily established: the en- elements of an enlarged Galilei algebra g˜,cf.[6].The larged Lagrangian is (quasi-) invariant w.r.t. enlarged (constant) magnetic field, B, belongs, together with space translations, rotations and boosts, implemented m and κ =−θm2, to the center of g˜. The additional as nonzero brackets are {E , J }=− E , {E , K }=B . (2.7) δXi = ai,δPi = 0,δEi = 0,δπi = eait, i ij j i j ij Our enlarged Galilei group has two independent δXi =−ϕij Xj ,δPi =−ϕij Pj , Casimirs, namely, δEi =−ϕij Ej ,δπi =−ϕij πj , m 2 δX = b t, δP = mb , C = eθ BH − ij PiEj + E , (2.8) i i i i 2B 2 t P2 δEi =−Bij bj ,δπi = ebi . (2.1) e meθ 2 2 C = − H − (KiEi + J B)− E . (2.9) 2m m 2B P.A. Horváthy et al. / Physics Letters B 615 (2005) 87–92 89

In the representation of the enlarged Galilei algebra (which is here a constant), also an anomalous term given in terms of phase space variables proportional to θP × E. 2 Such a theory is still symmetric w.r.t. the enlarged eθB m Galilei group by construction. The equation of motion, C = Pi − ij Ej , 2m B reminiscent to Eq. (5.3) of [9],is es0B C =−C − . (2.10) ∗ ˙ g g m m Xi = 1 − eθB Pi − 1 − emθij Ej , 2 2 C generalizes one of the two Casimirs of the planar (3.3) Galilei group, namely, the internal energy [7]. C + C supplemented with the Lorentz force law (1.3). ˙ is in turn proportional to the second Casimir identified When g = 2 and eθB = 1, mXi = Pi , so that as the spin; these two quantities are linked, just like our equations describe an ordinary charged particle for the model of [2]. The relation of the enlarged and in an electromagnetic field. For g = 2 and eθB = 1, ordinary Galilean algebras can be clarified by a sub- Eq. (3.3) is identically satisfied. tle group contraction. Though constants of the motion, We assume henceforth that g = 2. Then Eq. (3.3) our Casimirs are not fixed constant: the representation describes an “exotic” particle with anomalous moment of our enlarged Galilei group is, in general reducible. coupling with gyromagnetic ratio g,cf.[9], which If the fields become non-dynamical, those transfor- generalizes the g = 0 theory of Ref. [3]. mations which are consistent with the constant fields Let us now consider Hall motions, i.e., such that remain symmetries. For example, (2.2) becomes the Ej familiar magnetic translation. X˙ = . (3.4) i ij B For

3. Algebraic construction of the coupled anyon 2 1 B = B = , (3.5) plus electromagnetic ﬁeld system and Hall effects crit g eθ

Having established our enlarged planar Galilei al- this is a solution of the equations of motion (3.3) and gebra, now we build a new theory of anyons inter- (1.3) when the momentum satisfies acting with (constant) external fields. Generalizing a Ej P = m . (3.6) formula of Bacry [8] who argued that the Hamiltonian i ij B should be constructed from generators of the symme- The constraint (3.6) is clearly equivalent to the vanish- try group, we consider ing of the Casimir, C = 0. Conversely, from (3.6) we ˙ P2 infer that Pi = 0, so that the Lorentz force on the r.h.s. e meθ 2 H = − (KiEi + J B)− E , (3.1) of (1.3) is necessarily zero and the motion follows the 2m m 2B = Hall law. Thus, when B Bcrit, the Hall motions are where that last term is dictated by boost invariance. H characterized by the constraint (3.6), in turn equiva- is indeed H = H + C . Choosing a real parameter g, C = lent to 0. The condition (3.6) is invariant w.r.t. the H can be further generalized as enlarged Galilei transformations and, when restricted to such motions, the representation of the enlarged = + g C Hanom H Galilei algebra becomes irreducible. 2 P2 g The generic motions have the familiar cycloidal = 1 − eθB − eE · X − µB form, made of the Hall drift of the guiding center, com- 2m 2 posed with uniform rotations with frequency geθ mgeθ 2 + P × E − E , (3.2) eB g 2 4B Ω = 1 − eθB . (3.7) 2m∗ 2 where µ = ges0/2m. The kinetic term gets hence a field-dependent factor; our Hamiltonian contains, For g = 2 this reduces to the usual Larmor frequency together with the usual magnetic moment term µB eB/m.TheCasimirC measures the extent the actual 90 P.A. Horváthy et al. / Physics Letters B 615 (2005) 87–92 motion fails to be a Hall motion. For m∗ = 0, i.e., for Berry curvature, B = B = (eθ)−1, (3.3) implies (3.6), and hence the crit ∂ only allowed motions are the Hall motions [3,9]. θ(P)= ij Aj (P), (4.2) ∂P Interestingly, this is also what happens for B = B i crit ˙ (which plainly requires g = 0). Then the momentum was a constant. Xi , physically the group velocity of drops out from (3.3).Forg = 2 (3.3) holds identically, the Bloch electron, satisfies hence the equation but for g = 2 it becomes 1 − eBθ(P) X˙ = ∂ E − eθ(P) E , (4.3) g i Pi ij j X˙ = eθ E , (3.8) i 2 ij j supplemented with the Lorentz equation (1.3). which is once again the Hall law (3.4) with B = Eq. (4.3) has the same structure as our original = B [9]. Now the constraint (3.6) is not enforced: by (g 0) velocity relation (1.2). The Berry curvature crit provides us with a momentum dependent effective non- (1.3), the momentum is an arbitrary constant. The two critical values correspond to the frequen- commutative parameter θ(P), which yields in turn P - ∗ = − cies Ω =∞and Ω = 0, respectively. In the first case dependent effective mass m m(1 eθ(P )B) and m∗ = 0, only those initial conditions are consistent anomalous velocity terms, cf. [5,10]. which satisfy (3.6). The system is singular and re- In a Hamiltonian framework, the system is de- Bloch = E − quires reduction [3]. In the second case the initial mo- scribed by the Bloch Hamiltonian H (P) mentum can be arbitrary, since it has no influence on eEiXi , and by formally the same Poisson brackets the motion. The system acquires an extra translational (1.4), except for the momentum dependence of θ. symmetry in momentum space. In particular, the mean band position coordinates do not commute, as it had been observed a long time ago [10].TheP dependence is consistent with the 4. Planar Bloch electron in external fields Jacobi identity, [11], even for a position-dependent B [3]. While our theory may seem to be rather specula- Once again, we can consider the enlarged frame- tive, it has interesting analogies in solid state physics, work of Section 2. This yields the previous equations namely in the theory of a Bloch electron in a crystal. of motion and commutation relations, supplemented ˙ = { }= Restricting ourselves to a single band, the band en- with Ei 0 and Ei,πi δij , respectively. The ergy and the background fields provide in fact effective P -dependence of NC parameter breaks the Galilean terms for the semiclassical dynamics of the electronic symmetry down to the magnetic translations (times wave packet [5]. The mean Bloch wave vector (quasi- time translation) alone. Assuming θ and E only de- 2 momentum) we denote here by P varies in a Brillouin pend on P , rotational symmetry is restored, though. 2 zone. In terms of P and the mean band position co- J 1 P 2 It is generated by in (2.4), with 2 θ(P ) re- ordinates, Xi , the system is described by the effective placing the “exotic” contribution θP2/2. Hence, we Lagrangian [5] are left with a residual symmetry with generators Bloch ˙ ˙ Pi , H , J , Ei and B; the latter belongs to the cen- L = PiXi − E + e Ai(X,t)Xi + A0 ter. The enlarged euclidean algebra has the Casimirs + A ˙ Bloch i(P)Pi (4.1) C0 = BH − ij PiEj and the infinite tower Cn = =−1 = · (E2g)n, n = 1,.... with Ai 2 ij BXj , A0 E X. The expression Let us now inquire about the Hall motions. Insert- E(P)= E0(P)− M(P)B (where E0(P) is the energy of the band and M(P) is the mean magnetic moment) ing the Hall law into the equation of motion (4.3) gen- yields a kinetic energy term for the effective dynamics. eralizes the condition (3.6) as A The (effective) vector potential i is the Berry ij Ej ∂ E = . (4.4) connection; it can arise, e.g., in a crystal with no Pi B spatial inversion symmetry as in GaAs [5].Thelast Then (4.4) satisfies C = 0 for the Casimir term in (4.1) is indeed analogous to our “exotic” term ˙ C = E 2 − E 2 − +| | (θ/2)ij PiPj in (1.1), to which it would reduce if the B P P0 ij PiEj E P0, (4.5) P.A. Horváthy et al. / Physics Letters B 615 (2005) 87–92 91

2 E 2 2 = = where P0 is a solution of the equation P0 ( (P0 )) [θ θ(P)], supplemented with the Lorentz equation (E/ 2B)2. That C = 0 is equivalent to (4.4) can be (1.3). For a Hall motion the momentum is constant, proved, e.g., in the generalized parabolic case E ∼ P = P 0. Putting θ 0 = θ(P 0) and inserting into the (P2)α,1/2 <α<3/2. It appears, however, that the equation of motion vanishing of C is a mere coincidence, and (4.4) is bet- g 0 Ej C = 1 − eθ B ∂ E 0 − ter interpreted as the extremum condition ∂Pi 0for Pi P ij 2 B P0 the Casimir. (For (2.10) we obviously have a minimum.) Eq. (4.4) can have more than just one solution. = g C 0 e P ∂Pi θ . (4.9) This happens, e.g., for the energy expression 2 P0 = Let us assume that ∂Pi θ 0. Although (4.9) has E = a2P2 + b2 1 + c2P2 more general solutions, we observe that if the Hall condition (4.4) holds, then we have again C = 0. At considered by Culcer et al. in Ref. [4] for the anom- last, 1 − (g/2)eθ 0B = 0 can also yield Hall motions alous Hall effect. Then the vanishing of C is clearly of the second type for some particular value of the mo- irrelevant, but the extremum property is still valid. mentum. If the effective non-commutative parameter θ = θ(P) is genuinely momentum-dependent as it happens in Ref. [4,5], the magnetic field cannot be tuned to ei- 5. Conclusion ther of the critical values Bcrit or Bcrit. Hence, the Hall motions cannot be made mandatory. In this Letter we presented, following Ref. [6],a Writing either (1.2) or (4.3) as framework which unifies phase space and field variables, and used it to introduce “enlarged Galilean ˙ = − ˙ Xi (kinetic term) eθij Pi, (4.6) symmetry”. Then we built a theory along the lines we recognize the anomalous velocity used in the de- put forward by Bacry [8]. Adding a Casimir to the scription of the anomalous Hall effect in ferromagnet- Hamiltonian, our theory can accommodate anomalous moment coupling, cf. [9]. We derived an algebraic ics [4]. The additional contribution (g/2)emθij Ej in characterization of the Hall motions and have shown Eq. (3.3) comes from the anomalous term (g/2)θ E × that for B = B or B the Hall motions become P in the Hamiltonian (3.2). This latter can be viewed crit crit mandatory. The physical interpretation is provided by as the “Jackiw–Nair” [12] limit of the spin–orbit cou- the semiclassical theory of a Bloch electron, where a pling momentum-dependent effective non-commutative pa- 1 rameter is derived by a Berry phase calculation [5,11]. σ × E · P, (4.7) m2c2 advocated by Karplus and Luttinger half a century 2 2 Acknowledgements ago [4]. Putting σ = sσ3, θ =−s/c m , (4.7) becomes indeed proportional to our anomalous term. P.A.H. and P.C.S. would like to thank for hospitality Such a term also arises in the non-relativistic limit of the University of Lecce. We are indebted to W. Za- charged Dirac particle in a constant electric field [13]. krzewski for discussions. Adding (geθ/2)C to H Bloch yields an anomalous extension of the semiclassical Bloch model with (4.3) generalized to References (1 − eBθ)X˙ i [1] J.-M. Lévy-Leblond, Galilei group and Galilean invariance, in: g g E. Loebl (Ed.), Group Theory and Applications, vol. 2, Acad- = 1 − eθB ∂ E − 1 − eθ E 2 Pi 2 ij j emic Press, New York, 1972, p. 222; Y. Brihaye, C. Gonera, S. Giller, P. Kosinski,´ hep-th/9503046; eg D.R. Grigore, J. Math. Phys. 37 (1996) 240; − C∂P θ (4.8) 2 i D.R. Grigore, J. Math. Phys. 37 (1996) 460. 92 P.A. Horváthy et al. / Physics Letters B 615 (2005) 87–92

[2] J. Lukierski, P.C. Stichel, W.J. Zakrzewski, Ann. Phys. [6] J. Negro, M.A. del Olmo, J. Math. Phys. 31 (1990) 2811. (N.Y.) 260 (1997) 224; [7] P.A. Horváthy, M.S. Plyushchay, Phys. Lett. B 595 (2004) 547, P.A. Horváthy, M.S. Plyushchay, JHEP 0206 (2002) 033, hep- hep-th/0404137; th/0201228. C. Duval, P.A. Horváthy, Phys. Lett. B 547 (2002) 306, hep- [3] C. Duval, P.A. Horváthy, Phys. Lett. B 479 (2000) 284, hep- th/0209166. th/0002233; [8] H. Bacry, Lett. Math. Phys. 1 (1976) 295. C. Duval, P.A. Horváthy, J. Phys. A 34 (2001) 10097, hep- [9] C. Duval, P.A. Horváthy, Phys. Lett. B 594 (2004) 402, hep- th/0106089. th/0402191; [4] R. Karplus, J.M. Luttinger, Phys. Rev. 95 (1954) 1154; See also: C. Duval, Thèse de 3e cycle, Marseille, 1972; T. Jungwirth, Q. Niu, A.H. MacDonald, Phys. Rev. Lett. 90 B.S. Skagerstam, A. Stern, Phys. Scr. 24 (1981) 493; (2002) 207208; G. Grignani, M. Plyushchay, P. Sodano, Nucl. Phys. B 464 D. Culcer, A.H. MacDonald, Q. Niu, Phys. Rev. B 68 (2003) (1996) 189, hep-th/9511072. 045327. [10] E.N. Adams, E.I. Blount, J. Phys. Chem. Solids 10 (1959) 286. [5] M.C. Chang, Q. Niu, Phys. Rev. Lett. 75 (1995) 1348; [11] A. Bérard, H. Mohrbach, Phys. Rev. D 69 (2004) 127701, hep- M.C. Chang, Q. Niu, Phys. Rev. B 53 (1996) 7010; th/0310167. G. Sundaram, Q. Niu, Phys. Rev. B 59 (1999) 14915; [12] R. Jackiw, V.P. Nair, Phys. Lett. B 480 (2000) 237, hep- See also: A. Bohm, A. Mostafazadeh, H. Koizumi, Q. Niu, th/0003130. J. Zwanziger, The Geometric Phase in Quantum Systems, [13] A. Bérard, H. Mohrbach, hep-th/0404165. Springer-Verlag, Berlin, 2003, Chapter 12. Physics Letters B 615 (2005) 93–101 www.elsevier.com/locate/physletb

Phase structure of Nambu–Jona-Lasinio model at ﬁnite isospin density

Lianyi He, Pengfei Zhuang

Physics Department, Tsinghua University, Beijing 100084, China Received 5 January 2005; received in revised form 28 March 2005; accepted 28 March 2005 Available online 7 April 2005 Editor: W. Haxton

Abstract

In the frame of ﬂavor SU(2) Nambu–Jona-Lasinio model with UA(1) breaking term we found that, the structure of two chiral phase transition lines does not exist at low isospin density, and the critical isospin chemical potential for pion superﬂuidity is exactly the pion mass in the vacuum.  2005 Elsevier B.V. All rights reserved.

PACS: 11.10.Wx; 12.38.-t; 25.75.Nq

1. Introduction

It is generally believed that there exists a rich phase structure of quantum chromodynamics (QCD) at finite temperature and baryon density, for instance, the deconfinement process from hadron gas to quark–gluon plasma, the transition from chiral symmetry breaking phase to the symmetry restoration phase [1], and the color superconductivity [2] at low temperature and high baryon density. Recently, the study on the QCD phase structure is extended to finite isospin density. The physical motivation to study isospin spontaneous breaking and the corresponding pion superfluidity is related to the investigation of compact stars, isospin asymmetric nuclear matter and heavy ion collisions at intermediate energies. While the perturbation theory of QCD can describe well the properties of the new phases at high temperatures and/or high densities, the study on the phase structure at moderate (baryon or isospin) density depends on lattice QCD calculation and effective models with QCD symmetries. While there is not yet precise lattice result at finite baryon density due to the fermion sign problem [3], it is in principle no problem to do lattice simulation at finite isospin density [4]. It is found [5] that the critical isospin chemical potential for pion condensation is about the pion

E-mail address: [email protected] (L. He).

c mass in the vacuum, µI mπ . The QCD phase structure at finite isospin density is also investigated in many low energy effective models, such as chiral perturbation theory [4,6,7], ladder QCD [8], random matrix method [9], strong coupling lattice QCD [10] and Nambu–Jona-Lasinio (NJL) model [11–13]. One of the models that enables us to see directly how the dynamic mechanisms of chiral symmetry breaking and restoration operate is the NJL model [14] applied to quarks [15]. Within this model, one can obtain the hadronic mass spectrum and the static properties of mesons remarkably well [15,16], and the chiral phase transition line [15–17] in the temperature and baryon chemical potential (T –µB ) plane is very close to the one calculated with lattice QCD. Recently, this model is also used to investigate the color superconductivity at moderate baryon density [18–22]. In the study at finite isospin density, it is predicted [11] in this model that the chiral phase transition line in T –µB plane splits into two branches. This phenomena is also found in random matrix method [9] and ladder QCD [8]. Since the NJL Lagrangian used in [11] does not contain the determinant term which breaks the UA(1) symmetry and leads to reasonable meson mass splitting, it is pointed out [13] that the presence of the UA(1) breaking term will cancel the structure of the two chiral phase transition lines, if the coupling constant describing the UA(1) breaking term is large enough. However, this coupling constant is considered as a free parameter and not yet determined in [13]. Another problem in the NJL calculation at finite isospin density is that the critical isospin c = chemical potential for pion condensation µI mπ is not recovered in the model [12]. In this Letter we will focus on these two problems in the frame of NJL model with UA(1) breaking term. We hope to derive the critical isospin chemical potential exactly and try to fix the chiral structure by fitting the meson masses in the vacuum. The Letter is organized as follows. We present at finite temperature and baryon and isospin densities the chiral and pion condensates in mean field approximation and meson masses in random phase approximation (RPA) in Section 2, determine the coupling constant of the UA(1) breaking term in Section 3, and then analytically prove c = the relation µI mπ in Section 4. We conclude in Section 5.

2. NJL model at ﬁnite temperature and baryon and isospin densities

We start with the flavor SU(2) NJL model defined by ¯ µ L = ψ iγ ∂µ − m0 + µγ0 ψ + Lint, (1) where m0 is the current quark mass, µ the chemical potential matrix in flavor space, µ µ µ 0 B + I 0 µ = u = 3 2 (2) 0 µ µB − µI d 0 3 2 1 with µB and µI being the baryon and isospin chemical potential, respectively, and the interaction part includes [13] the normal four Fermion couplings corresponding to scalar mesons σ, a0,a+ and a− and pseudoscalar mesons η , π0, π+ and π− excitations, and the ’t Hooft [23] determinant term for UA(1) breaking, G 3 K L = (ψτ¯ ψ)2 + (ψiγ¯ τ ψ)2 + det ψ(¯ 1 + γ )ψ + det ψ(¯ 1 − γ )ψ int 2 a 5 a 2 5 5 a=0 1 1 = (G + K) (ψψ)¯ 2 + (ψiγ¯ τψ) 2 + (G − K) (ψ¯ τψ) 2 + (ψiγ¯ ψ)2 . (3) 2 5 2 5 For K = 0 and m0 = 0, the Lagrangian is invariant under UB (1) ⊗ UA(1) ⊗ SUV (2) ⊗ SUA(2) transformations, but for K = 0, the symmetry is reduced to UB (1) ⊗ SUV (2) ⊗ SUA(2) and the UA(1) breaking leads to σ and a mass splitting and π and η mass splitting. If G = K, we come back to the standard NJL model [15] with only σ, π0,π+ and π− mesons.

1 The isospin chemical potential here is deﬁned as the double of the one in [11–13]. L. He, P. Zhuang / Physics Letters B 615 (2005) 93–101 95

We introduce the quark condensates ¯ σu =¯uu,σd =dd, (4) or, equivalently, the σ and a0 condensates ¯ ¯ ¯ ¯ σ =ψψ=¯uu + dd=σu + σd ,a0 =ψτ3ψ=¯uu − dd=σu − σd , (5) and the pion condensate

π ¯ ¯ 1 ¯ √ =ψiγ5τ+ψ=ψiγ5τ−ψ=√ ψiτ1γ5ψ, (6) 2 2 where we have chosen the pion condensate to be real. The quark condensate and pion condensate are, respectively, the order parameter of chiral phase transition and pion superfluidity. By separating each Lorentz scalar in the Lagrangian (3) into the classical condensate and the quantum fluctuation, and keeping only the linear terms in the fluctuations, one obtains the Lagrangian in mean field approximation,

− G + K L = ψ¯ S 1 ψ − G σ 2 + σ 2 − 2Kσ σ − π 2, (7) MF MF u d u d 2 S−1 where MF is the inverse of the mean ﬁeld quark propagator, in momentum space it reads γ µk + µ γ − M i(G + K)πγ S−1 = µ u 0 u 5 MF(k) µ (8) i(G + K)πγ5 γ kµ + µd γ0 − Md with the effective quark masses

Mu = m0 − 2Gσu − 2Kσd ,Md = m0 − 2Gσd − 2Kσu. (9) The thermodynamic potential of the system in mean ﬁeld approximation can be expressed in terms of the effective quark propagator,

G + K T − Ω(T,µ ,µ ; σ ,σ ,π)= G σ 2 + σ 2 + 2Kσ σ + π 2 − ln det S 1 (k). (10) B I u d u d u d 2 V MF

The condensates σu,σd and π as functions of temperature and baryon and isospin chemical potentials are determined by the minimum thermodynamic potential, ∂Ω ∂Ω ∂Ω = 0, = 0, = 0. (11) ∂σu ∂σd ∂π

It is easy to see from the chemical potential matrix and the quark propagator matrix that for µB = 0orµI = 0the gap equations for σu and σd are symmetric, and one has σ σ = σ = ,a= 0,M= M = M = m − (G + K)σ. (12) u d 2 0 u d q 0 The quark propagator in ﬂavor space can be formally expressed as Suu(k) Sud (k) SMF(k) = . (13) Sdu(k) Sdd(k) From the comparison with the deﬁnition of the condensates (4) and (6), one can express the condensates in terms of the matrix elements of SMF, d4k d4k σu =− Tr iSuu(k) ,σd =− Tr iSdd(k) , (2π)4 (2π)4 96 L. He, P. Zhuang / Physics Letters B 615 (2005) 93–101 d4k π = Tr Sud (k) + Sdu(k) γ . (14) (2π)4 5

In the self-consistent mean ﬁeld approximation, it is well known that the meson masses MM as the bound states of the colliding quark–antiquark pairs are determined as the poles of the meson propagators in RPA at zero momentum,

1 − (G + K)ΠM (k0 = MM , k = 0) = 0 (15) for σ and π, and

1 − (G − K)ΠM (k0 = MM , k = 0) = 0 (16) for η and a, where ΠM is the meson polarization function 4 d p ∗ −iΠM (k) =− Tr Γ iS (p + k)ΓM iS (p) , (17) (2π)4 M MF MF ∗ and ΓM and ΓM are the interaction vertexes   = =  1,Mσ,  1,Mσ,    τ ,M= a ,  τ ,M= a ,  3 0  3 0  =  =  τ+,Ma+,  τ−,Ma+,   τ−,M= a−, ∗ τ+,M= a−, ΓM = Γ = (18)  iγ ,M= η, M  iγ ,M= η,  5  5  =  =  iγ5τ3,Mπ0,  iγ5τ3,Mπ0,    iγ τ+,M= π+,  iγ τ−,M= π+,  5  5 iγ5τ−,M= π−, iγ5τ+,M= π−, √ with τ± = (τ1 ± iτ2)/ 2. Doing the trace in ﬂavor and color spaces, one has d4p d4p Πa+ (k) = 6i Tr Suu(p + k)Sdd(p) ,Πa− (k) = 6i Tr Sdd(p + k)Suu(p) , (2π)4 (2π)4 d4p Ππ+ (k) =−6i Tr γ Suu(p + k)γ Sdd(p) , (2π)4 5 5 d4p Ππ− (k) =−6i Tr γ Sdd(p + k)γ Suu(p) , (2π)4 5 5 d4p Π (k) = i S (p + k)S (p) + S (p + k)S (p) σ 3 4 Tr uu uu ud du (2π) + Sdu(p + k)Sud (p) + Sdd(p + k)Sdd(p) , d4p Π (k) = i S (p + k)S (p) − S (p + k)S (p) a0 3 4 Tr uu uu ud du (2π) − Sdu(p + k)Sud (p) + Sdd(p + k)Sdd(p) , d4p Π (k) =− i γ S (p + k)γ S (p) − γ S (p + k)γ S (p) π0 3 4 Tr 5 uu 5 uu 5 ud 5 du (2π) − γ5Sdu(p + k)γ5Sud (p) + γ5Sdd(p + k)γ5Sdd(p) , L. He, P. Zhuang / Physics Letters B 615 (2005) 93–101 97 d4p Π (k) =− i γ S (p + k)γ S (p) + γ S (p + k)γ S (p) η 3 4 Tr 5 uu 5 uu 5 ud 5 du (2π) + γ5Sdu(p + k)γ5Sud (p) + γ5Sdd(p + k)γ5Sdd(p) , (19) now the trace is taken only in spin space.

3. Chiral phase structure

c We now consider the QCD phase structure below the minimum isospin chemical potential µI for pion superfluidity. Since the pion condensate is zero, there is only chiral phase structure in this region. S−1 The simple diagonal matrix MF(k) in this region makes it easy to calculate the matrix elements of the effective quark propagator, they can be expressed explicitly as u u d d Λ+γ0 Λ−γ0 Λ+γ0 Λ−γ0 Suu(k) = + , Sdd(k) = + , Sud (k) = Sdu(k) = 0, (20) k0 − E1 k0 + E2 k0 − E3 k0 + E4 with quasiparticle energies = − = + = − = + E1 Eu µu,E2 Eu µu,E3 Ed µd ,E4 Ed µd , = 2 + 2 = 2 + 2 Eu k Mu,Ed k Md , (21) and energy projectors · + u,d 1 γ0(γ k Mu,d ) Λ± = 1 ± . (22) 2 Eu,d After performing the Matsubara frequency summation in (14), the gap equations determining the two quark condensates as functions of temperature and baryon and isospin chemical potentials read 3 d k Mu σu =−6 1 − f(E1) − f(E2) , (2π)3 k2 + M2 u d3k M σ =− d − f(E ) − f(E ) , d 6 3 1 3 4 (23) (2π) 2 + 2 k Md with the Fermi–Dirac distribution function 1 f(x)= . (24) ex/T + 1 It is well known that in the absence of isospin degree of freedom the temperature and baryon density effects on chiral symmetry restoration are different [15]: the chiral condensate drops down continuously with increasing temperature, which means a second order phase transition, but jumps down suddenly at a critical baryon density, which indicates a first order phase transition. At finite baryon density and finite isospin density, while the density behavior of the u- and d-quark condensates are different to each other, they may jump down at the same critical baryon chemical potential or at two different critical points. In the case without considering the UA(1) breaking term, the two critical points do exist, and therefore, there are two chiral phase transition lines [11] in the temperature and baryon chemical potential plane at fixed isospin chemical potential. What is the effect of the UA(1) breaking term on the QCD phase structure? The calculation in the NJL model showed [13] that if there exists a structure of two chiral phase transition lines depends on the coupling constant K of the determinant term. Let 1 G − K K α = 1 − = , (25) 2 G + K G + K 98 L. He, P. Zhuang / Physics Letters B 615 (2005) 93–101 it was found [13] that at µI = 60 MeV the two-line structure disappears for α>0.11. This means that the two-line structure is true only at small ratio α. However, α is a free parameter in [13] and the two coupling constants G and K are not yet separately determined. Before the discussion of the problem in real world with nonzero current quark mass, it is instructive to analyze the phase structure in chiral limit with m0 = 0. From the gap equations (23) one can clearly see that when one of the quark condensates becomes zero, the other one is forced to be zero for any coupling constants G and K = 0. Therefore, there is only one chiral phase transition line at any K = 0 in chiral limit. The two-line structure happens only at K = 0. In this case, the two gap equations decouple, the two chiral phase transition lines are determined by d3k 1 µ µ µ µ 1 − 12G 1 − f |k|− B − I − f |k|+ B + I = 0, (2π)3 |k| 3 2 3 2 3 d k 1 µB µI µB µI 1 − 12G 1 − f |k|− − µI − − f |k|+ − µI + = 0, (26) (2π)3 |k| 3 2 3 2 the difference in baryon chemical potential between the two critical points at fixed T and µI is

µB (T , µI ) = 3µI . (27)

In real world there are four parameters in the NJL model, the current quark mass m0, the three-momentum cutoff Λ, and the two coupling constants G and K. Among them m0,Λand the combination G + K can be determined by ﬁtting the chiral condensate σ , the pion mass mπ and the pion decay constant fπ in the vacuum. In [13] 2 3 m0 = 6MeV,Λ = 590 MeV, and (G + K)Λ /2 = 2.435, for which σ = 2(−241.5MeV) , mπ = 140.2MeV, and fπ = 92.6 MeV. To determine the two coupling constants separately or the ratio α, one needs to know the η - or a-meson properties in the vacuum. In the vacuum the two quark condensates degenerate, and the pole equations determining the meson masses are much simpliﬁed as 3 2 − 2 d k 1 Eq Mq 1 − 12(G + K) = 0, 3 2 − 2 (2π) Eq Eq mσ /4 3 2 − 2 d k 1 Eq Mq 1 − 12(G − K) = 0, 3 2 − 2 (2π) Eq Eq ma/4 3 2 d k 1 Eq 1 − 12(G + K) = 0, 3 2 − 2 (2π) Eq Eq mπ /4 3 2 d k 1 Eq 1 − 12(G − K) = 0, (28) ( π)3 E 2 − 2 2 q Eq mη /4 = 2 + 2 = with quark energy Eq k Mq . It is easy to see that if K 0, the masses of σ - and a-mesons degenerate and the masses of η- and π-mesons degenerate, and if K = G as in the standard NJL Lagrangian, the a- and η-mesons disappear in the model. The mass equation for π-meson can be used to determine the combination G + K, which is already considered in [13], and the mass equation for η-meson is related to the combination G − K and then to the ratio 1 1 1 = − α 1 3 . (29) 2 G + K 12 d k Eq (2π)3 E2−m2 /4 q η

Combining with the above known parameters m0, Λ and G + K obtained in [13] and choosing mη = 958 MeV, we have α = 0.29 which is much larger than the critical value 0.11 [13] for the two-line structure at µI = 60 MeV. L. He, P. Zhuang / Physics Letters B 615 (2005) 93–101 99

In fact, for a wide mass region 540 MeV 0.11, and there is no two-line structure. To fully answer the question if the two-line structure exists before the pion condensation happens, we should = c consider the limit µI µI where the difference in baryon chemical potential between the two critical points is the maximum, if the two-line structure happens. From the approximate result of lattice simulation [5] and the exact result of NJL analyze in next section, we know that the critical isospin chemical potential for pion superfluidity c = is equal to the pion mass in the vacuum, µI mπ . From the analysis above, at this value the isospin chemical potential difference between the two critical points corresponding, respectively, to σu = 0 and σd = 0 in chiral limit and with K = 0is µB = 3mπ .Inrealworldwithm0 = 0 and K = 0, with increasing coupling constant K or the ratio α the two lines approach to each other and finally coincide at about α = 0.21 which is still less than the value 0.29 calculated by fitting mη = 958 MeV. Therefore, the two-line structure disappears if we choose mη = 958 MeV. In fact, for 720 MeV 0.21 and the two phase transition lines are cancelled in this wide mass region. When the η -meson mass is outside this region, the UA(1) breaking term is not strong enough to cancel the two-line structure, but the two lines are already very close to each other.

4. Critical isospin chemical potential for pion condensation

Since both thermal motion and nonzero baryon number will increase the critical isospin density for pion super- c = = ﬂuidity, the minimum isospin chemical potential µI corresponds to the parameters T 0 and µB 0. In this case the pion condensate is zero and the two quark condensates degenerate, the pole equations for the meson masses are then reduced to 3 2 − 2 d k 1 Eq Mq 1 − 12(G + K) = 0, 3 2 − 2 (2π) Eq Eq Mσ /4 3 2 − 2 d k 1 Eq Mq 1 − 12(G − K) = 0, (2π)3 E E2 − M2 /4 q q a0 3 2 − 2 d k 1 Eq Mq 1 − 12(G − K) = 0, 3 2 − ± 2 (2π) Eq Eq (Ma± µI ) /4 3 2 d k 1 Eq 1 − 12(G + K) = 0, (2π)3 E E2 − M2 /4 q q π0 3 2 d k 1 Eq 1 − 12(G + K) = 0, 3 2 − ± 2 (2π) Eq Eq (Mπ± µI ) /4 3 2 d k 1 Eq 1 − 12(G − K) = 0, (30) ( π)3 E 2 − 2 2 q Eq Mη /4 where the quark mass Mq satisﬁes the same gap equation as in the vacuum, d3k 1 M − m − (G + K)M = , q 0 12 q 3 0 (31) (2π) Eq namely, = = c = Mq T 0,µB 0,µI µI Mq (0, 0, 0). (32) 100 L. He, P. Zhuang / Physics Letters B 615 (2005) 93–101

Therefore, from the comparison with the meson mass equations (16) in the vacuum, we derive the isospin depen- c dence of the meson masses for µI µI , = = = ∓ Mσ (µI ) mσ ,Ma0 (µI ) ma,Ma± (µI ) ma± µI , = = ∓ = Mπ0 (µI ) mπ ,Mπ± (µI ) mπ± µI ,Mη (µI ) mη . (33) We see that the mesons with zero isospin charge keep their vacuum values, the mesons with positive isospin charge drop down linearly in the isospin chemical potential, and the mesons with negative isospin charge go up linearly in the isospin chemical potential. Since ma >mπ , the above isospin dependence will firstly break down at µI = mπ . Beyond this value the mass of π+ becomes negative and it makes no sense. This gives us a strong hint that the end point of the normal phase without pion condensation or the starting point of the pion superfluidity phase is at c = µI mπ . To prove this hint we should consider directly the isospin behavior of the order parameter of the pion superfluidity, namely, the pion condensate. Deriving the matrix elements Sud and Sdu from (8) and then performing the Matsubara frequency summation, the gap equation determining self-consistently the pion condensate as a function of isospin chemical potential at T = 0 and µB = 0 reads d3k 1 1 π − (G + K) + = . 1 6 3 0 (2π) 2 2 2 2 2 2 (Eq − µI /2) + (G + K) π (Eq + µI /2) + (G + K) π (34) c Obviously, the minimum isospin chemical potential µI where the pion superfluidity starts is controlled by the square bracket with π = 0, 3 2 d k 1 Eq 1 − 12(G + K) = 0. (35) 3 2 − c 2 (2π) Eq Eq (µI ) /4 From the comparison of (35) with the pion mass equation (28) in the vacuum, we find explicitly that the critical c = = isospin chemical potential µI for the pion superfluidity phase transition at T µB 0 is exactly the vacuum pion mass mπ , c = µI mπ . (36) We emphasize that this conclusion is independent of the model parameters and the regularization scheme, it is a general conclusion in mean field approximation for quarks and RPA approximation for mesons.

5. Conclusions

We have investigated the two flavor NJL model with UA(1) breaking term at finite isospin density, as well as at finite temperature and baryon density. With a self-consistent treatment of quarks in mean field approximation and mesons in RPA, we determined the two coupling constants separately by fitting the meson masses in the vacuum, and then investigated the phase structure for chiral symmetry restoration and pion superfluidity, especially the isospin effect on the chiral phase transition line and the minimum isospin chemical potential for pion superfluidity. The main conclusions are:

(1) The two chiral phase transition lines in the T –µB plane predicated by the NJL model without UA(1) breaking term is cancelled by the strong UA(1) breaking term at low isospin chemical potential. Only when the isospin c chemical potential µI is close to the critical value µI of pion condensation, there is probably the two-line chiral structure, depending on the η-meson mass. Therefore, in relativistic heavy ion collisions where the c typical µI value is much less than µI , it looks impossible to realize the two-line structure. L. He, P. Zhuang / Physics Letters B 615 (2005) 93–101 101

(2) The critical isospin chemical potential for pion condensation in NJL model is exactly the pion mass in the c = vacuum, µI mπ , independent of the model parameters, the regularization scheme, and the UA(1) breaking term.

The temperature and baryon and isospin density dependence of the chiral and pion condensates as well as the meson masses, and the extension to ﬂavor SU(3) NJL model are under way.

Acknowledgements

We thank Dr. Meng Jin for helpful discussions. The work was supported in part by the grants NSFC10135030, 10428510, 10435080, SRFDP20040003103 and G2000077407.

References

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How do chiral condensates affect color superconducting quark matter under charge neutrality constraints?

Hiroaki Abuki a, Masakiyo Kitazawa a,b, Teiji Kunihiro a

a Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan b Department of Physics, Kyoto University, Kyoto 606-8502, Japan Received 27 December 2004; received in revised form 11 February 2005; accepted 6 April 2005 Available online 18 April 2005 Editor: J.-P. Blaizot

Abstract We investigate the effects of the dynamical formation of the chiral condensates on color superconducting phases under the electric and color neutrality constraints at vanishing temperature. We shall show that the phase appearing next to the color-ﬂavor locked (CFL) phase down in density depends on the strength of the diquark coupling. In particular, the gapless CFL (gCFL) phase is realized only in a weak coupling regime. We give a qualitative argument on why the gCFL phase in the weak coupling region is replaced by some other phases in the strong coupling, once the competition between dynamical chiral symmetry breaking and the Cooper pair formation is taken into account.  2005 Elsevier B.V. All rights reserved.

PACS: 12.38.-t; 25.75.Nq

On the basis of the asymptotic-free nature of QCD (CFL) phase where all the quarks equally participate and the attraction between quarks due to gluon ex- in pairing [3,4]. changes, we now believe that the ground state of the In reality, nature may not, however, allow such an quark matter composed of u, d and s quarks at ex- extremely high-density matter to exist, even in the tremely high densities is a special type of color super- core of neutron stars and in possible quark stars. In conducting phases [1,2]; that is the color-ﬂavor locked such systems at relatively low density corresponding to the quark chemical potential of, say, 500 MeV, the following two ingredients become important for the fate of the CFL phase and determining the pattern of E-mail addresses: [email protected] (H. Abuki), color superconductivities [5–7]: ﬁrstly, one cannot ne- [email protected] (M. Kitazawa), glect the effect of the strange quark mass Ms which [email protected] (T. Kunihiro). ranges from around 100 to 500 MeV depending on the

0370-2693/$ – see front matter  2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physletb.2005.04.017 H. Abuki et al. / Physics Letters B 615 (2005) 102–110 103 quark density. Secondly, the constraints of the color tween the chiral and diquark condensations makes the and electric neutrality must be satisfied as well as β- quark masses depend on the realized phases [19–21], equilibrium under the weak interaction. The former and hence some phases in turn become disfavored or causes Fermi-momentum mismatch [8–10], and the favored. (ii) The gapless system might become unsta- latter pulls up or down the Fermi momentum of each ble against the phase separation into the phases with a species of quarks [6,7]; as the density goes lower, the different chiral condensate ¯qq since the scalar con- symmetric CFL pairing would cease to be the ground densate has no gauged charges; recall the fate of the state at some critical density, and some phases other possible gapless state in the electronic superconduc- than the CFL phase would appear. tors mentioned above [14]. We shall show that the next One of the recent findings of such novel pairing pat- phase of the CFL phase down in the density is not nec- terns is the gapless CFL (gCFL) phase [11,12], which essarily the gCFL phase, but strongly depends on the is a non-BCS state having some quarks with gap- coupling constant in the scalar diquark channel even less dispersions despite the same symmetry breaking at zero temperature: the gCFL phase is found to ap- pattern as the CFL phase. Historically, a possible re- pear only in the small coupling regime; this fragileness alization of the stable gapless state was first discussed of the gCFL phase with the dynamical quark conden- for the two-flavor color superconducting phase [13]:it sate will be shown to be understandable in a rather was shown that the local charge neutrality gives a so model-independent way. We shall show that the most strong constraint that such an exotic state, called the favorable phases realized in a wide parameter window g2SC phase, exists stably; this is in contrast with the are the g2SC, 2SC, and unpaired neutral phases. case of the electronic superconductivity in metals [14], We start from the following extended three fla- where the possible gapless state is unstable against vor Nambu–Jona-Lasinio (NJL) Lagrangian density the spatial separation into the Pauli-paramagnetic and with the diquark coupling Gd , and the scalar coupling superconducting phases because of the absence of a Gs [13,21]. long-range force mediated by gauge fields. The gCFL G 3 phase is the three flavor analogue of the g2SC phase. L =¯q(i/∂ − m + µγ )q + d qP¯ t q¯ t qP¯ q 0 0 16 η η Successive detailed studies have revealed a rich phase η=1 structure of superconducting quark matter at zero and Gs non-zero temperatures [15,16]. It should be also noted + (q¯λ q)2 + (qiγ¯ λ q)2 . (1) 8N F 5 F that the possible existence of the gCFL phase in a com- c √ pact star may lead to astronomically interesting conse- Here, λF ={ 2/3 1, λF } are the unit matrix and the quences because of the existence of the gapless quarks Gell-Mann matrices in the flavor space. Pη is defined [17]. Thus, examining the robustness of the gapless as in Ref. [12] phases as well as exploring their nature has become (P )ab = iγ Cηab no sum over index η (2) one of the central subjects in the study of QCD mat- η ij 5 ηij ¯ † ter in extreme conditions. In fact, it has been recently and Pη = γ0(Pη) γ0. a,b,... and i,j,... represent indicated [18] that the gluons in the gapless phases ac- the color and flavor indices, respectively. The sec- quire an imaginary Meissner mass, which may signal ond term in (1) simulates the attractive interaction in an instability of the system to a yet unknown state. the color anti-triplet, the flavor anti-triplet and in the P + In this Letter, we investigate how the supercon- J = 0 channel in QCD. m0 = diag{mu,md ,ms} is ducting orders including the gapless phases are af- the current-quark mass matrix; the full lattice QCD fected by the incorporation of the dynamical chiral simulation shows that mu,d (2GeV) = 3–4 MeV and condensation. This incorporation should be important ms(2GeV) = 80–100 MeV [22]. We take the chiral when the color superconductivity in a compact star SU(2) limit for the u, d sector (mu = md = 0) and is considered, where a change of the chiral conden- ms = 80 MeV throughout this Letter. These values sate ¯qq is also expected. In fact, the dynamically might slightly underestimate the effect of the current generated chiral condensate may greatly affect the sta- masses. bility of the gapped superconducting phases, leading In order to impose the color and electric neutrality, to a quite novel phase structure: (i) The interplay be- we introduce the chemical potential matrix µ in the 104 H. Abuki et al. / Physics Letters B 615 (2005) 102–110 color-flavor space as with p/ = iωnγ0 − p · γ . Finally, Ωe is the contribution from the massless electrons ab = − ab + ab + ab µij µδabBij µeδ Qij µ3δij T3 µ8δij T8 . 4 µe (3) Ωe =− 12π2 Bij = δij and Qij = diag{2/3, −1/3, −1/3} count dp −| − | − T + e µe ξp /T . baryon number and electric charge of quarks, respec- 2 3 log 1 ab = { − } ab ={ (2π) =± tively. T3 diag 1/2, 1/2, 0 and T8 1/3, 1/3, ξ −2/3} are the diagonal generators of the color SU(3). (8) In the numerical analysis, we shall adopt the three- The functional determinant in Eq. (4) can be evaluated momentum cutoff Λ = 800 MeV and the scalar cou- using the method given in the literature [12,16].The optimal values of the variational parameters ∆ , M pling constant Gs giving the vacuum constituent quark η mass 400 MeV in the chiral limit, for comparison with and Ms must satisfy the stationary condition (the gap the results in Refs. [11,12,16]; these parameter val- equations) ues give larger condensates than those used in [23] ∂Ω = ∂Ω = ∂Ω = and [13,21,24]. 0, 0 and 0. (9) ∂∆η ∂M ∂Ms We treat the diquark coupling constant Gd as a simple parameter, although the perturbative one-gluon In order to clarify the effects of the chiral condensation 2 µ on the diquark pairing, we shall also reexamine the exchange vertex Lint =−(g /2)qγ¯ µ(λa/2)qqγ¯ × case in which the dynamical chiral condensates are not (λa/2)q, which is valid at extremely high density, tells = = incorporated [11,12,15,16] us that Gd /Gs 1/2 with Nc 3 [21,25]. Further- more, we shall use, instead of G , the gap energy (∆ ) d 0 ∂Ω = in the pure CFL phase at µ = 500 MeV and T = 0in 0. (10) ∂∆η m =M the chiral SU(3) limit, as was done in Refs. [11,12, 0 16]. We evaluate the thermodynamic potential in the Here the quark mass Ms is regarded as the in-medium mean-field approximation strange quark mass and will be varied by hand. Our task is to search the minimum of the effective potential 3 3 through solving these gap equations under the local = 4 2 + Nc − 2 Ω ∆η (Mi mi) electric and color charge neutrality conditions Gd = Gs = η 1 i 1 ∂Ω = ∂Ω = ∂Ω = T dp −1 0, 0 and 0. (11) − Tr Log S (iωn, p) + Ωe, (4) ∂µ ∂µ ∂µ 2 (2π)3 e 3 8 Before going into a numerical computation, one where must specify the phase characterized by the various G patterns of the mean fields and the chemical potentials ∆ = d t qP q , (5) η η (µ ,µ ,µ ); comparing the value of Ω obtained for 8 e 3 8 M 00 each phase, one can determine which phase is real- M = 0 M 0 ized for given µ and ∆0. In the present analysis, we 00M consider the states listed in Table 1 as possible phases s ¯uu 00 to be realized. These phases are described in the fol- Gs ¯ = m0 + 0 dd 0 , (6) lowing sub-model spaces defined by the parameter sets 2 00¯ss in the parenthesis, respectively; within which the gap equation for each phase is solved. are the gap parameter matrix and the constituent quark mass matrix, respectively, and S denotes the 72 × 72 Set 1 (µe,M,Ms). This parameter space can model Nambu–Gor’kov propagator defined by the χSB phase and the neutral unpaired quark mat-

+ − ter (UQM). The dynamical condensates in all ﬂavor − p/ µγ0 M Pη∆η ¯ S 1(iω ,p)= η , sectors (¯uu, dd, ¯ss) are accompanied in the χSB n ¯ t − + η Pη∆η p/ µγ0 M phase, while in the UQM phase, only ¯ss is non- (7) vanishing. H. Abuki et al. / Physics Letters B 615 (2005) 102–110 105

Table 1 The non-zero gap parameters, some conditions between gaps and chemical potentials, and the gapless quarks in each phase. We abbreviate the unpaired neutral quark matter with non-zero ¯ss to “UQM”, and the chiral-symmetry broken phase as “χSB”, respectively. (g) means the gapless version of the pairing state. The number in the parenthesis in the ﬁrst column represents the number of gapped quasi-quark modes. “gd–bs (1)” means that one of the linear combination of gd and bs quarks remains gapless. δµgd–bs denotes the chemical potential difference ≡ (µgd − µbs)/2, and so forth. The equations for chemical potentials which are necessarily satisﬁed in the given phase are written in a bracket in the third column Name of Gap and mass parameters Conditions for chemical potentials Gapless quarks phase ∆1(ds) ∆2(us) ∆3(ud) M Ms (ru–gd–bs)(bd–gs)(bu–rs)(gu–rd) CFL (9) ∆1 ∆2 ∆3 Ms [µe = 0] All quark modes are fully gapped 2 + Ms gCFL (7) ∆1 ∆2 ∆3 Ms δµbd–gs(bu–rs) 4µ ∆1(2) bd bu uSC (6) ∆2 ∆3 Ms [µe = 0] gd–bs (1) (bd, gs) 2 + Ms guSC (5) ∆2 ∆3 Ms δµbu–rs 4µ ∆2 gd–bs (1) (bd, gs) bu 2SC (4) ∆3 Ms [µ3 = 0] bs (bd, gs)(bu, rs) g2SC (2) ∆3 Ms [µ3 = 0],δµgd–ru = δµrd–gu >∆3 gd, bs (bd, gs)(bu, rs) rd dSC (6) ∆1 ∆3 Ms ru–bs (1) (bu, rs) 2 + Ms gdSC (5) ∆1 ∆3 Ms δµbd–gs 4µ ∆1 ru–bs (1) bd (bu, rs) 2SCus (4) ∆2 Ms gd (bd, gs)(bu, rs) UQM (0) Ms [µ3 = µ8 = 0] All quarks are ungapped χSB (0) MMs [µ3 = µ8 = 0] All quarks are massive

Set 2 (∆3,µe,µ8,Ms). In this parameter space, the in [16], in which the phase structure only for several (g)2SC and UQM phases are described. values of ∆0 is given. One may notice the following Set 3 (∆2,µe,µ3,µ8,Ms). The 2SCus and UQM points: phases are described in this parameter space. (1) The gCFL phase always exists as the next phase Set 4 (∆2,∆3,µe,µ3,µ8,Ms). The (g)uSC, (g)2SC of the CFL phase down in density, irrespective of and UQM phases are described in this space. the value of the diquark coupling ∆0. In addition, 2 Set 5 (∆1,∆3,µe,µ3,µ8,Ms). The (g)dSC, (g)2SC the parameter region of Ms /µ for accommodating and UQM phases are described in this space [26]. the gCFL phase grows as the diquark coupling con- Set 6 (∆1,∆2,∆3,µe,µ3,µ8,Ms). The (g)CFL, stant increases. (g)uSC and UQM phases are described in this space. (2) The stronger the coupling ∆0, the richer the phase structure: The CFL phase turns into the UQM phase We numerically confirmed that other phases such as through successive transitions; gCFL → guSC → the 2SCds or sSC phases, which are described with the 2SC → g2SC. Accordingly, the number of frozen de- parameter set (∆1,µe,µ3,µ8) or (∆1,∆2,µe,µ3, grees of freedom (gapped quarks) decreases as 9 → µ8), respectively, never get realized as the ground 7 → 5 → 4 → 2 → 0 towards lower density. state at T = 0. In this Letter, we restrict ourselves to Now let us examine how the above features are the zero temperature case, leaving an analysis on the changed once the strange quark mass is determined T = 0 case for a future publication. dynamically through Eq. (9). The resulting phase di- We first present the phase diagram without the agram in the (µ,∆0) plane is shown in Fig. 1(b); Ms dynamical chiral condensates with Ms being varied is now determined dynamically and thus becomes a = by hand for a fixed quark chemical potential µ function of µ and ∆0. The following should be no- 500 MeV; the ground state is searched with the aid table from the figure: of the gap equation Eq. (10) which gives a candi- (1) Although the CFL phase is present still in all the date of the ground state. Fig. 1(a) shows an entire diquark coupling region, the chemical potential win- 2 phase diagram in the (Ms /µ, ∆0) plane; we notice dow for realizing the gCFL phase shrinks with the that this phase diagram is consistent with the one given increasing coupling constant, and eventually closes at 106 H. Abuki et al. / Physics Letters B 615 (2005) 102–110

(a) (b)

2 = Fig. 1. (a) Phase diagram in (∆0,Ms /µ) plane: the gap equations Eq. (10) under the neutrality constraints are solved with varying Ms ms by hand, while the quark chemical potential is fixed at µ = 500 MeV as in Refs. [11,12,15,16]. The scale of the vertical axis on the right-hand side represents the value of η = Gd /Gs for the corresponding diquark coupling ∆0. The number in a parenthesis attached to each phase name in the figure denotes the number of gapped quasi-quarks. (b) Phase diagram in (∆0,µ)plane: continuous (first order) transition lines are represented by thin (bold) lines. Although we did not determine the phase boundaries in the dark-shaded area where a rather better precision is required for the comparison of the potentials, we confirmed that any new phase structure does not appear in this region.

∆0 ∼ 50 MeV. This is highly in contrast with the case (140 MeV ∆0); any gapless superconductivity van- described above. ishes and the fully-gapped 2SC phase is realized. It (2) As the coupling is increased, the following phases is worth mentioning that the g2SC → 2SC transition appear successively as the next phase of the CFL phase with the increasing diquark coupling near the CFL down in density; the gCFL phase, the UQM phase, the phase seen in Fig. 1(b) is not of a second order but g2SC phase, and finally the 2SC phase. of a first order with a small jump in the strange quark We can clearly divide the entire phase diagram mass and density. The first order phase transition ends into four distinct regions according to which phase at the point denoted by a large dot in Fig. 1(b). For comes in as the next phase down from the CFL phase. smaller chemical potential than that at this point, the (i) The weak coupling regime (∆0 50 MeV); the 2SC → g2SC transition is continuous and essentially gCFL phase exists between the UQM and CFL phases. the same as that found in the original paper [13], be- (ii) The moderate coupling regime (50 MeV ∆0 cause the on-shell strange quarks are not present in this 90 MeV); the gCFL phase ceases to be the secondly region due to a large dynamical mass of the strange densest phase, and is taken over by the UQM phase, quark. which is nearly two-flavor quark matter with large So far the global structure of the phase diagram. ¯ss condensate. We stress that the superconductiv- Next let us discuss in detail how the gCFL phase ity itself is destroyed before the gCFL phase sets in disappears for the stronger diquark coupling, examin- when the density is decreased. More words will be ing closely two typical coupling cases; ∆0 = 25 MeV given on this matter later on. (iii) The strong cou- (weak coupling), ∆0 = 60 MeV (moderate coupling). pling regime (90 MeV ∆0 140 MeV); a gapless We shall also show the µ dependence of gaps in the superconducting phase reappear but only with the two phases which are actually not realized as the ground flavors being involved, which is called the g2SC phase. state, for completeness. Also a large chiral condensate ¯ss is accompanied in Fig. 2(a) shows the gap parameters as functions of this phase. (iv) The extremely strong coupling regime µ for several phases. At high chemical potentials µ H. Abuki et al. / Physics Letters B 615 (2005) 102–110 107

(a) (b)

Fig. 2. (a) The gap parameters for each state at a weak coupling ∆0 = 25 MeV. We remark that the states other than the (g)CFL and UQM phases are realized only as metastable states for all the region of µ; nevertheless the 2SC, 2SCus and (g)dSC phases give the global minimum in the corresponding sub-model spaces, namely, Set 2, Set 3, and Set 5, respectively. The vertical dashed line represents the point µ =∼ 558 MeV where the UQM phase takes over the gCFL phase, blow which the UQM phase with massive strange quarks accordingly becomes the ground state of the system. The CFL phase turns into the gapless CFL phase at the critical chemical potential µ = µ∗ = 581.1 MeV. (b) The state-dependent strange quark masses and the electron chemical potentials for the (g)CFL and UQM phases at the same coupling as in (a).

580 MeV, the ground state is in the CFL phase. As the what shifted to a lower Ms(µ) (higher µ) than is ob- 2 density is decreased, the stress energy Ms (µ)/µ in- tained in these papers. creases in the symmetric CFL pairing; notice that as Fig. 2(b) shows the state-dependent strange quark the density is decreased, the in-medium strange quark mass as a function of µ in the (g)CFL and the UQM mass Ms(µ) increases, which causes a further increase phases. One should notice that the generation of the 2 of the stress Ms (µ)/µ. Accordingly, the phase suffers dynamical strange quark mass is suppressed in the from a slight distortion in the gaps (∆1 = ∆2 ∆3); (g)CFL phase in comparison with that in the UQM nevertheless the CFL phase persists down to the criti- phase. This is because the (g)CFL paring requires a ∼ cal chemical potential µ = µ∗ = 581.1 MeV, at which Fermi-momentum matching among all the species as the effective chemical potential difference in (bd–gs) much as possible, which is achieved by suppressing ≡ + 2 sector, δµeff(bd–gs) δµbd–gs Ms /(4µ), reaches the dynamical generation of the strange quark mass; it almost the magnitude of the gap ∆1 = ∆2. This tran- is also to be noticed that a better Fermi-momentum sition is essentially the Q˜ -insulator-to-metal transi- matching also lowers the energy cost for imposing tion discussed in [11,12], with Q˜ being the unbroken charge neutrality. On the other hand, the UQM phase U(1) charge in the CFL phase [3,4]. We notice that does not need such a Fermi-momentum matching, and the onset condition of this metal–insulator transition, thus can gain the condensation energy in the chiral 2 ∼ ¯ Ms (µ∗)/2µ∗ = ∆1(µ∗) [11], works well even when ss sector. In other words, the first-order unlocking the strange quark mass is treated as a dynamical vari- gCFL → UQM phase transition is brought about by able. As the density is decreased further, the UQM the competition between the following two factors; (i) phase with massive strange quarks finally takes over reducing the neutrality costs by matching the Fermi the gCFL phase at µ =∼ 558 MeV, which is denoted momenta of three species in the gCFL phase, and (ii) by the dashed line in Fig. 2. This unlocking transi- increasing the condensation energy gain in the chiral tion gCFL → UQM is a first order as was found in ¯ss sector in the UQM phase. The former effect (i) Refs. [11,12,16], although the transition point is some- is underestimated in the previous work [11,12,16] be- 108 H. Abuki et al. / Physics Letters B 615 (2005) 102–110

dynamic potential. In Fig. 3, we show the thermodynamic potentials for the UQM phase and the CFL phases with ∆0 = 25, 35, 50, and 70 MeV. We first = 2 notice that µ Ms (µ∗)/2∆1(µ∗) denoted by large dots on the horizontal axis gives a quite good guide for the critical chemical potential µ∗ for the CFL–gCFL transition. Accordingly, the transition point shifts to a lower µ as the diquark coupling is increased and eventually moves to the left of the UQM line which rapidly falls for µ 550 MeV because of the quite large chiral condensate realized in the UQM phase as is clearly seen in Fig. 2(b). In particular, for ∆0 = 70 MeV, the CFL phase becomes metastable against the UQM phase before the gCFL phase sets in. In other words, the UQM phase takes over the CFL phase before the Fig. 3. The thermodynamic potentials Ω for the UQM phase and gapless dispersions of bd and bu quarks get to be real- the CFL phases with various couplings as functions of µ.Wehave ized. In short, although the CFL phase becomes more chosen the zero of the potential such that the unpaired neutral quark stable with the larger diquark coupling, making the = = matter with bare (Ms 80 MeV) strange quarks has Ω 0. The gCFL–CFL transition point µ∗ lower, the UQM phase CFL phases turn into the gapless phases at the points where the overwhelms all the paired phases because of the large bold lines change into the dashed lines. These gCFL–CFL transi- ¯ tion points are linked by the dash-dotted line. In the horizontal axis, ss -condensation energy. We also remark that at the 2 we also put large dots, which denote the values Ms (µ∗)/2∆1(µ∗) end points of the (g)CFL lines in Fig. 3, the (g)CFL for corresponding ∆0. states cease to exist even as a metastable state, nor as a local maximum like the unstable Sarma state [14]. Finally let us see the reason why the (g)2SC phase cause the strange quark mass is treated as a simple comes as the next phase down in density as the di- parameter, i.e., Ms(UQM) = Ms(CFL). The latter ef- quark coupling is increased further. As the coupling fect (ii) is taken into account for the first time in the is increased, the UQM phase is expected to experi- present work. As a consequence, the transition den- ence successive second order phase transitions firstly sity (strange quark mass) is somewhat larger (smaller) to the g2SC phase and then to the 2SC phase if the in comparison with the results in [11,12,16]; in fact strange quarks are not present in the system [13]. 2 ∼ the transition takes place at Ms /µ 4∆1(µ∗) in the Our full calculation shows that the on-shell strange 2 ∼ present work, while Ms /µ 5∆1(µ∗) in Refs. [11, quarks are absent in the system at µ 450 MeV be- 12,16], with ∆1(µ∗) being the gap in the (bd–gs) sec- cause of a large Ms ; thus the series of transitions tor at the onset point of the gCFL phase. We emphasize UQM → g2SC → 2SC becomes essentially the same that this tendency of destabilizing the (g)CFL phase by as in [13]. At relatively higher chemical potential, the dynamical generation of chiral condensate should however, the small amount of the strange quarks are hold generically, not depending on a model used. present, which brings about a non-trivial tricritical How do the above features change when the di- point on the g2SC → 2SC transition line in the phase quark coupling is raised? One might naively expect diagram (see the large dot in Fig. 1(b)). At higher µ that the window in µ for the gCFL realization becomes than this point, the g2SC → 2SC transition is a first wide, thinking that the gCFL phase should become order with jumps in physical quantities as mentioned more robust in the stronger coupling regime. It is, how- before. We notice that this first order transition is also ever, not the case; in fact, we have no longer a window caused by the competition between the chiral ¯ss confor the gCFL phase in the moderate coupling, for in- densation in the g2SC phase and the pairing energy stance, at ∆0 = 60 MeV. (See Fig. 1(b).) This disap- gain in the fully gapped 2SC order. In fact, the transi- pearance of the gCFL phase in the stronger coupling tion is from a 2SC phase (2SC+s) with a small strange can be nicely understood with the aid of the thermo- quark density to a nearly two-flavor g2SC phase with H. Abuki et al. / Physics Letters B 615 (2005) 102–110 109 larger vacuum ¯ss condensate along the first order certainly a challenge but needed for our deeper under- line. The 2SC phase tends to lower the density mis- standing of the QCD phase diagram. match between nu and nd , and thus needs more strange quarks as well as electrons for the electric neutrality than in the g2SC phase. As a result, the 2SC + s phase Acknowledgements tends to reduce the dynamical strange quark mass. In contrast, the g2SC phase has extra d quarks accumu- The authors are grateful to M. Alford for his inter- lated in the momentum shell, thus requires less strange est in this work and valuable comments which led to quarks. For this reason, the g2SC phase can have a Fig. 3. One of the authors, H.A. thanks M. Asakawa larger dynamical chiral condensate ¯ss. for encouragement. H.A. is supported by the Fellow- In this Letter, we have made an analysis on the in- ship program, Grant-in-Aid for the 21COE, “Center terplay between the chiral and diquark condensations for Diversity and Universality in Physics” at Kyoto in the three-flavor neutral quark matter using an ex- University. M.K. is supported by Japan Society for tended NJL model. We have shown that which phase the Promotion of Science for Young Scientists. T.K. appears next to the CFL phase strongly depends on the is supported by Grant-in-Aide for Scientific Research strength of the diquark coupling; as the diquark cou- by Monbu-Kagaku-sho (No. 14540263). This work is pling is increased, the gCFL, the UQM, the g2SC and supported in part by a Grant-in-Aid for the 21st Cen- finally the 2SC phase appear as the next phase down tury COE “Center for Diversity and Universality in in density. 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Signals for CP violation in split supersymmetry

N.G. Deshpande, J. Jiang

Institute of Theoretical Science, University of Oregon, Eugene, OR 97403, USA Received 25 March 2005; accepted 5 April 2005 Available online 14 April 2005 Editor: M. Cveticˇ

Abstract Split supersymmetry is characterized by relatively light chargino and neutralino sector and very heavy sfermion sector. We study the consequence of CP violation in this scenario by evaluating two-loop contributions to electric dipole moments of fermions from Higgs-photon as well as W–W diagrams. These contributions add coherently and produce electron and neutron electric dipole moments close to present bounds. We then explore Higgs production at a photon–photon collider, and consider the feasibility of measuring CP violating hγ γ coupling induced by chargino loops. Methods of enhancing the sensitivity are discussed. For lower chargino masses and lower Higgs boson masses, the effect of the CP violation can be observed with 90% conﬁdence level signiﬁcance.  2005 Elsevier B.V. All rights reserved.

1. Introduction matter candidate are retained in split SUSY. By allowing the existence of fine-tuning, the SUSY breaking Supersymmetry (SUSY) has been one of the most scale can be relaxed to be much higher than 1 TeV. promising candidates for the extension of the Stan- Subsequently, the heavier sfermion masses help to dard Model (SM). It provides an elegant solution to eliminate several unpleasant aspects of SUSY, includ- the gauge hierarchy problem. Recently a new scenario ing excessive flavor and CP violation, fast dimension-5 of SUSY model was proposed, in which solution of proton decay and the non-observation of the lightest the naturalness problem is no longer required [1].This CP even Higgs boson. Various aspects of phenomenol- scenario is dubbed split SUSY because of the hier- ogy in the split SUSY scenario have been explored in archical mass difference between the scalar and the Refs. [2–4]. fermionic superpartners. The other two prominent fea- Split supersymmetry is characterized by relatively tures of SUSY, gauge coupling unification and dark light (100 GeV–1 TeV) charginos and neutralinos and much heavier squarks and sleptons. In this Letter we further explore some of the consequences of CP- E-mail address: [email protected] (J. Jiang). violation in split SUSY. We shall consider electric

Fig. 1. Feynman diagrams of the fermion EDMs at two-loop level. The (red) crosses indicate CP violating couplings. dipole moments (EDMs) of electron and quarks, and model with CP violating mixing of scalar and pseudo- arrive at their values in split SUSY versus in standard scalar Higgs bosons. We extend the work of Ref. [6] SUSY. Similarly, we consider CP violating coupling to a more realistic level and examine the sensitivity of of the Higgs boson to photons, and examine the fea- measuring CP violation at a future γγ collider. sibility of measuring this effect at a γγ collider. We After the introduction, we discuss the EDMs in shall allow CP violating phases in the SUSY potential Section 2 and hγ γ coupling in Section 3. Our con- to take values of O(1), and all suppression of one-loop clusions are presented in Section 4. contribution is attributed to higher masses of the supersymmetric particles. Electric dipole moments of fermions arise at one 2. Electric dipole moments loop in conventional SUSY. As squark and slepton masses exceed 5 TeV and charginos and neutrali- In split SUSY, as the sfermions get heavy, the nos remains light, the one-loop contributions become one-loop contributions to the fermion EDM get sup- comparable to the two-loop contributions. In split pressed due to the large sfermion mass. The neutrali- SUSY, the two-loop contributions arise from a set of nos, charginos and the lighter CP even Higgs boson Higgs-photon diagram considered before [2],aswell remain light. The CP phases in the gaugino sector can as the W–W diagram, that we consider here.1 Allow- induce EDM for fermions at 2-loop level. Study of the ing SUSY parameters to have arbitrary complex val- two-loop fermion EDMs in the SM and SUSY can be ues, we show that these two contributions always add found in Refs. [7–13]. In split SUSY, the diagrams coherently. The predicted values of the electron EDM involving charged Higgs bosons are suppressed due in particular set useful constraint on split SUSY mass to the very large charged Higgs boson masses. The scale, and further improvement in measurements [5] typical diagrams, shown in Fig. 1, include a set of di- can provide strong constraints on the theory. We simi- agrams that involve a Higgs boson and a photon (left) larly discuss neutron EDM. and those that involve two W bosons (right). The con- Another CP violation signal is through the study tributions from the Higgs diagrams have been studied of the hγ γ coupling. In the SM, the Higgs coupling in Ref. [2]. We focus on the contributions from the to photons arises predominantly through W -boson W–W diagram. and top-quark loops, and is CP conserving. In super- To specify our notation, we start with the chargino symmetry, a CP violating coupling can arise through and neutralino mass matrices. The chargino mass ma- chargino loop, provided the complex phases in the trix is √ chargino sector are non-zero [2,6]. The CP violat- ∗ M2 gv2 / 2 + = √ ing effect is similar to that in a two Higgs doublet Mχ ∗ , (1) gv1 / 2 µ where g is the weak coupling. In general, the gaug- 1 As we were preparing to submit this Letter, we noticed a sim- ino and higgsino mass parameter M2 and µ, and the ilar study by Chang, Chang and Keung [4] was submitted to vacuum expectation values v1 and v2 are all complex. the arXiv. After absorbing three of the complex phases through N.G. Deshpande, J. Jiang / Physics Letters B 615 (2005) 111–119 113

ﬁeld redeﬁnition, there is only one independent phase To evaluate the W–W diagram, it is necessary to φµ left. The chargino mass matrix can be diagonalized write out the Lagrangian involving W boson, neutrali- by unitary matrices U and V , nos and charginos † 1 1 − γ 1 + γ + − U Mχ+ V = diag(m + ,m + ), (2) L = √ 0 µ 5 + 5 χ1 χ2 gχ γ Lij Rij χ Wµ 2 i 2 2 j with the chargino masses satisfying m +

W Fig. 2. Left: W–W diagram contribution d as a function of the complex phase φ1 for φµ = 0 (dotted), π/4 (dashes) and π/2 (solid). Right: h h W the dominant 2-loop contributions to the electron EDM d (black) and d + d (red) as functions of m + ,forφ1 = 0, φµ = π/2, tanβ = 1, χ1 mh = 120 GeV and m + /m + = 2. (For interpretation of the references to colour in this ﬁgure legend, the reader is referred to the web version χ2 χ1 of this Letter.)

h h as mχ+ increases, while d is also reduced as mh larger mh, d will be reduced, hence the relative im- gets larger. With our choice of independent complex portance of dW increases. The dash line in the plot h W phases, d depends only on φµ and d depends on shows the current 95% confidence level upper bound −27 both φµ and φ1. on the electron EDM, |de| < 1.7 × 10 e cm [14]. To show the dependence of dW on the complex If the CP phases are indeed of order O(1), the elec- phases φµ and φ1, we choose the following parame- tron EDM bound constrains the chargino masses in ters for illustrative purpose split SUSY to be m + 150 GeV. The next genera- χ1 tion EDM experiments can improve the sensitivity by | |= = M1 100 GeV,M2 200 GeV, a few orders of magnitude [5]. Again assuming order |µ|=300 GeV, tan β = 1.0. (9) O(1) CP phases, these measurements will either observe the electron EDM or put stronger constraints on W Although d depends on both φ1 and φµ, the effect of chargino masses in split SUSY. W varying φµ is more important. We show d as a func- If the sfermion masses are of the order of TeV, the tion of φ1 for φµ = 0,π/4, and π/2 in the left panel one-loop diagrams involving sfermions and gauginos W of Fig. 2. The variation of d due to φ1 is an order of or gluinos will dominate the EDM contribution. In the magnitude smaller than the variation due to φµ.Nu- MSSM, the predicted EDM values of the fermions can merical evaluation also show that dh has the same sign be much larger then the current experimental bounds. as dW and is about twice in magnitude. Thus, for large The fact that we have not observed large EDMs can h W enough φµ, independent of changes in φ1, d and d be explained by, small complex phases, larger super- always add constructively. symmetric particle masses, cancellation at work or a In the right panel of Fig. 2, we show both the contri- combination of the above [15]. If we assume that the butions form the Higgs-photon diagram and the W–W phases are of order O(1) and no large cancellation is diagram. Here, we use φ1 = 0, φµ = π/2, tanβ = 1, present, the remaining explanation is to adopt heavy mh = 120 GeV and the unification inspired mass re- sfermion masses. In Fig. 3, we plot the one-loop pre- 2 lation M1 = 5/3tan θW M2 to reduce the number of diction of electron EDM coming from the neutralino– variables. As we vary M2, we change µ accordingly selectron and chargino–sneutrino diagrams as a func- to maintain the chargino mass ratio m + /m − = 2. tion of the selectron mass, while the sneutrino mass is χ2 χ2 We see that the W–W diagram contribution is about set to be the same as the selectron mass. We see that, 25% to 50% of that of the Higgs-photon diagram for for tan β = 1, a selectron mass of me˜ ≈ 5 TeV is suf- chargino mass range from 100 GeV to 2 TeV. For ficient to suppress the electron EDM to be below the N.G. Deshpande, J. Jiang / Physics Letters B 615 (2005) 111–119 115

3. CP violation in γγ to h production

The loop induced hγ γ coupling in the SM is CP conserving. However, if there exists mixing of the CP even and the CP odd Higgs bosons, there would be CP violating hγ γ coupling in two Higgs doublet models. On the other hand, the chargino loop can induce CP violating hγ γ coupling due to the complex phases in the chargino mass matrix. In principal, this CP violation can manifest in both the Higgs boson decay into two photons and production of a Higgs boson in photon-photon collisions. It is, in practice, difﬁcult to Fig. 3. One-loop electron EDM values as a function of the selectron determine the helicities of the outgoing photons from mass. the Higgs decay. The mixing of Higgs bosons of different CP state has been discussed in Refs. [18,19]. Similar to these analysis, the chargino loop induced experimental bound. For tan β = 10, the correspond- CP violation can also be explored at a photon col- ing mass is me˜ ≈ 20 TeV. If we compare Fig. 3 to lider [2,6]. We study in more detail the experimen- the right panel of Fig. 2, it is interesting to note that, tal observables and the backgrounds and estimated in the SUSY parameter space where sfermion masses the sensitivity in determining the CP violating cou- are of a few TeV and the gauginos are light, both the pling. one-loop and the two-loop contributions are equally The Higgs production rate in γγ collision is related important. to h → γγ decay width at a given γγ center-of-mass The same two-loop diagrams can also generate energy Eγγ and the two colliding photon helicities, λ EDMs for up and down quarks, when the one-loop and λ [20] contributions are suppressed by the large squark → → masses. The quark EDMs manifest through the EDM σ(γγ h X) of neutron. In the conventional SUSY, when squark 8πΓ(h→ γγ)Γ(h→ X) masses are around 1 TeV, there are also chromoelec- = (1 + λλ ), (11) (E2 − m2)2 + Γ 2m2 tric dipole moments and gluonic dipole moments. As γγ h h h the squarks become heavy, both one-loop and two- where Γ(h→ X) is the partial width of Higgs bo- loop contributions from these sources are suppressed. son decay to X and Γh is the total decay width of the Lacking the full knowledge of the neutron wave func- Higgs boson. The h → γγ decay partial width is given tion, we use the chiral quark model approximation [16] by to estimate the neutron EDM from the quark EDMs, α2g2m3 Γ(h→ γγ)= h |e|2 +|o|2 , (12) ηe d u π 3m2 dn = 4d − d , (10) 1024 W 3 2 2 d u 4 4mt 4mW where d and d are the down quark and up quark e = F1/2 + F1 3 m2 m2 EDMs and ηe ≈ 1.53 is the QCD correction factors. h h We evaluate the two-loop induced neutron EDM for √ 2 the same set of parameters as in Eq. (9). The estimated + ∗ + ∗ mW 2Re cos βU2iV1i sin βU1iV2i neutron EDM is 4.0 × 10−26 e cm, which is close to m + i=1 χi the current experimental 90% conﬁdence level upper −26 2 bound of 6.3 × 10 e cm [17]. The predicted value 4m + χi will be smaller for larger tan β, m + and m ,asinthe × F1/2 , (13) χ h m2 case of electron EDM. h 116 N.G. Deshpande, J. Jiang / Physics Letters B 615 (2005) 111–119

Fig. 4. Left: The ratio of RCP as a function of mh. The solid curve is for m + = 100 GeV and m + = 200 GeV and the dashed curve for χ1 χ2 m + = 150 GeV and m + = 300 GeV. Right: the statistical signiﬁcance as a function of RCP for mh = 120 GeV and mh = 140 GeV. χ1 χ2

√ 2 = ∗ + ∗ mW In the current case, both e and o are real, and o is o 2Im cos βU2iV1i sin βU1iV2i m + small compared to e. Therefore, A1 is always 0 and the i=1 χi 2 deviation of A3 from ±1isoforder(o/e) . The devi- 2 4m + ation of A2 from 0 is of order (o/e), thus rendering A2 χi × F1/2 , (14) the most promising observable. The Higgs production m2 h rate can now be expanded in terms of the asymmetries where the integration functions for spin-1/2 and [6,18] spin-1 particles in the loop are = 1 | |2 +| |2 dN dLγγ dΓ e o 2 2 1 F1/2(x) =−2x 1 + (1 − x) arcsin √ , (15) × 1 +ζ2ζ + ζ3ζ +ζ1ζ A2 , (18) x 2 1 3 where, dL is the luminosity of the back-scattered 1 2 γγ F1(x) = 2 + 3x 1 + (2 − x) arcsin √ . (16) photons, dΓ is the phase space of the decay particles x and the ζi are the Stokes parameters, which indicate Note because of the tininess of the bottom quark the degree of linear and circular polarizations [21].In loop contribution, we ignore it here. The magnitude the above expression we have dropped the A1 term and of the CP violation can be characterized by the ra- we “turn off” the A3 term by setting the azimuthal tio RCP =|o/e|.WeshowRCP for different chargino angle between the maximum linear polarization di- masses as a function of the Higgs boson mass in the rection of the two back-scattered photons κ [21] to = left panel of Fig. 4. RCP stays rather constant for dif- satisfy cos 2κ 0. The quantity A2 can be accessed ferent values of mh until mh approaches the thresh- by measuring the difference between the production =− old for decay into two W bosons. Increasing m + to rates with sin 2κ 1 and 1. To accentuate the ef- χ1 fect of A2, it is preferable to make (ζ3ζ +ζ1ζ ) 150 GeV will reduce RCP by about a factor of 2. For 1 3 + = = = as large as possible. This is achieved by setting the ra- mχ 100 GeV, φµ π/2, and mh 120 GeV, RCP 1 tio of the emitted photon energy to the initial electron is about 0.135. energy to be close to it is maximal value [6,18]. Thus Three asymmetries can be constructed from e and o the electron–electron center of mass energy shall√ be ∗ ∗ = −2Im(eo ) −2Re(eo ) slightly higher than the Higgs threshold,√ e.g., see A = ,A= , = = 1 |e|2 +|o|2 2 |e|2 +|o|2 150 GeV for mh 120 GeV and see 175 GeV for m = 140 GeV. | |2 −| |2 h e o As the Higgs boson in the mass range of 120– A3 = . (17) |e|2 +|o|2 140 GeV decays signiﬁcantly into bb¯, we observe the N.G. Deshpande, J. Jiang / Physics Letters B 615 (2005) 111–119 117

Higgs boson production signal in the bb¯ final state. O(1), we find that fermion EDMs arise from two Since it is only necessary to tag one of the two b- loop diagrams in which gauginos are in the loops. − 2 ≈ jets, the tagging efficiency is 2b b 98%, with We have shown that apart from Higgs-photon di- b = 85% being the tagging efficiency of one b-jet agram already considered, W -boson diagram is of [22]. There exist large γγ → bb¯ and cc¯ backgrounds. comparable importance. Furthermore, the two dia- Assuming the rate of mistagging a c-jet as a b-jet is grams always add constructively, and sum of their c = 4.5% [22], then the overall mistagging rate is contributions are close to the present experimental − 2 ≈ 2c c 0.2%. These two sources of backgrounds bounds. In the case of the electron, we already see + can be significantly reduced by imposing the invariant that the present bound requires mχ 150 GeV, pro- | − | 1 mass cut, mbb mh 10 GeV and the angular cut vided that phase φµ ≈ π/2. An order of magnitude on outgoing b-jet direction relative to the beam line improvement in the electron EDM would definitely ◦ ◦ direction, 30 <θbz < 150 . With these cuts imposed, constrain the chargino masses and would thus be the background cross sections are σbb = 5.7 fb and competitive with accelerator bounds. We have also σcc = 9.1 fb. As a comparison, for mh = 120 GeV and observed that unlike one loop contribution, the two RCP = 0.10, the signal cross section with sin 2κ = 1 loop gaugino contribution is largest for small tan β. is σ+ = 5.03 fb and that with sin 2κ =−1isσ− = We have compared with the one-loop sfermion con- − 4.66 fb. The total of 1 ab 1 luminosity will be divided tribution, and we see that the contribution becomes − into 500 fb 1 for the each of the sin 2κ =−1 and 1 small as sfermion masses exceed several TeV, the runs. In our analysis, we use 80% initial electron po- precise value being a function of tan β.Wehave larization and 100% polarization for linearly polarized also estimated the neutron EDM from two-loop di- initial photons [23]. agrams and find the predicted value close to the The statistical significance is presented by present bound, again for choice of large complex − phase. 2 N+ N− χ = √ , (19) Another consequence of the CP phase in the gaug- N+ + N− + 2N BG ino sector is the loop induced CP violation in the where N+ and N− are the event number with hγ γ coupling. We have considered studying this CP = − sin 2κ 1 and 1 respectively and NBG is the sum violating coupling at a future γγ collider, by the mea- ¯ ¯ of the bb and cc background event numbers. In the surement of the Higgs production cross section for right panel of Fig. 4, we show the statistical signifi- different initial photon polarizations. We have opti- cance as a function of the ratio RCP, where the dash mized the signal by arranging the initial electron and line indicates the significance corresponding to a 90% photon polarization and minimized the background = confidence level measurement. For mh 120 GeV with kinematic cuts. We conclude that with a lumi- −1 − and with a 1 ab integrated luminosity, RCP ≈ 0.12 1 + = nosity of 1 ab ,formχ 150 GeV, the lower can be observed with 90% confidence level. Thus 1 RCP = 0.06 might be difficult to observe. A 90% for m + = 100 GeV and mh = 120 GeV, the pre- χ1 confidence level observation of CP violation can be dicted R = 0.135 can be observed with 90% con- CP achieved for RCP = 0.12. In the split SUSY the pre- fidence level significance at a future γγ collider. For + = = dicted value for mχ 100 GeV and mh 120 GeV = = 1 mh 140 GeV, the predicted RCP 0.13 is harder is about 0.135, thus it is hopeful that this effect can to observe because of the reduced branching ratio of ¯ be observed. For higher Higgs boson masses, it will Higgs boson decay to bb. Increasing luminosity will be necessary to increase the luminosity and to include improve the significance, as also including other chan- other decay channels. nels of Higgs boson decay.

4. Conclusion Acknowledgements

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A gauge-mediation model with a light gravitino of mass O(10) eV and the messenger dark matter

Masahiro Ibe a, Kazuhiro Tobe b, Tsutomu Yanagida a,c

a Department of Physics, University of Tokyo, Tokyo 113-0033, Japan b Department of Physics and Astronomy, Michigan State University, East Lansing, MI 48824, USA c Research Center for the Early Universe, University of Tokyo, Tokyo 113-0033, Japan Received 22 March 2005; accepted 1 April 2005 Available online 14 April 2005 Editor: M. Cveticˇ

Abstract In the light of recent experimental data on gaugino searches, we revisit the direct-transmission model of dynamical supersymmetry breaking with the gravitino mass mG˜ 16 eV, which does not have any cosmological or astrophysical problems. We ﬁnd that in the consistent regions of parameter space, the model predicts not only upper bounds on superparticle masses (1.1 TeV, 320 GeV, 160 GeV, 5 TeV, 1.5 TeV and 700 GeV for gluino, wino, bino, squarks, left-handed sleptons and right-handed sleptons, respectively), but also a mass of the lightest messenger particle in the range of 10–50 TeV. The lightest messenger particle can naturally be a messenger sneutrino. Therefore, this may suggest that the messenger sneutrino could be the dark matter, as proposed recently by Hooper and March-Russell to account for the gamma-ray spectrum from the galactic center observed by HESS experiment.  2005 Elsevier B.V. All rights reserved.

A light gravitino of mass 16 eV is very interesting, since it does not cause any cosmological or astrophysical problems [1]. In particular, we have no so-called gravitino problem [2,3], since the next lightest supersymmetry (SUSY) particle decays sufficiently before the Big Bang nucleosynthesis. Thus, the reheating temperature after inflation can be higher than 1010 GeV, which is the lowest temperature for the thermal leptogenesis to work [4,5]. However, it is very difficult to construct a consistent gauge-mediation model with such a light gravitino and only a quite few examples are known [6–9]. In this Letter, we discuss the model in Ref. [7] in the light of recent experimental data on the search for gauginos in a class of gauge-mediation models, and show that the minimal model in Ref. [7] is already excluded, but the next-to-minimal model still survives. We show that the model predicts not only upper bounds on superparticle masses (1.1 TeV, 320 GeV, 160 GeV, 5 TeV, 1.5 TeV and 700 GeV for

E-mail address: [email protected] (K. Tobe).

0370-2693/$ – see front matter  2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physletb.2005.04.008 M. Ibe et al. / Physics Letters B 615 (2005) 120–126 121 gluino, wino, bino, squarks, left-handed sleptons and right-handed sleptons, respectively), but also a mass of the lightest messenger particle in the range of 10–50 TeV. This may suggest that the lightest messenger particle is the dark matter proposed recently by Hooper and March-Russell [10] to account for the gamma-ray spectrum from the galactic center [11]. Let us briefly describe a model discussed in Ref. [7]. We assume a SUSY SU(2) gauge theory with four doublet i = ; = =− chiral superfields Qα (α 1, 2 i 1,...,4) and six singlet chiral superfields Zij Zji [12,13]. We impose, for simplicity, a flavor SP(2) symmetry in the superpotential

W = λij QiQj Zij , (1) 2 and assume the SP(2)-invariant vacuum, Q1Q2=Q3Q4=Λ . Here, Λ is the dynamical scale of the strong SU(2) gauge interactions. Then, the effective superpotential is given by 2 Weff λΛ Z. (2)

Here, the chiral superfield Z is a SP(2) singlet combination of Zij . We see that the Z acquires a non-vanishing F 2 term (FZ λΛ ) and the SUSY is spontaneously broken [12,13]. The low energy effective superpotential in the messenger sector is given as follows [7]: n 2 ¯ ¯ ¯ ¯ ¯ ¯ W = λΛ Z + Z(k d d + k l l ) + m d d + m ¯ d d + m l l + m¯ l l . (3) eff da a a la a a da a a da a a la a a la a a a=1 ¯ ¯ ¯ Here we introduce n sets of vector-like messenger quark multiplets da, da,da and d a and lepton multiplets la, la,la ¯ ¯ ¯ ¯ ¯ ¯ and l a (a = 1,...,n). We assume that the multiplets (d, l) and (d ,l ) transform as 5, and (d,l) and (d , l ) as 5 under SU(5), so that the gauge coupling unification remains in the models. Note that the perturbativity up to the 16 1 GUT scale (MG = 2 × 10 GeV) allows only cases with n = 1 and 2. We refer a case with n = 1 as the minimal model and a case with n = 2 as the next-to-minimal model. We also assume that the correction to Kähler potential for Z field induces the vacuum expectation value (vev) ZΛ.2 Under the following condition, 2 2 |m m ¯ | > k F (ψ = d and l), (4) ψa ψa ψa Z we find the SUSY-breaking vacuum is true one with vanishing vevs of the messenger squarks and sleptons: 2 =¯ = =¯ = = FZ λΛ , ψa ψa ψa ψa 0 (ψ d and l). (5) The mass terms of the messenger particles are represented as    ˜ ψa    n ˜  ψ  ψa   ¯ ¯ (ψa ) a ˜ ∗ ˜ ∗ ˜¯ ˜¯ ˜ 2(ψa)   L =− (ψa, ψ )M + h.c. + ψ , ψ , ψ , ψ M  ˜  , (6)  a ψ a a a a  ψ¯ ∗  a=1 ψ=d,l a a ˜¯ ∗ ψa where ψ and ψ˜ denote fermionic and bosonic components of the superfield ψ, respectively. The mass matrices M(ψa) and M˜ 2(ψa) are given by m(ψa) m ¯ M(ψa) = ψa , (7) mψa 0

1 5 Assuming α3(mZ) = 0.12(0.11) and Λ = 2.6 × 10 GeV, we get α3(MG) 6(1.1) at one-loop level in the model with n = 3. Therefore, the model with n = 3 might be marginally allowed in some particular parameter regions, but we do not discuss it further in this Letter. 2 The mass parameters m ,m¯ ,m and m¯ can be generated dynamically as discussed in Ref. [7].IfZ=0, we may introduce mass da da la la ¯ + ¯ terms a(Mda dada Mla lala) [7]. 122 M. Ibe et al. / Physics Letters B 615 (2005) 120–126

 ∗ ∗  |m(ψa)|2 +|m |2 m(ψa) m ¯ F (ψa) 0 ψa ψa  (ψa) | |2   m mψ¯ ∗ mψ¯ 00 ˜ 2(ψa ) =  a a  M ∗ . (8)  F (ψa) 0 |m(ψa )|2 +|m ¯ |2 m(ψa)m  ψa ψa ∗ (ψa) | |2 00m mψa mψa

(ψa) = (ψa) = = Here m kψa Z and F kψa FZ (ψ d and l). Diagonalizing the mass matrices, we obtain masses of the messenger particles. Once the messengers receive SUSY breaking masses, gaugino (sfermion) masses in the minimal SUSY standard model (MSSM) sector are generated by one-loop (two-loop) diagrams of the messenger particles via standard model gauge interactions. The gaugino masses are given by n α3 (d ) m ˜ = F a , (9) g3 2π a=1 n α2 (l ) m ˜ = F a , (10) g2 2π a=1 n α1 2 (d ) 3 (l ) m ˜ = F a + F a , (11) g1 2π 5 5 a=1 and the sfermion masses are 1 n α 2 α 2 3 α 2 2 3 m2 = Cf 3 G(da)2 + Cf 2 G(la)2 + Y 2 1 G(da)2 + G(la)2 , (12) f˜ 2 3 4π 2 4π 5 4π 5 5 a=1 = 5 f = 4 where we have adopted the SU(5) GUT normalization of the U(1)Y gauge coupling (α1 3 αY ), and C3 3 and f = 3 ˜ f = C2 4 when f is in the fundamental representation of SU(3)C and SU(2)L respectively, and C3,2 0forthe (ψ) (ψ) gauge singlets, and Y denotes the U(1)Y hypercharge (Y = Q − T3). Here F and G are functions of the messenger masses and mixings, and their explicit expressions can be found in Ref. [7]. Since SUSY is broken, the gravitino gets a mass √ 2 FZ FZ m ˜ = √ = . G 16 5 eV (13) 3M∗ 2.6 × 10 GeV

Here M∗ is the reduced Planck mass (M∗ = 2.4×1018 GeV). As pointed out in Ref. [1], the matter power spectrum inferred from large samples of Lyman-α forest data and the cosmic microwave background data of WMAP strongly constrain the gravitino mass. As a result, its current upper limit is 16 eV.3 From Eq. (13), the gravitino mass limit translates into a limit on the SUSY breaking scale: × 5 ←→ FZ < 2.6 10 GeV mG˜ 16 eV. (14)

Note that in the model discussed here, the gravitino mass and SUSY breaking vev FZ are related each other as shown in Eq. (13), because the SUSY breaking in the dynamical SUSY breaking sector is directly transmitted to the messenger sector. In most of gauge-mediation models in literatures [15], however, the SUSY breaking scale in the messenger sector is suppressed, compared to the original SUSY breaking scale, due to the transmission mechanism. Therefore we stress that the gravitino mass limit is quite severe in most of gauge-mediation models,

3 To evade the bound on the gravitino mass, one needs to consider a late-time entropy production, as suggested in Ref. [14]. M. Ibe et al. / Physics Letters B 615 (2005) 120–126 123

(ψ) Fig. 1. Mass spectrum of the gauginos (solid lines) and sfermions (dashed lines) in the MSSM sector as a function of a parameter F /(mψ mψ¯ ) (ψ) = = = (ψ) = = × 5 2 in the next-to-minimal model. Here we have assumed m mψ¯ mψ for ψ d,l.WealsofixedF FZ (2.6 10 GeV) for = = ψ d,l, which corresponds to be the maximal gravitino mass in Eq. (14), mG˜ 16 eV. The D0 limit on wino mass (195 GeV) is also shown. and it is crucial to have the direct-transmission of dynamical SUSY breaking in order to construct the gauge- mediation models with mG˜ 16 eV. Now we are in position to discuss the prediction of the model based on the limit in Eq. (14). Since the minimal model (the model with n = 1) has been excluded as we will see later, we mainly consider the next-to-minimal model (the model with n = 2). In Fig. 1, mass spectrum of the gauginos and sfermions in the MSSM sector are (ψ) 4 plotted as a function of F /(mψ mψ¯ ) in the case of the next-to-minimal model. Throughout our discussion, we assume F (ψa) ≡ F (ψ), m(ψa) ≡ m(ψ), m ≡ m , and m ¯ ≡ m ¯ for a = 1, 2, and hence we suppress the index a, ψa ψ ψa ψ 5 (l) (d) 5 2 for simplicity. In Fig. 1, we have assumed all F-parameters are equal, F = F = FZ = (2.6 × 10 GeV) , (ψ) which corresponds to m ˜ = 16 eV, and all mass parameters are equal, m = m = m ¯ for ψ = d,l.InFig. 2, G ψ ψ (ψ) we also show the mass spectrum of the gauginos and sfermions as a function of a parameter m / mψ mψ¯ .Here we assume m = m ¯ and F (ψ)/(m m ¯ ) = 0.98 as an example. We see that the gaugino masses are maximal for ψ ψ ψ ψ (ψ) (ψ) m / mψ mψ¯ 1 with fixed F /(mψ mψ¯ ). Because of the limit in Eq. (14), we find, in Fig. 1, the next-to- minimal model predicts upper limits on superparticle masses (1.1 TeV, 320 GeV, 160 GeV, 5 TeV, 1.5 TeV and 700 GeV for gluino, wino, bino, squarks, left-handed sleptons and right-handed sleptons, respectively). Important experimental bounds on gaugino masses have been set by D0 and CDF experiments [17,18].6 They have been searching for diphoton events induced by the lightest neutralino decay into a gravitino plus a photon, subsequent to wino pair production at Tevatron. The D0 lower limit on wino mass is about 195 GeV [17] and currently it is the strongest bound for the models considered here. In Figs. 1 and 2, the D0 limit is also shown. Note that in the minimal model (n = 1), the predicted upper limits on gaugino masses are half of those in the next-to-minimal model shown in Fig. 1, and hence the predicted wino mass in the minimal model is smaller than about 160 GeV. Therefore, the D0 bound has excluded the minimal model. Since models with n 3 are not

4 Within a particle content of MSSM, µ-term tends to be larger than wino mass because squarks are much heavier than the wino, as discussed in Ref. [7]. Therefore, higgsinos are heavier than winos. However, the Higgs sector can be modiﬁed, as we will discuss later, and hence it will be model-dependent. Thus we will not discuss the Higgs and higgsino sector in detail here. A more detail analysis will be given in Ref. [16]. 5 = = In our results, we assume the Yukawa couplings kda kla 1 because we expect these Yukawa couplings are of order one. One can easily estimate the change of our results when these Yukawa couplings are deviated from one. 6 LEP experiments also have some constraints on gaugino masses for gauge-mediation models. However, in the models considered here, sleptons are so heavy that their limits do not give a signiﬁcant constraint. 124 M. Ibe et al. / Physics Letters B 615 (2005) 120–126

Fig. 2. Mass spectrum of the gauginos (solid lines) and sfermions (dashed lines) in the MSSM sector as a function of a parameter (ψ) (ψ) = = = = m / mψ mψ¯ in the next-to-minimal model. Here we have assumed F /(mψ mψ¯ ) 0.98 and mψ¯ mψ for ψ d,l,andmG˜ 16 eV. The D0 limit on wino mass (195 GeV) is also shown. allowed by the perturbativity up to the GUT scale as we have mentioned before, the next-to-minimal model is the only viable model. The D0 bound constrains the parameter space in√ the next-to-minimal√ model. For example, the parameter (l) (l) = (l) F /(mlml¯) has to be larger than about 0.93 for m / mlml¯ 1, and m / mlml¯ should be in the range of 0.5–2 (l) = for F /(mlml¯) 0.98, as shown in Figs. 1 and 2. The experimental bound on gluino mass is about 200 GeV for (d) (l) the next-to-minimal model, and hence a constraint on F /(md md¯) is weaker than that of F /(mlml¯). (l) ∼ We√ ﬁnd that there is an interesting consequence in the√ consistent region with F /(mlml¯) 1 and (l) ∼ (l) ∼ (l) ∼ m / mlml¯ 1. In the region with F /(mlml¯) 1 and m / mlml¯ 1, one of messenger sleptons gets lighter (d) (l) (l) = − and it becomes the lightest messenger particle, provided that F /(md md¯)

ψ (ψ) = Fig. 3. The lightest messenger mass as a function of a parameter F /(mψ mψ¯ ). We show lines for m / mψ mψ¯ 0.5, 1 and 2. Here we = = have assumed that mψ mψ¯ and mG˜ 16 eV. superpotential [21]:7 λ W = λ Nψ¯ ψ + λ NH H + N N 3. (17) ψ 1 2 H u d 3 ψ=d,l The interaction of the singlet N could increase the annihilation cross-section of the messenger sneutrino σv and it 2 would provide a thermal relic density of the messenger sneutrino Ω ˜ h consistent with the measured dark matter ψν density [10,21]: −26 3 2 10 cm /s Ω ˜ h 0.1 × , ψν σv 4 4 4 4 6 y Λ − y Λ 30 TeV σv ∼ 10 26 cm3/s × , (18) 6 × 5 32πm ˜ 0.4 2.6 10 GeV mψ˜ ψν ν where y is a function of Yukawa couplings kψa ,λψ ,λH and λN , which is of order one. Furthermore, recently Hooper and March-Russell [10] has proposed that if the mass of messenger sneutrino dark matter is about 20–30 TeV, it could also account for the multi-TeV gamma-ray spectrum from the galactic center observed by HESS [11,22]. As can be seen from Fig. 3, the next-to-minimal model can predict such a messenger (l) dark matter if F /(mlml¯) is about 0.97–0.99. In this range of parameter space, the predicted wino mass is about (d) 250–290 GeV. The gluino (bino) is also predicted to be lighter than 1 TeV (280 GeV) since F /(md md¯)< (l) 8 F /(mlml¯) in order for the messenger sneutrino to be lighter than the messenger squarks. In this Letter, we have discussed the direct-transmission model of dynamical SUSY breaking proposed in Ref. [7]. We have found that in order to be consistent with the current cosmological bound on the gravitino mass (mG˜ < 16 eV), the direct-transmission of SUSY breaking is required. Combined with the current D0 limit on wino mass, the minimal model has been ruled out, and the next-to-minimal model is the only consistent model. We have shown that predictions of upper limits on superparticle masses (1.1 TeV, 320 GeV, 160 GeV, 5 TeV, 1.5 TeV and 700 GeV for gluino, wino, bino, squarks, left-handed sleptons and right-handed sleptons, respectively). We have also found that in the consistent region, the next-to-minimal model predicts the light messenger slepton in the

7 The superpotential in this model is consistent with R-symmetry, and hence it is natural. 8 We note that if we assume the uniﬁcation of mass parameters between messenger quarks and leptons at the GUT scale, we get a relation: (l) (d) F /(mlml¯) 2F /(md md¯). However, this relation can be easily changed by the GUT threshold corrections [7]. Also if the messenger quarks and leptons belong to different GUT multiplets, there is no such a uniﬁcation. 126 M. Ibe et al. / Physics Letters B 615 (2005) 120–126 mass range of 10–50 TeV, and hence it would be an interesting dark matter candidate to account for the multi-TeV gamma-ray spectrum observed by HESS experiment. Therefore, not only current and future sparticle searches but also more multi-TeV gamma-ray data from HESS and Cangaroo III in the coming years would provide a further important insight for the direct-transmission model of the dynamical SUSY breaking.

Acknowledgements

K.T. thanks Wayne Repko for a careful reading of the manuscript. K.T. acknowledges support from the National Science Foundation.

References

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Generalized curvature and the equations of D = 11 supergravity

Igor A. Bandos a,b, José A. de Azcárraga a,MoisésPicóna,c, Oscar Varela a,d

a Departamento de Física Teórica, Universidad de Valencia and IFIC (CSIC-UVEG), 46100-Burjassot (Valencia), Spain b Institute for Theoretical Physics, NSC “Kharkov Institute of Physics and Technology”, UA-61108 Kharkov, Ukraine c Department of Physics and Astronomy, University of Southern California, Los Angeles, CA 90089-2535, USA d Michigan Center for Theoretical Physics, Randall Laboratory, Department of Physics, University of Michigan, Ann Arbor, MI 48109-1120, USA Received 13 January 2005; accepted 18 March 2005 Available online 31 March 2005 Editor: P.V. Landshoff

Abstract α = It is known that, for zero fermionic sector, ψµ(x) 0, the bosonic equations of Cremmer–Julia–Scherk eleven-dimensional γ b β β supergravity can be collected in a compact expression, Rabα Γ γ = 0, which is a condition on the curvature Rα of the γ abc ˆ [abc d] ˆ generalized connection w. In this Letter we show that the equation Rbcα Γ γβ = 4i((Dψ)bcΓ )β (ψd Γ )α,whereD is the covariant derivative for the generalized connection w, collects all the bosonic equations of D = 11 supergravity when the α = gravitino is nonvanishing, ψµ(x) 0.  2005 Elsevier B.V. All rights reserved.

1. Introduction

Recently, the notion of generalized connection and generalized holonomy has been applied to the analysis of supersymmetric solutions of D = 10, 11 dimensional supergravity [1–7]. The generalized connection (see [8])

1 w α := ω α + t α = ωabΓ α + t α (1) β Lβ 1β 4 abβ 1β involves, in addition to the true Lorentz (or spin) connection, the Lorentz covariant part α i a [3] α 1 [4] α t = E F [ ]Γ + F Γ [ ] , (2) 1β 18 a 3 β 8 a 4 β

E-mail address: j.a.de.azcarraga@iﬁc.uv.es (J.A. de Azcárraga).

0370-2693/$ – see front matter  2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physletb.2005.03.045 128 I.A. Bandos et al. / Physics Letters B 615 (2005) 127–133 constructed from the tensor Fabcd, the ‘supersymmetric’ field strength of the antisymmetric tensor field Aµνρ(x) [ ] [ ] 3 := b1b2b3 4 α = b1...b4 α (see Eqs. (19)). In Eq. (2) Fa[3]Γ Fab1b2b3 Γ , F Γa[4]β F Γab1...b4β and we have denoted the µ a a a = µ a vielbein one-form dx eµ(x) by E , E dx eµ(x). The generalized connection allows for a simple expression of the supersymmetric transformation rules for the gravitino (hence the name of ‘supersymmetric’ connection frequently used). Denoting the gravitino one-form by α = µ α ψ dx ψµ(x), this variation is given by α α α β α α β α δεψ = Dε (x) := Dε (x) − ε (x)t1β (x) = dε (x) − ε (x)wβ (x). (3) It was already noticed in [8] that the gravitino equation of motion has also a compact form (see Eq. (25))in terms of its generalized covariant (or ‘supercovariant’) derivative ˆ α α β α α β α Dψ := dψ − ψ ∧ wβ ≡ Dψ − ψ ∧ t1β (4) defined for the generalized connection (1). Then the following observation (see [4,7]) holds: when the fermionic sector is set to zero, all the bosonic equations of the Cremmer–Julia–Scherk (CJS) eleven-dimensional supergravity can be collected in the simple expression α γ b α Naβ := Rabβ Γ γ = 0 (5) γ b α a γ b α or, equivalently, ibRβ Γ γ ≡ E Rbaβ Γ γ = 0, in terms of the generalized curvature R (see, e.g., [3]) 1 R α := dw α − w γ ∧ w α = Rab(Γ ) β + Dt β − t γ ∧ t β , (6) β β β γ 4 ab α 1α 1α 1γ which takes values in the Lie algebra of the generalized holonomy (holonomy of the generalized connection) group [1].1 A similarly concise equation in the case of the purely bosonic limit of massive type IIA supergravity was given recently in [13]. We present here the generalization of Eq. (5), to the case of nonzero gravitino, ψα = 0. It reads

γ ∧8 abc ˆ δ γ ∧7 [a1a2a3 a4] Rβ ∧ E Γ =−iDψ ∧ ψ ∧ E Γ Γ , (7) abc γα a1...a 4 δα βγ ⇒ R γ abc = Dˆ [abc d] bcα Γ γβ 4i ( ψ)bcΓ β ψd Γ α, (8) ˆ α a b ˆ α where Dψ = 1/2E ∧ E (Dψ)ba is deﬁned in (4) and

∧ − 1 E (11 k) := ε Eb1 ∧···∧Eb11−k . (9) a1...ak (11 − k)! a1...akb1...b11−k Eq. (7) (or (8)) collects all the bosonic equations when the gravitino is not zero, ψα = 0. Although the final result is formulated as a statement about dynamical equations of motion and, in this sense, refers to the second order approach to supergravity, we find it convenient to use the first order supergravity action of [10,11]. Our notation (which is explained in the text) is close to that in [11] and the same of [7,12].

2. First order action for D = 11 supergravity

The ﬁrst order action for D = 11 supergravity [10,11], = L a α ab S 11 E ,ψ ,ω ,A3,Fa1a2a3a4 , (10) M11

1 See [9] for further discussion on the generalized holonomy. I.A. Bandos et al. / Physics Letters B 615 (2005) 127–133 129

11 is the integral over eleven-dimensional spacetime M of the eleven form L11 which can be written as [10,11] 1 ∧ 1 L = Rab ∧ E 9 − Dψα ∧ ψβ ∧ Γ¯ (8) + ψα ∧ ψβ ∧ T a + i/2ψ ∧ ψΓ a ∧ E ∧ Γ¯ (6) 11 4 ab αβ 4 a αβ 1 1 1 + (dA − a ) ∧ (∗F + b ) + a ∧ b − F ∧∗F − A ∧ dA ∧ dA . (11) 3 4 4 7 2 4 7 2 4 4 3 3 3 3 Following [11] (see also [12]), we have introduced the notation 1 i a := ψα ∧ ψβ ∧ Γ¯ (2),b:= ψα ∧ ψβ ∧ Γ¯ (5) (12) 4 2 αβ 7 2 αβ for the bifermionic 4- and 7-forms and

1 1 ∧ F := Ea4 ∧···∧Ea1 F , ∗F := − E 7 F a1...a4 (13) 4 4! a1...a4 4 4! a1...a4 for the purely bosonic forms constructed from the antisymmetric tensor zero-form Fabcd. We also use the compact notation 1 Γ¯ (k) := Eak ∧···∧Ea1 Γ (14) αβ k! a1...akαβ ∧(11−k) = ¯ (k) = − k(k−1)/2 (k) and Eq. (9) [to be compared with the notation of [11], Ea1...ak Σa1...ak , Γαβ ( ) γαβ ]. The action (10) is invariant under the local supersymmetry transformations δε, which are given by a =− α a β δεE 2iψ Γαβ ε , (15) α α α β α δεψ = Dε (x) = Dε (x) − ε (x)t1β (x), (16) = α ∧ ¯ (2) β δεA3 ψ Γαβ ε , (17) ab plus more complicated expressions for δεω and δεFabcd, which can be found in [11] and that will not be needed below. Let us stress that, as shown in [11], the supersymmetry transformation rules of the physical ﬁelds are the same in the second and in the ﬁrst order formalisms.

3. Equations of motion

In the ﬁrst order action (10) one distinguishes between the true equations of motion and the algebraic (or nondynamical) equations a =− α ∧ β a T iψ ψ Γαβ , (18)

dA3 = a4 + F4 (19) (see Eqs. (12), (13) for the notation) which follow, respectively, from the variation with respect to the spin connec- ab tion ωL and the antisymmetric tensor Fabcd 1 ∧ δ L = E 8 ∧ T a + iψα ∧ ψβ Γ a ∧ δωbc + d(···), ω 11 4 abc αβ 1 ∧ δ L =− (dA − a − F ) ∧ E 7 δF a1...a4 . (20) F 11 4! 3 4 4 a1...a4 Notice that Eqs. (18), (19) are the counterparts of the superspace constraints of D = 11 supergravity (see [12] for a discussion and references), but for forms on eleven-dimensional spacetime. As far as the dynamical bosonic 130 I.A. Bandos et al. / Physics Letters B 615 (2005) 127–133 equations are concerned, one ﬁnds that the variation with respect to A3,

δAL11 = d(∗F4 + b7 − A3 ∧ dA3) ∧ δA3 + d(···) (21) results in

G8 := d(∗F4 + b7 − A3 ∧ dA3) = 0, (22) which becomes the standard CJS three-form gauge ﬁeld equations of motion once the algebraic equations (constraints) (18), (19) are taken into account. The more complicated variation δE with respect to the vielbein form, δEL11 =···, as well as the full expression of the Einstein equations

1 ∧ M := Rbc ∧ E 8 +···=0, (23) 10a 4 abc which follows from that variation, will not be needed here (see [11]). After some algebra, the fermionic variation δψ of the Lagrangian form L11,Eq.(11), reads (cf. [11]) L =− Dˆ α ∧ ¯ (8) ∧ β + − − ∧ ¯ (5) ∧ α ∧ β δψ 11 2 ψ Γαβ δψ i(dA3 a4 F4) Γαβ ψ δψ + ¯ (8) + ∧ ¯ (6) ∧ a + α ∧ β a ∧ α ∧ β − α ∧ ¯ (8) ∧ β iaΓαβ 1/2Ea Γαβ T iψ ψ Γαβ ψ δψ d ψ Γαβ δψ , (24) where Dˆ ψα is given by Eq. (4). Taking into account the algebraic equations (18), (19), and ignoring the (last) total derivative term in Eq. (24) one ﬁnds the gravitino equation of [8] written, as in [11], in the suggestive differential form := Dˆ α ∧ ¯ (8) = Ψ10β ψ Γαβ 0. (25)

4. Bosonic equations of D = 11 supergravity as a condition on the generalized curvature

4.1. A concise form of the bosonic equations from selfconsistency of the gravitino equations ˆ It is important that the above gravitino equation, Ψ10β = 0, is expressed in terms of the covariant derivative D, Eqs. (4), (1), (2). As a result, the integrability/selfconsistency condition for Eq. (25) may be written in terms of the Dˆ Dˆ α =− β ∧R α γ ∧ ¯ (8) = 2 D ¯ (8) = ¯ (8) = generalized curvature (6).Using ψ ψ β and t1[β Γ α]γ 0 which implies Γβα DΓβα a ∧ ¯ (8) T iaΓβα , we obtain Dˆ = Dˆ β ∧ a + ∧ a ∧ ¯ (8) Ψ10α ψ T iψ ψΓ iaΓβα i ∧ ∧ [ ] − ψβ ∧ R γ ∧ E 8 Γ abc + iDˆ ψδ ∧ ψγ ∧ E 7 Γ a1a2a3 Γ a4 = 0. (26) 6 β abc γα a1...a4 δα βγ The ﬁrst term in the second part of Eq. (26) vanishes due to the algebraic (constraint) equation (18). Hence on the surface of constraints the selfconsistency of the gravitino equation is guaranteed when ∧ ∧ [ ] M := R γ ∧ E 8 Γ abc + iDˆ ψδ ∧ ψγ ∧ E 7 Γ a1a2a3 Γ a4 = 0. (27) 10αβ β abc γα a1...a4 δα βγ Our main observation is that Eq. (27) (see Eq. (7) or (8)) collects all the bosonic equations of motion (22), (23) and the corresponding Bianchi identities for the A3 gauge ﬁeld and for the Riemann curvature tensor. Let us stress that we distinguish between the algebraic equations or constraints, Eqs. (18) and (19), from the true dynamical

2 γ ∧ ¯ (8) =−i ∧ ¯ (5) + 1 ∗ ∧ ¯ (2) This follows, e.g., from direct calculation of t1α Γγβ 2 F4 Γαβ 2 F4 Γαβ . I.A. Bandos et al. / Physics Letters B 615 (2005) 127–133 131 equations (22), (23), and that our statement above refers to the dynamical equations; thus it is also true for the second order formalism. To show this it is not necessary to make an explicit calculation. It is sufﬁcient to use the second Noether theorem and/or the fact that the purely bosonic limit of (27) implies Eq. (5) (see Section 4.3 below), which is equivalent to the set of all bosonic equations and Bianchi identities when ψα = 0.

4.2. Proof using the Noether identities for supersymmetry

In accordance with the second Noether theorem, the local supersymmetry under (15)–(17) reflects (and is re- flected by) the existence of an interdependence among the bosonic and fermionic equations of motion; such a relation is called a Noether identity. Furthermore, as the local supersymmetry variation of the gravitino is given by the covariant derivative Dˆ εα with generalized connection, Eq. (16), the gravitino equation Ψ should enter the corresponding Noether identity through Dˆ Ψ . Thus, Dˆ Ψ should be expressed in terms of the equations of motion for the bosonic fields, in our case including the algebraic equations for the auxiliary fields. Hence, in the light of (26), (18) the l.h.s. of Eq. (27) vanishes when all the bosonic equations are taken into account. Indeed, schematically, ignoring for simplicity the purely algebraic equations and neglecting the boundary contributions, the variation of the action (10), (11) (considered now in the second order formalism) reads α a δS = −2Ψ10 α ∧ δψ + G8 ∧ δA3 + M10a ∧ δE . (28)

For the local supersymmetry transformations δε,Eqs.(16), (15) and (17), one ﬁnds integrating by parts α a δεS = −2Ψ10α ∧ Dε + G8 ∧ δεA3 + M10a ∧ δεE =− − D − G ∧ β ∧ ¯ (2) + ∧ β a α = 2 Ψ10α 8 ψ Γβα 2iM10a ψ Γβα ε 0. (29)

α As δεS = 0 is satisﬁed for an arbitrary fermionic function ε (x), it follows that 1 DΨ =− ψβ ∧ −2iΓ a M + G ∧ Γ¯ (2) . (30) 10α 2 βα 10a 8 βα In the light of Eqs. (26) and (30), and after the algebraic equations (18), (19) are taken into account, ∧ ∧ [ ] M := R γ ∧ E 8 Γ abc + iDˆ ψδ ∧ ψγ ∧ E 7 Γ a1a2a3 Γ a4 =−3i −2iΓ a M + G ∧ Γ¯ (2) . (31) 10αβ β abc γα a1...a4 δα βγ βα 10a 8 βα

It then follows that M10αβ = 0, Eq. (27), is satisﬁed after the dynamical equations (23), (22) are used. Moreover, Eq. (31) also shows what Lorentz-irreducible parts of the concise bosonic equations M10αβ = 0 coincide with the Einstein and with the 3-form gauge ﬁeld equations. These are given, respectively, by 1 M =− tr(Γ M ), (32) 10a 192 a 10 i G ∧ Ea ∧ Eb = tr Γ abM . (33) 8 96 10

It is clear that all other Lorentz–irreducible parts in Eq. (27), M10 αβ = 0, are satisﬁed either identically or due to the Bianchi identities that are the integrability conditions for the algebraic equations (18), (19) used in the derivation of (31). Thus,wehaveproventhatEq.(27) collects all the dynamical bosonic equations of motion in the second order approach to supergravity. To see that it collects all the Bianchi identities as well, one may either perform a straightforward calculation or study the pure bosonic limit of Eq. (27). The latter way is simpler and it also provides an alternative proof of the above statement as we now show below. 132 I.A. Bandos et al. / Physics Letters B 615 (2005) 127–133

4.3. Proof using the purely bosonic limit of the equations

For bosonic conﬁgurations, ψα = 0, Eq. (27) takes the form

α = R γ ∧ ∧8 abc = ψ 0, β EabcΓγα 0. (34) We show here that this equation is another form of Eq. (5) [7],

α = R γ aα ≡ bR γ aα = ψ 0,ia β Γγ E abβ Γγ 0. (35) As the above Eq. (35) collects all the bosonic equations of standard CJS supergravity as well as all the Bianchi identities in the purely bosonic limit [4,7], the equivalence of Eqs. (35) and (34) will imply that M10αβ = 0, Eq. (27), does the same for the case of nonvanishing fermions, ψα = 0. β β a b β Decomposing Rα on the basis of bosonic vielbeins, Rα = 1/2E ∧ E Rbaα , one ﬁnds that Eq. (34) implies

R γ abc = abβ Γγα 0. (36)

αδ Contracting (36) with Γc one ﬁnds

γ ab δ Rabβ Γ γ = 0. (37) αδ ab = ab + [a b] Then, contracting again with the Dirac matrix Γd and using Γ Γd Γ d 2Γ δd as well as Eq. (36), one α recovers Eq. (5), Naβ = 0, which is an equivalent form of Eq. (35). ≡ ≡ N N The Bianchi identities R[abc]d 0 and dF4 0 appear as the irreducible parts tr(Γc1c2c3 a) and tr(Γc1...c5 a) N [ ] b1...b5 + of Eq. (5) [7] [more precisely, in the last case the relevant part in a is proportional to dF4 b1...b5 (Γa [b b ...b ] 10δa 1 Γ 2 5 ), but the two terms in the brackets are independent]. Knowing this, one may also reproduce the terms that include the Bianchi identities in the concise equation (7) (equivalent to (8) or (31)) with a nonvanishing gravitino.

5. Conclusion

We have shown that all the bosonic equations of D = 11 supergravity can be collected in a single equation, Eq. (27), written in terms of the generalized curvature (6) which takes values in the algebra of the generalized holonomy group. In the ﬁrst proof we used the shortcut provided by the second Noether theorem, which implies the Noether identity (30) for the local supersymmetry (15)–(17) relating the bosonic and fermionic equations. The second proof uses the purely bosonic limit (35) of the desired equation (27). This is simpler since the properties of the purely bosonic Eq. (35) are known [4,7] and may also be used to simplify the extraction of Bianchi identities from Eq. (27), although we do not do it here. The concise form (27) of all the bosonic equations is obtained by factoring out the fermionic one-form ψβ in the selfconsistency (or integrability) conditions DΨ10β = 0[Eq.(26)], for the gravitino equations Ψ10α = 0, Eq. (25). In this sense, one can say that in (the second order formalism of) D = 11 CJS supergravity all the equations of := Dˆ α ∧ ¯ (8) = motion and Bianchi identities are encoded in the fermionic gravitino equation Ψ10β ψ Γαβ 0[Eq.(25)]. Actually this should be expected for a supergravity theory including only one fermionic ﬁeld, the gravitino, and whose supersymmetry algebra closes on shell. As we have discussed, the basis for such an expectation is provided by the second Noether theorem. We hope that the explicit form (27) of the equation collecting all the bosonic equations of motion and Bianchi identities may be useful in a further understanding of the properties of D = 11 supergravity and in the analysis of its supersymmetric solutions, including those with nonvanishing fermionic sector (see [14] and references therein). I.A. Bandos et al. / Physics Letters B 615 (2005) 127–133 133

Acknowledgements

The authors thank Dima Sorokin for useful comments and Paul de Medeiros for correspondence. This work has been partially supported by the research grant BFM2002-03681 from the Ministerio de Educación y Ciencia and from EU FEDER funds, the Generalitat Valenciana (Grupos 03/124), the grant No. 383 of the Ukrainian State Fund for Fundamental Research, the INTAS Research Project No. 2000-254 and the EU network MRTN-CT-2004- 005104 ‘Forces Universe’. M.P. and O.V. wish to thank the Ministerio de Educación y Ciencia and the Generalitat Valenciana, respectively, for their FPU and FPI research grants, and I. Bars (M.P.) and M. Duff (O.V.) for their hospitality at the USC and the University of Michigan.

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Sigma model Lagrangian for the Heisenberg group

Belal E. Baaquie, Kok Kean Yim

Department of Physics, National University of Singapore, Singapore 119260, Singapore Received 6 January 2005; received in revised form 15 March 2005; accepted 22 March 2005 Available online 7 April 2005 Editor: T. Yanagida

Abstract We study the Lagrangian for a sigma model based on the non-compact Heisenberg group. A unique feature of this model— unlike the case for compact Lie groups—is that the Lagrangian has to be regulated since the trace over the Heisenberg group is otherwise divergent. The resulting theory is a real Lagrangian with a quartic interaction term. In particular, in D = 2 space–time dimensions, after a few non-trivial transformations, the Lagrangian is shown to be equivalent, at the classical level, to a complex cubic Lagrangian. A one-loop computation conﬁrms that the quartic and cubic Lagrangians are equivalent at the quantum level as well. The complex Lagrangian is known to be classically equivalent to the SU(2) sigma model, with the equivalence breaking down at the quantum level. An explanation of this well-known results emerges from the properties of the Heisenberg sigma model.  2005 Elsevier B.V. All rights reserved.

PACS: 11.10.Lm; 11.10.Ef; 11.30.Rd; 11.10.Hi

Keywords: Sigma models; Heisenberg Lagrangian; Duality transformation; Cubic Lagrangian; Chiral symmetry

1. Introduction These sigma models are usually based on compact Lie groups. In this Letter, we construct a sigma model based on Sigma models in two-dimensional space–time have the non-compact Heisenberg group. The present study a long history in theoretical physics [1]. They are ubiq- is motivated by the more complex case studied in [2] uitous in particle physics, with many applications and extensions in quantum field theory and string theory. which deals with a supersymmetric Yang–Mills theory having a local infinite-dimensional Kac–Moody group as its gauge group. The need for regulating the La- E-mail addresses: [email protected] (B.E. Baaquie), grangian of the theory was essential in obtaining local yimkk@pacific.net.sg (K.K. Yim). Kac–Moody gauge symmetry. The Heisenberg alge-

0370-2693/$ – see front matter  2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physletb.2005.03.063 B.E. Baaquie, K.K. Yim / Physics Letters B 615 (2005) 134–140 135 bra is an infinite-dimensional non-compact subalgebra field method to study the renormalizability of the the- of the well-known Kac–Moody algebra [3], and the ory up to the one-loop correction. A similar, but much Heisenberg sigma model is the simplest theory hav- more complex, calculation was carried out in [10] ing the new features that emerge from constructing to study the renormalizability of a U(1) gauge field quantum field theories based on infinite-dimensional with Kac–Moody gauge symmetry. We calculate the Lie algebras. Coupled to the fact that there is a central β-functions for both the quartic and cubic Lagrangian extension associated with the Heisenberg algebra, the realizations of the theory, and show that to one-loop sigma model obtained here will be different from the they are identical. In so doing we also verify the one- sigma model based on a compact Lie group. This is the loop renormalizability of two (apparently dissimilar) main motivation for studying the Lagrangian obtained bosonic theories. in this Letter. Given that the group element of the Heisenberg group is infinite-dimensional, the usual procedure of 2. The Heisenberg sigma model Lagrangian obtaining a Lagrangian by tracing over a representation of the group yields a divergent result. We regulate Consider the (non-compact) Heisenberg algebra the trace to obtain a finite Lagrangian, which turns out [x,p]=ik, where k is the central extension. In terms to be a real quartic Lagrangian L4. Since no restriction of the creation and destruction operators, it is given by was imposed in obtaining the Lagrangian, the result is [a,a†]=k. Since we would like to construct a sigma valid for arbitrary space–time dimensions D. model based on the Heisenberg group, thus let us start Furthermore, one can show that in two dimen- from the finite group elements of the Heisenberg alge- sions, after some straightforward calculations involv- bra. By the usual exponential mapping, we can write ing functional integrals, the quartic Lagrangian is such an element as L ∗ equivalent to a complex cubic Lagrangian 3. Interest- Ω = exp iφ + iωa + iω a† . (1) ingly enough, the relationship of the two Lagrangians is not that of a usual duality transformation since the Note that the field (group coordinate) φ is a real vari- mapping does not induce an inversion of the coupling able, whereas the field ω is a complex variable. The constant. The cubic Lagrangian L3 in turn is known field φ has to be introduced due to the existence of the to be classically equivalent to the sigma model based central extension of the Heisenberg algebra. on the SU(2) group [4] which has spontaneous parti- The simplest non-linear sigma model Lagrangian cle production [5]. Furthermore, it is also known that based on a space–time dependence of the group coor- this equivalence breaks down on quantizing the two dinates φ and ω is defined by theories [5]. The causes of this breakdown was ex- † L = Tr ∂µΩ ∂µΩ . (2) pounded in [6,7] and explored in [8,9]. Since we are dealing with a model based on an infinite-dimensional However, this approach fails since the trace over † Lie group, our work should provide an understanding the non-compact operators a, a diverges, yielding L =∞ of the quantum inequivalence from a different perspec- . A similar situation was encountered in defin- tive. ing the supersymmetric gauge fields with the infinite- It is well known that a pair of quantum theories are dimensional Kac–Moody symmetry [2]. equivalent if the corresponding correlation functions To successfully obtain a finite Lagrangian, one [···] of both theories are equal. Since our cubic Lagrangian must regularize the trace Tr over the infinite- is obtained from that of the quartic Lagrangian by per- dimensional operators. There is a wide variety of reg- forming an exact Gaussian integration and by constant ulators which one can choose, and we expect from the field rescalings which do not affect the path integral principle of universality that a whole range of regu- measure, the quantum theory of the two Lagrangians lators would lead to the same renormalizable theory 1 are equivalent. [2]. We make the natural choice for the regulator of To verify the quantum equivalence of the cubic and quartic theories, we compute the one-loop beta func- 1 Note that another approach of regularizing this Lagrangian may tion for the two theories. We employ the background be developed from the works of Wigner and Moyal [12]. 136 B.E. Baaquie, K.K. Yim / Physics Letters B 615 (2005) 134–140 the trace given by The Lagrangian given above describes a non-trivial quartic scalar field theory consisting of three mass- − † e ρa a: trace regulator. (3) less scalar fields. The kinetic terms of the two fields α(x), β(x) are scaled by the regularization factor ρ In direct analogy with the procedure given in [2],we (through the function f(ρ)) with respect to the field define a generalization of the non-linear sigma model φ(x). Since the highest-order interaction term is quar- by the following Lagrangian tic in the fields, we call this the quartic Lagrangian L4. 1 −ρa†a † Define the (bounded) coupling constant by L = Tr e ∂µΩ ∂µΩ , (4) 2λ kρ g = tanh ∈[0, 1]. (10) where a (normalization) constant of (2λ)−1 has been 2 included. We have added the regularization factor Now, rescale the fields by † e−ρa a in the Lagrangian, so that the trace over the φ(x) infinite-dimensional a and a† converges. Thus, we φ(x)→ ,α(x)→ f(ρ)α1(x), have a Lagrangian that resembles the Lagrangian of g the U(N)chiral model or non-linear sigma model, and β(x) → f(ρ)α2(x). (11) which is valid for the infinite-dimensional Heisenberg We then obtain the Lagrangian in the following com- group. pact form To write the Lagrangian in terms of the real fields, α and β, define 1 i i i j 2 L = ∂µα ∂µα + ∂µφ − εij α ∂µα , (12) 4 2g2 = + ω(x) α(x) iβ(x). (5) which no longer contains f(ρ) or k—they are ab- Using the Campbell–Baker–Hausdorff (CBH) formula sorbed into g and the fields. The corresponding action, in D space–time dimensions, is given by [13],wehave † = D L = + −iΩ ∂µΩ = ∂µφ + k(β∂µα − α∂µβ)+ ∂µαa S4 dx 4 Sα Squartic, (13) + † ∂µβa . (6) where Squartic contains the interaction terms of the the- Note that despite the complexity of the CBH formula, ory. the exact expression (6) can appear in such a simple The factor g is the coupling constant of the the- form because the commutator of the Heisenberg alge- ory. An important feature of this theory is that this bra gives a right-hand side which is proportional to the coupling constant is bounded. As expected, this cou- identity operator. pling constant depends on the central extension k of Choosing the normalization factor λ such that the Heisenberg algebra and the regularization factor ρ, g as what one would expect. This coupling constant is − † λ = Tr e ρa a , (7) non-trivially related to λ which was dropped from the theory earlier. the Lagrangian (4) yields, using Eq. (6), the following If we had started with the following Lagrangian form 1 − † L = Tr e ρa a∂ Ω∂ Ω† (14) 1 2 2λ µ µ L = ∂µφ − k(β∂µα − α∂µβ) 2 instead of Eq. (4), then we would get a Lagrangian + f (ρ)(∂µα∂µα + ∂µβ∂µβ) , (8) identical to Eq. (8) since the change in the ordering of the Ω field switches g to −g. where Dimensional analysis tells us that the coupling con- [ † + † −ρa†a] stant g is dimensionless in D = 2. When one quantizes = Tr (a a aa )e = k f(ρ) † , (9) the theory, one can expect the theory to be renormal- Tr[e−ρa a] tanh( kρ ) 2 izable for D = 2. Since the coupling constant is di- after some straightforward computations. mensionless in D = 2 and moreover bounded, at least B.E. Baaquie, K.K. Yim / Physics Letters B 615 (2005) 134–140 137 in the case of the regulator that we have chosen, it is Recall that the SU(2) sigma model is invariant un- possible that the renormalized coupling constant also der global left and right group multiplication; how- turns out to be bounded. Thus, if this theory needs ever, it is known that only one of the symmetries renormalization (which it does), then as we let the cou- of the SU(2) sigma model is present in the (classi- pling constant run, an educated guess is that it should cally) equivalent cubic Lagrangian [8,9]. An expla- hit a fixed point. On performing a one-loop correc- nation of this fact follows from the derivation of this tion calculation for the beta function, we will see that cubic Lagrangian from the Lagrangian of the Heisen- the coupling increases for short distances; hence, we berg group: due to the regulator, the Lagrangian given expect that the theory should have a non-trivial ultra- in Eq. (4) is invariant only under the following global violet fixed point with coupling constant g∗ = 1. transformation The quantum field theory is defined by the Feyn- Ω(t,x)→ ΦΩ(t, x), (21) man path integral where Φ is a constant element of the Heisenberg −S4 Z = Dαi Dφ e . (15) group. Multiplication on the right by Φ is not a symmetry due to the regulator, and hence explaining the Since the field φ appears only quadratically in the lack of invariance of the cubic Lagrangian under both L Lagrangian 4, one can integrate it out in the path left and right group multiplications. integral to get a simpler expression. We perform the We also found an explanation as to why the classi- integration in the following manner: cal equivalence of the SU(2) cubic (pseudo-chiral) La- − grangian with the principal SU(2) chiral model breaks = S4 Z Dαi Dφ e (16) down: our working shows that the cubic Lagrangian 1 results from a sigma model based on the non-compact −Sα 2 = Dαi e Dφ DAµ exp − A Heisenberg group rather than on the compact SU(2) 2 µ Lie group. In the earlier derivation [4–7] of the cubic i i j + Aµ ∂µφ − εij α ∂µα . (17) Lagrangian, the equation of motion and the Bianchi g identity were used and the issue of compactness/non- Performing the φ integrations yields compactness of the degrees of freedom was not addressed. This difference in the target spaces emerges = ⇒ = ∂µAµ 0 Aµ εµν∂νχ. (18) under renormalization, causing the classical equiva- Performing the Aµ Gaussian path integrations in lence to break down. D = 2 space–time, we can rewrite the Lagrangian as One can establish the equivalence between the cubic (19) and the quartic (12) Lagrangian (and hence 1 i i 2 i j k L3 = ∂µφ ∂µφ − i gεij k εµνφ ∂µφ ∂νφ , (19) the Heisenberg sigma model) using a very different 2 3 approach—by working out the canonical transforma- with the notations tions between the fields of the two models as was done in the literature (see [9,16] and references therein). α1 = gφ1,α2 = gφ2,χ= φ3, (20) However, our treatment here by-passes this lengthy where εij k is the standard totally antisymmetric tensor. process which in principle should exist and could also In this form, the Lagrangian is complex with the inter- be interesting, in view of the mapping between the action given by a single imaginary cubic term. Note dual sigma model to the chiral model [8,9]. that this conversion works only for D = 2 space–time The cubic Lagrangian (19) also corresponds to dimensions, unlike earlier parts of our working. One the first two non-trivial terms of the SU(2) WZW can also obtain the cubic Lagrangian (19), without the model [11] under a linear expansion. In fact, it was introduction of Aµ, using a more lengthy route. pointed out in [8] that by performing a Wigner– The cubic Lagrangian is identical to that of the Inönü contraction [15] on the symmetry manifold scalar-field theory model that is known to be classi- of the WZW model, one can obtain this cubic La- cally equivalent [5,6,8,9] to the SU(2) sigma model; grangian (19) from the WZW model. Furthermore, this model has spontaneous particle production [5]. the cubic Lagrangian is also almost identical to a La- 138 B.E. Baaquie, K.K. Yim / Physics Letters B 615 (2005) 134–140 grangian which describes the low-energy excitation their quantum field theories are radically different, of the one-dimensional quantum antiferromagnet with with the sigma model being asymptotically free and short-range Néel order [14]. The difference being that the cubic Lagrangian being non-asymptotically in the antiferromagnet case the fields must reside on a free [5]. Since—unlike equivalent theories related by sphere while in our case we have no such restriction. a duality transformation—both the quartic and cubic In particular, we note the following: Lagrangians have a common weak coupling sector, we can compare the quantum behavior of the two theories (1) In the original Lagrangian (4), the only symmetry using weak coupling perturbation theory. We conse- which is manifest is the invariance of the La- quently, separately compute the beta functions of both grangian under left multiplication of Ω. Although the quartic and cubic theories to check whether the not too obvious, this symmetry is present in the classical equivalence that we have derived remains quartic Lagrangian L4. However, this original valid when the fields are quantized. symmetry is hidden in the cubic Lagrangian L3. On the other hand, in the cubic Lagrangian L3,the classical symmetries of the sigma model are man- 3. One-loop renormalization of L4 and L3 ifest, and these in turn are only implicit in the L4. (2) We have transformed a Lagrangian with quartic The background field method is used to obtain the interaction to a Lagrangian with cubic interaction one-loop β-function. We will obtain the result that using Gaussian integrations. Hence, we expect the both L4 and L3 are one-loop renormalizable, and hav- two theories to be equivalent at the quantum level ing the same β-function, showing that the two theories as well. are equivalent as quantum fields theories. (3) The coupling constant g is identically transformed Consider the quartic Lagrangian from one Lagrangian into another Lagrangian, un- 1 changed. This contrasts with the usual duality L = ∂ αi∂ αi + ∂ φ − ε αi∂ αj 2 . 4 2 µ µ µ ij µ (22) transformation which inverts the strength of g. 2g The coupling constant is bounded: −1 g 1. The quantum field theory has both ultra-violet (UV) This bound naturally emerges from its formulation and infrared divergences. To perform the one-loop cal- as a Heisenberg sigma model, and does not ap- culation, we use a UV-regulated propagator via the pear in the formulation based on the SU(2) sigma replacement

model. 2 2 1 e−p /Λ (4) Since the interaction in the cubic Lagrangian is → . one order lower than the quartic Lagrangian (12), p2 p2 one can expect its renormalization to be relatively The infrared divergences are regulated by evaluating simpler. The limitation of the cubic Lagrangian is the Feynman diagrams in the dimension d = 2 + ε;we that it is defined only in the 2-dimensional space– consequently have a dimensionless coupling constant time, while the original Lagrangian (4) and the g2Λ−ε. quartic Lagrangian (12) are defined for all space– The action is given by time dimensions. i S4 φ; α Thus, we have constructed two classically equiva- = 1 lent Lagrangians, L and L , that appear to be quite 2 −ε 4 3 2g Λ different, with the explicit symmetries of one theory 2+ε i i i j 2 being implicit in the other. In particular, interaction × d x ∂µα ∂µα + ∂µφ − εij α ∂µα . vertices and hence the Feynman diagrams of the two (23) theories will be entirely different. Nevertheless, these Using the standard approach [17], we define the back- two Lagrangians share the same coupling constant. ground fields by Note that although the cubic Lagrangian L3 is clas- i i i sically identical to that of the SU(2) sigma model, φ → φ + Φ, α → α + A , B.E. Baaquie, K.K. Yim / Physics Letters B 615 (2005) 134–140 139 where φ, αi are the quantum fields and Φ, Ai are the The β-function is then given by background fields. The generating functional for the ε 2 3 ε ε Λ 3gR Λ 4 quartic Lagrangian is then given by β = gR + + O g 2 µ 2π µ R − [ + ; i + i ] = i S4 φ Φ α A 3 Z Dφ Dα e . → g3 + O(ε). (29) 2π R Let the contribution from the one-loop be denoted Thus we have a positive β-function for small g, and by L4. The one-loop calculation yields the coupling constant is increasing for short distance. The theory is clearly not asymptotically free. L + L 4 4 A similar one-loop calculation for the cubic La- 2 ε 1 i i g hΛ¯ grangian yields the renormalized Lagrangian = ∂µA ∂µA 1 + 2g2 πε LRN[ ]=L + L 3 ΦR,AR,gR 3 3, (30) g2hΛ¯ ε + ∂ Φ − ε Ai∂ Aj 2 1 − . (24) where µ ij µ πε g2 Λ ε 3g2 Λ ε Z = 1 − R ,Z= 1 + R . The renormalized Lagrangian is defined at some Φ πε µ g 2πε µ momentum scale µ with renormalized coupling con- (31) 2 −ε stant gRµ . The one-loop renormalized Lagrangian Note that while Zg remains unchanged in going from can be written as the quartic to the cubic case, ZΦ is different. Hence, we obtain the same β-function as for the quartic La- LRN[ ]=L + L 4 ΦR,AR,gR 4 4 (25) grangian, establishing its one-loop equivalence to the where renormalization constants are given by, quartic theory. The fact that g decreases with increasing distance 1 1 | | Φ = Z 2 Φ ,Ai = Z 2 Ai , is consistent with the condition that g 1. We con- φ R A R = ε jecture that g 1 is an UV fixed point. This means Λ 2 that the β-function will eventually bend down towards g = g Z . (26) µ R g zero. This needs to be studied, perhaps numerically. We also conjecture that the g>1 theory might renor- From the one-loop renormalized Lagrangian (24), one malize to strong coupling. However, in this case, the can deduce theory will no longer be so straightforward. 2g2 Λ ε g2 Λ ε Z = 1 + R ,Z= 1 + R , Φ πε µ A πε µ 2 ε 4. Conclusions 3gR Λ Zg = 1 + , (27) 2πε µ We have analyzed the Heisenberg sigma model La- grangian. The Lagrangian needed to be regularized, where we have used the vertex condition Z = Z2 Φ A and we found that, with a natural regulator, the La- which comes out explicitly in the Feynman diagram grangian has a real quartic interaction. calculations. One particularly interesting feature about this the- To compute the β-function, we adopt Wilson’s ory is when the dimension of space–time is two. Here, point of view that the bare coupling constant g is a we can reformulate the real quartic Lagrangian as an function of the UV-cutoff Λ, and varies in such a man- equivalent theory with an imaginary cubic interaction. ner, so that the renormalized coupling constant g is R Moreover, these two Lagrangians have very different independent of the cutoff. The β-function is defined manifest symmetries. The two equivalent theories are for the dimensionless coupling constant; we hence not related by a usual duality transformation in that have the coupling constants for the two theories are not in- ∂g ∂g β ≡ Λ , R = 0. (28) versely related. On the contrary, they share the same ∂Λ ∂Λ coupling constant. 140 B.E. Baaquie, K.K. Yim / Physics Letters B 615 (2005) 134–140

By computing the one-loop β function we explic- [2] B.E. Baaquie, Phys. Rev. D 61 (2000) 085009. itly demonstrated the one-loop quantum renormaliz- [3] R.V. Moody, Bull. Am. Math. Soc. 73 (1967) 217; ability and equivalence of the two theories. The equiv- V.G. Kac, Funct. Anal. Appl. 1 (1966) 328; P. Goddard, D. Olive, Int. J. Mod. Phys. A 1 (1986) 303. alence of the cubic and quartic Lagrangians is to our [4] V.E. Zakharov, A.V. Mikhailov, Sov. Phys. JETP 47 (1978) knowledge the first instance of two apparently differ- 1017. ent two-dimensional bosonic non-linear theories being [5] C. Nappi, Phys. Rev. D 21 (1980) 118. equivalent as quantum field theories. [6] E.S. Fradkin, A.A. Tseytlin, Ann. Phys. (N.Y.) 162 (1985) 49. We also found an explanation as to why the classi- [7] B.E. Fridling, A. Jevicki, Phys. Lett. B 134 (1984) 70. [8] T. Curtright, C. Zachos, Phys. Rev. D 49 (1994) 5408. cal equivalence of the SU(2) cubic (pseudo-chiral) La- [9] T. Curtright, C. Zachos, in: PASCOS 94, hep-th/9407044. grangian with the principal SU(2) chiral model breaks [10] B.E. Baaquie, R.R. Parwani, Phys. Rev. D 54 (1996) 5259, hep- down. The equivalence breaks down since the cubic th/9511165. Lagrangian is a sigma model based on the infinite- [11] E. Witten, Commun. Math. Phys. 92 (1984) 455; dimensional Heisenberg group rather than on a com- S. Novikov, Sov. Math. Dokl. 24 (1981) 222; J. Wess, B. Zumino, Phys. Lett. B 37 (1970) 95. pact Lie group. [12] J. Klauder, in: Proceedings of the Wigner Centennial Con- Finally, we note that the bound on g is seem to be ference, 2000, http://quantum.ttk.pte.hu/wigner/proceedings/ respected under renormalization. papers/w55.pdf. [13] J.F. Cornwell, Group Theory in Physics, vol. II, Academic Press, San Diego, 1984. Acknowledgements [14] E. Fradkin, Field Theories of Condensed Matter Systems, Addison–Wesley, Reading, MA, 1991, Chapter 5. We have benefited from many fruitful discussions [15] E. Inönü, E. Wigner, Proc. Natl. Acad. Sci. (U.S.A.) 39 (1953) with Rajesh Parwani. We would also like to thank the 510. anonymous referee who has given us many useful sug- [16] T. Curtright, T. Uematsu, C. Zachos, Nucl. Phys. B 469 (1996) gestions. 488. [17] G. ’t Hooft, Nucl. Phys. B 62 (1973) 444. References

[1] E. Abdalla, M. Cristina, B. Abdalla, K.D. Rothe, 2 Dimen- sional Quantum Field Theory, World Scientiﬁc, Singapore, 1991. Physics Letters B 615 (2005) 141–145 www.elsevier.com/locate/physletb

Noncommutativity in quantum cosmology and the hierarchy problem

F. Darabi, A. Rezaei-Aghdam, A.R. Rastkar

Department of Physics, Azarbaijan University of Tarbiat Moallem, 53714-161 Tabriz, Iran Received 12 February 2005; received in revised form 8 April 2005; accepted 15 April 2005 Available online 25 April 2005 Editor: M. Cveticˇ

Abstract We study the quantum cosmology of an empty (4 + 1)-dimensional Kaluza–Klein cosmology with a negative cosmological constant and a FRW type metric with two scale factors, one for 4D universe and the other for one compact extra dimension. By assuming the noncommutativity in the corresponding mini-superspace we suggest a solution for the hierarchy problem, at the level of Wheeler–DeWitt equation.  2005 Elsevier B.V. All rights reserved.

PACS: 98.80.Qc

Recently, the study of various physical theories We know that in the study of homogeneous uni- from noncommutative point of view, such as string verses, the metric depends only on the time parameter. theory [1], quantum field theory (see, for instance, [2]), Thus, one can find a model with a finite-dimensional quantum mechanics [3], and classical mechanics [4], configuration space, the so-called mini-superspace, has been of particular interest. In particular, a new whose variables are the three-metric components. The interest has been developed to study the noncommu- quantization of these models can be performed by tative quantum cosmology [5–7]. In these studies, the using of the rules of quantum mechanics. The mini- influence of noncommutativity at early universe was superspace construction is a procedure to define quan- explored by the formulation of a version of noncom- tum cosmological models in the search for describing mutative quantum cosmology in which a deformation the quantum features of the early universe. The influ- of mini-superspace is required instead of space–time ence of noncommutativity on the mini-superspace has deformation [5]. already been considered for the cosmological models with Kantowski–Sachs and FRW metrics [5,6]. Here, we consider the effect of noncommutativity on the E-mail addresses: [email protected] (F. Darabi), [email protected] (A. Rezaei-Aghdam). configuration space in the model which was previously

0370-2693/$ – see front matter  2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physletb.2005.04.036 142 F. Darabi et al. / Physics Letters B 615 (2005) 141–145 studied by one of the authors [8]. This model shows not claim to solve the problem in a fundamental way an empty (4 + 1)-dimensional Kaluza–Klein universe because we just solve the problem in a special and sim- with a negative cosmological constant and a FRW type ple model. However, this may shed light on the similar metric having two scale factors. Following the idea in approaches to solve the problem in a more fundamen- regarding the Wheeler–Dewitt equation more primi- tal way. tive than the classical Einstien equations1 we study the We start with the metric considered in [8] in which noncommutativity in the quantum cosmology of this the space–time is assumed to be of FRW type which model and try to propose a solution for the hierarchy has a compact space, namely the circle S1. We adopt problem, in the context of quantum cosmology. the chart {t,ri,ρ} with t, ri and ρ denoting the cos- The experimental upper bound on the value of cosmic time, the space coordinates and the compact space mological constant is extremely small. On the other coordinate, respectively. We therefore take hand, it is usually assumed that an effective cosmolog- dri dri ical constant describes the energy density of the vac- ds2 =−dt2 + R2(t) + a2(t) dρ2, (1) + kr2 2 uum ρvac . In fact, the vacuum energy density ρvac (1 4 ) is a quantum field theory contribution to the effective where k = 0, ±1 and R(t), a(t) are the scale factors cosmological constant of the universe and compact dimension, respectively. The curvature scalar corresponding to metric (1) is ob- Λ = λ + κρ , eff vac tained as where λ is a small bare cosmological constant. The R¨ k + R˙2 a¨ R˙ a˙ calculations show that, these contributions affect enor- R = 6 + + 2 + 6 , (2) R R2 a R a mously the value of effective cosmological constant ∼ 4 where a dot represents differentiation with respect to t. as ρvac MP, where MP is the Planck mass which defines the ultraviolet cutoff scale of the quantum Substituting this result into Einstein–Hilbert action gravity. This is the well-known cosmological constant with a cosmological constant Λ problem. However, there is another fundamental en- √ 3 ergy scale in nature, namely the electroweak scale I = −g(R − Λ) dt d rdρ, (3) MEW whose experimental investigation is of particular interest and the corresponding interaction has and integrating over spatial dimensions gives an effec- been probed successfully. Over the past two decades tive Lagrangian L in the mini-superspace (R, a)as there was a great interest to explain the smallness of 1 1 1 1 L = RaR˙2 + R2R˙a˙ − kRa + ΛR3a. (4) MEW/MP, which is known as the hierarchy problem. 2 2 2 6 Recently, a great amount of interest has been concen- 2 ≡−2Λ trated on solving this problem based on the existence By defining ω 3 and changing the variables as of large extra dimensions [9]. Some attempts have also 1 3k been done to solve this problem based on the noncom- u = √ R2 + Ra − , Λ mutativity in the space–time coordinates [10].Tothe 8 1 3k authors knowledge, this problem has not yet been paid v = √ R2 − Ra − , (5) attention by considering the noncommutativity in the 8 Λ mini-superspace coordinates of a quantum cosmology. L takes on the form In this Letter, we introduce a mechanism based 1 on the existence of noncommutativity in the mini- L = u˙2 − ω2u2 − v˙2 − ω2v2 . (6) superspace of a quantum cosmology corresponding to 2 an empty (4 + 1)-dimensional Kaluza–Klein cosmol- The assumption that the full (4 + 1)-dimensional Ein- ogy. It is to be noted that the present mechanism does stein equations hold, implies that the Hamiltonian corresponding to L in (6) must vanish, that is 1 1 This idea was followed by Hawking and Page investigating the H = u˙2 + ω2u2 − v˙2 + ω2v2 = 0, (7) wormholes [11]. 2 F. Darabi et al. / Physics Letters B 615 (2005) 141–145 143 which describes an isotropic oscillator–ghost–oscil- The above equation can be written in a more conve- lator system. nient form The corresponding quantum cosmology is de- 2 2 2 + 2 − B 2 − 2 + 2 − B 2 scribed by the Wheeler–DeWitt equation resulting pu ω u pv ω v from Hamiltonian (7) and can be written as 4 4 2 + 2 2 − 2 + 2 2 = + B(vp + up ) Ψ(u,v)= 0, (15) pu ω u pv ω v Ψ(u,v) 0. (8) u v

Now, we consider the effect of noncommutativity on 2 where B =− 4ω θ is derived through B = ∇ × A the configuration space in the above model. The non- 1−ω2θ2 commutative quantum mechanics is defined by the fol- from lowing commutators [3] B B A =− v, A = u. (16) u 2 v 2 [u, v]=iθ, [u, pu]=i, The B-term in Eq. (15) deserves more scrutiny so [ ]= [ ]= v,pv i, pu,pv 0, (9) that we can find the correct frequency for this system. where we use the natural units (h¯ = 1). The corre- To this end, it is useful to compare this oscillator– sponding noncommutative Wheeler–DeWitt equation ghost–oscillator system (15) with the corresponding can be written by use of the star product as [3,5] oscillator–oscillator system. By straightforward calculations we find for the latter system H ∗ Ψ = 0. (10) 1 We can always represent the noncommutative har- H = (p − A )2 + ω2x2 2 i i i monic oscillator in terms of anisotropic harmonic os- i=1,2 cillator in the commuting coordinates [3] 2 = 1 2 + 2 + B 2 pi ω xi [u, v]=0, [u, pu]=i, 2 4 i=1,2 [ ]= [ ]= v,pv i, pu,pv 0. (11) 1 − B(x p − x p ), (17) Therefore, we obtain anisotropic oscillator–ghost– 2 1 2 2 1 oscillator Wheeler–DeWitt equation where A = (−x ,x , 0) B .TheB-term in Eq. (17) is 2 1 2 2 2 2 pu 2 θpv 2 + B + ω u − the magnetic potential energy and (ω 4 ) is the 4 2 effective frequency of the system. It is obvious that p 2 θp 2 the B-term has an independent role and does not con- − v + ω2 v + u Ψ(u,v)= 0, (12) 4 2 tribute to the oscillator frequency, at all. Now, comparing the two systems we realize that each term in where the transformations u → u − θpv and v → + Eq. (17) has a corresponding term in Eq. (15).Forex- v θpu have been used. However, in two dimensions ample, the first two brackets in Eq. (17) correspond we can also represent the two-dimensional anisotropic to the first two brackets in Eq. (15), and the third oscillator–ghost–oscillator as the two-dimensional B-term in Eq. (17) corresponds to the third B-term isotropic oscillator–ghost–oscillator in the presence of in Eq. (15). Therefore, the corresponding terms have an effective magnetic field as follows the same roles with the difference that the terms in (p − A )2 + ω 2u2 Eq. (15) have the ghost characters. This occurs in other u u examples, as well. For instance, the two-dimensional − (p − A )2 + ω 2v2 Ψ(u,v)= 0, (13) v v oscillator with no external magnetic field is defined by where  2 2 2 2 2 2 H = p + p + ω x + y , (18)  ω 2 ≡ 4ω , x y  (1−ω2θ2)2  whereas the corresponding oscillator–ghost–oscillator ≡ 2ω2θ (14)  Au − 2 2 v, system is defined by  1 ω θ  2 A ≡− 2ω θ u. H = p2 − p2 + ω2 x2 − y2 . (19) v 1−ω2θ2 x y 144 F. Darabi et al. / Physics Letters B 615 (2005) 141–145

Here, the first parenthesis in both systems has the ki- cutoff in the original commutative model, the Planck netic term role and the second parenthesis plays the mass is then the cutoff in the noncommutative model. role of potential term, with the difference that the This solves the hierarchy problem at the level of quan- terms in Eq. (19) have the ghost characters. tum Wheeler–DeWitt equation by assuming that MEW Therefore, as the B-term in Eq. (17) is an inde- is the only fundamental mass scale in the model, and pendent potential term which does not contribute to MP is the mass scale which is appeared due to intro- 2 + B2 ducing the noncommutativity in the mini-superspace. the effective frequency (ω 4 ), the corresponding B-term in Eq. (15) should also be an independent Put another way, one may suppose that the universe, potential term which does not contribute to the effec- in principle, has just one fundamental energy scale for tive frequency, as well. all interactions, namely MEW. The quantum gravity In this regard, we can write sector of this universe must then be described by the Wheeler–DeWitt equation (8) with the vacuum energy 2 2 B 4ω density of the same scale ω2 =−2 Λ ∼ M4 .How- ω˜ 2 = ω 2 − = ,ω2θ 2 1, (20) 3 eff EW 4 (1 − ω2θ 2) ever, the energy scale of the quantum gravity which which defines the resultant frequency of the isotropic we expect to experience in the universe is defined by oscillator–ghost–oscillator in the presence of a con- the Planck mass MP and not MEW. This discrepancy stant magnetic field B. This frequency comes from a between MP and MEW, namely the hierarchy prob- ghost-like combination of the oscillator term ω 2 and lem is solved by the assumption of noncommutativity the cyclotron term B2/4. in the quantum gravity sector of the universe which In Eq. (8), the oscillator frequency ω was defined leads Eq. (8) to Eq. (15) with the vacuum energy den- 2 2 ˜ 4 =−3 2 sity of the Planck scale ω˜ =− Λeff ∼ M . It then by the effective cosmological constant Λeff 2 ω . 3 P Comparing with Eq. (15), we suppose ω˜ to be de- turns out that MP is not a new fundamental scale and ˜ fined by a new effective cosmological constant Λeff = its enormity MP MEW is simply a consequence of − 3 ˜ 2 the noncommutativity in the quantum gravity sector of 2 ω . Substituting these definitions into Eq. (20) we obtain the model, namely the Wheeler–DeWitt equation. It is worth noting that having the exact numeric co- ˜ 4Λeff Λeff = . (21) efficients in Eq. (22) is very important to get Eq. (23) ( − 2 θ 2|Λ |) 1 3 eff which solves the hierarchy problem. This in particu- This is a redefinition of the effective cosmological lar means that our proposed solution for the hierarchy constant due to the noncommutativity. However, we problem comes with a fine tunning. Although this fine know that the effective cosmological constant, in prin- tunning makes the proposed solution less appealing, ciple, is a measure of the ultraviolet cutoff in the the- however, it is still a novel method, as the hierarchy ory. Therefore, the above equation can also be consid- problem seems to be solved with tunning only one pa- ered as a redefinition of the cutoff in the theory due to rameter, θ. the noncommutativity. This is the main result which is considered to solve the hierarchy problem in the present model. To this end, we take Acknowledgements 3 M4 − 4M4 θ 2 = P EW , We would like to thank the referee very much for 4 4 (22) 2 MPMEW the useful and constructive comments. This work has been financially supported by the Research Depart- where MEW is the electroweak mass scale. Moreover, ∼ 4 ment of Azarbaijan University of Tarbiat Moallem, we take Λeff MEW representing MEW as the natural cutoff in the original commutative model. Therefore, Tabriz, Iran. we obtain from the above equation ˜ ∼ 4 References Λeff MP, (23) which defines the cutoff in the noncommutative model. [1] N. Seiberg, E. Witten, JHEP 9909 (1999) 032. In other words, if we assume MEW to be the natural [2] R. Szazbo, Phys. Rep. 378 (2003) 207. F. Darabi et al. / Physics Letters B 615 (2005) 141–145 145

[3] M. Chaichian, M.M. Sheikh-Jabbari, A. Tureanu, Eur. Phys. J. [6] G.D. Barbosa, N. Pinto-Neto, Phys. Rev. D 70 (2004) 103512. C 36 (2004) 251; [7] G.D. Barbosa, Phys. Rev. D 71 (2005) 063511. L. Mezincescu, hep-th/0007046; [8] F. Darabi, H.R. Sepangi, Class. Quantum Grav. 16 (1999) J. Gamboa, M. Loewe, J.C. Rojas, Phys. Rev. D 64 (2001) 1656. 067901; [9] N. Arkani-Hamed, S. Dimopoulos, G. Dvali, Phys. Lett. B 429 S. Bellucci, A. Nersessian, Phys. Lett. B 542 (2002) 295. (1998) 263; [4] J.M. Romero, J.A. Satiago, D. Vergara, Phys. Lett. A 310 L. Randall, R. Sundrum, Phys. Rev. Lett. 83 (1999) 3370. (2003) 9; [10] F. Lizzi, G. Mangano, G. Miele, Mod. Phys. Lett. A 16 (2001) A.E.F. Djemai, hep-th/0309034. 1; [5] H. García-Compeán, O. Obregón, C. Ramírez, Phys. Rev. X.-J. Wang, hep-th/0411248. Lett. 88 (2002) 161301. [11] S.W. Hawking, D.N. Page, Phys. Rev. D 42 (1990) 2655. Physics Letters B 615 (2005) 146–152 www.elsevier.com/locate/physletb

Large-scale inhomogeneities in modiﬁed Chaplygin gas cosmologies

Luis P. Chimento a, Ruth Lazkoz b

a Departamento de Física, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Ciudad Universitaria, Pabellón I, 1428 Buenos Aires, Argentina b Fisika Teorikoa, Zientzia eta Teknologia Fakultatea, Euskal Herriko Unibertsitatea, 644 Posta Kutxatila, 48080 Bilbao, Spain Received 15 December 2004; received in revised form 5 April 2005; accepted 13 April 2005 Available online 25 April 2005 Editor: M. Cveticˇ

Abstract We extend the homogeneous modiﬁed Chaplygin cosmologies to large-scale perturbations by formulating a Zeldovich-like approximation. We show that the model interpolates between an epoch with a soft equation of state and a de Sitter phase, and that in the intermediate regime its matter content is simply the sum of dust and a cosmological constant. We then study how the large-scale inhomogeneities evolve and compare the results with cold dark matter (CDM), CDM and generalized Chaplygin scenarios. We ﬁnd that unlike that like the latter, our models resemble CDM.  2005 Elsevier B.V. All rights reserved.

PACS: 98.80.Cq

1. Introduction ferent roles: dark matter would be responsible for matter clustering, whereas dark energy [2] would account According to increasing astrophysical indicia, the for accelerated expansion. Several candidates for dark evolution of the Universe seems to be largely gov- energy haven proposed and confronted with observa- erned by dark energy with negative pressure together tions: a purely cosmological constant, quintessence with pressureless cold dark matter (see [1] for the lat- with a single field (see [3] for earliest papers) or two k est review) in a two to one proportion. However, little coupled fields [4], -essence scalar fields, and phan- is know about the origin of either component, which in tom energy [5]. Interestingly, a bolder alternative pre- the standard cosmological model would play very dif- sented recently suggests that an effective dark energy- like equation of state could be due to averaged quantum effects [6]. E-mail addresses: [email protected] (L.P. Chimento), The lack of information regarding the provenance [email protected] (R. Lazkoz). of dark matter and dark energy allows for speculation

0370-2693/$ – see front matter  2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physletb.2005.04.029 L.P. Chimento, R. Lazkoz / Physics Letters B 615 (2005) 146–152 147 with the economical and aesthetic idea that a single arising from (3) in standard gravity, could alternatively component acted in fact as both dark matter and dark be obtained in the modified gravity picture for a pure energy. The unification of those two components has dust configuration under the modification risen a considerable theoretical interest, because on the 2 = + α/(α−1) one hand model building becomes considerable sim- 3H (A ρm) , (5) 3 pler, and on the other hand such unification implies the where ρm ∝ a . existence of an era during which the energy densities Modified Chaplygin cosmologies with α>1are of dark matter and dark energy are strikingly similar. transient models which interpolate between a ρ ∝ One possible way to achieve that unification is a−3α/(α−1) evolution law at early times and a de Sitter through a particular k-essence fluid, the generalized phase at late times, but interestingly the matter content Chaplygin gas [7], with the exotic equation of state at the intermediate stage is a mixture of dust and a cos- A mological constant. The sound speed for the modified p =− , (1) ρβ Chaplygin gas [10] becomes 1 − where constants β and A satisfy respectively 0 <β 1 c2 = 1 − Aρ (1 α)/α . (6) and A>0. Using the energy conservation equation s α − 1 2 and the Einstein equation 3H = ρ one obtains the The observational tests of traditional and gen- evolution eralized Chaplygin models are numerous. Several 1/(1+β) teams have analyzed the compatibility of those mod- 2 B 3H = A + , (2) els with the Cosmic Microwave Background Radiation a3(1+β) (CMBR) peak location and amplitude [15], super- =˙ where as usual a is the scale factor, H a/a and novae data [16] and gravitational lensing statistics B>0 is an integration constant. This model interpo- [17]. The main results of those papers can be summa- ∝ −3 lates between a ρ a evolution law at early times rized as follows: models with β>1 and some small and ρ const at late times (i.e., the model is domi- curvature (positive or negative) are favored over the nated by dust in its early stages and by vacuum energy CDM model, and Chaplygin cosmologies are much in its late history). In the intermediate regime the mat- likelier as dark energy models than as unified dark ter content of the model can be approximated by the matter models. sum of a cosmological constant an a fluid with a soft In what regards the modified Chaplygin gas, it has = equation of state p βρ. The traditional Chaplygin only been tested observationally in [18],usingthe = gas [8,9] corresponds to β 1 (stiff equation of state). most updated and reliable compilation of supernovae Another possibility which has emerged recently is data so far: the Gold dataset by Riess et al. [19].By the modified Chaplygin gas (MCG) [10]. It is charac- means of a statistical test which depends no only on terized by 2 χmin (as in usual procedures) bu also on the number of − B α/(α 1) parameters of the parametrization of the Hubble fac- ρ = A + , (3) tor as a function of redshift, it was concluded that the a3 modified Chaplygin gas cosmologies give better fits 1 p = ρ − αAρ1/α , (4) than usual and generalized Chaplygin cosmologies. α − 1 In this Letter we shall be concerned with the evolu- with α>0 a constant. tion of large-scale inhomogeneities in modified Chap- Alternatively, such evolution can be seen as com- lygin cosmologies. This is an issue of interest because ing from a modified gravity approach, along the lines candidates for the dark matter and dark matter unifica- of the Dvali–Gabadadze–Porrati [11], Cardassian [12] tion will only be valid if they ensure that initial pertur- and Dvali–Turner [13] models. In those works the bations can evolve into a deeply nonlinear regime to present acceleration of the universe is not attributed form a gravitational condensate of super-particles that to an exotic component in the Universe, but to modi- can act like cold dark matter. Here we will follow the fications in gravitational physics at subhorizon scales. covariant and sufficiently general Zeldovich-like non- Following the proposal by [14], an evolution like that perturbative approach given in [20], because it can be 148 L.P. Chimento, R. Lazkoz / Physics Letters B 615 (2005) 146–152 adapted to any balometric or parametric equation of The latter result suggests that the evolution of inhomo- state. Our results indicate that our model fits well in the geneities can be studied using the Zeldovich method standard structure formation scenarios, and we find, in through the deformation tensor [20,23,24]: general, a fairly similar behavior to generalized Chap- ∂2ϕ(q) lygin models [7]. Dj = a(t) δj − b(t) , (12) i i ∂qi∂qj where b(t) parametrizes the time evolution of the in- 2. The model homogeneities and q are generalized Lagrangian coordinates so that For the modified Chaplygin gas described by γ = δ DmDn, (13) Eqs. (3) and (4) the effective equation of state in the in- ij mn i j termediate regime between the dust dominated phase and h is a perturbation and the de Sitter phase can be obtained expanding = i Eqs. (4) and (3) in powers of Ba−3, we get h 2b(t)ϕ,i . (14) 2 Hence, using the equations above and Eqs. (7) and (8), − − αB B ρ = Aα/(α 1) + A1/(α 1) + O , (7) it follows that (α − 1)a3 a6 ¯ + 2 ρ ρ(1 δ), (15) − B p =−Aα/(α 1) + O , (8) 1 a6 p ρ¯ − Aαρ¯1/α + δ ρ¯ − Aρ¯1/α , (16) α − 1 which corresponds to a mixture of vacuum energy den- where ρ¯ is given by Eq. (3) and the density contrast δ α/(α−1) sity A , presureless dust and other perfect fluids is related to h through which dominate at the very beginning of the universe. h In the intermediate regime the modified Chaplygin gas δ = (1 + w), (17) behaves as dust at the time where the energy den- 2 sity satisfies the condition ρ = Aα/(α−1). At very early where w ≡¯p/ρ¯. Finally, after some algebra we get times the equation of state parameter w ≡ p/ρ be- (1 + w)δ comes p¯ =¯ρ w + . (18) α 1 w c2 , Now, the metric (13) leads to the following 00 com- s − (9) α 1 ponent of the Einstein equations: so that for very large α the dust-like behavior is recov- a¨ 1 ered. −3 + h¨ + H h˙ The next step is to investigate what sort of cos- a 2 3(1 + w) mological model arises when we consider a slight in- = 4πGρ¯ 1 + 3w + 1 + δ , (19) α homogeneous modified Chaplygin cosmologies.√ For a 0 general metric gµν , the proper time dτ = g00 dx , where the unperturbed part of this equation corre- and γ ≡−g/g00 as the determinant of the induced 3- sponds to the Raychaudhuri equation metric, one has a¨ −3 = 4πGρ(¯ 1 + 3w). (20) gi0gj0 a γij = − gij . (10) g00 Using the Friedmann equation for a flat spacetime In the first approximation it will be interesting to inves- H 2 = 8πGρ/¯ 3, one can rewrite Eq. (19) as a differen- tigate the contribution of inhomogeneities introduced tial equation for b(a): in the modified Chaplygin gas through the expression 2 a2b + (1 − w)ab − B α/(α 1) 3 ρ = A + √ . (11) 3(1 + w) γ − (1 + w) 1 + b = 0, (21) α L.P. Chimento, R. Lazkoz / Physics Letters B 615 (2005) 146–152 149

Fig. 1. Evolution of b(a)/b(a ) for the modified Chaplygin gas eq Fig. 2. Evolution of b(a)/b(a ) for the modified Chaplygin gas for α = 10, 11, 12, 12, 15, 20 (continuous lines) as compared with eq for α = 60, 80, 140 (continuous lines) as compared with CDM CDM (dashed line) and CDM (dashed-dotted line). Lower curves (dashed line) and CDM (dashed-dotted line). Lower curves corre- correspond to higher values of α. spond to higher values of α. where the primes denote derivatives with respect to the scale-factor, a. We find that modified Chaplygin scenarios start dif- An expression for w as a function of the scale- fering from the CDM only recently (z 1) and that, factor can be derived from Eqs. (3) and (4): in any case, they yield a density contrast that closely resembles, for any value of α>1, the standard CDM B − (α − 1)Aa3 w(a) = . (22) before the present. Notice that CDM corresponds ef- − + 3 (α 1)(B Aa ) fectively to using Eq. (23) and removing the factor The latter must be conveniently recast in terms of the (1 + 3(1 + w)/α) in Eq. (21). Figs. 1 and 2 show also fractional vacuum and matter energy densities. This that, for any value of α, b(a) saturates as in the CDM can be done by using case. B In what regards the density contrast, δ,using lim ρ = A + , (23) α→∞ a3 Eqs. (14), (17) and (25) one can deduce that the ra- combined with tio between this quantity in the modified Chaplygin and the CDM scenarios is simply given by 3 2 = 2 a0 + H H0 Ωm0 ΩΛ0 , (24) a δmChap bmChap α = , − (27) where H0 and a0 are, respectively, the current value δCDM bCDM α 1 of the Hubble and scale factor, and ΩΛ0 and Ωm0 are, and its behavior is depicted in Fig. 3. We find that it respectively, the fractional vacuum and matter energy asymptotically evolves to a constant value. = densities today. Setting a0 1 we obtain Now, in Fig. 4, we have plotted δ as a function of Ω − (α − 1)Ω a3 a for different values of α. As happens in the tradi- w(a) = m0 Λ0 , 3 (25) tional [20,26] and generalized Chaplygin models, in (α − 1)(Ωm0 + ΩΛ0a ) our models the density contrast decays for large a also. and consistently Ω a3 lim w(a) =− Λ0 . (26) →∞ 3 α Ωm0 + ΩΛ0a 3. Discussion and conclusions We have used this expression to integrate Eq. (21) numerically, for different values of α, and taking Using a Zeldovich-like approximation, we have Ωm = 0.27 and ΩΛ = 0.73 [25].Wehavesetaeq = studied the evolution of large-scale perturbations in a 10−4 for matter-radiation equilibrium (while keep- recently proposed theoretical framework for the uni- ing a0 = 1 at present), and our initial condition is fication of dark matter and dark energy: the so-called b (aeq) = 0. Our results are shown in Figs. 1 and 2. modified Chaplygin cosmologies [10], with equation 150 L.P. Chimento, R. Lazkoz / Physics Letters B 615 (2005) 146–152

models and in generalized Chaplygin cosmologies, and therefore our represent plausible alternatives. As usual, in modified Chaplygin cosmologies, the equation of state parameter w can be expressed in terms of the scale factor and a free parameter α, and the value of the latter can be chosen so that the model resembles as much as desired the CDM model. It would be very interesting to deepen in the comparison between modified and generalized Chaply- gin models, particularly from the observational point of view (as already done in [18]). It would also be Fig. 3. Evolution of δmChap/δCDM for the modified Chaplygin gas worth generalizing our study by going beyond the = for α 60, 65, 70, 75, 80, 85, 90, 95 (continuous lines) as compared Zeldovich approximation, to incorporate the effects of with CDM (dashed line). Lower curves correspond to higher values of α. finite sound speed. This can be done by generalizing the spherical model to incorporate the Jeans length as in [21]. Alternatively, following [22] one could investigate whether the modified Chaplygin admits an unique decomposition into dark energy and dark matter, and if that were the case then study structure formation and show that difficulties associated to unphysical oscillations or blow-up in the matter power spectrum can be circumvented. We hope this will be addressed in future works.

Acknowledgements

L.P.C. is partially funded by the University of Fig. 4. Evolution of δmChap for the modiﬁed Chaplygin gas for α = 50, 60, 70, 85, 105, 150 (continuous lines) as compared with Buenos Aires under project X224, and the Consejo CDM (dashed line). Lower curves correspond to higher values Nacional de Investigaciones Cientíﬁcas y Técnicas of α. under proyect 02205. R.L. is supported by the Uni- versity of the Basque Country through research grant of state UPV00172.310-14456/2002 and by the Spanish Min- istry of Education and Culture through research grant 1 FIS2004-01626. p = ρ − αAρ1/α , α − 1 with α>1. This model evolves from a phase that References is initially dominated by nonrelativistic matter to a phase that is asymptotically de Sitter. The intermedi- [1] V. Sahni, astro-ph/0403324. ate regime corresponds to a phase where the effective [2] P.J.E. Peebles, B. Ratra, Rev. Mod. Phys. 75 (2003) 559. = [3] B. Ratra, P.J.E. Peebles, Phys. Rev. D 37 (1988) 3406; equation of state is given by p 0 plus a cosmological B. Ratra, P.J.E. Peebles, Astrophys. J. Lett. 325 (1988) L17; constant. We have estimated the fate of the inhomo- C. Wetterich, Nucl. Phys. B 302 (1988) 668; geneities admitted in the model and shown that these J.A. Frieman, C.T. Hill, A. Stebbins, I. Waga, Phys. Rev. evolve consistently with the observations as the den- Lett. 75 (1995) 2077; sity contrast they introduce is smaller than the one K. Coble, S. Dodelson, J.A. Frieman, Phys. Rev. D 55 (1997) 1851; typical of CDM scenarios. M.S. Turner, M.J. White, Phys. Rev. D 56 (1997) 4439; On general grounds, the pattern of evolution of per- P.G. Ferreira, M. Joyce, Phys. Rev. Lett. 79 (1997) 4740; turbations follows is similar to the one in the CDM P.G. Ferreira, M. Joyce, Phys. Rev. D 58 (1998) 023503; L.P. Chimento, R. Lazkoz / Physics Letters B 615 (2005) 146–152 151

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Final results on the neutrino magnetic moment from the MUNU experiment

MUNU Collaboration Z. Daraktchieva a,C.Amslerb,M.Avenierc, C. Broggini d,J.Bustoa,C.Cernad, F. Juget a, D.H. Koang c,J.Lamblinc,D.Lebrunc, O. Link b, G. Puglierin d,A.Stutzc, A. Tadsen d, J.-L. Vuilleumier a, V. Zacek e

a Institut de physique, A.-L. Breguet 1, CH-2000 Neuchâtel, Switzerland b Physik-Institut, Winterthurerstr. 190, CH-8057 Zurich, Switzerland c Laboratoire de Physique Subatomique et de Cosmologie, IN2P3/CNRS-UJF, 53 Avenue des Martyrs, F-38026 Grenoble, France d INFN, Via Marzolo 8, I-35131 Padova, Italy e Université de Montréal, C.P. 6128, Montréal, PQ, H3C 3J7 Canada Received 19 February 2005; received in revised form 31 March 2005; accepted 5 April 2005 Available online 25 April 2005 Editor: L. Rolandi

Abstract − The MUNU detector was designed to study ν¯ee elastic scattering at low energy. The central component is a Time Projection Chamber ﬁlled with CF4 gas, surrounded by an anti-Compton detector. The experiment was carried out at the Bugey (France) nuclear reactor. In this Letter we present the ﬁnal analysis of the data recorded at 3 bar and 1 bar pressure. Both the energy and the scattering angle of the recoil electron are measured. From the 3 bar data a new upper limit on the neutrino magnetic moment short × −11 µe < 9 10 µB at 90% CL was derived. At 1 bar electron tracks down to 150 keV were reconstructed, demonstrating the potentiality of the experimental technique for future applications in low energy neutrino physics.  2005 Elsevier B.V. All rights reserved.

1. Introduction the existence of a magnetic moment of the electron antineutrino. The detector is located at 18 m from The MUNU experiment was designed to study the core of a 2.75 GWth commercial nuclear reactor ¯ − νee elastic scattering at low energy and to probe in Bugey (France). The central component is a CF4 gas Time Projection Chamber (TPC). In [1] we presented an analysis of 66.6 days live time of reactor-on E-mail address: [email protected] (J.-L. Vuilleumier). data, as well as 16.7 days of reactor-off data, taken

0370-2693/$ – see front matter  2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physletb.2005.04.030 154 MUNU Collaboration / Physics Letters B 615 (2005) 153–159 at a pressure of 3 bar. In this Letter we present the ﬁnal analysis, using the same data set, and the same event selection, but taking better advantage of the electron kinematics, extending the area in which the background is measured and achieving a more precise determination. Moreover we present the analysis of 5.3 days live time of reactor-on data taken at 1 bar pressure. Technical details of the MUNU detector have already been presented in Refs. [2,3]. Here we only describe the most essential features.

2. The experiment Fig. 1. The MUNU detector at the Bugey reactor.

The central part of the detector is a cylindrical TPC The acrylic vessel is immersed in a steel tank (2 m 3 filled with CF4 gas. Measurements were performed at diameter and 3.8 m long), filled with 10 m of liq- a pressure of 3 bar (11.4 kg of CF4) and 1 bar (3.8 kg). uid scintillator (NE235) and viewed by 48 photomul- The gas serves as target and detector medium for the tipliers (PMT). The liquid scintillator acts as an anti- recoil electrons. CF4 was chosen because of its high Compton detector and as a veto against cosmic muons. density, low atomic number, which reduces multiple The anti-Compton detector also sees the primary scin- scattering, and its absence of free protons, which elim- tillation light of heavily ionizing particles such as α’s, + inates the background from ν¯ep → e n.Asshownin and the secondary light emitted during the amplifica- Fig. 1 the gas is contained in a 1 m3 acrylic vessel tion process around the anode wires, which provides a 90 cm in diameter and 160 cm long. The drift volume second measurement of the total deposited energy. is defined on one end by a plain cathode and the other In addition, the detector is shielded against local one by a grid made from wires. An anode plane made activities by 8 cm of boron loaded polyethylene and of 20 µm wires with a pitch of 4.95 mm, separated 15 cm of Pb. The concrete and steel overburden of the by 100 µm potential wires, is placed behind the grid laboratory corresponds to the equivalent of 20 m of to collect and amplify the ionization charge. The inte- water. grated anode signal gives the total deposited energy. Events in the TPC not in coincidence with a signal An x–y read-out plane is located behind the anode above 22 MeV in the scintillator within an 80 µs time plane. It contains x strips on one side, and perpendic- window are recorded. The selection of good events ular y strips on the other one. The pitch is 3.5 mm. off-line proceeds in two steps. First a software filter The signals induced in these strips provide the spatial rejects the muon related events, Compton electrons, information in the x and y directions. The third projec- discharges and uncontained events. The selection is tion z is obtained from the time evolution of the signal. finalized in a visual scan. A neutrino scattering can- The drift field was selected such as to achieve a drift didate event is a continuous single electron track fully velocity of 2.15 cm/µs. contained in a 42 cm fiducial radius, with no energy It must be noted that the detector is installed under deposition above 90 keV in the anti-Compton in the the reactor core, at an angle of 45◦. The detector axis is preceding 200 µs. The initial direction of the electron perpendicular to the reactor core-detector axis. The an- track is obtained from a visual fit [1]. The scattering ode wires are parallel to the reactor core-detector axis, angles in the x–z and y–z projections are determined and the grid wires perpendicular to it. The x strips are first, and used to calculate the scattering angle θreac vertical and the y strips horizontal. The read-out plane with respect to the reactor-detector axis, which coin- is therefore by construction symmetric with respect to cides quite precisely with the scattering angle, as well four directions: reactor core to detector and opposite, as the angle θdet with respect to the detector axis (see as well as the two orthogonal directions. Fig. 2). The angle ϕdet between the projection of the MUNU Collaboration / Physics Letters B 615 (2005) 153–159 155

lect the good candidate events. For each electron track the neutrino energy Eν is reconstructed from the scattering angle, taken as θreac, and the measured electron recoil energy Te. Events with Eν > 0 are declared forward events since, effectively, this criteria selects electrons with an initial track direction within a forward cone, the axis of which coincides with the reactor core-detector axis. The opening angle depends on the energy, and is larger than that of the kinematic cone for recoil electrons (Eν >Te). Simulations taking into account the angular response of the detector show that nearly 100% of the recoil electrons fall in the forward category, which however also contains a contribution Fig. 2. The four kinematical cones. from the background, which is isotropic around the detector axis. initial track direction on the x–y plane and the vertical To estimate the background the same procedure is y axis is also determined. applied in the three directions equivalent, considering As described in more details in [1] we apply the an- the read-out plane, to the reactor core-detector direc- ◦ gular cut θdet < 90 to suppress the background from tion, taken as reference for the forward events. These activities on the read-out plane side of the TPC, which directions are thus also equivalent from the point of was found to be noisier. This is presumably due to the view of the angular response in ϕdet, which is not com- greater complexity of the anode side and the larger in- pletely linear, and of the acceptance. As depicted in active volume in the scintillator. Fig. 2 these directions define the backward cone, op- The reactor neutrino spectrum, necessary to inter- posite to the forward cone, and the two perpendicular pret the data, was calculated using the formalism de- upward and downward cones [8]. To avoid overlap of 2 scribed in [1,4]. Above 1.5 to 2 MeV neutrino spectra the cones, which can occur for Te < 2mec , we require ◦ reconstructed from the measured β spectra of the fis- in addition that the angle ϕdet is within less than 45 sion fragments were used. The uncertainty is of order with respect to the cone axis. This only reduces the ac- 5% or less in this energy range, in which the neutrino ceptance of the forward cone for recoil electrons in a spectrum was moreover thorougly probed in measure- negligible way. + ments of ν¯ep → e n scattering at reactors. At lower While the forward electrons contain recoil plus energies calculations only are available. We have used background events, the backward, upward and down- the calculated neutrino spectra of the fission fragments ward electrons are background events only. The en- from [5,6], and taken into account the neutron activa- ergy distributions of the upward, downward and back- tion of 238U, as discussed in [7], which is significant ward electrons (1154 ± 34 in total) are presented in below 1 MeV. The uncertainty in the neutrino spec- Fig. 3. The distributions are compatible within the er- trum in this energy range was estimated to 20%. rors, which confirms the isotropy of the background inside the TPC. We normalize the background to the forward cone 3. 3-bar forward-normalized background analysis by dividing by 3 the rates in the backward, upward and downward cones. This normalized background (NB) is The 3 bar data were taken with a TPC trigger then directly compared with the event rate in the for- threshold of 300 keV. Considering the entire event se- ward cone. The energy distributions of both forward lection procedure, the live time, limited primarily by (455 ± 21) and NB (384 ± 11) electrons are shown the data transfer rate and the anti-Compton, was 65%. in Fig. 3. There is a clear excess of forward events, At 3 bar the tracks are long enough to be scanned from the reactor direction. The total number of events with sufficient efficiency for electron kinetic energies forward minus NB above 700 keV is 71 ± 23 counts Te > 700 keV. As before we rely on kinematics to se- for 66.6 days live time reactor-on, corresponding to 156 MUNU Collaboration / Physics Letters B 615 (2005) 153–159

1.07 ± 0.34 counts per day (cpd). The forward minus NB spectrum representing the measured electron recoil spectrum is displayed in Fig. 4. We make the same analysis with the data taken during the reactor-off period as a cross check (16.7 days live time). The energy distributions of both forward (133 ± 11) and NB electrons (147 ± 7) are given in Fig. 5. The integrated forward minus NB rate above 700 keV is −0.8 ± 0.8 cpd, consistent with zero. We now turn to the comparison with expectations. Monte Carlo simulations (GEANT 3) were used to calculate the various acceptances of the event selection procedure. The detector containment efficiency in the 42 cm fiducial radius was found to vary from 63% at 700 keV, 50% at 1 MeV to 12% at 2 MeV, with a typical error of 2%. The errors on the global acceptance, including track reconstruction efficiency (4%), are of order of 7%. This leads to an expected event rate above 700 keV of 1.02 ± 0.10 cpd assuming a vanishing magnetic moment. This is in good agreement with the observed 1.07 ± 0.34 cpd. The measured and expected recoil spectra for no magnetic moment are compared in Fig. 4.Firstwe note that the large excess of events in the first two bins (700–900 keV) observed in our previous analysis [1] has to a large extend disappeared. It is thus most likely due to a statistical fluctuation in the background, more Fig. 3. 3 bar data, reactor-on, energy spectra; top: upward (dashed precisely determined in the present analysis. And sec- line), downward (solid line) and backward (dotted line) electrons; ond the measured and expected spectra are seen to be bottom: forward (solid line) and NB (dashed line) electrons. in agreement. For a more quantitative analysis the χ2 method was used as in [1], with the same binning for Gaussian statistics to apply (100 keV bins from 700 keV to 1400 keV, and then a bin from 1400 to 2000 keV). The magnetic moment is varied to find the best fit. We remind that the neutrinos travel over a short distance only so that the experiment probes the mag- short netic moment µe , as described in Ref. [1],tobe precise its square. The allowed range at the 90% CL short 2 = − ± × −20 2 2 = is (µe ) ( 0.72 1.25) 10 µB , with χ 11.5 for 7 degrees of freedom at the central value. This result is compatible with a vanishing magnetic mo- short ment. To obtain a limit on µe we renormalize to (µshort)2 > µshort < Fig. 4. Energy distribution of the forward minus NB electrons above the physical region ( e 0) and obtain e × −11 700 keV, 3 bar, reactor-on; comparison with the expected spectrum 9(7) 10 µB at 90(68)% CL. This constitutes for weak interaction alone (dashed line) and for a magnetic moment a small improvement over our previous analysis [1], × −11 of 9 10 µB (dotted line). restricted to recoil energies above 900 keV. The im- MUNU Collaboration / Physics Letters B 615 (2005) 153–159 157 provement results from the lower threshold and the 4. 1-bar forward-normalized background analysis better background estimation. We can compare this with results from other exper- The technical details of the 1 bar measurements iments. The TEXONO collaboration is performing an will be presented elsewhere. Here we restrict ourselves experiment near the Kuo-Sheng reactor in Taiwan [9]. to the most relevant parameters. The trigger threshold Thanks to the use of an Ultra Low Background High on the electron recoil was lowered to 100 keV. The Purity Germanium detector they achieve a very low trigger rate is 0.4 s−1. The deadtime, here also mostly threshold of 12 keV. The reactor-on minus reactor-off due to the data read-out and transfer time, and to the rates were found to be identical and from that the limit anti-Compton, is around 50%, somewhat less than for short × −10 µe < 1.3 10 µB at 90% CL was deduced. the measurements at 3 bar. The energy calibration of sol 137 54 Super-Kamiokande [5,10] reported the limit µe < the TPC is again determined with Cs and Mn ra- −10 1.5×10 µB at 90% CL from the study of the shape dioactive sources. The energy resolution is found to of the recoil spectrum of electrons in the scattering of vary from 10% (1σ ) at 200 keV to 6% at 600 keV fol- solar neutrinos. Depending on oscillation parameters, lowing an empirical E0.57 law. Data were accumulated short this quantity may however differ from µe . during 5.3 days live time reactor-on. The selection of good events proceeds in two steps, as with the 3 bar data, ending with a visual scan. Tracks can be reconstructed with reasonable efficiency for energies above 150 keV. Events above that energy are retained. As an example a 190 keV electron track in 1 bar of CF4 is shown in Fig. 6. The end of the track is easily identified from the increased energy deposition (blob). The containment efficiency of the TPC in the 42 cm fiducial radius for recoil electrons was calculated with the GEANT3 simulation code, and found to vary from 85% at 200 keV, 50% at 400 keV to 5%

Fig. 5. 3 bar data, reactor-off, energy spectra; top: upward (dotted line), downward (solid line) and backward (dashed line) electrons; Fig. 6. A 190 keV electron at 1 bar in the TPC; the xz, yz projections bottom: forward (solid line) and NB (dashed line) electrons. are displayed as well as the anode signal. 158 MUNU Collaboration / Physics Letters B 615 (2005) 153–159

at 1 MeV. The angular resolution was determined by scanning visually simulated electron tracks. It varies from 15◦ (1σ ) at 200 keV,12◦ at 400 keV,to 10◦ above 500 keV. ◦ The angular cut θdet < 90 is also applied in the 1 bar analysis. The electrons are selected in the same four kinematical cones. Here however the overlap of the cones is more critical. The constraint in ϕdet is done somewhat differently: downward electrons 190◦ to 270◦, forward electrons 80◦ to 190◦, upward electrons 0◦ to 80◦ and backward electrons 270◦ to 0◦. This makes the solid angle for the forward electrons somewhat larger. We expect from the Monte Carlo simulations that nearly 100% of the recoil events above 200 keV will fall in it. The normalized background NB in the three other cones has to be scaled accordingly, with a factor which remains close to 3 however. The energy distributions of the upward, downward and backward electrons (326 ± 18 in total) are presented in Fig. 7. They are seen to be compatible within the errors. The energy distributions of the forward (124 ± 11) and NB (109±6) electrons are displayed in Fig. 7.The difference is 15 ± 12 events, giving an indication of a signal, corresponding to 2.89 ± 2.39 counts per day. The energy distribution is shown in Fig. 8. This measured total rate above 200 keV agrees with the expected value 0.49 ± 0.12 counts per day, assuming Fig. 7. 1 bar data, reactor-on, energy spectra; top: upward (dotted no magnetic moment. The error comes mainly from line), downward (solid line) and backward (dashed line) electrons; the uncertainties in the low energy part of the neutrino bottom: forward (solid line) and NB (dashed line) electrons. spectrum, as described above. Due to the limited statistics the data do not have a competitive sensitivity to the neutrino magnetic moment. Nevertheless our experiment shows that a gas TPC can be used to measure the energy and direction of electrons with energies as low as 150 keV. Provided background problems can be solved, this opens the way, for instance, to online measurements of low energy solar neutrinos from the pp reaction.

5. Conclusions

− The MUNU experiment studied ν¯ee scattering at low energy near a nuclear reactor, measuring both Fig. 8. Energy distribution of the forward minus NB electrons above the energy and scattering angle of the recoil electron. 200 keV, 1 bar, reactor-on; comparison with the expected spectrum Thanks to a better estimation of the background in a for weak interaction alone (dashed line). larger kinematical domain it was possible to reduce MUNU Collaboration / Physics Letters B 615 (2005) 153–159 159 the statistical uncertainties, analyzing data taken at 3 References bar of CF4 during 66.6 days live time reactor-on with an energy threshold of 700 keV. Good agreement is [1] MUNU Collaboration, Z. Daraktchieva, et al., Phys. Lett. observed between the measured and expected recoil B 564 (2003) 190. [2] MUNU Collaboration, C. Amsler, et al., Nucl. Instrum. Meth- spectra assuming weak interaction alone. From this ods A 396 (1997) 115. short the limit on the neutrino magnetic moment µe < [3] MUNU Collaboration, M. Avenier, et al., Nucl. Instrum. Meth- −11 9 × 10 µB at 90% CL was derived. Moreover ods A 482 (2002) 408. reactor-on data were taken at 1 bar of CF4 during 5.3 [4] G. Zacek, et al., Phys. Rev. D 34 (1986) 2621. days live time. Electron tracks were reconstructed ef- [5] J.F. Beacom, P. Vogel, Phys. Rev. Lett. 83 (1999) 5222. [6] P. Vogel, J. Engel, Phys. Rev. D 39 (1989) 3378. ﬁciently down to 150 keV recoil energy. This demon- [7] V.I. Kopeikin, et al., Phys. At. Nucl. 60 (1997) 172. strates the potentiality of gas TPC’s for possible future [8] Z. Daraktchieva, Thesis, University of Neuchatel, 2004. applications in low energy neutrino physics. [9] TEXONO Collaboration, H.B. Li, et al., Phys. Rev. Lett. 90 (2003) 131802. [10] Super-Kamiokande Collaboration, Y. Fukuda, et al., Phys. Rev. Acknowledgements Lett. 82 (1999) 2430.

The authors want to thank the directors and the staff of EDF-CNPE Bugey for the hospitality. This work was supported by IN2P3, INFN, and SNF. Physics Letters B 615 (2005) 160–166 www.elsevier.com/locate/physletb

Is the K-quantum number conserved in the order-to-chaos transition region? G. Benzoni a, A. Bracco a, S. Leoni a,N.Blasia,F.Cameraa, C. Grassi a, B. Million a, A. Paleni a, M. Pignanelli a, E. Vigezzi a,O.Wielanda,M.Matsuob, T. Døssing c, B. Herskind c, G.B. Hagemann c,J.Wilsonc,A.Majd, M. Kmiecik d,G.LoBiancoe,f, C.M. Petrache e,f, M. Castoldi g, A. Zucchiati g,G.DeAngelish, D. Napoli h, P. Bednarczyk d,i,D.Curieni

a Dipartimento di Fisica, Università di Milano and INFN Sezione di Milano, Via Celoria 16, 20133 Milano, Italy b Graduate School of Science and Technology, Niigata University, Niigata 950-2181, Japan c The Niels Bohr Institute, Blegdamsvej 15-17, 2100 Copenhagen, Denmark d The Niewodniczanski Institute of Nuclear Physics, Polish Academy of Science, 31-342 Krakow, Poland e Dipartimento di Fisica, Università di Camerino, Via Madonna delle Carceri 9, 62032 Camerino, Italy f INFN Sezione di Perugia, Perugia, Italy g INFN Sezione di Genova, Genova, Italy h Laboratori Nazionali di Legnaro, Viale dell’Università 2, 35020 Legnaro (PD), Italy i Institut de Recherches Subatomic, F-67037 Strasbourg cedex 2, France Received 29 June 2004; received in revised form 25 October 2004; accepted 9 December 2004 Available online 12 April 2005 Editor: V. Metag

Abstract To study the order-to-chaos transition in nuclei we investigate the validity of the K-quantum number in the excited rapidly rotating 163Er nucleus, analyzing the variance and covariance of the spectrum ﬂuctuations of γ -cascades feeding into low-K and high-K bands. The data are compared to simulated spectra obtained using a microscopic cranked shell model. K-selection rules are found to be obeyed for decay along excited unresolved rotational bands of internal excitation energy up to around 1.2 MeV and angular momenta 20h¯ I 40h¯. At higher internal energy, from about 1.2 to 2.5 MeV, the selection rules are found to be only partially valid.  2005 Elsevier B.V. All rights reserved.

PACS: 21.10.Re; 21.60.-n; 23.20.Lv; 24.60.Lz; 25.70.Gh; 27.70.+q

Keywords: K-quantum number; Compound states; Excited rotating nuclei; Residual interaction; Quasi-continuum gamma spectra

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

The conditions under which K, the projection of rations. This differs from the case of low-spin neutron aligned nucleonic angular momentum on the symme- resonances, for which a Wigner level spacing distribu- try axis in deformed nuclei, is a good quantum number tion has been experimentally observed [8], and from remain a topic of much current interest, as testified by the case of levels near yrast, which were found to the extensive experimental work on high-K isomers follow the Poisson distribution typical of ordered sys- [1]. The study of nuclear states with high values of tems [9]. the K-quantum number is interesting not only from The present work is based on new data on the the point of view of the decay-out from such states but 163Er nucleus, which was already studied in a previous also in connection with their feeding, which allows to experiment [10]. Two novelties are presented. First, investigate the validity of the associated selection rules a more detailed experimental investigation is carried at higher internal excitation energies. As it was stated out for the highest region of internal energy. Secondly, by Mottelson [2], the question of K-quantum number and most important, a direct and rather realistic com- violation in excited states is a key issue in the study parison between experiment and theory is made for of the transition between ordered and chaotic motion the first time. This comparison is based on simulated in nuclei caused by the residual interaction and the spectra constructed using recent calculations on this high level density. In fact, as pointed out in Ref. [2], specific nucleus [11]. for a nuclear ordered system a complete set of sin- The experiment was carried out using the EURO- gle particle quantum numbers can be defined for each BALL array at the IReS Laboratory (France), employ- 18 150 given state, resulting in selection rules on the associ- ing the reaction O + Nd, at Ebeam = 87, 93 MeV. ated electromagnetic transitions. On the contrary, in The 150Nd target was made of a stack of two thin a chaotic regime, due to the complex nature of every foils for a total thickness of 740 µg/cm2. The cor- state, no precise definition of quantum numbers (be- responding maximum angular momentum reached in sides energy, spin and parity) is possible and selection the reaction has been calculated to be 40h¯ and 45h¯ rules lose their validity. at the two bombarding energies, respectively. Energy- The issue of the validity of the K-quantum num- dependent time gates on the Ge time signals were ber has been previously addressed by studying the used to suppress background from neutrons. A total of γ -decay from neutron resonances at excitation energy ≈ 3 × 109 events of triple and higher Ge-folds were of ≈ 8MeV[3,4]. In this Letter, we will focus on finally obtained, with 162,163Er as main evaporation γ -transitions of quasi-continuum nature emitted by residua. The data have been sorted into a number of nuclei formed in fusion reactions, which are populat- two-dimensional (2D) matrices in coincidence with ing states at high spins (up to I ≈ 40h¯ ) and are probing specific γ -transitions of 163Er [12]. First, a matrix col- the energy region extending up to excitation energy lecting the entire decay flow of 163Er (named total) has above yrast U ≈ 4MeV[5], with particular emphasis been constructed by gating on the two cleanest low on the lower energy interval up to U ≈ 2.5MeV. spin transitions. In addition, seven matrices gated by From theoretical investigations one expects that for transitions belonging to the low-K(K= 5/2) signa- U ≈ 1–3 MeV a transition between order and chaos ture and parity configurations labeled A = (1/2, +), takes place. This has been studied, as usual, in terms B = (−1/2, +), E = (1/2, −) and F = (−1/2, −), of statistical fluctuations of the energy levels. It was and by the high-K(K= 19/2) bands labeled K1 (neg- found that, as the intrinsic excitation energy U in- ative parity) and K2 and K4 (positive parity) [12] have creases, the level statistics shows a gradual transition been sorted together with their corresponding two- from order-to-chaos, reaching at U ≈ 2.5MeVthe dimensional backgrounds. In addition, in order to per- Wigner distribution typical of the Gaussian orthogonal form a proper analysis of the 2D spectra in terms of ensemble of random matrices [6,7]. Our investigation statistical fluctuations [13], all known peak–peak and of selection rules can provide an experimental probe peak–background coincidences have been subtracted of this theoretical prediction. In fact, an experimen- from each 2D spectrum using the Radware software tal investigation based on level statistics is extremely [14]. Finally, the separately gated matrices have also hard at these excitation energies and spins, due to the been summed into one low-K(A+ B + E + F) and rapidly increasing level density of interacting configu- one high-K(K1 + K2 + K4) matrix. Fig. 1 (left col- 162 G. Benzoni et al. / Physics Letters B 615 (2005) 160–166

region of strongly interacting bands (rotational damping regime). The asymmetry observed in the intensity of the spectra is due to the discrete line subtraction, which is here made only for Eγ1 Eγ2 . Two experimental observables are used to determine the validity of the K-quantum number at increas- (2) ing internal energy, namely, the number Npath of decay paths measured in coincidence with low-K/high-K discrete bands and the r correlation coefficient [13, (2) 15]. While Npath is related both to the level density and to the rotational damping width, r measures the similarity of the cascades recorded in coincidence with low-K or high-K bands. For example, a large value of r (≈ 1) indicates that many decay paths can lead both to low-K and high-K configurations, thus sug- (2) gesting a weakening of the selection rules. Both Npath and r can be obtained by a fluctuation analysis of γ –γ coincidence spectra. In the present case, the fluctuations of counts in each channel of the 2D matrices, expressed as variance and covariance, are evaluated by = the program STATFIT [13] and stored into 2D spectra. Fig. 1. 60 keV wide projections perpendicular to the Eγ1 Eγ2 diagonal of experimental and simulated 2D spectra of 163Er (left Because each rotational Eγ -cascade on the average and right panels, respectively), at the average transition energy 4h¯ 2 contributes one count in each (2) interval, the statisti- Eγ =900 keV. The spectra collect either the total γ -decay flow J ¯ 2 ¯ 2 (panels (a) and (d)) or the γ -decay in coincidence with low-K (pan- cal moments are evaluated over sectors of 4h × 4h , J (2) J (2) els (b) and (e)) or high-K (panels (c) and (f)) specific configurations. corresponding to 60 × 60 keV intervals for rare earth In the simulation, a state is defined as low-K (high-K)ifK 8 nuclei around 163Er. (K > 8). The ridge-valley structure typical of rotational nuclei is seen in all spectra, with a separation between the two most inner From the fluctuation spectra we first extract the ef- ridges equal to 8h¯ 2/J (2), as indicated by the arrows. The reduced fective number of decay paths, which eventually feed intensity observed in the Eγ1 Eγ2 region of the spectra is due to into the gate-selected band. The number of decay paths the subtraction of all discrete lines known from the level scheme, in (2) Npath having two γ -transitions with energies lying in the case of the experimental data, and of the yrast and first excited × bands, in the case of the simulation. a chosen 60 60 keV window in the γ –γ coincidence spectrum is obtained from the simple expression

(2) = N × (2) umn) shows examples of cuts perpendicular to the Npath P , (1) = µ2/µ1 − 1 Eγ1 Eγ2 diagonal, 60 keV wide, in the total, low- K and high-Kγ–γ matrices, at the average transition where N is the number of events, while µ1 and µ2 + = energy (Eγ1 Eγ2 )/2 900 keV. As one can see, are the first and second moments of the distribution of the spectra show the ridge-valley structure typical of counts. The superscript (2) indicates that the extrac- quasi-continuum spectra of rotational nuclei, with a tion of the number of paths is based on first and second separation between the two most inner ridges equal moments, while the P (2) factor corrects for the finite to 8h¯ 2/J (2), J (2) being the dynamical moment of in- resolution of the detector system [13]. ertia of the bands [13]. In particular, while the rather The number of paths obtained from the analysis of sharp ridges are due to the strong rotational correla- the first ridge of the 2D matrices gated by individ- tions associated with the decay along discrete rota- ual bands is found, in average, to be ≈ 10 for each tional bands up to ≈ 1 MeV excitation energy above of the four low-K and of the three high-K configura- = yrast, the rather smooth central valley at Eγ1 Eγ2 tions. Adding together the number of paths relative to mostly collects contributions from the more excited specific configurations, taking into account their rel- G. Benzoni et al. / Physics Letters B 615 (2005) 160–166 163

decay flow, that is including both low-K and high-K bands, one finds that a total of ≈ 45 discrete rotational bands exist in the 163Er nucleus, at internal energies below the onset of damping, as shown by triangles in Fig. 2(a). In contrast to the results of the ridge analysis, the number of paths obtained from the valley region is found to depend significantly on the nuclear configuration for Eγ 1 MeV. This result is shown in Fig. 2(b) together with the number of paths deduced from the total Eγ 1 × Eγ 2 spectrum. As the valley is probing the region in which the rotational bands are strongly mixed, this result intuitively suggests that the mixing process is indeed different for high-K and low-K states. To provide a better understanding of the mixing of states with different K-quantum numbers we have studied the correlations between the fluctuations of the spectra gated on specific low-K and high-K bands. These correlations are expressed by the covariance of counts, defined as [15] 1 µ (A, B) ≡ M (A) − M˜ (A) 2,cov N j j ch j ˜ × Mj (B) − Mj (B) , (2) where M(A) and M(B) refer to spectra gated by tran- Fig. 2. (a) The number of decay paths extracted from the fluctuation sitions from two different bands, A and B.Thesum analysis of the ridges structure of 163Er. The open circles (squares) is over a region spanning Nch channels (in this case refer to the number of unresolved rotational bands populating the 15 × 15) in a two-dimensional 60 × 60 keV win- ridges of γ –γ matrices gated by low-K (high-K) configurations, ˜ while the full triangles give the number of discrete paths obtained dow, and M denotes an average spectrum (which in from the total matrix. The corresponding values for simulated spec- our case is obtained by the routine STATFIT as a nu- tra are shown by dotted, dashed and full lines (low-K, high-K and merical smoothed 3rd order approximation to the 2D total, respectively). The total number of discrete bands directly ex- spectrum). To normalize the covariance and thereby tracted from the bands mixing (BM) calculations is also given for determine the degree of correlation between the two comparison (thin solid line). (b) The effective number of transitions among mixed bands obtained from the fluctuation analysis of the spectra, the correlation coefficient r(A,B) is calcu- valley region is shown by open circles (squares) for low-K (high-K) lated: gated spectra, while the full triangles show the results obtained from µ (A, B) the total γ –γ matrix. The dashed and solid lines give the theoretical r(A,B) ≡ 2,cov . expectations for total and high-K cascades, as obtained from simu- (µ2(A) −˜µ1(A))(µ2(B) −˜µ1(B)) lated spectra. (3)

Here, µ2 denotes the second moment defined for the same region Nch, related to the expression for the co- ative intensities and correlations in a similar way as variance by µ2(A) = µ2,cov(A, A). The first moment described in Ref. [15], a total number of ≈ 20 paths is µ˜ 1 is the average of M over the region Nch. The sub- found both for low-K and high-K states, as shown in traction of the first moments in the denominator of Fig. 2(a) by open circles and squares, respectively. For Eq. (3) corrects for the contribution to µ2 from count- the gates at the bottom of the bands, collecting the total ing statistics, which is linear in the number of events. 164 G. Benzoni et al. / Physics Letters B 615 (2005) 160–166

The more interesting fluctuations are due to the nature ing by a factor of ≈ 3. For U 2–2.5 MeV the average of the finite number of transitions available to each value of K converges towards K≈7 and the intrin- cascade, and their contribution to µ2 is quadratic in sic FWHM is larger than the spreading in the average the number of events. values. The distinction between low-K and high-K Fig. 3 shows the average values of the correlation configurations is blurred, corresponding to the statis- coefficient r extracted from the covariance analysis tical limit of strong K-mixing. For energies around of the ridge and valley structures of γ –γ coincidence U ≈ 1.5 MeV one finds an intermediate regime asso- spectra of 163Er. In each panel the solid lines indicate ciated with the onset of K-mixing. the two opposite limits expected in the case of a com- Starting from microscopically calculated bands, we plete conservation of selection rules (r = 0) and of have generated simulated γ –γ spectra by means of a a compound nucleus regime (r = 1). In the case of Monte Carlo code [16,17] describing the competition the ridge analysis, r is found to be of the order of 0.2 between E2 collective and E1 statistical transitions, for spectra gated on bands with similar low-K values which cool the nucleus. While the E2 transitions are (Fig. 3(a)), while it is approximately zero for com- calculated microscopically, the statistical E1 transi- binations of low-K and high-K spectra (Fig. 3(b)). tions are obtained using the calculated level density This shows that there are basically no cross-transitions and a GDR strength function corresponding to a pro- between the ≈ 20 bands feeding high-K states and late nucleus with quadrupole deformation β = 0.25 the ≈ 20 bands feeding low-K states. For the valley and rotating collectively. This β value coincides with fluctuations, the correlation coefficient between low- the deformation parameter of the Nilsson potential K configurations is again of the order of 0.2. The r used to produce the microscopic bands. In addition, an coefficient between low-K and high-K configurations, exponential quenching factor that takes into account instead, increases from 0.2, at Eγ ≈ 700 keV, up to the difference in K-quantum number between the ini- r ≈ 0.5forEγ ≈ 1 MeV. Together with the fact that tial and final states has been used in the simulation. also the number of paths in the valley gated by low- Such a factor is analogous to the one employed in K and high-K approaches each other, this represents the analysis of the E1 decay-out from isomeric states an experimental indication of a weakening of selec- [18,19]. tion rules associated with the K-quantum number with Each γ -cascade is started from initial values of in- increasing rotational frequency and internal excitation ternal energy U and spin I randomly chosen from a energy. two-dimensional entry distribution of Gaussian shape, To obtain further insights on the validity of the K- with centroids and widths reproducing the experi- quantum number in excited states, we have performed mental conditions (i.e., U=4MeV,FWHMU = simulations using band mixing calculations including 4MeV,I=44h¯ , FWHMI = 20h¯ ). The resulting both the residual interaction and a term that takes simulated intensities well reproduce the experimental into account the angular momentum carried by the values both for ridge structures and low spin yrast tran- K-quantum number [11]. The calculations are made sitions. diagonalizing the Hamiltonian in a basis of np–nh The right column of Fig. 1 shows examples of = excitations in a cranked Nilsson potential. The low- 60 keV wide projections perpendicular to the Eγ1 est eigenstates are retained, covering an interval above Eγ2 diagonal of simulated γ –γ matrices collecting all yrast of approximately 2.5 MeV. Inspecting the states cascades (total, panel (d)) and cascades finally feeding resulting from the band mixing calculations, one finds into low-K (panel (e)) and high-K (panel (f)) bands. that every state is characterized by a Gaussian dis- The projections are taken at the average transition en- tribution of K with a FWHM which is found to in- ergy (Eγ 1 + Eγ 2)/2 = 900 keV, as in the correspond- crease with U.BelowU ≈ 1.5MeVtheFWHMof ing experimental spectra (left column of Fig. 1). The the Gaussian distribution is found to be smaller than asymmetry in the spectra is due to the subtraction of the typical spreading in the average value of K, so that discrete lines which also in the simulated spectra is it makes sense to define two different sets of states performed only for Eγ1 Eγ2 . It is worth noticing (i.e., low-K with K 8 and high-K with K>8), that in the simulated matrices only the yrast and the which turn out to correspond to level densities differ- first excited band (for each parity and signature con- G. Benzoni et al. / Physics Letters B 615 (2005) 160–166 165

Fig. 3. The results of the covariance analysis on ridge (bottom panels) and valley (top panels) structures of 163Er. Panels (a) and (c) show by open squares the correlation coefficient r obtained experimentally by averaging over pairs of γ –γ spectra gated by low-K configurations, while the correlation coefficient obtained from the experimental analysis of the low-K versus the high-K matrices is shown by full circles in panels (b) and (d). The theoretical values, as obtained from the covariance analysis of simulated spectra, are represented by dashed lines in all the panels.

figuration) have been subtracted, resulting in a more Turning now to the correlation coefficient in the pronounced ridge than in the data, where all discrete ridge, the rather low value r = 0.2 obtained for the transitions known from the level scheme have been re- low-K versus low-K ridge analysis may be under- moved. stood from the fact that at most two E1 transitions (2) The number of paths Npath and the correlation coef- cool down the nucleus from the excited unresolved ficient r have been extracted from the simulated spec- bands around U≈0.6 MeV to the low-lying re- tra using the same statistical analysis employed for the solved bands. A simple quantitative estimate of the data, and the results obtained for the ridge and valley correlation coefficient can be deduced from the ratio structures are shown with lines in Figs. 2 and 3.There- between path probabilities, as described in Ref. [15]. sults of the band mixing model are in good agreement In the case of bands with same parity and different with the data, both for the number of paths (Fig. 2) and signature, this path probability depends on the rela- the correlation coefficient (Fig. 3). tive probability f for emitting an unstretched versus The number of paths in the ridge region corre- a stretched E1 transition, with f ≈ U 4/(U + ω)4, be- sponds to the total number of discrete bands which ing ω the rotational frequency. Altogether, for ridges branch-out to less than 2 states [20] (shown by the thin including also bands with different parities, one ob- solid line in Fig. 2(a)). This confirms that the nucleus tains the simple expression r = 2f/(1 + f 2). Insert- 163Er contains about 45 discrete bands at low inter- ing now U = 0.6 MeV and ω = 0.4 MeV, one finds nal energies before damping sets in [20]. Such number r ≈ 0.25. One may note that such small correlation is higher than the typical 20 to 25 discrete bands ob- coefficients support the specific choice of cooling tran- tained for 164,167,168Yb [13,15], and can be attributed sitions applied in the simulations. Thus, for example, to the existence of the additional 20 high-K bands in an even competition between M1 and E1 transitions 163Er, which do not exist at such low energies in the would lead to much higher correlation coefficients, other nuclei. which is excluded by the experimental results. The 166 G. Benzoni et al. / Physics Letters B 615 (2005) 160–166 combined effect of selection rules on E1s together of the K-quantum number is found, as deduced by with the straightforward parity-signature rules leads the analysis of the ridge structure which is formed by to a smaller value of r relative to the typical value transitions along discrete unresolved bands. At higher r ≈ 0.2. This is seen for simulations as well as data internal energy (1.2 U 2.5MeV), probed by the for low-K versus high-K ridge correlations. weaker and more numerous transitions forming the Considering the r coefficient in the valley, a reason- valley, a partial conservation of the K-quantum num- able agreement between simulations and experimental ber is found, as shown by both the fluctuation and data is generally found. In the case of low-K versus covariance analysis. low-K the analysis gives a low value r ≈ 0.25. This in- Further progress in the interesting topic of the dicates similar probabilities for E1s crossing as found order-to-chaos transition in nuclei will benefit from a in the ridge, although one could expect, on the ba- better understanding of the internal excitation energy sis of the previous estimates, slightly higher values. dependence of the K-mixing problem. For this pur- The r coefficient between low-K and high-K tends pose, future experiments focusing on high K-bands of instead to increase with Eγ , suggesting a progressive larger internal energies are envisaged. weakening of K-selection rules, due to the mixing of K-states. The flattening observed for E 1MeVin γ Acknowledgements the simulated coefficient between low-K and high-K is associated with the fact that the excitation energy The work has been supported by the Italian In- of the γ -cascades at high spins extends up to values stitute of Nuclear Physics, by the Danish Natural larger than the energy range covered by the simulated Science Foundation Research Council, by the Polish bands (U 2.5MeV). Committee for Scientific Research (KBN Grant No. 2 From the fluctuation analysis in the valley P03B 118 22), by EU Transnational Access to Ma- (Fig. 2(b)), a clear difference in the number of paths jor Research Infrastructures (Contract No. HPRI-CT- (2) Npath between low-K and high-K gated spectra is seen 1999-00078) and by the EU TMR project (Contract ≈ ≈ ≈ up to Eγ 1 MeV (i.e., I 38h¯ and U 2MeV), No. ERBFMRXCT970123). while the two quantities tend to converge at higher transition energies. This can be understood in terms of the onset of K-mixing, described above. In the lower References energy region, not only the level density for high-K [1] P. Walker, G. Dracoulis, Nature 399 (1999) 35. states is ≈ 3 times lower than for the low-K ones [11], [2] B.R. Mottelson, Nucl. Phys. A 557 (1993) 717c. but also the rotational damping width has been mea- [3] I. Huseby, et al., Phys. Rev. C 55 (1997) 1805. suredtobe≈ 30% reduced for high-K states [21].In [4] V.G. Soloviev, Phys. Lett. B 317 (1993) 501. schematic evaluations of the number of paths [13],the [5] A. Bracco, S. Leoni, Rep. Prog. Phys. 65 (2002) 299. [6] M. Matsuo, et al., Nucl. Phys. A 620 (1997) 296. level density and the rotational damping width enter as [7] S. Åberg, Phys. Rev. Lett. 64 (1990) 3119. quadratic terms, thus explaining roughly the factor of [8] R.U. Haq, A. Pandey, O. Bohigas, Phys. Rev. Lett. 48 (1982) 10 separating the number of high-K and low-K gated 1086. paths. [9] J.D. Garrett, et al., Phys. Lett. B 392 (1997) 24. In conclusion, we have discussed the onset of K- [10] P. Bosetti, et al., Phys. Rev. Lett. 76 (1996) 1204. mixing in excited rapidly rotating nuclei by means of [11] M. Matsuo, et al., Nucl. Phys. A 736 (2004) 241. [12] G.B. Hagemann, et al., Nucl. Phys. A 618 (1997) 199. a comparison of results from high statistics experimen- [13] T. Døssing, et al., Phys. Rep. 268 (1996) 1. 163 tal data on Er with recently developed band mixing [14] D.C. Radford, Nucl. Instrum. Methods A 361 (1995) 297. calculations for this specific nucleus. The experimen- [15] S. Leoni, et al., Nucl. Phys. A 671 (2000) 71. tal results from the fluctuation and covariance analy- [16] A. Bracco, et al., Phys. Rev. Lett. 76 (1996) 4484. sis on γ –γ coincidences address the region of spin [17] A. Bracco, et al., Nucl. Phys. A 673 (2000) 64. [18] P. Walker, et al., Phys. Lett. B 408 (1997) 242. 20h¯–40h¯ and internal excitation energies up to around [19] F.G. Kondev, et al., Nucl. Phys. A 632 (1998) 473. 2.5 MeV. For the lower interval of internal energy up [20] M. Matsuo, et al., Nucl. Phys. A 617 (1997) 1. to approximately 1.2 MeV, a rather strict conservation [21] S. Leoni, et al., Phys. Rev. Lett. 93 (2004) 022501. Physics Letters B 615 (2005) 167–174 www.elsevier.com/locate/physletb

16O Coulomb dissociation: towards a new means to determine the 12C + α fusion rate in stars F. Fleurot a,1, A.M. van den Berg a, B. Davids a,2, M.N. Harakeh a, V.L. Kravchuk a,3, H.W. Wilschut a, J. Guillot b,H.Laurentb, A. Willis b, M. Assunção c,4, J. Kiener c, A. Lefebvre c, N. de Séréville c,5, V. Tatischeff c

a Kernfysisch Versneller Instituut, Zernikelaan 25, 9747 AA Groningen, The Netherlands b IPN, 15 rue Georges Clemenceau, 91406 Orsay cedex, France c CSNSM, bat. 104-108, 91405 Orsay Campus, France Received 21 April 2004; received in revised form 8 March 2005; accepted 13 April 2005 Available online 25 April 2005 Editor: V. Metag

Abstract A feasibility study was made of an important aspect of the Coulomb-dissociation method, which has been proposed for the determination of the rate of the astrophysically important 12C(α, γ )16O reaction. A crucial aspect is the disentanglement of nuclear and Coulomb interactions on one hand and the separation of dipole and quadrupole contributions on the other. + As a ﬁrst step the resonant breakup via two well-known 2 states of 16O was measured. The differential cross section of 208Pb(16O, 16O*)208Pb and the angular correlations of the fragments 12Candα in the center of mass were measured and compared to theoretical predictions calculated in DWBA and the coupled-channel method. The best agreement was found for the state at 11.52 MeV associated to a one-step excitation from the ground state, while the 9.84 MeV requires coupling to the + ﬁrst-excited 2 state and is not well described.  2005 Elsevier B.V. All rights reserved.

PACS: 97.10.Cv; 26.20.+f; 24.10.Eq; 27.20.+n; 25.70.De; 25.60.Gc

Keywords: Stellar structure, interiors, evolution, nucleosynthesis, ages; Hydrostatic stellar nucleosynthesis; Coupled-channel and distorted-wave models; Nuclei with 6

E-mail addresses: ﬂ[email protected] (F. Fleurot), [email protected] (H.W. Wilschut). 1 Present address: Department of Physics, Laurentian University, Sudbury, ON, P3E 2C6, Canada. 2 Present address: TRIUMF, 4004 Wesbrook Mall, Vancouver, BC, V6T 2A3, Canada. 3 Present address: INFN, Laboratori Nazionali di Legnaro, Viale dell’Universita 2, 35020 Legnaro (PD), Italy. 4 With ﬁnancial support from CNPq, Brazilian Government Agency; present address: Departamento de Fisica Nuclear, Institute de Fisica da Universidade de São Paulo, São Paulo, Brazil. 5 Present address: Centre de Recherche du Cyclotron, UCL, 2, B-1348 Louvain-la-Neuve, Belgium.

The 12C(α, γ )16O reaction is known as one of the gular correlation of the fragments is very sensitive to most important nuclear reactions in astrophysics, and the interference between the contributions of the vari- remains one of the most challenging to study. Despite ous multipolarities involved in the excitation process. many experimental efforts to measure direct radiative Therefore, a measurement of the angular correlation of capture (see in particular [1–7]), our knowledge of the fragments is an important requirement to separate its cross section is still unsatisfactory. Measurements the E1 and E2 contributions in the 16O continuum. have been carried out down to a center-of-mass energy Two experiments were carried out. For the first of = 1 MeV with relatively low statistics, requiring experiment, the superconducting cyclotron AGOR at uncertain extrapolation of the fusion excitation func- KVI provided a 60 MeV/u 16O beam to bombard a tiondownto = 300 keV, i.e., the astrophysically 7.8 mg/cm2 208Pb target. The final experiment used relevant region. The uncertainty is due to the many dif- an 80 MeV/u 16O beam, anda4mg/cm2 208Pb tar- ferent mechanisms that contribute. get to reduce straggling effects. At these energies there A promising alternative to direct measurements is is a sufficiently high yield of virtual photons for a the Coulomb-dissociation method (see in particular high excitation cross section. The experimental detec- [8–12] and [13] for a complete review). This method tion technique exploits the fact that the relative energy highly favors the E2 component and appears to be of the α and 12C particles in the moving 16O* frame complementary to the 16N β-delayed α-decay tech- is small. In the laboratory frame their velocities are nique that allowed the estimation of the E1 contribu- approximately equal to the beam velocity. The mag- tion [14,15]. netic rigidities of the particles are also close because A complicating aspect is the uncertainty in the of the same mass-to-charge ratio A/q = 2 for both nuclear interaction amplitude interfering with the α and 12C(seeFig. 1). Both particles could then be Coulomb part of the interaction. Moreover, one must detected in the spectrometer where their relative mo- disentangle the dipole and quadrupole contributions. mentum could be measured. In nearly all cases of Measuring Coulomb breakup at ≈ 300 keV all com- elastic breakups (without target excitation), the mag- plications occur simultaneously. Therefore, as an in- netic rigidity of one fragment is slightly above the termediate step we have studied the dissociation via magnetic rigidity of elastically scattered 16O particles, two well-known 2+ states in 16O. This restricts the while the other is slightly below. When the nominal problem to the nuclear-Coulomb interference. How- rigidity of the spectrometer is set to the rigidity of ever, such measurements at larger center-of-mass en- elastically scattered 16O particles, the fragments can ergies ∼ 4 MeV require a spectrograph with large simultaneously be detected in the scintillator paddles angular opening and momentum bite, as we will dis- on the left and right sides of the nominal rigidity, re- cuss below. The Big Bite Spectrometer (BBS) at KVI spectively. fulfills this requirement. Our result clearly indicates As shown in Fig. 1, a high beam velocity reduces the crucial role of the nuclear part of the interac- the angular opening between the fragments in the lab- tion and the additional steps required for a reliable oratory frame, as well as the momentum bite, and measurement in the astrophysically relevant energy re- increases the chance to detect both fragments simul- gion. taneously. Moreover, an important aspect of the ex- The Coulomb-dissociation experiment was car- periment is the measurement of the fragment angu- ried out on the well-known 2+ states at 9.84 MeV lar correlations in the center-of-mass frame. A large + + (2 ; = 2.68 MeV) and 11.52 MeV (2 ; = acceptance, therefore, allows the measurement of the 2 3 + 4.36 MeV) in 16O. The aim was twofold: first, to correlations following the breakup via the 2 states lo- test the possibility to fit the angular distributions cated at relatively high excitation energy. The B-mode with distorted-wave Born approximation (DWBA) or of the BBS has large acceptances in both angular open- coupled-channel (CC) calculations that include the nu- ing (10 msr) and momentum bite (19%) and is there- clear and Coulomb interactions and the interferences fore well suited for this type of measurement [16]. between their contributions; second, to measure the The BBS ion-detection system [17] consisted of angular correlations of the fragments in the breakup two cathode-strip chambers (CSC’s) for position and center of mass. It has been shown [8,10,11] that the an- angle measurements. These were placed in a vacuum F. Fleurot et al. / Physics Letters B 615 (2005) 167–174 169

Fig. 1. For a given oxygen scattering angle, the limiting factors are the spectrometer angular and momentum (or equivalently velocity) accep- 12 tances, vC and vα are the velocities in the breakup center of mass of Candα, VC and Vα are their velocities in the laboratory frame, ζ is 16 the angle between VC and Vα, VO is the velocity of O in the laboratory, i.e., the velocity of the center of mass and θCcm and θαcm are the 16 * breakup polar angles of the fragments in the O centerofmass.(φCcm and φαcm, not shown for clarity, are the out of plane angles.) chamber, with the first one positioned at the BBS focal For natural parity states, the excited 16O nucleus plane and the second 30 cm further downstream. Out- decays into α and 12C in their 0+ ground states. How- side the vacuum chamber, scintillator paddles placed ever, two 2− states in 16O, lying at 12.53 MeV and downstream measured the time of flight and energy 12.97 MeV, can only decay to the first-excited state of loss of the light ions. A coincidence between the scin- 12C(2+, 4.44 MeV), because a transition to the ground tillator paddles was used as the trigger condition for state is parity-forbidden. In this case, these events ap- the experiment. In this configuration, the elastically pear in the region of interest of the relative-energy scattered 16O particles arrived near the center of the spectrum. However, we could remove these events be- focal plane, where they were stopped with a narrow cause the sum of the kinetic energies of the fragments brass finger to avoid high count rates and random co- plus the Q-value of the reaction is nearly equal to the incidences. beam energy for decays to ground states, while this is The time of flight through the BBS and the energy 4.44 MeV lower for decays to the first-excited state of loss of each fragment in the scintillator paddles pro- 12C. The 208Pb recoil energy is about 200 keV. The vided full particle identification. In this way 12C–α sum-energy spectrum also allowed the discrimination coincidences were distinguished from other coincident of mutual-excitation events, i.e., when the target is also fragmentation products. The measurement of the posi- excited. Thus to measure breakup at relative high en- tions and angles of the fragments in the spectrometer ergies is not as demanding as for the astrophysically focal plane by the CSC’s allowed the calculation of relevant small energies. In the BBS configuration, the their trajectories through the spectrometer. The recon- optimal resolution in is determined by the angular struction of the angular correlations of the fragments resolution in the opening angle ζ . An average reso- in the breakup center of mass and of the angular dis- lution of 5 mrad can be achieved; at = 400 keV tribution of the 16O* particles before breakup in the this means a resolution of 40 keV. However, at these laboratory was carried out via a ray-tracing matrix small energies other issues such as track separation be- containing the empirical ion-optical parameters of the come increasingly important, and were not part of the BBS. present study. 170 F. Fleurot et al. / Physics Letters B 615 (2005) 167–174

+ Fig. 2. The calculated angular differential cross section of 16O excited to the 2 resonance at 11.52 MeV for a beam energy of 80 MeV/nucleon. The dotted line is a pure nuclear calculation, the dashed line is pure Coulomb, and the solid line includes both interfering interactions.

To elucidate the role of the nuclear and Coulomb cific efficiency and acceptance of the setup (details are processes and their interference, separate calculations in Ref. [19]). The 2+ state at 9.84 MeV is known for either contribution were made, the details will be to be populated via a coupling with the 2+ state at discussed later. The result is shown in Fig. 2.The 6.92 MeV [20]. Therefore, coupled-channel calcula- maximum Coulomb contribution occurs near the 16O tions were necessary to calculate the differential cross scattering angle of 3◦, where a strong interference oc- section for the state at 9.84 MeV. The DWBA and CC curs with the nuclear contribution that dominates else- calculations were carried out with the optical-model where. The BBS was set at an angle of 3◦. potential parameters determined in Ref. [21] from To make comparisons between the experimental elastic scattering of 94 MeV/u 16Oon208Pb. We made data and calculations, three different projections were the usual assumption of equal deformation lengths for made in view of the limited statistics. These are the each potential. The sampling took into account the angular distribution of the reconstructed 16O* and the angular correlations of the fragments calculated from correlations in the scattering plane WθCcm (tobere- the m-substate population with the S-matrix elements ferred to as Wθ ) and out of the scattering plane WφCcm computed by ECIS. The obtained differential distribu- 208 16 16 * 208 + (to be referred to as Wφ). Here θ = 0 corresponds to tions of Pb( O, O ) Pb for both 2 resonances the direction of the 16O* particle and φ = 0 to its reac- are shown in Fig. 3. The simulations also provide pre- tion plane (left side). In particular the strongly varying dictions for the angular correlations of the fragments acceptance for the reconstructed 16O*, further com- in the center of mass. The results are plotted for both plicated by some poorly functioning sections of the states in Figs. 4 and 5, respectively. focal plane detectors, did not allow to construct an In the following we discuss the comparison be- acceptance and efficiency corrected 16O* angular dis- tween data and calculations. First we note the impact tribution. Instead the following strategy was adopted: of the nuclear contribution in the calculated angular the results of DWBA and CC calculations performed distributions. The CC and DWBA calculations predict with the code ECIS [18] were sampled in a Monte different interference patterns, indicating the impor- Carlo event simulation that takes into account the spe- tance of a correct description. The overall agreement is F. Fleurot et al. / Physics Letters B 615 (2005) 167–174 171

+ Fig. 3. Angular distributions of 16O* for both 2 states. The data points are from the ﬁnal experiment, the lines are from the Monte Carlo sampling of the ECIS calculations described in the text. The lines are marked the same way as in Fig. 2. The differential cross sections for the ◦ states at 9.84 MeV and 11.52 MeV were calculated with CC and DWBA, respectively. The average angular resolution is about 0.3 . best for the state at 11.52 MeV (bottom Figs. 3 and 5). at 55◦ and 65◦ in both cases. Both angular corre- 2 The minimum χ in ﬁtting the angular distribution lations Wφ display the typical quadrupole distribu- gave a deformation length of δ = 0.083 ± 0.003 fm, tion pattern. The state at 11.52 MeV is nearly per- only about 5% lower than the value extracted from lit- fectly reproduced. The state at 9.84 MeV shows a erature [22]. This would correspond to a 10% error in yield twice too strong for three points around 180◦, the reduced transition probability and thus in the cross while it is relatively good at other angles. The fact section. This is actually comparable to the systematic that populating this state requires a complex excita- error in the normalization of the data, which was es- tion process might be the reason why this is not per- timated to be 7%. Therefore, the integrated cross sec- fectly reproduced by the calculations. However, the tion over the investigated region is well measured. It low-energy continuum of 16O is excited directly from also supports strongly that the DWBA calculations de- the ground state with little contributions from any scribe the excitation process including both Coulomb coupled channels [20], therefore this should not be and nuclear interactions and their interferences. There- an issue for future experiments aiming to measure fore, this implies that the method can be used to mea- at excitation energies of relevance for stellar burn- sure the deformation parameters and to deduce the re- ing. duced electromagnetic transition probabilities B(EL). As a further check calculations for both states were Figs. 4 and 5 show that the angular correlations Wθ carried out using the folded-potential model. In this are reproduced if not very accurately for the points model a collective L = 2 transition density of the 172 F. Fleurot et al. / Physics Letters B 615 (2005) 167–174

12 Fig. 4. The angular correlations Wθ (upper panel) and Wφ (lower panel) of the C fragment for the 9.84-MeV resonance in the center-of-mass frame of the decaying 16O and oriented in the direction of its velocity. (The α-particle correlation is at complementary angles.) The average θ ◦ ◦ and φ resolutions are about 15 and 20 , respectively.

1st derivative type is used that reproduces the mea- The current procedure is tailored to the case of sured B(E2) value for the respective transitions. This 2+ states and therefore no interference with E1 exci- is folded with a projectile-nucleon interaction [19] to tations is taken into account. For a continuum mea- obtain the transition potential, which is used in the surement, the Coulomb-excitation process and the fact DWBA calculations. These results were also sampled that 16O, 12C and α are self-conjugate nuclei favor the with the setup efﬁciency but give yields more than quadrupole contribution. Nevertheless, the dipole part 50% below the experimental values. Thus, we ob- is present and must be measured and extracted. Our serve that the amplitude of the nuclear component experiments show that in principle the E1–E2 interfer- depends on whether the deformed-potential or the ence pattern, which should lead to forward–backward folded-potential model is used. The data imply that the asymmetry in the angular correlations, should also be phenomenological deformed-potential model as im- measurable in the continuum via the fragment angu- plemented in ECIS gives a better description of the lar correlations, and thus permits a separation of both data in this case than the folding-potential model. In contributions. A better test could be done by measur- the future, this issue should be resolved by an indepen- ing the two-dimensional correlation of the fragments + 16 dent test of the models on the 21 bound state in Oat instead of its projections on the θcm and φcm axes. This 6.92 MeV, i.e., a measurement of the differential cross will require high statistics. section under the same experimental conditions as for The next generation experiments could measure + + the unbound 22 and 23 states. the continuum cross section and separate the E1 F. Fleurot et al. / Physics Letters B 615 (2005) 167–174 173

◦ Fig. 5. Same as Fig. 4 but for the 11.52-MeV resonance. In the top figure, the data point at 75 has a value 14.3 ± 10.1. and E2 components. A comparison with the direct- References measurement data that exist at >1 MeV would be an ultimate verification of the Coulomb-dissociation [1] P. Dyer, C.A. Barnes, Nucl. Phys. A 233 (1974) 495. method applied to the 12C(α, γ )16O reaction. A mea- [2] A. Redder, H.W. Becker, C. Rolfs, H.-P. Trautvetter, T.R. surement at <1 MeV where almost no data exist, Donoghue, T.C. Rinckel, J.W. Hammer, K. Langanke, Nucl. would be accepted with more confidence. If the E1 Phys. A 462 (1987) 385. [3] R.M. Kremer, C.A. Barnes, K.H. Chang, H.C. Evans, B.W. Fil- component is sufficiently strong at low energy, also ippone, K.H. Hahn, L.W. Mitchell, Phys. Rev. Lett. 60 (1988) 16 matching with the N decay data [14,15] can be ver- 1475. ified. In all cases the nuclear contribution will have [4] J.M.L. Ouellet, M.N. Butler, H.C. Evans, H.W. Lee, to be checked by observing the differential cross sec- J.R. Leslie, J.D. MacArthur, W. McLatchie, H.-B. Mak, tion and correlations at scattering angles larger than 5◦ P. Skensved, J.L. Whitton, X. Zhao, T.K. Alexander, Phys. Rev. C 54 (1996) 1982. where the nuclear contribution is largely predominant. [5] G. Roters, C. Rolfs, F. Strieder, H.-P. Trautvetter, Eur. Phys. J. A 6 (1999) 451. [6] L. Gialanella, D. Rogalla, F. Strieder, S. Theis, G. Gyrky, Acknowledgements C. Agodi, M. Aliotta, L. Campajola, A. Del Zoppo, A. D’Onofrio, P. Figuera, U. Greife, G. Imbriani, A. Ordine, This work has been performed as part of the re- V. Roca, C. Rolfs, M. Romano, C. Sabbarese, P. Sapienza, search program of the Stichting voor Fundamenteel F. Schmann, E. Somorjai, F. Terrasi, H.-P. Trautvetter, Eur. Phys. J. A 11 (2001) 357. Onderzoek der Materie (FOM), with financial support [7] R. Kunz, M. Jaeger, A. Mayer, J.W. Hammer, G. Staudt, from the Nederlandse Organisatie voor Wetenschap- S. Harissopulos, T. Paradellis, Phys. Rev. Lett. 86 (2001) 3244. pelijk Onderzoek (NWO). [8] G. Baur, M. Weber, Nucl. Phys. A 504 (1989) 352. 174 F. Fleurot et al. / Physics Letters B 615 (2005) 167–174

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Resonant tunneling through the triple-humped ﬁssion barrier of 236U

M. Csatlós a, A. Krasznahorkay a,P.G.Thirolfb, D. Habs b,Y.Eisermannb, T. Faestermann c,G.Grawb,J.Gulyása, M.N. Harakeh d, R. Hertenberger b, M. Hunyadi a,d,H.J.Maierb,Z.Mátéa,O.Schaileb,H.-F.Wirthb

a Institute of Nuclear Research of the Hungarian Academy of Sciences, P.O. Box 51, H-4001 Debrecen, Hungary b Department für Physik, Universität München, D-85748 Garching, Germany c Technische Universität München, D-85748 Garching, Germany d Kernfysisch Versneller Instituut, 9747 AA Groningen, The Netherlands Received 21 August 2004; received in revised form 14 March 2005; accepted 19 April 2005 Available online 28 April 2005 Editor: V. Metag

Abstract The fission probability of 236U as a function of the excitation energy has been measured with high energy resolution using the 235U(d, pf ) reaction in order to study hyperdeformed (HD) rotational bands. Rotational band structures with a moment of inertia of θ = 217 ± 38 h¯ 2/MeV have been observed, corresponding to hyperdeformed configurations. From the level density of the rotational bands the excitation energy of the ground state in the third minimum was determined to be 2.7 ± 0.4MeV. The excitation energy of the lowest hyperdeformed transmission resonance and the energy dependence of the fission isomer population probability enabled the determination of the height of the inner fission barrier EA = 5.05 ± 0.20 MeV and its curvature parameter hω¯ A = 1.2 MeV. Using this new method the long-standing uncertainties in determining the height of the inner potential barrier in uranium isotopes could be resolved.  2005 Elsevier B.V. All rights reserved.

PACS: 21.10.Re; 24.30.Gd; 25.85.Ge; 27.90.+b

1. Introduction in modern nuclear structure physics. Superdeformed (SD) nuclei in the second minimum have shapes with The study of nuclei with exotic shapes (super- an axis ratio (c/a) of about 2 : 1, whilst hyperdeformed and hyperdeformation) is one of the most vital ﬁelds (HD) nuclei in the third minimum correspond to even more elongated shapes with axis ratio (c/a) of about 236 E-mail address: [email protected] 3:1 [1,2]. This is where the nucleus U is excep- (P.G. Thirolf). tional, because it is the only isotope where hyperde-

Fig. 1. The triple-humped potential-energy surface of 236U. Also damped class-I, class-II and class-III compound states are shown in the three minima. For strongly mixed class-I and class-II states, transmission resonances of class-III states can occur in fission. formed transmission resonances have been observed in general will not overlap with the isolated class-III [3] and at the same time a superdeformed fission iso- states. The well-established idea of transmission res- mer is well-established [4]. Fig. 1 schematically shows onances in the second minimum [10] could be car- the potential-energy surface (PES) of 236U as a func- ried over to the third minimum for mixed class-I and tion of deformation (axis ratio) with the triple-humped class-II states by simply replacing the former reso- fission barrier as deduced in this publication. Promi- nant class-II states by new resonant class-III states. nent new features are the lower inner barrier EA and The previous penetrabilities PA and PB now can be the large depth of the third minimum (EB1 +EB2)/2− replaced by PB1 and PB2 corresponding to the two EIII. All peaks of the barriers are saddle points in a barriers enclosing the third minimum. multi-dimensional deformation space [5]. Hyperdeformed resonances were also observed in Starting from this general picture we first focus on uranium isotopes [3,11]. In a previous measurement the properties of the third minimum. The presence of a we identified three transmission-resonance groups at third minimum in the fission barrier of light actinides, 5.27, 5.34 and 5.43 MeV in 236U as groups of hy- obtained from PES calculations, was first proposed by perdeformed K = 4 bands [3]. Although the energy Möller and Nix [5,6]. According to more recent cal- resolution was limited to 20 keV and individual mem- culations, the third minimum in these nuclei appears bers of the rotational bands could not be resolved, a ∼ 2 = +1.0 at large quadrupole (β2 0.9) and octupole deforma- rotational parameter h¯ /2θ 1.6−0.4 keV was deter- tions (β3 ∼ 0.35) and the depth is predicted to be much mined in good agreement with the value predicted for 2 larger than previously believed [7]. Since these hyper- a hyperdeformed state h¯ /2θtheor = 2.0keV.Inthe deformed actinide nuclei are also octupole deformed new measurement we achieved an energy resolution with reflection-asymmetric shapes, they show a possi- of ≈ 5 keV and were able to resolve the individual ble energy splitting between different parity members members of the rotational bands. For a group of lower of the rotational bands. The third minimum was first transmission resonances around 5.1 MeV, which for- established experimentally for thorium nuclei, study- merly was assumed to be a transmission resonance in ing the microstructure in transmission resonances with the second minimum, a good fit of fission probability (n, f ), (t, pf ) and (d, pf ) reactions [8,9]. For hyper- and angular correlation data could only be obtained by deformed transmission resonances to occur it is neces- assigning it to a transmission resonance in the third sary that the class-I and class-II compound states are minimum. The observation of rotational bands in the strongly mixed. Otherwise, the narrow class-II states third minimum requires a rather complete damping of M. Csatlós et al. / Physics Letters B 615 (2005) 175–185 177 states in the second minimum and thus allows to de- olution obtained from the 208Pb(d, p) reaction using 2 2 termine an upper limit for the inner-barrier height EA. a thin target (57 µg/cm on a 7 µg/cm carbon back- Combining this with results from the fission-isomer ing) was 3 keV (FWHM) at 8.7 MeV proton energy. population probability [12,13], we can for the first Due to the different target properties and the longterm time now determine the inner barrier height in 236U, drifts during the one week measurement time an exper- where formerly values differing by 1 MeV were ob- imental energy resolution of about 5 keV (FWHM) can tained. The uncertainty in the height of the inner bar- be estimated for the 235U(d, pf ) reaction. It is mainly rier occurs for all light actinides and was called the due to this excellent resolution that qualitatively new ‘thorium anomaly’ [4] as it was first observed in tho- information could be extracted from an already well- rium nuclei. It is now resolved in the picture of the studied reaction. Fission fragments were detected in triple-humped barrier and we reach a much better un- coincidence with the outgoing protons by two parallel derstanding of the potential-energy landscape of light plate avalanche counters (PPAC) [20] with active areas actinides in agreement with more recent predictions of 16 × 16 cm2, each positioned in a distance of 23 cm [7,14]. from the target, covering a solid angle of about 4% and an angular range of 55 ± 20◦ and 125 ± 20◦ relative to the beam direction (corresponding to 65◦–90◦ relative 2. Experimental method to the recoil axis). This limitation in angular coverage resulted from an optimization of the setup for time-of- The experiment was carried out at the Munich flight rather than for angular correlation measurements Accelerator Laboratory. We investigated the thus preventing the extraction of angular correlation 235U(d, pf )236U reaction in the excitation energy information from the present data. During 100 hours range 4.9

Fig. 2. Excitation energy spectrum inferred from the proton kinetic energies following the 235U(d, pf ) reaction measured in coincidence with the fission fragments. calibration agrees within 20 keV with the present measurement. From the angular distribution of the fission fragments the angular correlation coefficient A2 was determined as a function of excitation energy. This is shown in Fig. 3(b) for the energy range covered by our measurement.

3.1. Hyperdeformed transmission resonances in 236U

The resonances at 5.27, 5.34 and 5.43 MeV had previously been identified as hyperdeformed resonances [3], however without resolving any rotational structure. For the first time, this was achieved in the present experiment. The resonance structure around 5.1 MeV excitation energy was formerly interpreted by Goldstone et al. [22] and Just et al. [13] as a vibrational resonance in the second minimum in analogy to the 5.1 MeV resonance in 240Pu [20]. We will later interpret it as a group of hyperdeformed resonances. 235 The excitation energy region containing HD reso- Fig. 3. (a) Prompt fission probability, Pf ,for U(d, pf ) measured nances was analyzed in two steps. We start with the with high energy resolution in this work (random coincidences sub- resonance structure between 5.2 and 5.5 MeV, because tracted); (b) and (c): angular correlation coefficient A2 for prompt 235 236 here the hyperdeformed structure was already known fission following the reaction U(d, p) U → f and prompt fission probability of 236U from Just et al. [13]. The energy resolution [3], thus allowing for a test of our analysis procedure. of both measurements is indicated by a horizontal bar. In order to describe the rotational structure, we assumed overlapping rotational bands with the same moment of inertia, inversion-splitting parameter and intensity ratio for the band members. Gaussians were In case of the upper resonance region around 5.3 MeV used for describing the different band members. The an exponential background component was included relative excitation probabilities for the members of the in the fit procedure. In the lower resonance region rotational bands were taken from DWBA calculations only a negligible background contribution was iden- for the (d, p) reaction by Back et al. [23], calculated tified. During the fitting procedure the energy of the around 5.4 MeV excitation energy for the upper and band head and the absolute intensity of the band were 5.1 MeV for the lower resonance region (see Table 1). used as free parameters. A common rotational parame- M. Csatlós et al. / Physics Letters B 615 (2005) 175–185 179

Table 1 Relative excitation probabilities for rotational band members following 235U(d, p) according to DWBA calculations by Back et al. [23] for ∗ E = 5.1 and 5.4 MeV J 234567 π −+−+−+−+−+−+ Ref. [23] 0.033 0.008 0.117 0.083 0.157 0.132 0.083 0.074 0.074 0.074 0.025 0.050 (5.4 MeV) Ref. [23] 0.050 0.008 0.121 0.059 0.150 0.100 0.096 0.075 0.092 0.058 0.025 0.058 (5.1 MeV)

ter (h¯ 2/2θ) and inversion-splitting parameter (E±) tational band members, whilst the numbers to the left were adopted for each band. denote the corresponding K values which equal the The angular distribution of the fission fragments is lowest J value of the respective band. The correspond- defined by the total spin J and its projection K onto ing normalized χ2 values are presented in Fig. 4(c) as the deformation axis of the fissioning nucleus. The afunctionofh¯ 2/2θ. The horizontal line represents the height of the outer fission barrier increases with in- 99.9% (3σ ) confidence level for the χ2 test. creasing K value. The deduced rotational parameter is h¯ 2/2θ = +0.3 Since the ground-state spin and parity of the target 2.3− keV and for the inversion parameter a value − 0.5 7 π = − π = − = +8.8 is 2 , mainly J 3 and J 4 states are excited of E± 0−5.5 keV was derived, which is consistent in the case of l = 0 transfer, in this way the fission bar- with the small inversion parameters obtained by Blons rier is relatively high. In the case of l>0 transfer, the et al. [11]. appearance of lower K values is also possible. Due Fitting the fission probability with rotational bands to the limited energy resolution of the angular distri- assuming a reversed parity assignment for the band bution measured by Just et al. [13],theA2 angular members resulted in a fit of similar quality as the correlation coefficients for states with different spins one shown in Fig. 4(a), thus indicating that the analy- in a rotational band of a given K value cannot be ex- sis is not sensitive to parity. In addition to the spin- tracted. Assuming an octupole rotational band with a dependent excitation probabilities of Back et al. also certain K value, a theoretical A2 coefficient was de- values calculated by Goerlach et al. [12] were used termined by averaging over the calculated A(J, K) and found to produce almost identical results. Alter- values weighted by the excitation probabilities for dif- natively the assumption of a superdeformed rotational ferent J values taken from Ref. [23]. The calculated band structure was tested using spin sequences with + + + A2 coefficients increase for K 3, reaching a max- J = 2 ,4 ,6 based on the same set of excitation imum at K = 3 before decreasing rapidly. At K = 4 probabilities as used for the HD bands. As shown by anegativeA2 coefficient was obtained. Using the cal- the dotted line in Fig. 4(c) no minimum for the nor- 2 culated A2(K) function, we derived the trend of the malized χ was found in this case. With the series of K values from the experimental A2 coefficients as a 15 rotational bands shown in Fig. 4(a) not only the function of the excitation energy. In this way we could fission probability could be reproduced, but also the determine the K values independently from fitting the angular correlation coefficient A2 measured by Just et fission probabilities. al. [13] (data points in Fig. 4(c)) could be reproduced The HD states are characterized by the presence of by bands with high K values (K = 3, 4 and 5, solid alternating parity bands with very large moments of line in Fig. 4(b)). Again the assumption of SD rota- inertia because of the very large quadrupole and octu- tional bands with J = 2+,4+,6+ fails to reproduce pole moments [11]. Assuming alternating parity bands the data, as indicated by the dashed line in Fig. 4(b). with J = K = 3, 4 and 5 band heads, the fission proba- The extracted rotational parameter corresponds to bility between 5.2 and 5.5 MeV was fitted, resulting in a value characterizing HD rotational resonances. The the fit curve superimposed on the data in Fig. 4(a). The moment of inertia θ = 217 ± 38 h¯ 2/MeV is in good picket fence structures indicate the positions of the ro- agreement with the values calculated by Shneidman et 180 M. Csatlós et al. / Physics Letters B 615 (2005) 175–185

235 Fig. 4. (a) Fission probability (Pf )forthe U(d, pf ) reaction in the excitation-energy region above 5.2 MeV. The solid line shows a fit to the data assuming alternating-parity rotational bands starting with J = K, the dotted line represents the exponential background component included in the fitting procedure. The picket fence structures indicate the positions of the rotational band members used in the fit with K values as indicated for each band by the left-sided numbers. (b) Angular-correlation coefficients A2 from Ref. [13] (data points), superimposed by the ∗ A2 coefficients inferred from the fit curve from a) (solid line). In addition the curve for A2(E ) resulting from the assumption of superdeformed bands (dotted curve in (c)) is shown as a dashed line. (c) Normalized χ2 values for the fit to the spectrum of panel (a)) (solid line). The dotted line represents the fit quality for the alternative scenario of superdeformed rotational bands. The best fit has been obtained by hyperdeformed ¯ 2 = +0.3 2 rotational bands with h /2θ 2.3−0.5 keV. The horizontal line represents the 99.9% (3σ ) confidence level for the χ test. al. [24] for 234U and 232Th, who assumed dinuclear band members were based on DWBA calculations by systems suggesting the possibility of an exotic heavy Back et al. [23] for E∗ = 5.1MeV. clustering as predicted by Cwiok´ et al. [7]. In order to allow for a consistent description of the In the second step of the analysis, the proton energy fission probability and the angular correlation coef- spectrum below 5.2 MeV was investigated, where so ficient A2 from Just et al. [13], a distribution of K far no conclusive high-resolution data were available. values rising from K = 1toK = 4 between 5.05 and In the 234U(t, pf ) reaction, Back et al. [25] observed 5.2 MeV had to be chosen, thus differing from the a weak, narrow resonance at 5.0 MeV and a distinct excitation-energy region between 5.2 and 5.4 MeV shoulder (or resonance) around 5.15 MeV. Goldstone discussed above, where higher K values were domi- et al. [26] and Just et al. [13] reported the first clear ob- nating. servation of a series of narrow sub-barrier fission res- Again the picket fence structures indicate the po- onances in 236U produced in the (d,pf) reaction. The sitions of the rotational band members, together with measured resonance energies are given in Fig. 3(c). In the K value of the corresponding band. While the fit their analysis the underlying states of these resonances of the fission probability showed little influence of the were assumed to originate from the second well, close J = 3 rotational band members, a consistent descrip- to the top of the inner barrier. tion together with the angular correlation data required The result of the analysis in the excitation en- to include J = 3 with an intensity reduced by a factor ergy region between 5.05 and 5.2 MeV is shown in of four with respect to the intensity listed in Table 1. Fig. 5, where the prompt fission probability is dis- The sign change of the A2 coefficient from neg- played (panel (a)) together with the results of a fit ative values in the upper resonance region to largely by rotational bands similar to the procedure described positive values in the lower resonance region already above for the upper resonance region. Again the spin- indicates the tendency towards lower K values with dependent excitation probabilities of the rotational- decreasing excitation energy. The resulting curve for M. Csatlós et al. / Physics Letters B 615 (2005) 175–185 181

235 Fig. 5. (a) Fission probability (Pf )forthe U(d, pf ) reaction in the excitation-energy region below 5.2 MeV. The superimposed solid line shows a fit to the data using rotational bands with a variable rotational constant h¯ 2/2θ. The picket fences indicate the positions of the rotational + band members, whilst the numbers to the left denote the K value of the corresponding band. In each case the band starts with a J π = 3 member − with reduced intensity (see text) and a dominant J π = 4 state. The best fit has been obtained by using hyperdeformed rotational bands with a ¯ 2 = +0.6 2 rotational parameter h /2θ 2.4−0.3 keV, as indicated by the normalized χ values shown in panel (c) (solid line). The horizontal line marks the 99.9% (3σ ) confidence level of the χ2 test. The dotted line in (c) shows the resulting fit quality under the assumption of a superdeformed + + + spin sequence J = 2 ,4 ,6 . (b) Angular-correlation coefficients A2 from Ref. [13] (data points), reproduced by the A2 coefficients inferred ∗ from the fit curve from (a). In addition the curve for A2(E ) resulting from the assumption of superdeformed bands (dotted curve in (c)) is included as a dashed line. the A2 angular-correlation coefficient calculated based can be concluded. In the scenario of rotational bands on the fit function derived from the fission probability with J π = 2+,4+,6+ a pronounced minimum of the in panel (a) is displayed as a solid line in panel (b) normalized χ2 at a value of the rotational constant of Fig. 5. The quality of the fit can be judged from h¯ 2/2θ ∼ 3.3 keV (typical for SD bands) could only Fig. 5(c), where the normalized χ2 values are dis- be achieved by artificially increasing the 2+ excitation played with the horizontal solid line representing the probability by a factor of about 4, while again failing 99.9% confidence level. to reproduce the angular correlation data. The rotational parameter derived from the best fit (Fig. 5(c)) could be determined as h¯ 2/2θ = 3.2. The depth of the third minimum in 236U +0.6 2.4−0.3 keV, corresponding to a hyperdeformed configuration. This result is in contrast to the old assump- Similar to our previous work on the ground state ex- tion that the decaying vibrational excitations originate citation energy in the third well of 234U [27], the depth from the superdeformed second minimum. of the third well was determined by comparing the Assuming overlapping rotational bands in the sec- experimentally obtained level spacings of the J = 5 π = + + + ond well with a spin sequence of J 2 ,4 ,6 members of the rotational bands of Figs. 4(a) and 5(a) and using the same excitation probabilities from Back with the calculated ones using the back-shifted Fermi- 2 et al. [23] as before, the result of the χ analysis is gas description of the level density ρ as parametrized shown as a dotted line in panel (c) of Fig. 5. No unam- by Rauscher et al. [28], complemented by the K de- biguous minimum for either HD or SD configurations pendency as described by Bjørnholm et al. [29]: was found. However, when comparing the corresponding curve for A (E∗) (dotted line in Fig. 5(b) to the 1 2 ρ(U,K,J,π)= · F(U,J)· ρ(U)· G(K), (1) data points, a clear preference for HD configurations 2 182 M. Csatlós et al. / Physics Letters B 615 (2005) 175–185 where √ √ 1 π exp(2 aU) ρ(U)= √ · · , · 1/4 5/4 2π σ 12a U J + 1 −J(J + 1) F(U,J)= · exp , 2σ 2 2σ 2 θrigid U σ 2 = with U = E − δ, ¯ 2 a h 1 K2 G(K) = √ · − . exp 2 (2) K0 2π 2K0 Summing over all the allowed K values we get back the formula of Rauscher K=+J ρ(U,J,π)= ρ(U,J,K,π). (3) K=−J Thus the level density is dependent only on three parameters: the level density parameter a, the back- 2 shift δ and the cutoff parameter K0 . According to Fig. 6. Distances of the J = 5 spin states in the third minimum 2 of 236U as a function of the excitation energy as derived from the Ref. [30] the parameter K0 at an excitation energy of 5.2 MeV as obtained from fragment angular dis- fits in the two resonance regions discussed above. The solid curve 239 2 = (‘Rauscher 1’) shows values calculated by the parameterization of tributions for the Pu(n, f ) reaction is K0 27. In Rauscher et al. [28] (see text), the full points correspond to our ex- Ref. [28] an excitation-energy dependent parameteri- perimental values. The dashed curve represents a calculation of the zation of the level density parameter a was given by level density based on Eq. (2), using the experimentally determined fitting to experimental level density data (see Eq. (14) value of the rotational constant instead of the rigid-rotor value. The in Ref. [28]). The C(Z,N) “microscopic correction” triangles indicate the average level density in the two resonance regions, corrected for unobserved J = 5 states due to the K-dependent in this parameterization was taken from the calcula- filtering by the fission barrier. The dash-dotted curve originated from tions of Möller et al. [31]. The pairing gap , which is shifting the curve labeled ‘Rauscher 1’ through the corrected exper- related to the back-shift δ, was calculated from the dif- imental level density data. ferences in the binding energies of neighboring nuclei applying Eq. (19) in Ref. [28]. In order to determine the excitation energy EIII of EIII = 2.7 MeV was determined (indicated by the ar- the ground state in the third minimum, the excitation row in Fig. 6). energy UIII = E − δ − EIII relative to the ground state From our fits to the fission probability we obtained in the third well was used in Eq. (2). Varying the value a weighted average value of h¯ 2/2θ = 2.3 ± 0.4MeV. of EIII, the level spacings in the third well were cal- Using this value instead of the rigid-rotor value for culated according to Eq. (2) and compared to our ex- the determination of the spin-cutoff parameter σ in perimental data for the J = 5 levels (circles and curve Eq. (2), we repeated the calculation of the level spac- labeled ‘Rauscher 1’ in Fig. 6). Due to the K-selective ings (curve labeled ‘Rauscher 2’ in Fig. 6). In this case filtering of the fission barrier the unobserved J = 5 the analysis indicated a minimum at EIII = 2.55 MeV. levels from K>3(< 4) in the low (high) resonance Thus the uncertainty of EIII, composed of the uncer- region has to be corrected for. Taking into account the tainty as obtained from our analysis and of the uncer- G(K) weighting factors from Eq. (1), corrected aver- tainty introduced by the level density parameterization age level distances were derived from the experimental as quoted by the authors in Ref. [28], was estimated to data points for the two resonance regions (triangles be ±0.4MeV. in Fig. 6). Varying the backshift δ in order to de- Cwiok´ et al. [7] predicted two different HD minima 236 scribe the level density in the third well, a value of for U isotopes. In the case of U, EIII = 3.8MeV M. Csatlós et al. / Physics Letters B 615 (2005) 175–185 183 corresponds to the less reflection-asymmetric HD minimum, while EIII = 2.4 MeV belongs to the more reflection-asymmetric HD minimum. Our result is in good agreement with the prediction for the more reflection-asymmetric hyperdeformed minimum and with the so far only experimentally determined value 234 of EIII = (3.1 ± 0.4) MeV for U as obtained in our previous work [27]. These two results represent the first experimental determination of the potential- energy landscape for hyperdeformed configurations.

3.3. The vibrational energy hω¯ III in the third minimum

Fig. 7. Isomer population probability from Goerlach et al. [12,13]. A further important quantity is the β-vibrational phonon√ energy in the third minimum. Looking at h¯ C/B with mass parameter B and stiffness C, from terpretation as a resonance in the second minimum led the oscillatory behaviour of B as a function of the de- to the requirement that the inner barrier height EA was formation [32] a trend can be inferred indicating that B approximately equal to the outer barrier EB , because decreases from the second minimum to the third min- a strong resonance requires penetrabilities of both bar- imum to a value around the one known from the first riers with comparable values [21]. Using our new in- minimum, whilst the potential gets softer (see Fig. 1) terpretation the inner barrier has to be reduced to the and C gets smaller. In this way a somewhat smaller excitation energy of the lowest transmission resonance value for hω¯ compared to the second minimum of in order to achieve a good mixing between the first and about 600 keV is expected for the third minimum. second minima. In addition, there is the general feature that the With a back-shifted Fermi-gas formula the level fission probability of 234, 236U shows about twice as spacing for 0+ compound states in the second mini- many resonances compared to the one for 240, 238Pu. mum at a total excitation energy of 5.1 MeV is cal- This seems to be a general difference between the res- culated to be about 80 keV. The damping width of onances in the second and the third minimum. Perhaps these states due to tunneling through the inner bar- the spacing of these resonances can be explained by rier Γ = hω¯ II/2π · PA is about 100 keV at the top of the fact that theoretically two hyperdeformed minima the barrier with PA ≈ 1. Therefore the occurrence of a with β2 = 0.6 but different β3 values of 0.3 and 0.6 are transmission resonance in the third well at 5.1 MeV re- predicted [7]. In this way in each minimum vibrational quires a rather complete damping and an inner-barrier states with a spacing of about 0.5 MeV could occur, height EA lower than 5.2 MeV. On the other hand, we leading to a doubling of the number of resonances. can obtain a lower limit for the inner-barrier height EA Compared to the situation in the second minimum from the measurement of the isomer population proba- of 240Pu [20] transmission-resonance spectroscopy in bility. This is shown in Fig. 7 with data from Goerlach the third well can be performed reaching down in en- et al. [12,13]. For excitation energies between ∼ 4.4 ergy to lower phonon numbers due to the thinner barri- and ∼ 4.9 MeV, the weak coupling limit is valid and ers and lower excitation energies within the third well. the isomer population probability is proportional to the square root of the penetrability PA of the inner barrier 3.4. The inner barrier of 236U [33]. The data show a clear exponential increase. At an excitation energy of 5.1 MeV the isomer population The new interpretation of the 5.1 MeV transmis- probability starts to saturate because the compound sion resonance in 236U as a resonance in the third and states of the first and second minima become largely not in the second minimum leads to a determination of mixed. Above 6 MeV we observe a drop of the isomer the parameters of the inner barrier. Previously the in- population probability because the competing fission 184 M. Csatlós et al. / Physics Letters B 615 (2005) 175–185 and neutron channels open up. From the exponential version electron spectroscopy [36]. Thus we obtain: −15 rise of the isomer population probability for low en- PB1 · PB2 = (4.5 ± 0.4) × 10 . From the half-life of ergies we can deduce that the inner barrier EA has to the γ back-decay we can deduce a rough estimate of be larger than 4.9 MeV. From both limits we obtain the penetrability PA of the inner barrier: PA = tIγ /tIIγ . EA = 5.05 ± 0.20 MeV. Furthermore, from the expo- The typical half-life tIγ of the γ decay in the first min- nential rise and the estimate of the penetrability from imum with an energy close to the isomer energy can the γ back-decay we obtain hω¯ A = 1.2MeV. be obtained from the Weisskopf estimate: TW (E1) = −2/3 −3 [ ]× −15 It is interesting to note that in the data of Goerlach 6.76A Eγ MeV 10 s combined with a typ- et al. [12] in the very weak coupling region a reso- ical Weisskopf hindrance factor FW = 100 leading to: −15 nance in the isomer population probability could be tIγ ≈ 10 s. In this way we obtain a penetrability of −8 observed at an excitation energy of 4.20 ± 0.05 MeV the inner barrier PA ≈ 1 × 10 , which is about 7 or- (see Fig. 7). It nicely agrees with the sum of the ders of magnitude larger than the one of the two outer isomer energy [34,35] and the energy of the second barriers. phonon EII + 2 · hω¯ II = (2.814 + 2 × 0.685) MeV = For the inner barrier largely varying parameters 4.184 MeV. This again shows that the second well at have been published: (EA = 5.98 ± 0.15 MeV, hω¯ A = lower excitation energies is rather harmonic. A simi- 1.32 ± 0.1MeV[13,37])or(EA = 5.6 ± 0.2MeV, 240 lar observation was made for the second well of Pu hω¯ A = 1.04 MeV [4]). Here, we proved that a bet- [20]. The third phonon may be identified with the res- ter choice of parameters is: (EA = 5.05 ± 0.2MeV, onance observed in the prompt fission probability at hω¯ A = 1.2 MeV) allowing for a full damping of the 4.75 MeV (Fig. 3(c)). The energy of the fourth phonon class-II compound states in the region where transmis- may be reduced close to the top of the barrier EA sion resonances for the third minimum are observed. similar to the situation in 240Pu [20]. In this way a From the saturation of the prompt fission probabil- vibrational enhancement of the coupling via the sec- ity an outer barrier height EB = 6.0 ± 0.1 MeV and ond minimum may occur for the 5.1 MeV resonance; a common curvature hω¯ B = 0.68 ± 0.05 MeV was but then the close-lying 5.27 MeV resonance would deduced [13]. With these parameters we obtain a pen- −13 require a barrier energy below 5.27 MeV for complete etrability of the outer barrier of PB = 1.6 × 10 at damping. Allowing also for this special situation we the excitation energy of the fission isomer. Adjust- have increased the error for determining the barrier ing the values within errors to EB = 6.1 MeV and −15 EA = 5.05 ± 0.2MeV. hω¯ B = 0.63 MeV we obtain PB = 5.6 × 10 ,in good agreement with the lifetime measurement. The 3.5. The triple-humped fission barrier of 236U strong occurrence of transmission resonances in the third minimum requires barriers B1 and B2 to have In Fig. 1 we showed to scale the triple-humped fis- similar penetrabilities. Therefore, we have split the sion barrier of 236U with our newly determined values. outer barrier into two symmetric barriers with the Significant information on the barriers is obtained also same height EB1 = EB2 = 6.1 MeV and curvatures from the decay of the ground state of the second min- hω¯ B1 = hω¯ B2 = 2 · hω¯ B = 1.26 MeV. The depth of imum. It has an excitation energy of 2814 ± 17 keV the enclosed third minimum is EIII = 2.7 ± 0.4MeV. and a total half-life of 116 ± 3ns[34,35]. The partial For the deformation of the second minimum a = ± half-life for delayed fission is tIIf = 870 ± 95 ns and quadrupole moment Q (32 5) eb [38] was ob- the partial half-life for the γ back-decay to the first tained, corresponding to an axis ratio of 1.9 ± 0.1. minimum is tIIγ = 134 ± 4 ns. From the fission half- From the ground-state rotational band in the second 2 life the common penetrability of the outer two barriers minimum a rotational parameter h¯ /2θII = 3.36 ± PB1 ∗ PB2 can be calculated: 0.01 keV [39] was determined. For the third minimum we obtained h¯ 2/2 · θ = 2.3 ± 0.4 keV as the ln 2 · 2πh¯ III t = . weighted average of our two resonance regions. The IIf · · hω¯ II PB1 PB2 barrier heights and the potential minima of Fig. 1 The β-vibrational phonon energy in the second min- approximately agree with theoretically predicted val- imum hω¯ II = 685 keV has been measured in con- ues of EA = 5.46 MeV, EB1 = 6.0MeV,EB2 = M. Csatlós et al. / Physics Letters B 615 (2005) 175–185 185

6.0MeV,EII = 2.2 MeV and EIII = 3.8 MeV from [3] A. Krasznahorkay, et al., Phys. Rev. Lett. 80 (1998) 2073. Cwiok´ et al. [7] and Howard et al. [14]. [4] S.B. Bjørnholm, J.E. Lynn, Rev. Mod. Phys. 52 (1980) 725. [5] P. Möller, J.R. Nix, in: Proceedings of the International Sym- posium on the Physics and Chemistry of Fission, Rochester, 1973, IAEA, Vienna, 1974, p. 103. 4. Summary [6] P. Möller, et al., Phys. Lett. B 40 (1972) 329. [7] S. Cwiok,´ et al., Phys. Lett. B 322 (1994) 304. In summary, we have measured the prompt fission [8] J. Blons, et al., Nucl. Phys. A 414 (1984) 1. probability of 236U as a function of the excitation en- [9] J. Blons, Nucl. Phys. A 502 (1989) 121c. [10] S. Bjørnholm, V.M. Strutinsky, Nucl. Phys. A 136 (1969) 1. ergy using the (d, pf ) reaction with high resolution [11] J. Blons, et al., Nucl. Phys. A 477 (1988) 231. in order to study high-lying excited states in the third [12] U. Goerlach, Diploma thesis, University of Heidelberg/MPI well. From the analysis of the rotational band structure Heidelberg, 1978, unpublished. in the two resonance regions around 5.1 and 5.3 MeV [13] M. Just, et al., in: Proceedings of the International Symposium a weighted average for the rotational parameter could on the Physics and Chemistry of Fission, Jülich, 1979, IAEA, ¯ 2 = ± Vienna, 1979, p. 71. be extracted as h /2θ 2.3 0.4keV. [14] W.M. Howard, P. Möller, At. Data Nucl. Data Tables 25 (1980) The corresponding moment of inertia (θ = 217 ± 219. 38 h¯ 2/MeV) agrees with the calculated value of Shnei- [15] P. Ekström, http://nucleardata.nuclear.lu.se/database/masses. dman et al.. The hyperdeformed rotational-band struc- [16] G. Audi, et al., Nucl. Phys. A 729 (2003) 1. ture observed in the rather low excitation energy re- [17] A.H. Wapstra, G. Audi, C. Thibault, Nucl. Phys. A 729 (2003) 129. gion around 5.1 MeV independently supports our [18] H.A. Enge, S.B. Kowalsky, in: Proceedings of the 3rd Interna- experimental finding of a rather deep third minimum, tional Conference on Magnet Technology, Hamburg, 1970. which is in agreement with theoretical predictions. We [19] H.F. Wirth, Ph.D. Thesis, TU Munich, 2001 http://tumb1. furthermore used the lowest transmission resonance biblio.tu-muenchen.de/publ/diss/ph/2001/wirth.html. in the third well to determine the height of the inner [20] M. Hunyadi, et al., Phys. Lett. B 505 (2001) 27. = [21] A.V. Ignatyuk, N.S. Rabotnov, G.N. Smirenkin, Phys. Lett. barrier EA 5.05(20) MeV. In the double-humped B 29 (1969) 209. barrier picture the inner barrier for light actinide nuclei [22] P.D. Goldstone, et al., Phys. Rev. C 18 (1978) 1706. always had to be adjusted above the highest observed [23] B.B. Back, et al., Nucl. Phys. A 165 (1971) 449. resonance, while now the existence of a deep third [24] T.M. Shneidman, et al., Nucl. Phys. A 671 (2000). minimum only requires that the inner barrier lies above [25] B.B. Back, et al., Phys. Rev. C 9 (1974) 1924. [26] P.D. Goldstone, et al., Phys. Rev. Lett. 35 (1975) 1141. the highest resonance in the second well. This repre- [27] A. Krasznahorkay, et al., Phys. Lett. B 461 (1999) 15. sents a new trend in fission barriers for light actinides [28] T. Rauscher, et al., Phys. Rev. C 56 (1997) 1613. which is different from former assignments [4],butin [29] S. Bjørnholm, A. Bohr, B. Mottelson, in: Proceedings of the good agreement with more recent theoretical predic- International Symposium on the Physics and Chemistry of Fis- tions [14]. sion, IAEA, Vienna, 1974, p. 367. [30] R. Vandenbosch, J.R. Huizenga, Nuclear Fission, Academic Press, San Diego, 1973, p. 202. [31] P. Möller, et al., At. Data Nucl. Data Tables 59 (1995) 185. Acknowledgements [32] T. Ledergerber, H.C. Pauli, Nucl. Phys. A 207 (1973) 1. [33] U. Goerlach, et al., Z. Phys. A 287 (1978) 171. The work has been supported by DFG under [34] P. Reiter, Ph.D. Thesis, University of Heidelberg, 1993, MPI- H-V10-93. HA 1101/6-3 and 436 UNG 113/129/0, the Hungarian [35] P. Reiter, et al., in: Yu. Oganessian, et al. (Eds.), Proceedings Academy of Sciences under HA 1101/6-1, the Hun- of the Conference on Low Energy Nuclear Dynamics, World garian OTKA Foundation No. T038404. Scientific, Singapore, 1995, p. 200. [36] U. Goerlach, et al., Phys. Rev. Lett. 48 (1982) 1160. [37] M. Just, Ph.D. Thesis, University of Heidelberg, 1978, unpublished. References [38] V. Metag, Habilitation thesis, Heidelberg, 1974. [39] J. Borggreen, et al., Nucl. Phys. A 279 (1977) 189. [1] V. Metag, et al., Phys. Rep. 65 (1980) 1. [2] P.G. Thirolf, D. Habs, Prog. Part. Nucl. Phys. 49 (2002) 245. Physics Letters B 615 (2005) 186–192 www.elsevier.com/locate/physletb

30 32 Magnetic moments of 13Al17 and 13Al19 H. Ueno a, D. Kameda b, G. Kijima b,K.Asahia,b, A. Yoshimi a, H. Miyoshi b, K. Shimada b,G.Katob, D. Nagae b, S. Emori b, T. Haseyama a, H. Watanabe a,c, M. Tsukui b

a RIKEN, 2-1 Hirosawa, Wako, Saitama 351-0198, Japan b Department of Physics, Tokyo Institute of Technology, 2-12-1 Oh-okayama, Meguro-ku, Tokyo 152-8551, Japan c Department of Nuclear Physics, Research School of Physical Sciences and Engineering, The Australian National University, Canberra, AT 0200, Australia 1 Received 2 March 2005; received in revised form 29 March 2005; accepted 1 April 2005 Available online 25 April 2005 Editor: D.F. Geesaman

Abstract Ground-state magnetic moments of 30Al and 32Al were measured with the β-NMR method using radioactive-isotope beams spin-polarized in the projectile-fragmentation reaction. Polarization of sizes |P |=0.5–1% were obtained in spite of the large numbers of nucleons that are removed from the projectile 40Ar, providing a promising prospect that substantial polarizations 30 are obtained even in fragment nuclei that are far removed from the projectile nucleus. The obtained µ moments, |µexp( Al)|= 32 3.010(7)µN and |µexp( Al)|=1.959(9)µN, are in agreement with shell model calculations within the sd valence space, although a reduction in the energy-gap between the sd and pf states is predicted for 32Al in recent theoretical studies.  2005 Elsevier B.V. All rights reserved.

PACS: 21.10.Ky; 21.60.Cs; 24.70.+s; 25.70.Mn; 27.30.+t; 29.27.Hj; 76.60.-k

In the region of neutron-rich sd-shell nuclei, in- observed only in a localized area consisting of Ne, Na, triguing phenomena have been reported. Anomalously and Mg isotopes around the neutron number N = 20. tight binding was discovered in the nuclei around A These observations are considered as manifestations 32 [1,2]. Lowering of excitation energies of the ﬁrst of the deformation induced by the inversion of am- 2+ excited states [3,4] and large B(E2) values for their plitudes between sd-normal and pf -intruder conﬁg- excitation [5] were also found. Such phenomena were urations [6]; the region is known as the “island of inversion”. Central to this issue is to clarify what is the governing parameter whose variation brings about E-mail address: [email protected] (H. Ueno). the sudden amplitude inversion at a certain value of the 1 Present address. N/Z ratio. Microscopic studies of nuclei close to the

“island of inversion”, as well as those inside it, would offer a clue to this question. The mixing of the intruder configurations from the pf orbits in neutron-rich N = 20 isotopes has been studied in the Monte Carlo shell model [7]. These studies show that the ground state, which is normally occupied by the 0p–0h configurations as in 34Si, is taken over by the intruder 2p–2h or 4p–4h configurations in 30Ne, 31Na, and 32Mg. In 33Al, the predicted admixture of the intruder configurations is not as large as in the above nuclei but still substantial. These calculations indicate that the neutron-rich aluminum isotopes are located close to the borderline of the “island of inversion”, so that they would be important nuclei in the investigation of the “inversion” region. The evolution of the inversion along the neutron-rich aluminum isotopes has been studied in other sd isotopes on the basis of large-scale shell model calculations in terms of the energy difference between the sd and pf Fig. 1. Arrangement of fragment separator RIPS for the production configurations [8]. The predicted energy difference de- of spin-polarized RIBs, and the schematic layout of the β-NMR ap- creases from 31Al to 33Al, which suggests that the paratus. admixture of the intruder configurations already occurs near 33Al. This picture well accounts for recent tive nucleus is produced [11]. The mechanism of the experimental observations for neutron-rich aluminum fragmentation-induced spin-polarization is essentially isotopes. Through the measurement of the ground- related to the fact that a portion of the projectile to be state magnetic moment of 31Al, it was concluded that removed through the fragmentation process has non- 31 Al is described by a rather pure d5/2-proton and vanishing angular momentum due to the Fermi mo- I π =0+ (sd)10 -neutron configuration [9], suggesting that tion of nucleons. Because the polarization mechanism 31Al is located outside of the “island of inversion”. For relies on the general feature of the fragmentation re- the next isotope 32Al, an unusual ordering of the ex- action, it is suggested that essentially any fragments cited states was reported, which might be associated would be polarized irrespective of their chemical prop- with the reduced sd–pf shell gap [10]. erties. In the present work, the magnetic moments µ for Fig. 1 shows the arrangement of the RIKEN pro the ground states of 30Al (I π = 3+) and 32Al (I π = jectile-fragment separator RIPS [13] for the produc- 1+) have been measured. The µ moment is an observ- tion of spin-polarized RIBs. In the present experiment, able which sensitively reflects in which orbitals the a beam of 30Al (32Al) was obtained from the frag- valence nucleons reside. According to the above pic- mentation of 40Ar projectiles at an energy of E = ture, 30Al is located outside of the island, whereas 32Al 95 A MeV on a 93Nb target of 0.13 (0.37) g/cm2 is somewhere close to the border. From this point of thickness. In order to have the RIB spin-polarized, view, the magnetic moments of these two nuclei would the emission angle and the outgoing momentum be interesting for microscopic studies of the evolution were selected. Thus, fragments emitted at angles ◦ ◦ of the “inversion”. θL = 1.3 –5.7 were accepted by RIPS using a beam The availability of spin-polarized radioactive-iso- swinger installed upstream of the target. A range of tope beams (RIBs) [11] offers us an opportunity to momentum p = 12.4–12.7GeV/c (12.2–13.0GeV/c) measure the electromagnetic moments of nuclei far was selected by a slit at the momentum-dispersive fo- from the β-stability line [12]. In this method a radioac- cal plane F1. This momentum range corresponds to tive nucleus is spin-polarized through the projectile- 1.006–1.026p0 (0.975–1.034p0), where p0 is a peak fragmentation reaction itself by which the radioac- in the momentum distribution of the 30Al (32Al) beam. 188 H. Ueno et al. / Physics Letters B 615 (2005) 186–192

The isotope separation was provided by combined was kept in the vacuum chamber and cooled to a tem- analyses of the magnetic rigidity and momentum loss perature of T<100 K to assure the preservation of [13]. The purity of the 30Al (32Al) secondary beam spin polarization during the β-decay counting period. was 93% (86%). Subsequently the spin-polarized frag- For the preservation of polarization the static magnetic ments are introduced into the NMR apparatus located field B0 ∼ 0.5 T was applied to the stopper [14]. at the final focus. We employed the β-NMR method [15] to deter- The adiabatic fast passage (AFP) technique of mine the µ moment. In this method, β-rays emitted NMR [16] was used in the present experiment. A radio- from the implanted fragments are detected with plas- frequency oscillating field B1 of amplitude ∼ 1mT tic scintillator telescopes located above and below the was applied perpendicular to an external static mag- stopper. The up/down ratio R of the β-ray yields is netic field B0 with a pair of coils located outside a given by vacuum jacket in which a stopper material was placed. + v (1 c Aβ P) The frequency of the oscillating field B1 was swept R = a a(1 + 2A P), (2) ( − v A P) β over a certain region, and when the region includes 1 c β the Larmor frequency spin reversal takes place. where a is a constant factor representing asymmetries The 30Al and 32Al nuclei were implanted in an in the counter solid angles and efficiencies, v and c α-Al2O3 (corundum) single-crystal stopper mounted the velocities of the β particles and light, respectively. at the center of the β-NMR system. The stopper ma- In Eq. (2), we took an approximation that v/c 1, terial must be chosen so as to provide a spin-lattice since only a high-energy portion of the β spectra was relaxation time T1 that is long compared with the β- included in the analysis. When the polarization P is decay lifetime. Up to now, β-NMR signals for 28Al, altered due to the resonant spin reversal, a change ap- whose β-decay mean lifetime τ = 3.2 m is much pears in the ratio R. Thus the resonance frequency ν0 longer than those of 30Al and 32Al, have been obtained is derived from the observed peak or dip in the R spec- in an Al2O3 crystal [14]. This fact indicates that T1 for trum. 30 32 30 32 Al and Al in Al2O3 should be longer than their The beam in the Al ( Al) experiment was pulsed β-decay lifetimes. The corundum sample was cut, so with beam-on and off periods of 5 s (46 ms) and that the c-axis of the hexagonal crystal structure was 5.012 s (58 ms), respectively. At the beginning of the oriented parallel to a surface of the sample. The ori- beam-off period, the (B1) field was applied for a 6 ms entation was confirmed by X-ray diffraction analysis. (6 ms) duration, with its frequency swept from ν1 to The crystal c-axis was oriented at the magic angle ν1 + δν. Then the β rays were counted during the ◦ (54.7 )totheB0 field, as shown in the inset of Fig. 1, following 5 s (46 ms). In the remaining 6 ms (6 ms) where the quadrupolar splitting of the magnetic sub- the B1 field was applied again in the same scheme in levels vanishes. In first order perturbation theory, the order to restore the spin direction, so that the R ra- transition frequency between the magnetic sub-levels tio in the succeeding cycle might not be affected by m and m ± 1 of nuclear spins I under the combined the surviving activities. In the following n − 1cy- Zeeman and electric-quadrupole interactions is given cles, the above procedure was repeated with frequency by ranges νi to νi + δν (i = 2, 3,...,n), and then a cycle without the application of B was performed. These 2 − + 1 eqQ 3(3 cos θc-axis 1)(2m 1) n + 1 cycles, as a unit of sequence for the n-point νm,m±1 = ν0 ∓ . h 8I(2I − 1) spectrum measurement, were repeated many times to (1) obtain a sufficient counting statistics. The number n Here ν0 denotes the Larmor frequency, q the electric of the frequency points typically took values 1–7. In field gradient along the c-axis (axial symmetry about this scheme, the effect of the long-term fluctuation of the c-axis is assumed for the electric field gradient), the beam profile on the asymmetry should be removed. and θc-axis the angle between the c-axis and the B0 Whole control of the time sequence and frequency 2 field. The second term vanishes when cos θc-axis = sweep was given by the PSG module [17]. 1/3. The actual tilt angle of the α-Al2O3 crystal was The time spectrum obtained for the β-ray events ◦ 30 measured as θc-axis = 54.4(5) .Theα-Al2O3 stopper accumulated in the Al experiment is shown in H. Ueno et al. / Physics Letters B 615 (2005) 186–192 189

Table 1 Uncertainties and corrections taken into account for the determination of µ moments. The B0 ﬁelds and resonance frequencies are also shown. For chemical shift and νQ/ν0 values, see the text 30Al 32Al

(Static magnetic ﬁeld B0 500.59 mT 498.61 mT) Stability of B0 0.06% 0.4% Inhomogeneity of B0 < 0.02% < 0.02% Chemical shift < |−0.00002|% < |−0.00002|% (Resonance frequency ν0 3829.0 kHz 7445.7kHz) Fitting error 0.21% 0.08% Error in the frequency setting 0.014% 0.012% νQ/ν0 0.09% 0.15% Total 0.24% 0.44% µexp 3.010(7)µN 1.959(9)µN

states of 30Al and 32Al. As seen in Eq. (2), a deviation of the R ratio from that obtained without the B1 field (open circles in the figures) indicates the occurrence of the spin alteration by the AFP-NMR. We found in the 30Al spectra that, within a searched region of frequency, only the intervals including a common frequency ν 3829 kHz exhibited the up/down ratios deviating from those without the B1 field. Similar observation applies to the 32Al spectra: only the intervals including a common frequency ν 7446 kHz exhibited the deviation. The resonance frequency ν was obtained by the β 30 0 Fig. 2. Time spectra obtained for the -ray events in the (a) Al and 2 (b) 32Al experiments. Curve shows the result of a least-χ2 fitting of least-χ fitting analysis. Reflecting the amplitude a function to the data. Statistical error for each data point is much modulation and frequency sweep of the B1 field, smaller than the size of the circle. the NMR spectra are known to show a characteristic shape. Experimental µ moments were derived by Fig. 2(a). The spectrum was fitted with an expo- fitting the NMR spectra shown in Figs. 3(a2) and nential function plus a constant background. The (b2) with a function obtained from a computer sim- obtained halflife, t1/2 = 3.63(4) s, agrees with the ulation of the AFP spin reversal process. As a re- 30 weighted average of the two reported values t1/2 = sult, the resonance frequencies ν0( Al) = 3829.0(79) 32 3.69(3) s [18] and t1/2 = 3.56(2) s [19]. Similar kHz and ν0( Al) = 7445.7(58) kHz were obtained analysis was made for 32Al with the obtained spec- for 30Al and 32Al, respectively. To evaluate the er- trum shown in Fig. 2(b). The obtained halflife, t1/2 = ror in the experimental µ moments, the uncertainties 33.0(2) ms, also agrees with the reported value t1/2 = and corrections listed in Table 1 were taken into ac- 35(5) ms [20], but the present accuracy is higher. count. 30 32 These results confirmed that the Al and Al iso- In Table 1, the uncertainty in νQ due to the topes were correctly identified in the implantation quadrupolar interaction is listed as νQ/ν0, consid- procedure. ering the adjustment accuracy of the c-axis angle, ◦ In the actual measurement of the µ moments, sev- θc-axis = 54.4(5) , to the magic angle. Here, we define eral runs of resonance scan with progressively nar- νQ as the deviation of the m = 0 →+1 transition rower frequency windows were carried out. Fig. 3 frequency νm=0,1 from ν0 due to the misalignment shows the β-NMR spectra obtained for the ground θc-axis from the magic angle. According to Eq. (1), 190 H. Ueno et al. / Physics Letters B 615 (2005) 186–192

Fig. 3. NMR spectra obtained for 30Al in (a1) a resonance scan with wider frequency windows and (a2) the precision measurement. Also those obtained for 32Al are shown in (b1) and (b2). Up/down ratios of the β-rays measured for the respective frequency windows are indicated by solid circles, whereas those obtained without the B1 ﬁeld are shown by open circles. They are accompanied by the statistical errors (vertical bars) and the widths of the swept frequency (horizontal bars). The dotted curves shown in the lower panels (a2) and (b2) are results of the least-χ2 ﬁttings.

27 νQ is written as Al in α-Al2O3 [21],showninTable 1, is negligibly small compared with the errors of the other origins. ∂νm=0,1 30 νQ = θc-axis Within an sd model space, it is expected for Al ∂θc- axis that the valence d5/2 proton and d3/2 neutron cou- π + 9sin2θc-axis eqQ pletoformI = 3 in the main configuration of = θc-axis. (3) 8I(2I − 1) h the ground state. The corresponding theoretical µ moment is 4.59 µN. Hereafter, the theoretical µ mo- To evaluate νQ, we adopted the reported value of quadrupolar coupling constant for 27Al in α-Al O , ments are obtained with the effective µ-moment op- 2 3 ˆ |eqQ/h|=2.30(4) MHz [21], and the Q moment erator µ of Ref. [25], which has been empirically 27 27 determined to reproduce the M1 static and transi- of Al Qexp( Al) = 140.02(10) mb [22]. In addition, shell-model calculations were carried out for Q tion moments in the sd-shell nuclei close to the β- moments [23,24], since no experimental data have stability line. Also, consideration of the admixture of | ⊗ I π =3+ been reported. Thus the obtained theoretical values, the (πd3/2) (νd3/2) configuration should be 30 32 important, since its µˆ expectation value 1.42 µ is dis- QSM( Al) = 210 mb and QSM( Al) = 45 mb, were N taken. Inserting these values into Eq. (3), we obtain tinctly small and might bring the theoretical µ moment 32 values ν /ν = 0.09% and 0.15% for 30Al and 32Al, down to the experimental value. In the case of Al, Q 0 30 respectively. Taking all the uncertainties listed in Ta- the addition of two neutrons to Al just changes the ble 1 into account, we have determined the experi- role played by a d3/2-neutron particle state into that 30 by the corresponding hole state. In the 32Al ground mental µ moments, |µexp( Al)|=3.010(7)µN and 32 30 32 state the valence d proton and d -neutron hole are |µexp( Al)|=1.959(9)µN,for Al and Al, re- 5/2 3/2 coupled to form I π = 1+.Again,theµˆ expectation spectively, where µN is the nuclear magneton. No diamagnetic-shielding correction was made for the value for this configuration, 2.76 µN, and that for the I π =1+ present values, since the reported chemical shift of admixing |(πd3/2) ⊗ (νd3/2)−1 configuration, H. Ueno et al. / Physics Letters B 615 (2005) 186–192 191

Table 2 the fragmentation reactions that lead to production Comparison of experimental magnetic moments (µexp)and shell- of 30Al and 32Al fragments from the 40Ar projec- model predictions (µSM) for aluminum isotopes. The differences tile imply removals of as many as 8 and 10 nucleare calculated assuming positive signs for the µexp values if they are not assigned ons. 30 π In summary, the ground-state µ moments of Al Isotope I µexp (µN)µSM (µN)µSM–µexp (µN) 32 + and Al have been determined by the β-NMR method 26Al 5 +2.804(4) +2.95 0.14 (5.0%) + taking advantage of spin-polarized RI beams from the 27Al 5/2 +3.6415069(7) +3.75 0.11 (3.0%) + projectile-fragmentation reaction. Contrary to expec- 28Al 3 3.242(5) +3.29 0.04 (1.3%) + 30 32 30Al 3 3.010(7) +3.19 0.18 (6.1%) tation, the observed polarizations in Al and Al + 40 31Al 5/2 3.793(50) +3.98 0.19 (5.0%) fragments reached from Ar projectile via 10- and 8- + 32Al 1 1.959(9) +2.06 0.10 (5.0%) nucleon removals were as large as |P |=0.5–1%. The presently obtained polarizations provide a promising prospect that substantial magnitude of spin polariza- 1.42 µN are very different. The experimental µ lies in tion would be obtained in projectile-fragmentation re- between. actions involving many-nucleon removals, thus mak- The experimental µ values were compared with ing the µ-moment measurements feasible for the shell-model calculations. The calculations were made sd-shell nuclei around and beyond the neutron number using a code OXBASH [23] with the USD interac- N = 20. tion [24] and the above mentioned effective µˆ oper- The µ moments obtained for 30Al and 32Al are ator. The theoretical µ moments thus calculated are compared with shell-model calculations within the sd 30 32 +3.19 µN and +2.06 µN for Al and Al, respec- model space. In both cases, agreements within 6% tively, as listed in Table 2. In both cases, theoretical were obtained. Although a larger reduction of the sd– values are in agreement with the present experimen- pf energy gap has been predicted for 32Al than 30Al, tal values within ∼ 6%. Comparison was also made the observed deviations for 30Al and 32Al between the- with experimental values of other aluminum isotopes ory and experiment are of the same order. For more [9,26]. As listed in Table 2, they agree well with each quantitative discussion of the admixture of intruder other. Thus no prominent deviation was observed in all pf configurations in the 32Al ground state, however, of these aluminum isotopes. further study is needed. Although we have investi- With respect to 32Al, an isomer state has been found gated deviations of the standard shell-model calcu- recently at the excitation energy Ex = 956 keV [10], lation within the sd model space from the obtained 32 above the excited state at Ex = 734 keV [10,27]. Spin µexp value for Al, detailed discussion could be made and parity I π = 2+ and 4+ have been assigned to by comparing with the recent large-scale calculation the 734 and 956 keV levels, respectively, from the that takes the admixture of pf configurations into ac- study of their γ -decay properties. It is pointed out in count. Ref. [10] that this ordering is unusual and the USD interaction fails to reproduce it. Since such disagree- ments between calculations carried out with the USD Acknowledgements interaction and experimental observations are rare, this unusual level ordering of 32Al might stem from the presence of the “island of inversion”. In the present The authors are grateful to staffs at the RIKEN study, however, we found that the USD interaction is Ring Cyclotron for their support during the running of able to reproduce the experimental ground-state µ mo- the experiments. They would like to thank Dr. E. Yagi ment of 32Al as well as that of 30Al. for useful help and advice with crystallographic analy- Returning to the β-NMR spectra in Fig. 3,the sis of the corundum sample. This work was supported depths of the resonance dips indicate that the result- in part by a Grant-in-Aid for Scientific Research from ing polarization of 30Al and 32Al fragments were the Ministry of Education, Science, Sports and Cul- P(30Al) =−0.47(12)% and P(32Al) =−0.82(19)%. ture. The experiment was performed at RIKEN under These polarizations are unexpectedly large, because the Experimental-Program-No. R384n. 192 H. Ueno et al. / Physics Letters B 615 (2005) 186–192

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Determination of the Gamow–Teller quenching factor from charge exchange reactions on 90Zr

K. Yako a,H.Sakaia,b, M.B. Greenﬁeld c, K. Hatanaka d, M. Hatano a,J.Kamiyae, H. Kato a,Y.Kitamurad,Y.Maedaa, C.L. Morris f, H. Okamura g, J. Rapaport h, T. Saito a,Y.Sakemid, K. Sekiguchi i,Y.Shimizud,K.Sudab,A.Tamiid, N. Uchigashima a, T. Wakasa j

a Department of Physics, University of Tokyo, Bunkyo, Tokyo 133-0033, Japan b Center for Nuclear Study, University of Tokyo, Bunkyo, Tokyo 133-0033, Japan c International Christian University, Mitaka, Tokyo 181-8585, Japan d Research Center for Nuclear Physics, Osaka University, Ibaraki, Osaka 567-0047, Japan e Japan Atomic Energy Research Institute, Tokai, Ibaraki 319-1195, Japan f Los Alamos National Laboratory, Los Alamos, NM 87545, USA g Cyclotron and Radioisotope Center, Tohoku University, Sendai, Miyagi 980-8578, Japan h Department of Physics, Ohio University, Athens, OH 45701, USA i The Institute of Physical and Chemical Research (RIKEN), Wako, Saitama 351-0198, Japan j Department of Physics, Kyushu University, Higashi, Fukuoka 812-8581, Japan

Received 11 August 2004; received in revised form 4 April 2005; accepted 14 April 2005

Available online 22 April 2005

Editor: D.F. Geesaman

Abstract ◦ ◦ Double differential cross sections between 0 –12 were measured for the 90Zr(n, p) reaction at 293 MeV over a wide excitation energy range of 0–70 MeV. A multipole decomposition technique was applied to the present data as well as the previously obtained 90Zr(p, n) data to extract the Gamow–Teller (GT) component from the continuum. The GT quenching factor Q was derived by using the obtained total GT strengths. The result is Q = 0.88 ± 0.06, not including an overall normalization uncertainty in the GT unit cross section of 16%.  2005 Elsevier B.V. All rights reserved.

PACS: 24.30.Cz; 25.40.Kv; 27.60.+j

Keywords: Charge exchange reaction; Gamow–Teller strength; Gamow–Teller sum rule

E-mail address: [email protected] (K. Yako).

The (p, n) reaction at intermediate energies (Tp > σˆGT data at 295 MeV [12]. On the other hand, the 100 MeV) provides a highly selective probe of spin– uncertainty in the Sβ+ value is difficult to properly isospin excitations in nuclei due to the energy de- assess. A Sβ+ value of 1.0 ± 0.3usedinRef.[10] pendence of the isovector part of nucleon–nucleon was obtained from a similar, but simpler MD analy- (NN) t-matrices [1].The0◦ spectrum of this reac- sis of 90Zr(n, p) data at 198 MeV by Raywood et al. tion is marked by the dominance of the Gamow– [13]. Raywood et al. found a significant amount of the Teller (GT) giant resonance (GTGR), which is the στ± monopole (L = 0) cross sections in the continuum in mode [2,3]. There exists a model-independent sum the region of Ex = 8–31 MeV, which corresponds to rule, Sβ− − Sβ+ = 3(N − Z), where Sβ± is the total a GT strength of ∼5, but attributed all to the isovec- GT strength observed for the β± type [4]. Surpris- tor spin-monopole (IVSM) strength [13]. Thus, the ingly, however, only a half of the GT sum rule value quenching factor was subject to uncertainties of both was identified from (p, n) measurements in the 1980s the β+ strength in the continuum and contributions of on targets throughout the periodic table [5]. This prob- the IVSM component. To reduce those systematic un- lem, the so-called quenching of the GT strength, has certainties, it is essential to have accurate (n, p) data been one of the most interesting phenomena in nuclear at the same energy as the (p, n) data, and to perform physics because it could be related to non-nucleonic consistent analyses on both sets of data. In this Let- degrees of freedom in nuclei. At that time it was often ter the measurement of the 90Zr(n, p)90Y reaction at assumed that the missing strength was shifted to the 293 MeV is reported. Performing a consistent analy- energy region of the excitation due to coupling be- sis on both the (p, n) and (n, p) data, we have derived a tween nucleon particle hole (ph) and -isobar nucleon reliable GT quenching factor. hole (h) states [6,7]. However, part of the quench- The measurement was performed with the (n, p) fa- ing is due to the nuclear configuration mixing between cility [14] at the Research Center for Nuclear Physics 1p1h and 2p2h states [8,9]. To discuss the contribution (RCNP). A schematic view of the (n, p) facility is of 2p2h states quantitatively, one should search for shown in Fig. 1. A nearly mono-energetic neutron missing GT strength in the continuum, excitation en- beam was produced by the 7Li(p, n) reaction at 295 ergy region of 20–50 MeV, where a significant amount MeV. The primary proton beam, after going through of the 2p2h component is predicted [8,9]. the 7Li target, was deflected away by the clearing mag- In 1997, Wakasa et al. [10] accurately measured net to a Faraday cup in the floor. The typical beam the 90Zr(p, n) spectra at 295 MeV, the energy at which intensity was 450 nA and the thickness of the 7Li tar- spin-flip cross sections are large, distortion effects are get was 320 mg/cm2. About 2 × 106 s−1 neutrons minimal [11] and therefore the characteristic shapes of the angular distributions for each angular momentum transfer (L) are most distinct. They successfully identified the GT strength in the continuum region through multipole decomposition (MD) analysis which extracted the L = 0 component from the cross sections [10]. They obtained a GT quenching factor, defined as

S − − S + Q ≡ β β , 3(N − Z) of 0.90±0.05, where the error is due to the uncertainty of the MD analysis [10]. As discussed in Ref. [10], the main source of the systematic uncertainties is the overall normalization, i.e., the GT unit cross section and the Sβ+ value. The uncertainty of the GT unit cross section amounts to 16%, but will be reduced to ∼5% by the ongoing systematic analysis of new Fig. 1. A schematic view of the (n, p) facility at RCNP. K. Yako et al. / Physics Letters B 615 (2005) 193–199 195 bombarded the target area of 30W × 20H mm2 located 95 cm downstream from the 7Li target. Three 90Zr targets with thicknesses of 485, 233, and 215 mg/cm2 and a polyethylene (CH2) target with a thickness of 46 mg/cm2 were mounted in a multiwire drift chamber (MWDC). Wire planes placed between the targets detected outgoing protons and enabled us to determine the target in which the reaction had occurred. Charged particles coming from the beam line were rejected by a veto scintillator with a thickness of 1 mm. The scattering angle of the (n, p) reaction was determined by the information from the target MWDC and another MWDC installed at the entrance of the Large Acceptance Spectrometer (LAS) [15]. The outgoing protons were momentum-analyzed by LAS and were detected by the focal plane detectors [16].The 1 Fig. 2. Double differential cross sections for the 90Zr(n, p) (left number of H(n, p) events from the CH2 target was 90 1 panel) [10] and Zr(p, n) (right panel) reactions. The histograms compared to the SAID [17] calculated H(n, p) cross show the results of the MD analyses. sections for normalization of the neutron beam flux. Blank target data were also taken for background subtraction. of calculated distributions, We have obtained double differential cross sections calc = calc up to 70 MeV excitation energy over an angular range σ (θcm,Ex) aJ π σph;J π (θcm,Ex), (1) π of 0◦–12◦ in the laboratory frame. The data have been J 90 analyzed in 1-degree bins. The Zr(n, p) spectra at where the variables aJ π are fitting coefficients all three of the twelve angles are shown in the left panel in of which have positive values. The calculated an- Fig. 2 by the solid dots. The overall energy resolution gular distributions for each spin and parity transfer calc is 1.5 MeV, mainly originating from the target thick- σph;J π (θcm,Ex) have been obtained using the dis- nesses and the energy spread of the beam. The angular torted wave impulse approximation (DWIA) calcula- resolution is 10 mr which is dominated by multiple tions described below. scattering effects in the 90Zr targets. In addition to the The DWIA calculations were performed with the statistical uncertainty of ∼ 2% per 2-MeV excitation computer code DW81 [18] for the following J π trans- energy bin, there is a systematic uncertainty of 5%, fers: 1+ (L = 0),0−, 1−, 2− (L = 1),3+ (L = where the main contributions are that of target thick- 2), and 4− (L = 3). The one-body transition den- nesses (4%) and the angular distribution of the n + p sities were calculated from pure 1p1h configurations. −1 −1 cross section taken from the phase-shift analysis (2%). The (1g7/2, 1g9/2) and (1g9/2, 1g9/2) configurations The right panel in Fig. 2 shows the 90Zr(p, n) spectra were used to calculate the GT transitions in the analy- [10].At0◦ the (p, n) cross sections are larger than the ses of both the (p, n) and the (n, p) spectra. For the (n, p) cross sections not only in the GT resonance re- transitions with L 1 in the (p, n) channel, the ac- gion but also in the high excitation energy region of tive proton particles were restricted to the 1g9/2,1g7/2, 90 Ex = 70 MeV due to excess neutrons in Zr. 2d5/2,2d3/2,1h11/2,or3s1/2 shells, while the ac- The MD analyses were performed on the (p, n) tive neutron holes were restricted to the 1g9/2,2p1/2, 40 [10] and (n, p) excitation energy spectra to obtain 2p3/2,1f5/2,or1f7/2 shells by assuming Ca to be GT strengths. Details of the MD analysis are given the core. In the analysis of the (n, p) spectra, the active in Ref. [10]. For each excitation energy bin between neutron particles were restricted to the 1g7/2,2d5/2, 0 MeV and 70 MeV, the experimentally obtained an- 2d3/2,1h11/2,or3s1/2 shells, while the active pro- exp gular distribution σ (θcm,Ex) has been fitted using ton holes to 2p1/2,2p3/2,1f5/2,or1f7/2. The optical the least-squares method with the linear combination model potential (OMP) parameters for proton were 196 K. Yako et al. / Physics Letters B 615 (2005) 193–199 taken from Ref. [19]. The OMP parameters for neutron were also taken from Ref. [19], but without the Coulomb term. The effective NN interaction was taken from the t-matrix parameterization of the free NN interaction by Franey and Love at 325 MeV [1].It should be noted that DWIA calculations using this parameter set better reproduce the polarization trans- ◦ 90 fer DNN(0 ) for the Zr(p, n) reaction than those at 270 MeV [10]. The radial wave functions were generated from a Woods–Saxon (WS) potential [20], adjusting the depth of central potential V0 to reproduce the binding energies [21–24]. The unbound particle states were assumed to have a shallow binding energy to simplify the calculations. For a given J π transfer, the shapes of the angular distributions depend on the 1p1h configurations. Thus, all combinations of 1p1h 2 configurations were examined by the χ -minimization Fig. 3. GT plus IVSM strength distributions obtained by the MD program and the optimal combination was obtained for analysis of the 90Zr(p, n) and 90Zr(n, p) reactions (in GT unit). The each excitation energy bin. 90Zr(n, p) spectrum is shifted by +18 MeV. The curves are taken Results of the MD analyses are shown in Fig. 2. from Ref. [29]. The energy regions of IVSM excitation are indicated The obtained L = 0 component in the (p, n) spec- by braces. See text for details. tra has a large contribution not only in the GTGR region, but also in the high excitation energy region has not improved the stability. Therefore, we set the up to 50 MeV [10].TheL = 0 component of the upper limit of the excitation energy to 50 MeV. The cross section, σL=0(q, ω), is proportional to the GT integrated strength thus obtained is S(p,n);GT+IVSM = strength B(GT) [5] such that 33.5 ± 0.6(stat.) ± 0.4(MD) ± 4.7(σˆGT) up to 50 MeV excitation energy, neglecting the uncertainty in the σ = (q, ω) =ˆσ F(q,ω)B(GT), (2) L 0 GT subtraction of the IAS contribution. The first error where σˆGT is the GT unit cross section [10] and is the statistical error of the MD analysis. The sys- F(q,ω) is the kinematical correction factor [25].The tematic uncertainty due to the input parameters for GT unit cross section has been determined so that the DWIA calculations has been evaluated by using the Sβ− value up to the GTGR region of Ex < 16 MeV wave functions generated from a potential by the rel- becomes 18.3 ± 3.0 [5] as described in Ref. [10].The ativistic Hartree approach [27] or with other OMPs obtained value is σˆGT = 3.5 ± 0.6mb/sr, which is [11,27,28], and it is estimated to be ±0.4. The third consistent with 3.6 ± 0.6mb/sr, the value used in error reflects the error in the GT unit cross section. Ref. [10]. The distribution of β+ strength in Fig. 3 is shifted The strength distributions are shown in Fig. 3.Here by +18 MeV, accounting for the Coulomb displace- the contribution from the isobaric analogue state (IAS) ment energy and the nuclear mass difference. The at 5.1 MeV, corresponding to 0.7 ± 0.1 in GT unit strength integrated up to 32 MeV excitation of 90Y, [10], is already subtracted. The strength is denoted as or Ex = 50 MeV in Fig. 3,isS(n,p);GT+IVSM = 5.4 ± B(GT + IVSM) because it contains the IVSM com- 0.4(stat.) ± 0.3(MD) ± 0.9(σˆGT). ponent [13]. The error bars are the ±1σ confidence The curves in Fig. 3 show the theoretical predic- limits obtained by a Monte Carlo simulation, where tions of GT strength distribution by employing the the χ2 minimization is performed for synthetic data dressed particle random phase approximation (DRPA) sets generated by replacing the actual data set in accor- model [29], folded by a Gaussian distribution to sim- dance with the statistical errors [26]. The MD analysis ulate the energy resolution of the measurement. The of the (p, n) spectra becomes unstable above 50 MeV agreement between the experiment and the theory excitation [10] and reanalysis with larger bin width is excellent, except in the excitation energy region K. Yako et al. / Physics Letters B 615 (2005) 193–199 197

around 30–40 MeV, where the IVSM resonance is im- obtained Sβ+ = 1.7 ± 0.2 up to 10 MeV excitation portant [13,30]. The IVSM resonance is the 2hω¯ ex- at Tn = 98 MeV while Raywood et al. [13] obtained 2 citation via the r στ± operator and has the same spin 1.0 ± 0.3upto∼ 8 MeV at Tn = 198 MeV. The and parity transfer as the GT resonance. Since the an- errors given above do not include the uncertainty gular distribution of the IVSM transition has a forward in the GT unit cross section. The β+ strengths ob- peaking shape similar to that of the GT transition [31, tained in this work up to 8 MeV and 10 MeV ex- 32], the present MD analysis cannot discriminate these citation are Sβ+ = 0.4 ± 0.1(stat.) ± 0.1(σˆGT) and two components. 0.7 ± 0.1(stat.) ± 0.1(σˆGT), respectively. These Sβ+ Hamamoto and Sagawa studied the IVSM modes values are significantly smaller than those obtained by for both the 90Zr(p, n) and 90Zr(n, p) reactions [30]. Raywood et al. or by Condé et al. We note that the They pointed out that the response function to the op- analysis of the Tn = 98 MeV data may suffer from erator r2στ± calculated for the 2hω¯ excitation con- ambiguities in the reaction mechanism due to multi- tains a small GT component because of the difference step processes while the MD analysis at 200 MeV may of the neutron and proton one-particle wave functions suffer from the ambiguity due to the distortion effects with the same quantum numbers. This GT component which are larger than those at 300 MeV. has to be subtracted to obtain the pure IVSM strength The consistent analyses of both (p, n) and (n, p) [30]. Although interference exists between the GT and spectra yield a quenching factor of Q = 0.88 ± the IVSM modes, the cross section associated with the 0.02(stat.) ± 0.05(syst.) ± 0.01(MD) ± 0.02(IVSM), pure IVSM component is estimated in this work and where the systematic uncertainty of the normaliza- its contribution is subtracted incoherently from each tion in the cross section (5%) is also indicated. An spectrum since the distribution of the GT strength in uncertainty in the GT unit cross section of 16% is the IVSM resonance region is unknown. The DWIA not included. It should be noted that since the er- calculations have been performed by assuming that the rors are correlated, the combined systematic errors strengths are fully exhausted in the state with central are smaller than their geometric sum. Thus quenching energies reported to be 35 MeV in the (p, n) spectrum of GT strength up to 50 MeV due to coupling be- [32], corresponding to 19 MeV excitation of 90Y, and tween ph and h states becomes significantly smaller Ex = 37 MeV in Fig. 3, in the (n, p) spectrum [30]. when 2p2h contributions are properly accounted for The transition densities are obtained by the procedure [9,10]. of Condé et al. [33]. The small GT components are ex- The interpretation of this small quenching in terms plicitly eliminated by modifying the radial wave func- of the short range correlation in nuclei is particularly tions of the final states. The obtained IVSM strengths interesting. Assuming that the missing GT strength of are within 10% of the theoretical prediction [30].The ∼10% is attributed to the excitation, one can de- ◦ calculated IVSM cross sections at 0 for the (p, n) and rive the Landau–Migdal (LM) parameter gN which the (n, p) channels are 6.9 ± 1.5mb/sr (4.2 ± 0.9GT describes the short range correlations for ph → h = ± ± + + units with F(q,ω) 0.47) and 5.3 0.6mb/sr (2.5 transitions in the π ρ g model [34].ThegN 0.3 GT units with F(q,ω)= 0.61), respectively. The value deduced by using an RPA model in ph and h uncertainties are mainly due to the choice of OMP pa- spaces [34] with the Chew–Low [35] coupling con- rameters. By subtracting these values from SGT+IVSM, stant (f/f π = 2) is shown in Fig. 4 as a function of Q. the actual GT strengths of Sβ− = 29.3 ± 0.5(stat.) ± If we take here Q = 0.88±0.06, combining the uncer- 0.4(MD) ± 0.9(IVSM) ± 4.7(σˆGT) and Sβ+ = 2.9 ± tainty of MD analysis and that of IVSM contribution 0.4(stat.) ± 0.3(MD) ± 0.3(IVSM) ± 0.5(σˆGT) have in quadrature, then we derive g = 0.18 ± 0.09 as- N − = been obtained. This new Sβ value is consistent with suming g 0.5 [36]. Arima et al. have examined but slightly higher than the one reported in Ref. [10]. the finite size effects of the 90Zr nucleus by taking the The Sβ+ value agrees well with the DRPA prediction finite range interaction due to π-orρ-exchange into of Sβ+ = 3.2 [29]. account [37]. If their argument is employed here, the The present β+ strength may be compared with g value increases by 0.07 for the same Q. There- N previously reported results that employed similar MD fore, it is reasonable to assume the gN valuetobe techniques on the 90Zr(n, p) spectra. Condé et al. [33] 0.25 ± 0.09, which is significantly smaller than that 198 K. Yako et al. / Physics Letters B 615 (2005) 193–199

Acknowledgements

We wish to acknowledge the outstanding support of the accelerator group of RCNP. We thank I. Hamamoto, H. Sagawa and T. Suzuki for valuable discussions. This work was supported ﬁnancially in part by the Grant-in-Aid for Scientiﬁc Research No. 10304018 of Ministry of Education, Science, Cul- ture and Sports of Japan.

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Azimuthal asymmetry in unpolarized πN Drell–Yan process

Zhun Lu a, Bo-Qiang Ma b,a,c

a Department of Physics, Peking University, Beijing 100871, China b CCAST (World Laboratory), P.O. Box 8730, Beijing 100080, China c DiSTA, Università del Piemonte Orientale “A. Avogadro” and INFN, Gruppo Collegato di Alessandria, 15100 Alessandria, Italy Received 2 January 2005; received in revised form 3 April 2005; accepted 3 April 2005 Available online 13 April 2005 Editor: H. Georgi

Abstract Taking into account the effect of ﬁnal-state interaction, we calculate the non-zero (naïve) T -odd transverse momentum ⊥ 2 dependent distribution h (x, k⊥) of the pion in a quark-spectator-antiquark model with effective pion-quark-antiquark coupling 1 − as a dipole form factor. Using the model result we estimate the cos 2φ asymmetries in the unpolarized π N Drell–Yan process ⊥ × ¯⊥ ⊥ 2 which can be expressed as h1 h1 . We ﬁnd that the resulting h1π (x, k⊥) has the advantage to reproduce the asymmetry that agrees with the experimental data measured by NA10 Collaboration. We estimate the cos 2φ asymmetries averaged over the − kinematics of NA10 experiments for 140, 194 and 286 GeV π beam and compare them with relevant experimental data.  2005 Elsevier B.V. All rights reserved.

PACS: 12.38.Bx; 13.85.-t; 13.85.Qk; 14.40.Aq

Keywords: T -odd distribution function; Final-/initial-state interaction; Unpolarized Drell–Yan process; Azimuthal asymmetry

1. Introduction processes [2] from FSI/ISI via the exchange of a gluon, have been explored and are recognized as previ- Recently it is demonstrated that the effect of ously known Sivers effect [3,4]. This effect, formerly final-state interaction (FSI) or initial-state interaction thought to be forbidden by the time-reversal prop- (ISI) can lead to significant azimuthal asymmetries erty of QCD [5], can be survived from time-reversal in various high energy scattering processes involv- invariance due to the presence of the path-ordered ing hadrons [1,2]. Among these asymmetries, single exponential (Wilson line) in the gauge-invariant de- spin asymmetry (SSA) in semi-inclusive deeply in- finition of the transverse momentum dependent par- elastic scattering (SIDIS) [1] and that in Drell–Yan ton distributions [6–8]. Along this direction some phenomenological studies [9–11] have been carried out on transverse single-spin asymmetries in SIDIS E-mail address: [email protected] (B.-Q. Ma). process, which is under investigation by current ex-

⊥ periment [12]. Analogously the exchange of a gluon 2. Non-zero h1π of the pion in spectator model can also lead to another leading twist (naive) T -odd ⊥ 2 distribution h1 (x, k⊥): the covariant transversely po- In this section, we will show how to calculate ⊥ 2 larization density of quarks inside an unpolarized h1π (x, k⊥) in a quark-spectator antiquark model. We hadron. This chiral-odd partner of Sivers effect func- follow Ref. [1] to work in Abelian case at first and then tion, introduced first in Ref. [13] and is referred to generalize the result to QCD. There are pion-quark- as Boer–Mulders function, has been proposed [14] to antiquark interaction and gluon-spectator antiquark in- account for the large cos 2φ asymmetries in the un- teraction in the model: polarized pion-nucleon Drell–Yan process that were L =−g ψγ¯ ψϕ − e ψγ¯ µψA + h.c., (1) measured more than 10 years ago [15,16]. Recently I π 5 π 2 µ ⊥ 2 h1 (x, k⊥) of the proton has been computed in a quark- in which gπ is the pion-quark-antiquark effective cou- scalar diquark model [9,17] and also used to analyze pling, and e2 is the charge of the antiquark. When the the consequent cos 2φ azimuthal asymmetries in both intrinsic transverse momentum of the quark is taken unpolarized ep SIDIS process [9] and unpolarized pp¯ into account, as required by T -odd distributions, the Drell–Yan process [17], respectively. quark correlation function of the pion in Feynman The same mechanism producing T -odd distribu- gauge (we perform calculation in this gauge) is [7,8]: tion functions can be applied to other hadrons such as Φαβ (x, k⊥) mesons. In a previous paper [18] we reported that non- − ⊥ ⊥ 2 zero h of the quark inside the pion (denoted as h ) dξ d ξ⊥ ik·ξ ¯ − − 1 1π = e Pπ |ψβ (0)L0(0 , ∞ ) can also arise from final-state interaction, by applying (2π)3 † − − × L ∞ | | + a simple quark spectator-antiquark model. Among the ξ (ξ , )ψα(ξ) Pπ ξ =0, (2) phenomenological implications of the function h⊥ is 1π L − ∞− an important result for the cos 2φ azimuthal asym- where a(a , ) is the path-ordered exponential metry in the unpolarized π −N Drell–Yan process (Wilson line) accompanied with the quark field which [15,16], which can be produced by the product of h⊥ has the form 1 − of the pion and that of the nucleon. Therefore, one ∞ + − − can investigate how the theoretical prediction of the L0(0, ∞) = P exp −ig A (0,ξ , 0⊥)dξ , asymmetry is comparable with the experimental re- 0− sult, as a test of the theory and the model. In the (3) ⊥ etc. The Wilson line has the importance to make present Letter, based on h1π from our model calculation, we analyze the cos 2φ azimuthal asymmetry in the definition of the distribution/correlation func- the unpolarized π −N Drell–Yan process by consider- tion gauge-invariant. Without the constraint of time- ing the kinematical region of NA10 experiments [15]. reversal invariance, in leading twist the quark correla- To obtain the right QT dependence of the asymme- tion function of the pion can be parameterized into a ⊥ 2 set of leading twist transverse momentum dependent try, we recalculate h1π (x, k⊥) in a spectator model similar to the model used in Ref. [18]. The differ- distribution functions as follows [13,19] ence is that here we treat the effective pion-quark- ⊥ 2 Φ(x,k ) antiquark coupling gπ as a dipole form factor gπ (k ), σ kµ nν = 1 2 + ⊥ 2 µν ⊥ in contrary to the treatment in Ref. [18] where we f1π x,k⊥ n/ h1π x,k⊥ , (4) ⊥ 2 2 Mπ take gπ as a constant. We find that h1π (x, k⊥) result- ⊥ 2 ing from the new treatment together with h (x, k⊥) where n is the light-like vector with components 1 + − = = i [ ] for the nucleon in a similar treatment [10] can repro- (n ,n , n⊥) (1, 0, 0⊥), σµν 2 γµ,γν and Mπ duce the cos 2φ asymmetry which agrees with NA10 is the pion mass. Knowing Φπ (x, k⊥), one can obtain data. We give the asymmetries predicted by our model these distributions from equations averaged over the kinematics of NA10 experiments 2 = + − f1π x,k⊥ Tr Φ(x,k⊥)γ , (5) for 140, 194 and 286 GeV π beam and find that ⊥ 2 i 2h1π (x, k⊥)k⊥ i+ the energy dependence of these asymmetries is not = Tr Φ(x,k⊥)σ . (6) strong. Mπ 202 Z. Lu, B.-Q. Ma / Physics Letters B 615 (2005) 200–206

The calculation of unpolarized distribution function f1π in the antiquark spectator model can be done [20] from the lowest order (without Wilson line) correlation function in Eq. (2). However it cannot lead ⊥ to any T -odd distribution function such as h1π .As demonstrated in Ref. [1], the non-zero T -odd distribution requires final-state interaction from gluon exchange between the struck quark and target spectator. Here we follow the observation in Refs. [6,21] that Fig. 1. Effective correlation function Φ in the antiquark spectator final-state interaction in an initial hadron state can be model with final-state interaction modeled by one gluon exchange. taken into account effectively by introducing an appropriate Wilson line in the gauge-invariant definition + − light-cone vector n¯µ = (n¯ , n¯ , n¯ ⊥) = (0, 1, 0⊥).The of the transverse momentum dependent distribution eikonal line gives rise the final-state interaction effect function, or equivalently, quark correlation function of between the fast moving struck quark and the gluon the hadron. The Wilson line can provide non-trivial field from target spectator system [6,21]. The Feyn- phase needed for T -odd distribution functions. Since man rule for the eikonal line is 1/(q+ + iε) [22] (see we have defined such a correlation function in Eq. (2), ⊥ also appendix in Ref. [17]), where q is the momentum we can start from Eq. (2) to calculate h with the 1π of the gluon attached to the eikonal line. The Feynman explicit presence of the Wilson line. We expand the rule for the eikonal line-gluon vertex is e n¯µ [22].The Wilson line to first order corresponding to one gluon 1 straight line cut by the vertical dashed line denotes the exchange. Therefore, according to Eq. (6) and Eq. (2), ⊥ on-shell spectator antiquark state vs or v¯s . h can be calculated in the antiquark spectator model 1π Usually there are two choices of the pion-quark- from the expression antiquark coupling gπ : ⊥ 2 i 2h (x, k⊥)k⊥ 1π • Case 1: g as a normalization constant which M π π is used in Ref. [18]. A similar treatment for − 1 dξ dξ⊥ ik·ξ ¯ s the proton-quark-diquark coupling g has been = e Pπ |ψβ (0) q¯ 2 (2π)3 ⊥ 2 q¯s adopted in Refs. [1,17] to estimate f1T (x, k⊥) and h⊥(x, k2 ) of the proton. ∞− 1 ⊥ • Case 2: gπ as a dipole form factor [20] s + − − × q¯ −ie1 A (0,ξ , 0⊥)dξ 2 − 2 2 k m ξ − g k = N π π 2 − 2 2 + (Λ k ) × i | | + + σβαψα(ξ) Pπ ξ =0 h.c., (7) k2 − m2 = N (1 − x)2 , (8) |¯s π 2 + 2 2 in which q represents the antiquark spectator state (k⊥ Lπ ) with spin s, and e is the charge of the struck quark. 1 with Fig. 1 is the diagram equivalent to Eq. (7). The fig- 2 = − 2 + 2 − − 2 ure shows the effective correlation function in the anti- Lπ (1 x)Λ xm x(1 x)Mπ , (9) quark spectator model with the Wilson line expanding 2 =− − 2 − 2 + − 2 ⊥ k⊥ (1 x)k xm x(1 x)Mπ , (10) to the first order. h1π can be obtained from the diagram i+ by inserting σ , according to Eq. (6). Fig. 1 is similar and Nπ is the normalization constant, m is the to the diagram used by Ji and Yuan [21] to calculate mass of the quark/antiquark inside the pion, Λ ⊥ is the cut off parameter of the quark momen- f1T of the proton in scalar diquark model. The γ5 inside the circle denotes that the pion-quark-antiquark tum. This kind of treatment has been applied to coupling is pseudoscalar coupling. The double line model T -even nucleon distribution functions [20], ⊥ 2 represents the eikonalized quark propagator (eikonal and recently in the calculations of f1T (x, k⊥) and ⊥ 2 line), which is produced by the Wilson line along the h1 (x, k⊥) of the proton [10] in order to eliminate Z. Lu, B.-Q. Ma / Physics Letters B 615 (2005) 200–206 203

the divergences in the k⊥-moments of these k⊥- line of Eq. (15) comes from the contraction of the dependent distribution functions. eikonal line-gluon vertex and the gluon-antiquark ver- µ ν + tex: n¯ gµνγ = γ . The loop integral over the gluon In the previous paper [18] we performed a computa- momentum q is similar to the integral for calculat- ⊥ ⊥ − tion on h1π in case 1 which yields: ing f1T in [17,21].Theq integral is realized from contour method, and q+ integral can be done by 2 ⊥ A k + Bπ h x, 2 = π ⊥ , taking the imaginal part of the eikonal propagator: 1π k⊥ 2 2 ln (11) + + k⊥(k⊥ + Bπ ) Bπ 1/(q + iε) →−iπδ(q ), since the real part of the and the corresponding unpolarized distribution is propagator is canceled by the Hermitian conjugate term. After performing the integral we yield h⊥ with 2 + 1π 2 k⊥ Dπ a form different from Eq. (11): f1π x,k⊥ = Cπ , (12) (k2 + B )2 ⊥ π | | 2 − 3 ⊥ 2 e1e2 Nπ (1 x) mMπ h x,k⊥ = . (16) where 1π 3 2 2 + 2 3 4π 2(2π) Lπ (k⊥ Lπ ) 2 | | gπ e1e2 Aπ = mMπ (1 − x), The corresponding unpolarized distribution is 2(2π)3 4π = 2 − − 2 N 2 (1 − x)3(k2 + D ) Bπ m x(1 x)Mπ , (13) 2 π ⊥ π f1π x,k⊥ = . (17) 2(2π)3(k2 + L2 )4 = − 2 3 ⊥ π Cπ (1 x)gπ / 2(2π) , To calculate the trace in the nominator of Eq. (15) D = (1 + x)2m2. (14) π we take the spin sum of the antiquark state as s ¯s = − − An interesting result is that the transverse momentum s v v (P/ π k/ m), which is a little different ⊥ 2 from the spin sum adopted in Ref. [20]. One can find dependence of h1π (x, k⊥) in this model is the same ⊥ ⊥ as that of h of the proton in the quark-scalar diquark that the form of Eq. (16) is similar to h1 of the pro- 1 ¯⊥ model [17]. ton computed in Ref. [10]. We also calculate h1π ,the Now we perform the computation of h⊥ in case 2, T -odd distribution of the valence antiquark inside the 1π ¯⊥ = ⊥ that is, in the situation of gπ as a dipole form factor. pion, and yield h1π h1π . Comparing the two ver- ⊥ According to Eq. (7), also with the help of Fig. 1 and sions of h1π in Eq. (16) and Eq. (11), we find that each the Feynman rules introduced above, we can calculate one has a significant magnitude, which means both of ⊥ h1π from the integral: them can give unsuppressed cos 2φ asymmetry. How- ever, the transverse momentum dependence of the two ⊥ 2 i 2h (x, k⊥)k⊥ 1π versions are very different, that is to say, the QT be- Mπ havior of the cos 2φ asymmetry predicted by the two i|e1e2| cases should be different. One may expect experiments = + 8(2π)3(1 − x)P to make a discrimination between the two versions of π ⊥ 4 h . We will give a further comparison with available d q k/ + m + 1π × v¯sg k2 γ σ i experimental data in next section. ( π)4 π 5 k2 − m2 s 2 + + k/ q/ m 2 × g (k + q) γ − (k + q)2 − m2 π 5 3. The cos 2φ asymmetry in the unpolarized π N + − + Drell–Yan process × k/ q/ P/ π m 2 2 (k + q − Pπ ) − m + iε The unpolarized Drell–Yan process cross section × + s 1 1 + γ v + h.c. (15) has been measured in muon pair production by pion- q + iε 2 − − + − q iε nucleon collision: π N → µ µ X, with N denoting ¯ s − In above equation we have used Pπ |ψ(0)|¯q = a nucleon in deuterium or tungsten and a π beam s 2 2 2 v¯ gπ (k )γ5i(/k + m)/(k − m ), etc., which is a re- with energy of 140, 194, 286 GeV [15] and 252 GeV sult of the spectator model [20].Theγ + in the last [16]. The general form of the angular differential cross 204 Z. Lu, B.-Q. Ma / Physics Letters B 615 (2005) 200–206

cross section expressed in the Collins–Soper frame is [14] ¯ dσ(h1h2 → llX) dΩ dx dx d2 q⊥ 1 2 2 α ¯ = em A(y)F f af¯a + B(y)cos 2φF Q2 1 1 3 a ⊥,a ¯⊥,a¯ ˆ ˆ h1 h1 × (2h · p⊥h · k⊥ − p⊥ · k⊥) , Fig. 2. Angular definitions of the unpolarized Drell–Yan process in M M the lepton pair center of mass frame. 1 2 (19) section for the unpolarized Drell–Yan process is where Q2 = q2 is the invariance mass square of the lepton pair, q⊥ is the transverse momentum of the pair, 1 dσ 3 1 2 ˆ = + λ θ + µ θ φ and the vector h = q⊥/QT . We have used the nota- + 1 cos sin 2 cos σ dΩ 4π λ 3 tion + ν 2 sin θ cos 2φ , (18) ¯ 2 2 2 2 F[f1f1]= d p⊥ d k⊥ δ (p⊥ + k⊥ − q⊥) where φ is the angle between the lepton plane and × f x,p2 f¯ x,¯ k2 . (20) the plane of the incident hadrons in the lepton pair 1 ⊥ 1 ⊥ center of mass frame (see Fig. 2). The definition of From Eq. (19) one can give the expression for the the lepton plane depends on the choice of axes zˆ in asymmetry coefficient ν [14]: the lepton pair center of mass system. In our calcula- ⊥ ⊥ ¯ ˆ h ,ah¯ ,a tion we choose z parallel to the bisector of Pπ and = 2F ˆ · ˆ · − · 1 1 ν 2 ea (2h p⊥h k⊥ p⊥ k⊥) −PN , which is referred to as Collins–Soper frame M M a 1 2 [23]. The experimental data show large value of ν near −1 to 30% in the Collins–Soper frame. The asymmetry × 2F a ¯a¯ ea f1 f1 . (21) predicted by perturbative QCD is expected to be small a [15,24]. Several theoretical approaches have been suggested to interpret the experimental data, such as high- The cos 2φ dependence as observed by the NA10 twist effect [25,26] and factorization breaking mecha- Collaboration does not show a strong dependence nism [24]. A natural explanation has been proposed on A [15], i.e., the asymmetry is unlikely associated with nuclear effect. The leading contribution which by Boer [14] that the product of two T -odd chiral-odd ⊥ ¯ ⊥ ⊥ comes from the valence quarks is h¯ ,u × h ,u, there- h1 can give cos 2φ asymmetry without suppression 1π 1 by the momentum of the lepton pair. In that paper, fore we can adopt the u-quark dominance, i.e., we do ⊥ not include sea quark contribution which is expected to a parametrization of h1 in a similar form of Collins ⊥ fragmentation function [5] has been given to fit the ex- be small. We use h1π given in Eq. (16) (case 2) to esti- ⊥ mate the asymmetry. We also need h⊥ of the nucleon periment data. Recently it is found that non-zero h1 1 can arise from final-state interaction without violation in a similar treatment with effective coupling as a di- of time-reversal invariance, and has been used to esti- pole form factor. This has been done in Ref. [10].We ¯ ⊥ mate cos 2φ asymmetry in the unpolarized pp¯ → llX use this version h1 but only include contribution from ⊥ ⊥ Drell–Yan process [17]. the scalar diquark (h ,u = h ,S , with S denoting the ⊥ 1 1 Encouraged by this proposal, we apply h1π given scalar diquark). There are two considerations: the first by our model calculation to estimate the consequent is to reduce the number of the parameters, and the sec- cos 2φ asymmetry in the unpolarized π −N Drell–Yan ond is because of the u-quark dominance assumption. process measured by NA10 Collaboration. In case the Based on Eq. (21) with the denominator from the same vector boson that produces the lepton pair is a vir- model result, we give the numerical estimation of the tual photon, the leading order unpolarized Drell–Yan asymmetry at x¯ = x = 0.5inFig. 3 (shown by the Z. Lu, B.-Q. Ma / Physics Letters B 615 (2005) 200–206 205 solid curve) with experiment data from NA10 Collab- (the error in QT is chosen to be the bin size). We find ⊥ oration. For the parameters in the expressions of h1π that the estimated asymmetry agrees with the experi- ⊥ = = ment data fairly well, although our estimation is crude and h1 we choose: Λ 0.6GeV,Mπ 0.137 GeV, m = 0.1GeV, MN = 0.94 GeV, λS = 0.8 GeV, and since some approximations are adopted. In contrast, mN = 0.3 GeV, where MN , λS and mN are the nu- as shown by the dashed line in Fig. 3,theQT shape of ⊥ cleon mass, the scalar diquark mass and the mass of the asymmetry produced by h1π denoted in Eq. (11) the quark inside the nucleon, respectively. For the cou- (case 1 with the pion-quark-antiquark coupling as a pling constant |e1e2|/4π we extrapolate |e1e2|/4π → constant) is not consistent with experimental data (see CF αs , and take αs = 0.3 and CF = 4/3 which are Ref. [18]). ⊥ adopted in Ref. [1]. We choose the data at 194 GeV We further use h1π given in Eq. (16) and the same of Ref. [15], since the error bars of them are smallest parameters adopted above to estimate the asymmetries averaged over the kinematics of NA10 exper- 2 2 iments. For QT Q , the momentum fractions of the quarks inside the pion and√ the nucleon satisfy the relation: xx¯ = Q2/s, where s is the center of mass energy of the pion-nucleon√ system, for instance, for 194 GeV beam s = (194 + 0.94)2 − 194√2 = 19.1 GeV, and for 140 GeV, 286 GeV beam s = 16.2 GeV, 23.2 GeV respectively [15]. The data- selecting condition of the NA10 experiments is: x<¯ 0.7, 4.0GeV Q 8.5GeV(4.05 GeV Q 8.5 GeV for the 194 GeV data) and Q 11 GeV. We use above kinematical constrains to evaluate the averaged asymmetries for π − beam with different energy. In Fig. 4 we plot the asymmetries in the Collins–Soper frame versus QT for 140, 194 and 286 GeV beam together with the data of Ref. [15]. The estimated asymmetries for pion beam with different energy are Fig. 3. The cos 2φ asymmetry (solid line) in the unpolarized − still consistent with experimental data. Our estimation π N Drell–Yan process defined in the Collins–Soper frame at x¯ = x = 0.5. The dashed line represents asymmetry given by the shows that the energy dependence of the asymmetries ⊥ is not strong, and this agrees with the experimental previous model result [18] (with h1π calculated in case 1). The data are taken from Ref. [15] at 194 GeV. observation.

(a) (b) (c)

Fig. 4. The cos 2φ asymmetries in the Collins–Soper frame for pion beam with different energy averaged over kinematics region: x<¯ 0.7, 4.0GeV Q 8.5GeV(4.05 GeV Q 8.5 GeV for the 194 GeV data) and Q 11 GeV. The data are taken from Ref. [15]. 206 Z. Lu, B.-Q. Ma / Physics Letters B 615 (2005) 200–206

4. Summary References

The observed large cos 2φ azimuthal asymmetry [1] S.J. Brodsky, D.S. Hwang, I. Schmidt, Phys. Lett. B 530 (2002) in the unpolarized Drell–Yan process indicates a sub- 99. stantial non-zero value for leading twist T -odd dis- [2] S.J. Brodsky, D.S. Hwang, I. Schmidt, Nucl. Phys. B 642 ⊥ 2 (2002) 344. tribution function h1 (x, k⊥) (which is referred to as [3] D. Sivers, Phys. Rev. D 41 (1990) 83; Boer–Mulders function) from phenomenological as- D. Sivers, Phys. Rev. D 43 (1991) 261. ⊥ pects. Theoretically it has been demonstrated that h1 [4] M. Anselmino, M. Boglione, F. Murgia, Phys. Lett. B 362 of the nucleon and the meson can arise from final- (1995) 164. or initial-state interaction. In this connection, we have [5] J.C. Collins, Nucl. Phys. B 396 (1993) 161. performed a calculation of h⊥ of the pion in a sim- [6] J.C. Collins, Phys. Lett. B 536 (2002) 43. 1π [7] A.V. Belitsky, X. Ji, F. Yuan, Nucl. Phys. B 656 (2003) 165. ple antiquark spectator model by taking into account [8] D. Boer, P.J. Mulders, F. Pijlman, Nucl. Phys. B 667 (2003) final-state interaction, and estimated the consequent 201. − cos 2φ azimuthal asymmetry in the unpolarized π N [9] L.P. Gamberg, G.R. Goldstein, K.A. Oganessyan, Phys. Rev. Drell–Yan process which is then compared with ex- D 67 (2003) 071504(R). perimental data measured by NA10 Collaboration. In [10] A. Bacchetta, A. Schäfer, J.-J. Yang, Phys. Lett. B 578 (2004) 109. the calculation we adopt the pion-quark-antiquark ef- [11] Z. Lu, B.-Q. Ma, Nucl. Phys. A 741 (2004) 200. fective coupling as a dipole form factor. We find that [12] HERMES Collaboration, A. Airapetian, et al., Phys. Rev. ⊥ ⊥ Lett. 94 (2005) 012002. the resulting h1π , together with h1 of the nucleon resulting from a similar treatment with nucleon-quark- [13] D. Boer, P.J. Mulders, Phys. Rev. D 57 (1998) 5780. diquark coupling as a dipole form factor, can give [14] D. Boer, Phys. Rev. D 60 (1999) 014012. [15] NA10 Collaboration, S. Falciano, et al., Z. Phys. C 31 (1986) a good agreement of the estimated cos 2φ azimuthal 513; asymmetry with experimental data from NA10 Collab- NA10 Collaboration, M. Guanziroli, et al., Z. Phys. C 37 oration. This provides a new indication on the role of (1988) 545. ⊥ [16] J.S. Conway, et al., Phys. Rev. D 39 (1989) 92. T -odd distribution h1 to the cos 2φ asymmetry in the unpolarized Drell–Yan process from initial-state inter- [17] D. Boer, S.J. Brodsky, D.S. Hwang, Phys. Rev. D 67 (2003) 054003. action. [18] Z. Lu, B.-Q. Ma, Phys. Rev. D 70 (2004) 094044. [19] P.J. Mulders, R.D. Tangerman, Nucl. Phys. B 461 (1996) 197. [20] R. Jakob, P.J. Mulders, J. Rodrigues, Nucl. Phys. A 626 (1997) 937. Acknowledgements [21] X. Ji, F. Yuan, Phys. Lett. B 543 (2002) 66. [22] J.C. Collins, D.E. Soper, Nucl. Phys. B 194 (1982) 445. We acknowledge the helpful discussion with Vin- [23] J.C. Collins, D.E. Soper, Phys. Rev. D 16 (1977) 2219. [24] A. Brandenburg, O. Nachtmann, E. Mirkes, Z. Phys. C 60 cenzo Barone. This work is partially supported by (1993) 697. National Natural Science Foundation of China (Nos. [25] A. Brandenburg, S.J. Brodsky, V.V. Khoze, D. Müller, Phys. 10025523, 90103007, and 10421003), by the Key Rev. Lett. 73 (1994) 939. Grant Project of Chinese Ministry of Education (No. [26] K.J. Eskola, P. Hoyer, M. Vänttinen, R. Vogt, Phys. Lett. B 333 305001), and by the Italian Ministry of Education, (1994) 526. University and Research (MIUR). Physics Letters B 615 (2005) 207–212 www.elsevier.com/locate/physletb

The quark mixing matrix with manifest Cabibbo substructure and an angle of the unitarity triangle as one of its parameters

C. Jarlskog

Division of Mathematical Physics, LTH, Lund University, Box 118, S-22100 Lund, Sweden Received 21 March 2005; accepted 14 April 2005 Available online 25 April 2005 Editor: N. Glover

Abstract The quark mixing matrix is parameterised such that its “Cabibbo substructure” is emphasised. One can choose one of the parameters to be an arbitrarily chosen angle of the unitarity triangle, for example, the angle β (also called Φ1).  2005 Elsevier B.V. All rights reserved.

1. Introduction

The question of fermion masses and mixings has been among the most central issues in theoretical particle physics since a long time. Within the three family version of the Standard Model [1] many specific forms for the quark mass matrices have been proposed in the past with the hope that some insight may be gained into the flavour problem. For example, already in 1978 Fritzsch [2] proposed a structure which became quite popular as it could be realised in some grand unified theories (see, for example, Ref. [3]). Since then possible zeros in the quark mass matrices (usually called texture zeros) have enjoyed special popularity as these make the computations more transparent and generally lead to specific predictions. Again one has hoped that clues to the solution of the flavour problem may emerge. Another approach has been to “derive” quark mass matrices from experiments, see, for example, Ref. [4] where it was found that the two quark mass matrices are highly “aligned”. A troubling factor in all such studies is that the mass matrices are not uniquely defined but are “frame” dependent. In other words, given any set of three-by-three quark mass matrices Mu and Md , for the up-type and down-type quarks respectively, one can obtain other sets by unitary rotations without affecting the physics. The measurables are, of course, frame-independent and therefore they must be invariant functions under such unitary

E-mail address: [email protected] (C. Jarlskog).

0370-2693/$ – see front matter  2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physletb.2005.04.033 208 C. Jarlskog / Physics Letters B 615 (2005) 207–212 rotations. These functions were introduced in [5] and studied in detail in [6]. Furthermore, it has been shown recently [7] that this formalism can be extended to the case of neutrino oscillations. For the quarks what enters, in the Standard Model, is the pair ≡ † ≡ † Su MuMu,Sd Md Md . (1) The original motivation for the work presented here was to look for “the golden mean” mass matrices, to be deﬁned shortly. First we note that there are two “extreme frames”, one in which the up-type quark mass matrix is diagonal, i.e.,     2 2 mu 00 md 00 =  2  =  2  † Su 0 mc 0 ,Sd V 0 ms 0 V , (2) 2 2 00mt 00mb where the m’s refer to the quark masses and V is the quark mixing matrix. The other extreme frame is one in which the down-type quark mass matrix is diagonal, i.e.,     2 2 md 00 mu 00 =  2  = †  2  Sd 0 ms 0 ,Su V 0 mc 0 V. (3) 2 2 00mb 00mt One may then wonder how the mass matrices would look like in the “golden mean frame”, i.e., the frame right in the middle of the two extremes, where     2 2 mu 00 md 00 = †  2  =  2  † Su W 0 mc 0 W, Sd W 0 ms 0 W , (4) 2 2 00mt 00mb W is the square root of the quark mixing matrix, V = W 2. (5) In order to go to this frame one needs to compute the square root of the quark mixing matrix. The speciﬁc parameterisation of V turns out to be of paramount importance for achieving this goal. In spite of the fact that all valid parameterisations are physically equivalent, most of them are “nasty” and do not allow their roots to be taken so easily. After several attempts and having got stopped by heavy calculations, we have found a particularly convenient parameterisation, presented here below. It turns out that this parameterisation by itself is more interesting than the answer to our original question, which will be dealt with in a future publication.

2. A parameterisation with manifest Cabibbo substructure

The quark mixing matrix is usually parameterised as a function of three rotation angles and one phase, generally denoted by the set θ1,θ2,θ3 and δ. However there are many ways in which these parameters can be introduced (for a review see, for example, [8]) and the meaning of these quantities depends on how they are introduced. A speciﬁc parameterisation may have some beautiful features as well as short-comings. For example, a special feature of the seminal Kobayashi–Maskawa parameterisation [9] is that in the limit θ1 → 0 the ﬁrst family decouples from the other two. The parameterisation preferred by the Particle Data Group [10] has as its special feature that its phase δ is locked to the smallest angle θ3 but none of the families decouples if only one of the angles goes to zero. A most important and easy to remember empirical parameterisation has been given by Wolfenstein [13], where the matrix is expanded in powers of a parameter denoted by λ, where λ 0.22. In this Letter, we introduce an (exact) parameterisation of the quark mixing matrix in terms of four parameters denoted by Φ,θ3,δα and δβ . The reason for calling one of the angles θ3 when we have no other θ’s is to stay as C. Jarlskog / Physics Letters B 615 (2005) 207–212 209 close as possible to the usual nomenclature. Our angles δ are often somewhat different from what is commonly used and thus, in order not to confuse the reader, we do not denote them with θ. We write the quark mixing matrix (exactly) in a form such that its Cabibbo substructure is emphasised from the very beginning,

V = V0 + s3V1 + (1 − c3)V2, (6) where s3 = sin θ3, c3 = cos θ3 and the matrices Vj , j = 0 − 2, are given by   cos Φ sin Φ 0 0  R2(Φ)  V0 = − sin Φ cos Φ 0 = 0 , (7) 001 00 1 00a1 | = ≡ 0 A V1 00a2 | , (8) B 0 b1 b2 0  | | 0  A B  V2 = 0 . (9) 00 −1 Here a b |A= 1 , |B= 1 (10) a2 b2 | | ≡ and ( A B )ij aibj . We will impose the following conditions on A and B: A|A=B|B=1 (11) and

|A=−R2(Φ)|B, |B=−R2(−Φ)|A. (12)

By these conditions, the vector A represents two real parameters, for example, the magnitude of a1 and the relative phase of a1 and a2. These will provide the two remaining parameters (δα,δβ ) that together with Φ and θ3 add up to the four parameters needed to get the most general quark mixing matrix. Because of Eq. (12) B introduces no further parameters. Note that = = = = V13 a1s3,V23 a2s3,V31 b1s3,V32 b2s3. (13) We will also introduce the invariant J defined by = Im Vαj VβkVαkVβj J αβγ jkl. (14) γ,l In the above parameterisation we find = 2 = 2 J s3 c3 sin Φ cos Φ Im a1a2 s3 c3 sin Φ cos Φ Im b1b2 , (15) where the last equality follows from Eq. (12). We can check the unitarity of the matrix V without specifying what A (or equivalently B) looks like. We find   0 1 |AA| V V † + V V † = V V † + V V † = 0,VV † = V V † =− V V † + V V † =  0  . 0 1 1 0 1 2 2 1 2 2 1 1 2 0 2 2 0 00 1 These identities are derived trivially by using the relation between A and B,Eq.(12). 210 C. Jarlskog / Physics Letters B 615 (2005) 207–212

Given any A or B we have the freedom to rephase it, for example,

|A→eiη|A (16) whereby the vector B is also rephased by the same amount (see Eq. (12)). From the form of the matrix V we see immediately that the elements V11, V12, V21, V22 and V33 remain invariant under this rephasing. In this parameterisation, the usual unitarity triangle, obtained from Eq. (12), is a consequence of

a1 cos Φ − a2 sin Φ + b1 = 0. (17) Thus the three angles of the triangle are given by the phases of b1a2, a2a1 and a1b1. We can choose our A or B such that one of these angles enters directly as a parameter in the matrix V . The simplest one to incorporate is the angle usually denoted by γ , i.e., the phase of a2a1. We could choose − sin δ e iδα |A= β (18) cos δβ whereby

= = 2 sin δα sin γ, J s3 c3 sin Φ cos Φ sin δβ cos δβ sin γ. We would then compute B using Eq. (12). To incorporate the angle β (also denoted by φ1) of the unitarity triangle we could take a2 to be real and b1 to have the phase δβ = β.FromEq.(12), the reality condition on a2 implies that − sin Φb1 + cos Φb2 be real. This ﬁxes the vector B and thereby also the vector A. We ﬁnd iδβ | = 1 cos Φ sin δαe B −iδ , (19) σ − sin Φ sin δβ e α where

2 2 2 2 2 σ = cos Φ sin δα + sin Φ sin δβ . (20) The vector A thus obtained is given by − 1 −[cos 2Φ sin(δ + δ ) + sin δ eiδβ − sin δ e iδα ] |A= α β α β . (21) 2σ sin 2Φ sin(δα + δβ ) Here

sin δβ = sin β(BaBar) = sin Φ1 (Belle), (22) where BaBar [11] and Belle [12] Collaborations have determined this angle in their study of the B–B¯ system but use different notations for it. With this choice, J is given by

2 sin (2Φ)sin δα sin δβ sin(δα + δβ ) J = s2c . (23) 3 3 4σ 2

Finally in order to utilise the third angle, α also known as φ2, as a parameter we may take it to be the phase of b1 and require that a1 be real. The procedure to be followed to achieve this goal is exactly as depicted above. The above expressions may look somewhat complicated but they are generally quite easy to work with as we often only need their closed forms and not their details. C. Jarlskog / Physics Letters B 615 (2005) 207–212 211

3. Special features and an estimation of the parameters

The above parameterisation, Eq. (6),isanexact form and not a perturbative expansion. It has several special features as follows: (1) In the limit θ3 → 0 the third family decouples from the ﬁrst two and the exact Cabibbo substructure, with the mixing angle Φ between the ﬁrst two families, emerges. (2) Since the matrices Vj , j = 0–2, do not depend on θ3, this parameterisation provides a convenient framework 2 for perturbative expansion in powers of θ3 which is indeed small, of order λ . (3) We have seen that we can incorporate any one of the angles of the unitarity triangle as one of the four parameters of the mixing matrix. We now estimate the value of our parameters Φ, θ3, δα, δβ for the choice Eq. (19) by comparing them with Wolfenstein’s parameters [13]. Comparing the matrix elements V12 and V33 yields that the angles Φ and θ3 are or order λ and λ2, respectively, 2 Φ λ, θ3 Aλ . (24)

Next, from the moduli of the matrix elements V13, V23, V31, and V32 we ﬁnd that the angle δα is much smaller than the angle δβ , η sin δβ , (25) (1 − ρ)2 + η2 1 − ρ cos δβ , (26) (1 − ρ)2 + η2 2 sin δα ηλ . (27) Finally, the invariant J is given by 2 = 2 4 J θ3 sin δα A λ sin δα. (28) There is a somewhat subtle issue about this parameterisation that merits to be discussed even though it is hypothetical. It concerns the case with CP conservation while we know that CP is violated and therefore the parameters δα and δβ are both non-vanishing. Nonetheless, we are used to parameterisations with three rotation angles and a phase such that when the phase approaches zero one immediately obtains a mixing matrix with three rotation angles. The converse is not necessarily true that when one of the angles vanishes so does the phase. To remove the phase one often needs to expend some effort. The parameterisation here is more like having two rotation angles and two phases; both of the latter vanish when there is no CP violation. It would seem that we would end up with only two angles, Φ and θ3. How do we then recover the third angle, which should be there? The answer is that even though in the CP conserving limit δα and δβ both approach zero their ratio needs to be deﬁned. We may introduce two angles, θ1 and θ2, by putting

Φ = θ1 + θ2, (29) sin δα = tan θ1 tan(θ1 + θ2). (30) sin δβ Taking the limits carefully as the two δ’s approach zero, we ﬁnd sin θ1 sin θ |B= , |A= 2 (31) − cos θ1 cos θ2 and thus we end up with a mixing matrix with just three rotation angles. Furthermore, in this limit the invariant J contains three powers of sin δ (δ being δα or δβ ) in its numerator but only two in its denominator and thus vanishes as it should. 212 C. Jarlskog / Physics Letters B 615 (2005) 207–212

References

[1] S.L. Glashow, Nucl. Phys. 22 (1961) 579; S. Weinberg, Phys. Rev. Lett. 19 (1967) 1264; A. Salam, in: N. Svartholm (Ed.), Elementary Particle Theory, Almqvist and Wiksell, Stockholm, 1968. [2] H. Fritzsch, Phys. Lett. B 73 (1978) 317. [3] H. Georgi, D.V. Nanopoulos, Nucl. Phys. B 155 (1979) 52. [4] P.H. Frampton, C. Jarlskog, Phys. Lett. B 154 (1985) 421. [5] C. Jarlskog, Z. Phys. C 29 (1985) 491; C. Jarlskog, Phys. Rev. Lett. 55 (1985) 1039. [6] C. Jarlskog, Phys. Rev. D 35 (1987) 1685; C. Jarlskog, Phys. Rev. D 36 (1987) 2128. [7] C. Jarlskog, Phys. Lett. B 609 (2005) 323. [8] C. Jarlskog, in: C. Jarlskog (Ed.), CP Violation, World Scientiﬁc, Singapore, 1989, p. 3. [9] M. Kobayashi, T. Maskawa, Prog. Theor. Phys. 49 (1973) 652. [10] Particle Data Group, S. Eidelman, et al., Phys. Lett. B 592 (2004) 1. [11] BaBar Collaboration, B. Aubert, et al., Phys. Rev. Lett. 89 (2002) 201802. [12] Belle Collaboration, K. Abe, et al., Phys. Rev. D 66 (2002) 071102(R). [13] L. Wolfenstein, Phys. Rev. Lett. 51 (1983) 1945. Physics Letters B 615 (2005) 213–220 www.elsevier.com/locate/physletb

Chiral symmetry and exclusive B decays in the SCET

Benjamín Grinstein a, Dan Pirjol b

a Department of Physics, University of California at San Diego, La Jolla, CA 92093, USA b Center for Theoretical Physics, MIT, Cambridge, MA 02139, USA Received 17 February 2005; accepted 10 April 2005 Available online 20 April 2005 Editor: H. Georgi

Abstract We construct a chiral formalism for processes involving both energetic hadrons and soft Goldstone bosons, which extends the application of soft-collinear effective theory to multibody B decays. The nonfactorizable helicity amplitudes for heavy meson decays into multibody ﬁnal states satisfy symmetry relations analogous to the large energy form factor relations, which are broken at leading order in Λ/mb by calculable factorizable terms. We use the chiral effective theory to compute the leading + − corrections to these symmetry relations in B → Mnπν¯ and B → Mnπ decays, with one energetic meson Mn and one soft pion.  2005 Elsevier B.V. All rights reserved.

PACS: 12.39.Fe; 14.20.-c; 13.60.-r

1. Introduction In this Letter we present a combined application of the SCET with chiral perturbation theory which The study of processes involving energetic quarks can be used to study exclusive processes involving and gluons is simplified greatly by going over to both energetic light hadrons and soft pseudo Gold- an effective theory which separates the relevant en- stone bosons and photons. The main observation is ergy scales. The soft-collinear effective theory (SCET) that once the dynamics of the collinear degrees of free- [1] simplifies the proof of factorization theorems and dom has been factorized from that of the soft modes, allows a systematic treatment of power corrections. usual chiral perturbation theory methods can be ap- SCET has been applied to both inclusive and exclu- plied to the latter, unhampered by the presence of the sive hard processes with energetic final state particles. energetic collinear particles which might have upset the momentum power counting in p/Λχ . The chiral formalism has been applied previously to compute ma- E-mail address: [email protected] (D. Pirjol). trix elements of operators appearing in hard scattering

0370-2693/$ – see front matter  2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physletb.2005.04.024 214 B. Grinstein, D. Pirjol / Physics Letters B 615 (2005) 213–220 processes, such as DIS and DVCS [2–4]. Our Letter relations for the B → M form factors [5,14,16].We extends these results to processes with both soft and show that similar relations exist for B decays into collinear hadrons. multibody final states containing one collinear hadron We focus here on exclusive B decays, which are Mn plus soft hadrons XS , B → MnXS . Section 3 de- described by three√ well-separated scales: hard Q ∼ velops a chiral formalism for computing the matrix el- mb, hard-collinear ΛQ and the QCD scale Λ ∼ ements of the soft operators in (2) XS|OS|B with XS 500 MeV. This requires the introduction of a sequence containing only soft Goldstone bosons. As an applica- of effective theories QCD → SCETI → SCETII, con- tion we discuss in Section 4 the semileptonic and rare + − taining degrees of freedom of successively lower vir- radiative decays B → MnπSν¯ and B → MnπS . tuality [5]. The intermediate theory SCETI contains µ hard-collinear quarks ξn and gluons An with virtuality 2 ∼ 2. Symmetry relations phc ΛQ and ultrasoft quarks and gluons q, Aµ with 2 virtuality Λ . Finally, one matches onto SCETII which The most general SCET operator appearing in the µ I includes only soft q,Aµ and collinear ξn,An modes matching of SM currents qΓ¯ b for b → uν¯ or b → with virtuality p2 ∼ Λ2. The expansion parameter in sγ decays has the form (we neglect here light quark 2 both effective theories can be chosen as λ ∼ Λ/mb. masses, which can be included as in [15]) In the low energy theory SCETII the soft and ⊥ J eff = c (ω)q¯ γ P b collinear modes decouple at leading order and the ef- µ 1 n,ω µ L v fective Lagrangian is simply a sum of the kinetic terms + c2(ω)vµ + c3(ω)nµ q¯n,ωPRbv for each mode + (1L) + (1R) b1L(ωi)Jµ (ωi) b1R(ωi)Jµ (ωi) L(0) = L(0) + ¯ − + L(0) ξ q(iD/ mq )q A . (1) (10) n + b1v(ωi)vµ + b1n(ωi)nµ J (ωi). (3) q These are the most general operators allowed by The matching of an arbitrary operator O onto SCETII can be written symbolically as [5] power counting and which contain a left-handed collinear quark. We neglect O(λ) operators of the → ⊗ ⊗ + +··· † O T OS OC Onf , (2) form q¯nP⊥Γbv which do not contribute below. The where the ellipses denote power suppressed contribu- relevant modes are soft quarks and gluons with mo- ∼ tions. The first term is a ‘factorizable’ contribution, menta ks Λ and collinear quarks and gluons moving along n. n , n¯ are unit light-cone vectors satisfying with OS,OC soft and collinear operators convolved µ µ 2 =¯2 = ·¯= with a Wilson coefficient T depending on the argu- n n 0, n n 2. ments of O ,O . O denote ‘nonfactorizable’ oper- The O(λ) operators are defined as S C nf ators. Their precise form depends on the IR regulator (1L,1R) (1L,1R) 1 α J (ω ,ω ) =¯q Γ igB⊥ b , adopted for SCETII; for example, in dimensional reg- µ 1 2 n,ω1 µα n¯ · P n v ularization they might take the form of T products of ω2 1 ⊥ operators involving messenger modes [6]. J (10)(ω ,ω ) =¯q igB/ P b , (4) 1 2 n,ω1 n¯ · P n L v This formalism has been used to study exclusive ω2 B decays into energetic light hadrons (e.g., B → πν { (1L) (1R)}={ ⊥ ⊥ ⊥ ⊥ } and B → K∗γ ) [5,7–9], and nonleptonic decays into 2 with Γµα , Γµα γµ γα PR,γα γµ PR . The ac- energetic light hadrons such as B → ππ [10,11].This tion of the collinear derivative i∂µ on collinear fields P = 1 ¯ · Letter presents an extension of this formalism to de- is given by the momentum label operator µ 2 nµn P + P⊥ B = scribe multi-body B decays to one energetic hadron µ . The collinear gluon field tensor is ig µ †[¯ · ⊥ ] plus multiple soft pions and photons. Such decays re- W n iDc,iDcµ W . The Wilson coefficients ci,bi ceived increased attention recently [12,13] due to their depend on the Dirac structure of the QCD current Γ ability to extend the reach of existing methods for de- and are presently known to next-to-leading order in termining weak parameters. matching [1,17,18]. In Section 2 we introduce the SCET formalism and After matching onto SCETII, the effective current review the derivation of the large energy symmetry (4) contains the factorizable operators B. Grinstein, D. Pirjol / Physics Letters B 615 (2005) 213–220 215 fact 1 J =− dxdzdk+ b1L(z)J⊥(x,z,k+) helicity amplitudes. The most important one is the µ 2ω ¯ vanishing of the right-handed (nonfactorizable) helic- ⊥ λ † /n λ × (qY)¯ + /nγ γ⊥P Y bv q¯n,ω γ⊥qn,ω ity amplitudes at leading order in 1/m , for any cur- k µ R 1 2 2 b rent Γ coupling only to left chiral collinear quarks 1 − dxdzdk+ b (z)J (x,z,k+) 2ω 1R nf ¯ H+ (B → MnXS) = 0. (8) ¯ ⊥ † /n × (qY)¯ /nγ P Y bv q¯n,ω P qn,ω k+ µ R 1 2 L 2 For decays to one-body states, this constraint leads 1 to the well known large energy form factor relations − dxdzdk+ ω mB /(mB + mV )V (E) = (mB + mV )/(2E)A1(E) (for = = × b1v(z)vµ + b1n(z)nµ ΓV −A γµPL) and T1(E) mB /(2E)T2(E) (for ν ¯ ΓT = iσµνq PR) [14,16,18]. The argument above ex- † /n × J (x,z,k+) (qY)¯ /nP Y bv s¯n,ω P qn,ω , k+ L 1 2 L 2 tends this result to hadrons of arbitrary spin and multi- (5) body states MnXS . Another prediction is a relation between the time- where J⊥, are jet functions defined as in [11]. We de- like and longitudinal nonfactorizable contributions to ω = xω ω =−ω( − x) ω = ω − ω noted here 1 , 2 1 , 1 2. the helicity amplitudes for an arbitrary current Γ This has the factorized form of Eq. (2), with the Wil- ⊗ nf → · ∗ + · ∗ son coefficient T given by bi J ,⊥. Ht (B MnXS) c2(v εt ) c3(n εt ) = ∗ ∗ The nonfactorizable operator Onf in Eq. (2) arises nf → · + · H0 (B MnXS) c2(v ε0) c3(n ε0) from matching the LO SCET operators onto SCET I II Λ [5]. The precise form of the latter operators is not es- + O QCD . (9) sential for our argument, which depends only on the mb Dirac structure of the SCETI operators. Before pro- Finally, SCET predicts also the ratio of helicity am- ceeding to write down the SCET predictions for these plitudes mediated by different currents, into any state matrix elements, we define more precisely the kine- MnXS containing one energetic collinear particle, e.g., matics of the process. − − H V A(B → M X ) c(V A)(E ) The transition B → MnXS induced by the current − n S = 1 M T (T ) Jµ =¯qΓµb can be parameterized in QCD in terms of H− (B → MnXS) c (E ) 1 M 4 helicity amplitudes defined as Λ + O QCD , (10) (Γ ) = |¯ ∗µ | Hλ (Mn,XS) MnXS qΓµελ b B (6) mb − µ H V A(B → M X ) with ε±,0,t a set of four orthogonal unit vectors defined 0 n S µ = √1 ∓ µ = T → in the rest frame of v as ε± (0, 1, i, 0), ε H0 (B MnXS) 2 0 √1 | | µ = √1 | | (V −A) ∗ (V −A) ∗ ( q , 0, 0,q0), εt (q0, 0, 0, q ). These de- c (v · ε ) + c (n · ε ) Λ q2 q2 = 2 0 3 0 + O QCD . (T ) ∗ (T ) ∗ n = ( , , , ) n¯ = · + · mb finitions correspond to the choice 1 0 0 1 , c2 (v ε0) c3 (n ε0) (1, 0, 0, −1). (11) In the language of helicity amplitudes, the most These relations are in general broken by the fac- general matrix elements of the nonfactorizable oper- torizable contributions from Eq. (5). For example, the ators are given in terms of the 2 parameters helicity zeros (8) could disappear if the b1R term gives ∗ ¯ a nonvanishing contribution (note that the b term MnXS|¯qn,ωε/−PLbv|B=2EM ζ⊥(EM ,XS), 1R(L) in Eq. (5) contributes only to the H+ − helicity ampli- |¯ | ¯ = ( ) MnXS qn,ωPRbv B 2EM ζ0(EM ,XS), (7) tude). For a 1-body state, this is forbidden by angular ζ⊥,0(EM ,XS) are complex quantities depending on momentum conservation since the collinear part of the the momenta, spins and flavor of the particles in the operator can only produce a longitudinally polarized final state. meson. However, this constraint does not apply for The relations Eq. (7) imply several types of SCET multibody final states MnXS (except in channels of predictions for the nonfactorizable contributions to the well defined J P quantum numbers). In particular, this 216 B. Grinstein, D. Pirjol / Physics Letters B 615 (2005) 213–220

Table 1 energy chiral theory. Since we are interested in B de- Counting the independent hadronic parameters required for a gen- cays, the appropriate tool is the heavy hadron chiral → eral B MnXS decays in QCD, SCET and for a 1-body hadronic perturbation theory developed in Ref. [20].Themain state XS = 0 result is that the matrix elements of OS depend only Constraints Parameters # of indep. on the B meson light cone wave function. parameters The effective Lagrangian that describes the low V −A T momentum interactions of the B mesons with the QCD – H±,0,t , H±,0 7 V −A ∝ T ∝ (L,R) + pseudo-Goldstone bosons π,K and η is invariant un- SCET Hλ Hλ , H0 Ht ζ⊥,0, S 2 2 × V −A T Λ (0) der chiral SU(3)L SU(3)R symmetry and under 1-body H+ , H+ ∼ O( ), ζ⊥ , S 2 + 1 mb ,0 heavy quark spin symmetry. This requires the introduction of the heavy quark doublet (B, B∗) as the means that the helicity zero Eq. (8) receives correc- relevant matter field. The chiral Lagrangian for matter fields such as the B(∗) must be written in terms of tions at leading order in 1/mb. These corrections are computed in Section 4. velocity dependent fields, to preserve the validity of The factorizable corrections to these relations are the chiral expansion. parameterized in terms of the soft functions The chiral effective Lagrangian describing the ground state mesons containing a heavy quark Q (R) ∗ † ¯ S (k+,SX) =XS|(qY)¯ k+n//ε+PR Y bv |B, is [20] (L) = | ¯ ∗ ⊥ † | ¯ Sλ (k+,SX) XS (qY)k+n//ε−γλ PR Y bv B 2 f µ † † L = Tr ∂ Σ∂µΣ + λ0 Tr mq Σ + mq Σ 1 (L) λ 8 ≡− S (k+,XS)ε+, 2 − ¯ (Q)a µ (Q) (0) † ¯ i Tr H vµ∂ Ha S (k+,SX) =XS|(qY)¯ k+nP/ L Y bv |B. (12) i + Tr H¯ (Q)aH (Q)vµ ξ †∂ ξ + ξ∂ ξ † Parity invariance of the strong interactions gives one 2 b µ µ ba JΠ relation among these functions in channels with XS ig − Π + ¯ (Q)a (Q) † ν − ν † of well-defined spin J and intrinsic parity ( ) Tr H Hb γνγ5 ξ ∂ ξ ξ∂ ξ ba 2 (L) JΠ +··· S k+,S , (14) X JΠ † ¯ = X (qY)¯ k+nP/ R Y bv |B where the ellipsis denote light quark mass terms, S J +Π−1 (0) ˆ ˆ JΠ O(1/mb) operators associated with the breaking of = (−) S k+, Rπ PS , (13) X heavy quark spin symmetry, and terms of higher or- ˆ ˆ where P is the parity operator and Rπ the rotation op- der in the derivative expansion. The pseudoscalar and ◦ (Q) ∗(Q) erator by 180 around the y axis. vector heavy meson fields Pa and Paµ form the Compared with the decays into one-body hadronic matrix states, for which only the soft function S(0) is required, + this represents an increase in the number of indepen- (Q) 1 v/ ∗(Q) µ (Q) H = P γ − P γ5 . (15) dent parameters. However, the total number is still less a 2 aµ a than in QCD (see Table 1), such that predictive power = (b) (b) (b) = − ¯ 0 ¯ is retained. In the next section we construct a chiral For Q b, (P1 ,P2 ,P3 ) (B , B , Bs), and ∗(b) (Q) formalism which can be used to compute these matrix similarly for Paµ . The field Ha transforms as a ¯ elements for any state XS containing only soft pions. 3 under flavor SU(3)V ,

(Q) → (Q) † Ha Hb Uba (16) 3. Chiral formalism and describes B¯ and B¯ ∗ mesons with deﬁnite ve- We construct here the representation of the soft op- locity v. For simplicity of notation we will omit the ∗ erator OS giving the soft functions in (12) in the low subscript v on H , P and Pµ. The pseudo-Goldstone B. Grinstein, D. Pirjol / Physics Letters B 615 (2005) 213–220 217

iM/f bosons appear in the Lagrangian through ξ = e αˆ L,R(k+) = a1L,R + a2L,Rn/ + a3L,Rv/ 2 (Σ = ξ ) where 1   + a4L,R[n,/ v /]. (23) √1 π 0 + √1 ηπ+ K+ 2  2 6   − √1 0 √1 0  The heavy quark symmetry constraint H (Q)v/ =−H (Q) M =  π − π + ηK 2 6 reduces the number of these functions to four. Taking K− K¯ 0 − √2 η 6 the vacuum to B meson matrix element fixes the re- (17) maining functions as and the pion decay constant f 135 MeV. These αˆ L(k+) =ˆαR(k+) fields transform as √ ¯ = fB mB nφ/ +(k+) + nφ/ −(k+) , (24) Σ → LΣR†,ξ→ LξU † = UξR†. (18) where φ±(k+) are the usual light-cone wave functions The Lagrangian Eq. (14) is the most general La- of a B meson, defined by [19] grangian invariant under both the heavy quark and chiral symmetries to leading order in mq and 1/mQ. dz− − i k+z− i j ¯ e 2 0|¯q (z−)Y (z−, 0)b (0) B(v) The symmetries of the theory constrain also the 4π n v form of operators such as currents. For example, the i √ ν =¯ ν =− left handed current La qaγ PLQ in QCD can be fB mB 4 written in the low energy chiral theory as [20] + 1 v/ ¯ × nn/ · vφ+(k+) + n/n¯ · vφ−(k+) γ5 . iα (Q) † ν = ν +··· 2 ji La Tr γ PLHb ξba , (19) 2 (25) where the ellipsis denote higher dimension operators We find thus the remarkable result that the B meson in the chiral and heavy quark expansions. The para- light-cone wave functions are sufficient to fix the pion meter α is obtained by taking the vacuum to√B matrix matrix elements of the nonlocal operators Oa (k+). = L,R element of the current, which gives α fB mB (we The same result can be obtained also by consider- | (∗) use a nonrelativistic normalization for the B states ing only local operators. Let us consider the operator as in [20]). a a OL(k+) (the same results are obtained for OR(k+)). In the SCET we require also the matrix elements Expanding in a power series of the distance along the of nonlocal operators OS , which appear in Eq. (2).To light cone one is led to consider the matrix elements leading order in 1/mb these operators are quark bilin- of the operator symmetric and traceless in its indices ears ··· ←− { ←− } a,µ1µ2 µN =¯a − µ1 ··· − µN a OL q ( iD) ( iD) PRΓbv OL,R(k+) − (traces). (26) dx− − i k+x− a = e 2 q¯ (x−)Yn(x−, 0)PR,LΓbv(0). 4π Heavy quark and chiral symmetry constrain the chiral (20) effective representation of this operator to be of the

Under the chiral group they transform as (3L, 1R) and form (1L, 3R). In analogy with the local current (19) we a,µ1µ2···µN write for Oa (k+) in the chiral theory OL L,R i ··· = µ1µ2 µN (Q) † a i (Q) † Tr αN,jXj PRΓHb ξba , (27) O (k+) = Tr αˆ (k+)P ΓH ξ , (21) 4 L 4 L R b ba j i a = ˆ (Q) where the sum over j includes the most general OR(k+) Tr αR(k+)PLΓHb ξba , (22) 4 symmetric and traceless structures X formed from where the most general form for αˆL,R(k+) depends on γµ,vµ,gµν . There are many such structures, but only ··· eight unknown functions ai(k+) 2 of them survive when contracted with nµ1 nµN 218 B. Grinstein, D. Pirjol / Physics Letters B 615 (2005) 213–220

··· Xµ1µ2 µN = vµ1 vµ2 ···vµN − gµi µj − terms , space with one energetic meson M = π,ρ,K∗,etc. 0 n µ µ ···µ { } plus one soft pion. X 1 2 N = γ µ1 vµ2 ···vµN − gµi µj − terms . 1 The factorizable contributions to the transverse he- (28) licity amplitudes for B → Mnπ are given by the ma- This gives the chiral representation of the projection trix elements of Eq. (5). Specifically, one has for Mn a of the operators (26) on the light-cone pseudoscalar meson ←− q¯a(−in · D)N P Γb R v fact ¯ → H+ B Pn(k)XS → i + (Q) † Tr (αN,0 nα/ N,1)PRΓHb ξba , (29) 1 4 = C f f m S (X )b J φ , (32) 2 B P B R S 1R P which makes it clear that the constants αN,0,αN,1 are fact ¯ uniquely fixed in terms of the B → vacuum matrix H− B → Pn(k)XS = 0 (33) elements of the operators (26). Assuming that the B light-cone wave functions are well behaved at large and for Mn a vector meson k+, these matrix elements are related to the moments fact ¯ → of φ±(k+). Specifically, one finds H+ B Vn(k, η)XS √ fB fV mB mV ∗ N = ¯ · α =−2f m dk+ (k+) φ+(k+). (30) C SR(XS) n η b1RJ φV , (34) N,0 B B 2n¯ · pV √ fact ¯ = =−8 ¯ H− B → Vn(k, η)XS In particular, for N 1 this gives α1,0 3 fB mB Λ, which agrees with Ref. [21]. =− ⊥ ∗ · ∗ ⊥ CfB fV mB SL(XS) ε− η b1LJ⊥φV . (35) Beyond leading order in 1/mb many more opera- a tors can be written. For example, the matrix elements We used here the short notation biJaφ = a + + + + a of OL with one insertion of the chromomagnetic term dxdzdk bi(z)Ja(x,z,k )φ (k )φ (x). The iso- in the HQET Lagrangian L = gb¯ σ Gµνb gives spin factor C√ depends on the collinear meson, e.g., m v µν v ± structures of the form C(ρ0) = 1/ 2,C(ρ ) = 1. The corresponding re- + sults for the 1-body factorizable decay amplitudes are a µν 1 v/ (Q) † T O ,iL → Tr X P Γ iσ H ξ , obtained from these expressions by taking SR → 0, L m R 2 µν b ba SL → 1. (31) Inspection of the results (32)–(35) gives the follow- µν = [ µ ν]+ µν + [ µ ν]+ with X β1 n ,γ β2iσ nβ/ 3 n ,γ ing conclusions, valid to all orders in αs . µν β4niσ/ . The proliferation of unknown constants (see (i) The null result in Eq. (33) means that the sym- ¯ also [23]) spoils the simple leading order result that metry relation (10) for B → PnXS transitions is not knowledge of the B → vacuum matrix element is suf- broken by factorizable corrections and is thus ex- ficient to fix all low energy constants. act to leading order in 1/mb. This leads, e.g., to a ¯ ¯ + − ¯ The operators in Eqs. (21), (22) (together with (24)) relation between B → (KnπS)h=−1e e and B → − give the desired representation of the soft operators (πnπS)h=−1e ν¯. OL,R in the chiral effective theory, and can be used (ii) The vanishing of the right-handed nonfactor- ¯ to compute their matrix elements on states with a B izable helicity amplitudes H+ in B decay Eq. (8) meson and any number of pseudo-Goldstone bosons. is violated by the factorizable terms Eqs. (32), (34). These terms are however calculable in chiral perturbation theory for XS containing only soft pions. For 4. Application both Mn = P,V, the pion carries m3 =+1 angular momentum; the Vn collinear meson is emitted longi- ¯ ¯ + − B → Mnπν¯ and B → Mnπ . As an appli- tudinally polarized. cation we compute the factorizable corrections to the The soft functions SR,L(pπ ) in Eqs. (32), (35) can symmetry relations (8), (10) for the transverse helic- be computed explicitly in terms of the chiral perturba- ¯ ity amplitudes in B → Mnπ in the region of the phase tion theory diagrams in Fig. 1. We find B. Grinstein, D. Pirjol / Physics Letters B 615 (2005) 213–220 219

are invalidated when the final hadronic state contains more than one hadron (see Eq. (8)), already at leading order in the 1/mb expansion. The chiral formalism ¯ Fig. 1. Feynman diagrams for B → Mnπ. The filled circle denotes presented here allows the systematic computation of one insertion of the soft operator OS . The collinear hadron Mn is these effects. We point out the existence of an exact re- not shown. lation Eq. (10) among left-handed helicity amplitudes ∗ ¯ ε · p in B → PnXS transitions induced by different b → qn = g + π SR(pπ ) , (36) currents. fπ v · pπ + ∆ − iΓB∗ /2 These results extend the applicability of SCET 1 e3 · pπ SL(pπ ) = 1 − g , (37) to B decays into multibody states MnXS contain- · + − ∗ fπ v pπ ∆ iΓB /2 ing one energetic particle. It is interesting to note with ∆ = mB∗ − mB 50 MeV and ΓB∗ the width that the corrections to these predictions scale like ∗ of the B meson. While the soft matrix elements max(Λ/EM ,pS/ΛχpT ), rather than mX/EM .This (i) in Eq. (12) have a factorized form S (k+,SX) = suggests that the range of validity of factorization in φ+(k+)Si(SX), the total factorizable amplitude is not these decays might be wider than previously thought, ∗ ∗ simply the product B → B π times B → Mn, due a fact noted empirically in Ref. [22] in the context of to the direct graph in Fig. 1 (left) (nonvanishing only the B → DX decays. Many more problems can be for SL). At threshold, the relation Eq. (37) gives a studied using the formalism described here, e.g., the soft pion theorem which fixes the soft function in leading SU(3) violating contributions to the factoriz- B → Mnπ in terms of the factorizable contribution able contributions, analogous to the effects considered ∗ to the B → Mn transition. Note that the B width in in Refs. [3,25]. the propagator is a source of strong phases at leading order in 1/mb. These results can be extended to final states containing multiple soft pions, without introduc- Acknowledgements ing any new unknown hadronic parameters. We are grateful to Martin Savage for useful discussions. B.G. was supported in part by the DOE under 5. Conclusions Grant DE-FG03-97ER40546. D.P. was supported by the US Department of Energy under cooperative re- We presented in this Letter the application of the search agreement DOE-FC02-94ER40818. soft-collinear effective theory to B decays into multibody final states, containing one energetic meson plus soft pseudo-Goldstone bosons. The additional ingredi- References ent is the application of heavy hadron chiral perturbation theory [20] to compute the matrix elements with [1] C.W. Bauer, S. Fleming, M.E. Luke, Phys. Rev. D 63 (2001) 014006; Goldstone bosons of the nonlocal soft operators ob- C.W. Bauer, S. Fleming, D. Pirjol, I.W. Stewart, Phys. Rev. tained after factorization. (This assumes that the only D 63 (2001) 114020; SCET operators contributing to these decays are the C.W. Bauer, I.W. Stewart, Phys. Lett. B 516 (2001) 134; same as those describing B → Mn transitions [5].) C.W. Bauer, D. Pirjol, I.W. Stewart, Phys. Rev. D 65 (2002) Heavy quark and chiral symmetry are powerful con- 054022. [2] J.W. Chen, X.d. Ji, Phys. Rev. Lett. 87 (2001) 152002; straints which fix all these couplings in terms of the J.W. Chen, X.d. Ji, Phys. Rev. Lett. 88 (2002) 249901, Erratum. usual B light-cone wave functions. A similar connec- [3] J.W. Chen, I.W. Stewart, Phys. Rev. Lett. 92 (2004) 202001. tion between the nucleon structure functions and the [4] J.W. Chen, M.J. Savage, Nucl. Phys. A 735 (2004) 441. pion couplings induced by twist-2 operators has been [5] C.W. Bauer, D. Pirjol, I.W. Stewart, Phys. Rev. D 67 (2003) found in Ref. [24] (see also [4]) in the context of DIS 071502. [6] T. Becher, R.J. Hill, M. Neubert, Phys. Rev. D 69 (2004) and DVCS. 054017. Some of the symmetry predictions of SCET rely [7] M. Beneke, T. Feldmann, Nucl. Phys. B 685 (2004) 249; on angular momentum conservation arguments which B.O. Lange, M. Neubert, Nucl. Phys. B 690 (2004) 249. 220 B. Grinstein, D. Pirjol / Physics Letters B 615 (2005) 213–220

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Duality and pomeron effective theory for QCD at high energy and large Nc

J.-P. Blaizot a,1, E. Iancu b,1, K. Itakura b, D.N. Triantafyllopoulos b

a ECT*, Villa Tambosi, Strada delle Tabarelle 286, I-38050 Villazzano (TN), Italy b Service de Physique Théorique, CEA/DSM/SPhT, Unité de recherche associée au CNRS (URA D2306), CE Saclay, F-91191 Gif-sur-Yvette, France Received 28 February 2005; accepted 6 April 2005 Available online 13 April 2005 Editor: P.V. Landshoff

Abstract We propose an effective theory which governs pomeron dynamics in QCD at high energy, in the leading logarithmic approximation, and in the limit where Nc, the number of colors, is large. In spite of its remarkably simple structure, this effective theory generates precisely the evolution equations for scattering amplitudes that have been recently deduced from a more complete microscopic analysis. It accounts for the BFKL evolution of the pomerons together with their interactions: dissociation (one pomeron splitting into two) and recombination (two pomerons merging into one). It is constructed by exploiting a duality principle relating the evolutions in the target and the projectile, more precisely, splitting and merging processes, or ﬂuctuations in the dilute regime and saturation effects in the dense regime. The simplest pomeron loop calculated with the effective theory is free of both ultraviolet or infrared singularities.  2005 Elsevier B.V. All rights reserved.

There has been recently signiﬁcant progress in our understanding of high energy hadronic scattering, and in particular of the processes occurring at large parton densities and which are believed to be responsible for the unitarization of the scattering amplitudes and the saturation of the parton distributions. Non-linear evolution equations have been derived which describe the approach towards saturation and the unitarity limit, and which have the structure of stochastic evolution equations. However, it has been very recently recognized [1] that the equations which were considered as the most complete, namely the Balitsky–JIMWLK (Jalilian-Marian–Iancu–McLerran– Weigert–Leonidov–Kovner) equations [2–5], are in fact incomplete. This is manifest in the statistical language by the presence of ﬂuctuations at high momenta [6,7] which are not well accounted for by the JIMWLK evolution of

E-mail address: [email protected] (E. Iancu). 1 Membre du Centre National de la Recherche Scientiﬁque (CNRS), France.

0370-2693/$ – see front matter  2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physletb.2005.04.009 222 J.-P. Blaizot et al. / Physics Letters B 615 (2005) 221–230 the target wavefunction [1]. In the language of pomerons, the JIMWLK equation contains pomeron merging but not also pomeron splitting.2 Following this observation, two of us (E.I. and D.T.) have constructed a hierarchy of non-linear evolution equations for the dipole scattering amplitudes which include both gluon mergings and gluon splittings, and thus generate pomeron loops through iterations [1,8]. These equations have been argued to hold in the limit where the number of colors Nc is large (Nc 1), and indeed it has been checked explicitly in Ref. [8] that the vertices appearing in these equations are the same as the corresponding ‘triple pomeron vertices’ computed in perturbative QCD at large Nc [9–11]. A complementary approach has been developed by Mueller et al. [12] who proposed a generalization of the JIMWLK equation which includes the effects of pomeron splitting in the dilute regime and for large Nc. These two approaches follow the same general strategy—namely, they combine the non-linear JIMWLK equation at high density with the color dipole picture [13] in the dilute regime—and lead indeed to the same evolution equations for the scattering amplitudes, as demonstrated in Ref. [8]. (See also Refs. [14,15] for related recent developments.) It is our purpose in this Letter to show that the equations obtained in [1,8,12] can be reformulated in term of an effective theory for pomerons. By ‘pomeron’ we mean here the color singlet exchange which describes the interaction between an elementary color dipole and the field of a target in a single scattering approximation, and which reduces to two gluon exchanges in lowest order perturbation theory. The construction of the effective theory involves a projection onto restricted degrees of freedom, precisely the pomerons, and is expressed in terms of a simple Hamiltonian which describes the BFKL evolution of the pomerons together with their splitting and merging. By requiring that the evolution should lead to identical results whether it is viewed as the evolution of the target or that of the projectile, one arrives at a duality principle which is used to construct the effective Hamiltonian from the Hamiltonian derived in [12] in the dilute regime. The limitations of the effective theory, and the subtle mathematical problems that arises when one attempts to analyze its microscopic content will be briefly discussed at the end of this Letter. Most treatments of high energy scattering rely on an asymmetric approach: typically, the ‘projectile’ is viewed as a collection of test particles which probe the color field of the ‘target’. At high energy, the eikonal approximation is a good approximation, and the scattering of an elementary color charge is described by a Wilson line of the form †[ ]≡ − a − a Vx α Pexp ig dx α (x , x)t , (1) where x denotes the transverse coordinate of the particle, which is not affected by its interactions with the field of the target αa(x−, x), ta are the generators of the SU(3) algebra in the representation appropriate for the test − particle, and the symbol P indicates that, in the expansion of the exponential, the color matrices αa(x , x)ta √must be ordered from right to left in increasing order in x− (we are using light-cone vector notations, x± ≡ (t ± z)/ 2). For a more complex projectile, viewed as a collection of elementary color charges, the S-matrix is given by a product of Wilson lines like Eq. (1), one for each elementary color charge. In a frame in which most of the total rapidity Y is carried by the target, the target wavefunction can be described as a color glass condensate [4,16], and the corresponding S-matrix is obtained as:

SY = D[α]WY [α]S[α], (2) where α ≡ αa(x−, x) is a classical field randomly distributed with weight function W[α] (a functional probability distribution), and S[α] is the projectile S-matrix for a given configuration of this random field. With increasing Y ,

2 Note that the opposite terminology for what one means by ‘splitting’ and ‘merging’ would be more natural in relation with Balitsky equations, which refer to the evolution of the projectile. To avoid confusion on this point, in this Letter we shall systematically use the terminology appropriate to target evolution. J.-P. Blaizot et al. / Physics Letters B 615 (2005) 221–230 223 the weight function evolves according to a functional renormalization group equation, of the generic form: ∂ δ W [α]=H α, W [α], (3) ∂Y Y iδα Y where H is a functional differential operator commonly referred to as the ‘Hamiltonian’. Alternatively, one can view the same evolution as a change in the scattering operator, for a fixed weight function W[α]. To see that, take a derivative w.r.t. Y in Eq. (2),useEq.(3), and perform an integration by part in the functional integral: ∂ δ S = D[α]W [α]H † α, S[α]. (4) ∂Y Y Y iδα † This can be interpreted as describing the evolution of the scattering operator SY [α], with ‘Hamiltonian’ H : ∂ δ S [α]=H † α, S [α]. (5) ∂Y Y iδα Y Both points of view, somewhat reminiscent of, respectively, the Schrödinger and the Heisenberg pictures of quantum mechanics, will be used in the following discussion (although we shall refrain from introducing explicitly rapidity dependent operators). In the Schrödinger picture, one puts emphasis on the evolution of the state vector, whose role is played here by the weight functional WY [α]. In the Heisenberg picture, the state vector is a constant reference vector involved in the calculation of all expectation values, here W[α], and one puts all the evolution in the operators, here the scattering operators SY [α]. The Schrödinger picture corresponds to evolution equations which aim at providing a detailed microscopic description of the color field in the target, together with its complicated correlations. This is what the JIMWLK equation does. The Heisenberg picture rather describes how the test particles get dressed by color field fluctuations as they are boosted to higher rapidities. In this approach, the complicated color correlations in the target wavefunction are not immediately visible, and indeed the resulting equation of motion are established somewhat more easily. This second approach is essentially the one used by Balitsky to obtain his hierarchy of equations. The test particles that we shall consider are in fact elementary color dipoles, whose scattering amplitude reads: = − 1 † T(x, y) 1 tr Vx Vy , (6) Nc for a dipole with the quark leg at x and the antiquark leg at y. Here the Wilson lines are taken in the fundamental representation. We shall be interested in situations where the dipoles scatter off the color glass in the two-gluon exchange approximation (weak field limit) and we shall work in a large-Nc limit. In the weak field limit, the amplitude for a single dipole to scatter is obtained after expanding each of the Wilson lines to second order in α: †[ ]= + − a − a Vx α 1 ig dx α (x , x)t 2 g − − − − − − − − − dx dy αa(x , x)αb(y , x) θ(x − y )tatb + θ(y − x )tbta +···. (7) 2 Note that, to this order, the x−-ordering of the color matrices starts to play a role in Eq. (7). Still, this ordering is irrelevant for the computation of the dipole amplitude to lowest order, because of the symmetry of the color trace: a b = 1 ab = b a tr(t t ) 2 δ tr(t t ). Namely, one finds: g2 T(x, y) T (x, y) ≡ αa(x) − αa(y) 2, (8) 0 4N c which involves only the integrated field αa(x) ≡ dx− αa(x−, x). Similarly the amplitude for κ dipoles to scatter (κ) = ··· is given, within the same approximation, by T0 (x1, y1,...,xκ , yκ ) T0(x1, y1) T0(xκ , yκ ). In what follows, 224 J.-P. Blaizot et al. / Physics Letters B 615 (2005) 221–230 we shall refer to the amplitude (8) describing the single-scattering of an elementary dipole off a given color field (κ) as to a ‘pomeron exchange’. Similarly, T0 describes the exchange of κ pomerons. At this point we find it useful to digress on the linear evolution equation known as BFKL equation. This will allow a few observations which illuminate some of the mathematical subtleties involved in taking the large Nc limit when constructing our effective theory. Consider first the JIMWLK Hamiltonian. As shown in Ref. [17], when it is restricted to act on gauge-invariant observables, it can be given the simple form: 1 δ δ H = M(u, v, z) + V˜ †V˜ − V˜ †V˜ − V˜ †V˜ ab , JIMWLK 3 1 u v u z z v a b (9) 16π iδαY (u) iδα (v) u,v,z Y where M is the dipole kernel (x − y)2 M(x, y, z) = . (10) (x − z)2(z − y)2 Here the Wilson lines are in the adjoint representation. The derivatives can be freely moved across the bilinear form in Wilson lines, because they commute with the latter in the presence of the dipole kernel. That is, HJIMWLK is Hermitian. The BFKL limit is obtained by expanding the Wilson lines to lowest non-trivial order in α. One gets: g2 δ δ H =− M(u, v, z) αa(u) − αa(z) αb(z) − αb(v) f acf f bf d , BFKL 3 c d (11) 16π δαY (u) δα (v) u,v,z Y and it is not difficult to verify that HBFKL is again Hermitian. Let us now turn to the ‘large-Nc limit’. This is obtained by (i) restricting the action of HBFKL to the dipole (κ) operators T0 mentioned above and (ii) preserving only the dominant terms at large Nc in the action of the Hamil- tonian on these operators. When acting on the color fields inside a single factor T0 (i.e., on the same dipole), the cd acf bf c ab two functional derivatives in HBFKL yield a factor δ , and then f f =−Ncδ produces the expected Nc enhancement. On the other hand, the action on the color fields within two different factors T0 (i.e., upon two different dipoles) produces no such enhancement. Thus, at large Nc, HBFKL can be equivalently replaced by an effective † Hamiltonian in which the two functional derivatives are traced over color. This Hamiltonian, which we denote H0 for reason which will become clear shortly, is 1 α¯ δ δ H † = s M(u, v, z) αa(u) − αa(z) αa(z) − αa(v) , 0 2 b b (12) 2Nc 2π δα (u) δα (v) u,v,z where α¯s = αsNc/π. In the equation above, we have suppressed the subscript Y on the functional derivatives since they now act on functions which depend only upon the color field integrated over x−, so like in Eq. (8). Let us † emphasize that, as obvious from the construction we have given, the two derivatives in H0 are to act on the same † dipole. Note also that, as opposed to the original HBFKL, H0 is not Hermitian: in fact, it is readily seen that its † adjoint is ill-defined. This reflects the fact that the construction of H0 involves a projection on a specific set of degrees of freedom, and once this is done, one looses the possibility to integrate by part as in Eq. (4) in order to let H0 act on the weight functional W[α]. These special mathematical properties, restriction of the space on which the Hamiltonian is acting and loss of hermiticity, are general, and peculiar, mathematical features of the effective theory that we shall present. It is tempting to speculate that in doing the large Nc limit we are renouncing to follow the evolution of some color correlations (precisely those which are suppressed at large Nc), and that the corresponding loss of information may be responsible for the simpler Markovian stochastic theory that we shall arrive at. We now return to the main stream of our discussion and establish a useful property. In Ref. [18], a symmetric description was obtained for the scattering between two color glasses in the regime where both systems are in the J.-P. Blaizot et al. / Physics Letters B 615 (2005) 221–230 225 weak field regime. The final formula reads = [ ] [ ] [ ] [ ] 2 a a S Y D αR WY −y αR D αL Wy αL exp i d z ρL(z)αR(z) . (13)

a =−∇2 a [ ] In this expression, ρL(x) xαL(x) is the classical color charge density of the left-mover, and WY −y αR and Wy[αL] are the weight functions for the right-moving and, respectively, left-moving color glass (note that the rapidity of the left-mover is measured positively to the left, so that as we vary y, the total rapidity interval between projectile and target remains equal to Y ). The precise conditions for the validity of Eq. (13) are detailed in Ref. [18]. Let us emphasize here a non-trivial aspect of this formula. Although it is essentially a weak ﬁeld formula which assumes that the elementary dipoles interact only once, it contains the possibility that any number of dipoles of the projectile interact with an equivalent number of dipoles in the target. Thus Eq. (13) does account for multiple scattering, albeit in a restrictive way (each dipole interacting only once). These multiple scattering generate unitarity corrections if Y is large enough. At the same time, we require both color glasses to be unsaturated. This imposes an upper bound on Y and also limits the range of variation for y within which Eq. (13) is correct [18]. Lorentz invariance implies that, while SY may depend on the total rapidity interval Y , within the range of validity of Eq. (13) it cannot depend upon the rapidity y used to separate the system into a ‘projectile’ and a ‘target’, or equivalently on the frame which we choose to describe the collision. This implies (see also Ref. [19] for a similar argument): = ∂ S Y = [ ] [ ] 2 a a 0 D αR D αL exp i d z ρL(z)αR(z) ∂y ∂ ∂ × W − [α ] W [α ]+W − [α ] W [α ] . (14) ∂y Y y R y L Y y R ∂y y L The evolution of both weight functions are given by: ∂ ∂ δ ∂ δ WY −y[αR]=− WY −y[αR]=−H αR, WY −y[αR], Wy[αL]=H αL, Wy[αL]. ∂y ∂Y iδαR ∂y iδαL (15) We shall keep the evolution of the left-mover as shown in the above equation, but perform an integration by parts in the functional integral over αR in Eq. (14). Next, we note that † δ 2 a a = δ 2 a a H αR, exp i d z ρL(z)αR(z) H ,ρL exp i d z ρL(z)αR(z) . (16) iδαR iδρL a = Using this identity in Eq. (14) and performing a further integration by parts, now w.r.t. αL (recall that ρL(x) −∇2 a [ ]≡ [ ] xα (x)), one is left with a differential operator acting on Wy αL Wy ρL (with a slight abuse in the notation): L † δ H ,ρL Wy[ρL]. (17) iδρL For Eq. (14) to be satisﬁed, the contribution above should cancel against the term in Eq. (15) describing the evolution of the left-mover. This condition leads to the ‘self-duality’ condition: δ † δ H αL, Wy[αL]=H ,ρL Wy[ρL]. (18) iδαL iδρL

The same relation holds obviously for the ‘right’ variables αR,ρR. Going back to Eq. (16), one sees that what is involved in the duality operation3 is a matching of splitting processes in the left-movers, encoded by terms in the Hamiltonian of the form ρ2 δn/δρn, into merging process

3 To our knowledge, the duality between the roles of the operators ρ2 δn/δρn and αn δ2/δα2 has been ﬁrst recognized by L. McLerran. 226 J.-P. Blaizot et al. / Physics Letters B 615 (2005) 221–230

Fig. 1. An illustration of the relation (16). ρ denotes the color charge of the left-mover (L) and α is the color field of the right-mover (R). The same physical process can be represented by either of the two diagrams in the middle, and can be viewed either as a merging in R, or as a splitting † ∼ 2 4 4 in L. The first interpretation is natural when the diagram is produced by acting on the eikonal line with the Hamiltonian H1→2 ρ δ /δρ for † ∼ 4 2 2 L(cf.Eq.(19)). The second interpretation rather corresponds to the action of H2→1 α δ /δα for R (cf. Eq. (24)). in the right-movers, corresponding to terms of the form αn δ2/δα2. An example of such a matching is illustrated in Fig. 1. Splitting terms dominate in the dilute regime where they control the fluctuations, while merging terms become essential in the saturation regime where parton densities are large. This fluctuation–saturation duality is turned into a constraint on the evolution Hamiltonian of either the projectile or the target in Eq. (18). The self-duality constraint, which we expect to hold within the limited range of energies in which the factorization (13) is valid4 [18], will be used now to construct a simple Hamiltonian describing the pomeron dynamics in the dilute regime, starting from the known, dominant, contribution containing only splitting processes that has been constructed by Mueller et al. [12]. Quite remarkably, and somewhat unexpectedly, this Hamiltonian leads to equations of motion which reproduces the exact ones at large Nc [1,8], that is the effective theory appears to be valid beyond the dilute limit where it is established. The Hamiltonian constructed in [12] reads 2 g α¯ s H → =− M(u, v, z)G(u |u, z)G(v |u, z)G(u |z, v)G(v |z, v) 1 2 3 1 1 2 2 16Nc 2π × δ δ δ δ ∇2∇2 c c a a b b u v α (u)α (v). (19) δα (u1) δα (v1) δα (u2) δα (v2) In Eq. (19), the integration goes over all the transverse coordinates u, v, z, u1, v1, u2, v2. The function G(u1|u, z) a is,uptoafactorgt , the classical field created at u1 by the elementary dipole (u, z), and reads 1 (u − z)2 G(u |u, z) = 1 . 1 ln 2 (20) 4π (u1 − u) It is easy to understand (and was explicitly shown in [8]) that this Hamiltonian generates pomeron splittings.More † (2) precisely, the result of the operation of H1→2 on the two-pomeron exchange amplitude T0 is proportional to T0, (2) ; and thus generates the following, fluctuation, term in the evolution equation for T0 (x1, y1 x2, y2): α 2 α¯ H † T (2) = s s M(u, v, w)A (x , y |u, w)A (x , y |w, v)∇2∇2T (u, v), (21) 1→2 0 2π 2π 0 1 1 0 2 2 u v 0 u,v,w 2A where αs 0 is the amplitude for dipole–dipole scattering in the two-gluon exchange approximation and for large Nc: 1 (x − v)2(y − u)2 2 A (x, y|u, v) = ln . (22) 0 8 (x − u)2(y − v)2

4 The self-duality condition (18) has recently been claimed to hold in a much broader context [19]. J.-P. Blaizot et al. / Physics Letters B 615 (2005) 221–230 227

Clearly, this process corresponds to the splitting of one pomeron into two. In general, the Hamiltonian H1→2 can describe the transition n → n + 1, in which case n − 1 of the pomerons are simply “spectators”. Note that H1→2 is non-Hermitian, which we interpret as reﬂecting again the large-Nc approximation implicitly involved in its derivation. a To apply the duality transformation, it is convenient to reexpress H1→2 in terms of the sources ρ (x) of the a a =−∇2 a color ﬁeld α (x),byusingρ (x) xα (x). Then we obtain 2 ¯ 2 2 g αs δ δ δ δ c c H → =− M(u, v, z) − − ρ (u)ρ (v). 1 2 3 a a b b (23) 16Nc 2π δρ (u) δρ (z) δρ (z) δρ (v) u,v,z

At this point we force the Hamiltonian to be self-dual. This is done by adding to H1→2[δ/iδρ,ρ] its dual † [ ]≡ H1→2 α, δ/iδα H2→1 (this new notation will be justiﬁed shortly). The Hermitian conjugate of H2→1 reads g2 α¯ δ δ H † = s M(u, v, z) αa(u) − αa(z) 2 αb(z) − αb(v) 2 , 2→1 3 c c (24) 16Nc 2π δα (u) δα (v) u,v,z † and the action of H2→1 on the dipole scattering amplitude is α¯ g4 H † T (x, y) = s M(x, y, z) αa(x) − αa(z) 2 αb(z) − αb(y) 2 2→1 0 2 2π 16Nc z α¯ = s M(x, y, z)T (2)(x, z; z, y). (25) 2π 0 z

Thus H2→1 generates the non-linear term in the first Balitsky equation. Similarly, it is obvious to show that the (κ) operation on T0 (x1, y1,...,xκ , yκ ), will generate correctly the non-linear terms of κ—the Balitsky equation in the large-Nc limit (this is trivial; only one amplitude is “active”, and we need to take into account all the possible permutations). Therefore, the Hamiltonian in Eq. (24) generates in an effective way pomeron mergings (hence the notation H2→1); one has a transition of the form n + 1 → n where, again, n − 1 of the pomerons are spectators. †[ ] Note also that the BFKL piece H0 α, δ/iδα of the Hamiltonian is self-dual. Indeed, the dual conjugate of Eq. (12) is H0[δ/iδρ,ρ] with 1 α¯ δ δ δ δ H = s M(u, v, z) − − ρb(u)ρb(v). 0 2 a a a a (26) 2Nc 2π δρ (u) δρ (z) δρ (z) δρ (v) u,v,z † While H0 is not identical to H0 in Eq. (12), both Hamiltonians are equivalent, as we show now. To this aim we use Eq. (26) to deduce the evolution equation for the bilocal operator n(x, y) ≡ ρa(x)ρa(y), which, up to a normalization factor, can be identified with the dipole number density in the target wavefunction [1,18] (see also the discussion towards the end of this Letter). One then finds that n(x, y) obeys the BFKL equation in dipole form, that is, Eq. (5.7) in Ref. [1]. Thus, the Hamiltonian (26) describes the BFKL evolution of a system of dipoles, and † in that sense is equivalent to H0 ,Eq.(12). Thus the total Hamiltonian of our pomeron effective theory reads † = † + † + † H H0 H1→2 H2→1. (27) † The Hamiltonian H0 , describing the BFKL evolution, plays here the role of the free pomeron Hamiltonian. The † † other two pieces H2→1 and H1→2 correspond, respectively, to pomeron merging and splitting, and will naturally generate pomeron loops in the course of the evolution. The minimal pomeron loop, which is simply the one-loop correction to the scattering amplitude T(x, y)Y , can be isolated by the successive operation of these two parts of 228 J.-P. Blaizot et al. / Physics Letters B 615 (2005) 221–230

PL = † † the Hamiltonian, namely H1→2H2→1T0. The explicit results reads α¯ 2 α 2 PL =− s s M(x, y, z)M(u, v, w)A (x, z|u, w)A (z, y|w, v)∇2∇2 T (u, v) . (28) 2π 2π 0 0 u v 0 Y u,v,z,w Note that this result is free of any (ultraviolet or infrared) divergences. For instance, the pole in the dipole kernel at z = x is harmless because of A0(x, z = x|u, w) = 0. A simple physical picture of this result is obtained by assuming that this pomeron loop has been generated after the first two steps in the evolution starting with a target which is itself an elementary dipole (x0, y0). Then, Eq. (28) simplifies to: α¯ 2 PL0 =−2 s α4 M(x, y, z)M(x , y , w)A (x, z|x , w)A (z, y|w, y ). (29) 2π s 0 0 0 0 0 0 z,w This result has a clear physical interpretation: both original dipoles—in the projectile and the target—split into ¯ 2 new dipoles, processes which are represented by the two dipole kernels times αs . Then, the child dipoles from the A 4 two systems scatter with each other, by exchanging two pairs of gluons; this yields the two factors 0 times αs . Finally, note that this contribution is negative, as expected, leading to a decrease in the amplitude in the course of the evolution. As we have already emphasized, the Hamiltonian (27) reproduces the complete equations of motion established in [1,8,12]. One may gain some insight on how this works by analyzing how the merging processes in the effective Hamiltonian compare to those deduced from correct microscopic dynamics as described by JIMWLK. The action of HJIMWLK,Eq.(9), on the full dipole scattering amplitude T(x, y),Eq.(6),is 2 δ δ = g − b a † − a b † a b T(x, y) (δyv δxv) δux tr t t Vx Vy δuy tr t t Vx Vy . (30) δαY (u) δαY (v) Nc Simple algebra then easily yields the first Balitsky equation: α¯ H T(x, y) = s M(x, y, z) −T(x, y) + T(x, z) + T(z, y) − T(x, z)T (z, y) . (31) JIMWLK 2π z Then, after expanding the dipole operator T in the weak-field limit, and keeping terms up to the quartic order with respect to gauge field α, one finds an evolution equation which contains not only the BFKL dynamics, but also the lowest order mergings (four gluons merging into two). Consider now the action of the JIMWLK Hamiltonian on T0(x, y). Since: 2 δ δ = g ab − − a b T0(x, y) δ (δxu δyu)(δxv δyv), (32) δαY (u) δαY (v) 2Nc we have: g2 α¯ H T (x, y) = s M(x, y, z) + V˜ †V˜ − V˜ †V˜ − V˜ †V˜ . JIMWLK 0 2 Tr 1 x y x z z y (33) 2Nc 2π z When expanding the Wilson lines in powers of α, one obtains quadratic terms describing the BFKL evolution of T0(x, y) plus higher order terms which describe n → 2 gluon mergings. But at this level, it is easy to see that the 4 → 2 terms generated by this expansion are not the same as those in the r.h.s. of Eq. (25). For instance, while a a b b the merging term in Eq. (25) includes a piece containing three different transverse positions (i.e., αxαz αz αy ), the corresponding JIMWLK result in Eq. (33) cannot generate such terms. J.-P. Blaizot et al. / Physics Letters B 615 (2005) 221–230 229

Thus the actual, microscopic, dynamics of gluon merging in QCD is considerably more complicated than in the simple effective theory: cancellations occur in the action of HJIMWLK on Wilson lines, leaving a simple Hamiltonian acting on elementary pomerons. We interpret these cancellations as reflecting the fact that the dressing of pomerons by multiple scattering effects plays no role in their effective dynamics. This brings us to comment on the nature of the dynamics described by the effective theory. This theory generates (κ) evolution equations for the pomeron operators T0 which are formally identical to the equations satisfied by the (κ) complete dipole scattering operators T in QCD at large Nc. This means in particular that the solutions to the (κ) = equations for T0 Y will appear to saturate the unitarity (or ‘black disk’) limit T0 1 in the high energy limit, in spite of the fact that the respective operators describe single scatterings only! This indicates that one must be extremely careful in the physical interpretation of the effective theory. Let us then have a closer look at the microscopic dynamics that it describes. Effectively, the evolution of the target reduces to that of a system of dipoles subjected to a dynamics of a reaction–diffusion type: the dipoles undergo BFKL dynamics, they can split (one dipole into two dipoles), and they can also recombine with each other (two dipoles into one). The dynamics of such a system of dipoles is entirely coded in the k-body densities (k) nY (see Section 5 in Ref. [1] for a precise definition). Although we shall not work this out explicitly here, it is not hard, by using the results of Ref. [18] to relate these dipole densities to colorless correlation functions of the color charge density ρa. For instance, the dipole number operator n(x, y) can be identified with the bilocal operator ρa(x)ρa(y) of the effective theory. With such identifications, and by using the Hamiltonian in Eq. (27),it is straightforward to construct the evolution equations satisfied by the dipole densities. One thus finds that nY (x, y) (2) obeys the BFKL equation supplemented by a negative term proportional to nY , which is generated by the merging † (2) piece H2→1 of the Hamiltonian. Furthermore, the r.h.s. of the equation for ∂nY /∂Y includes the standard BFKL terms describing the individual evolutions of the two dipoles (x1, y1) and (x2, y2), but also a positive, fluctuation † term, proportional to nY —this is generated by the splitting piece H1→2 of the Hamiltonian, and is the same as the corresponding term deduced from the dipole picture in Refs. [1,8]—and, finally, a negative, recombination,term (3) proportional to nY . We thus obtain an infinite hierarchy, which describes a dipole reaction–diffusion dynamics, as 2 anticipated, and predicts the saturation of the dipole density at a value of order 1/αs . Now, it is clear that this is only an effective dynamics since, as well known, dipoles in real QCD do not simply recombine with each other: the interaction between two dipoles inside the target wavefunction goes beyond the large-Nc approximation and leads to more complicated color configurations, involving higher color multipoles [13]. The reason why it has been possible to simulate the non-linear effects responsible for unitarity corrections in the equations for the scattering amplitudes through simple ‘dipole recombination’ processes in the target wavefunction is because the same non-linear effects can be interpreted as projectile evolution, in which case they describe the splitting of a dipole in the projectile. Then, the 1 → 2 dipole splitting vertex from the projectile is simply reinterpreted, within the effective theory, as a 2 → 1 ‘dipole merging’ vertex in the target. Note finally that a similar dipole model including splitting and recombination has been recently used in Ref. [14] to generate evolution equations with pomeron loops. The present work shows how this effective dynamics may indeed emerge from the actual target dynamics in QCD, and points to numerous subtleties involved in this precise connection.

Acknowledgements

We would like to thank Larry McLerran for sharing with us his intuition about the potential importance of the ﬂuctuation–saturation duality. We acknowledge continuous conversations on this and related subjects with Al Mueller, and useful remarks and comments on the manuscript from Yoshitaka Hatta and Anna Stasto. One of the authors (K.I.) is supported by the program, JSPS Postdoctoral Fellowships for Research Abroad. 230 J.-P. Blaizot et al. / Physics Letters B 615 (2005) 221–230

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Leptogenesis, µ–τ symmetry and θ13

R.N. Mohapatra, S. Nasri, Hai-bo Yu

Department of Physics, University of Maryland, College Park, MD 20742, USA Received 2 February 2005; received in revised form 18 March 2005; accepted 28 March 2005 Available online 21 April 2005 Editor: M. Cveticˇ

Abstract We show that in theories where neutrino masses arise from type I see-saw formula with three right-handed neutrinos and where large atmospheric mixing angle owes its origin to an approximate leptonic µ–τ interchange symmetry, the primordial lepton asymmetry of the Universe, l can be expressed in a simple form in terms of low energy neutrino oscillation parameters as = 2 + 2 2 −2 −2 l (am bmAθ13),wherea and b are parameters characterizing high scale physics and are each of order 10 eV . ∝ 2 We also find that for the case of two right-handed neutrinos, l θ13 as a result of which, the observed value of baryon to photon ratio implies a lower limit on θ13. For specific choices of the CP phase δ we find θ13 is predicted to be between 0.10–0.15.  2005 Elsevier B.V. All rights reserved.

1. Introduction after electroweak symmetry breaking, lead to a primordial lepton asymmetry via the out of equilibrium → + There may be a deep connection between the origin decay NR H (where are the known lep- of matter in the Universe and the observed neutrino tons and H is the Standard Model Higgs ﬁeld). This oscillations. This speculation is inspired by the idea asymmetry subsequently gets converted to baryon– antibaryon asymmetry observed today via the elec- that the heavy right-handed Majorana neutrinos that are added to the Standard Model for understanding troweak sphaleron interactions [3], above T vwk small neutrino masses via the see-saw mechanism [1] (vwk being the weak scale). Since this mechanism in- can also explain the origin of matter via their decay. volves no new interactions beyond those needed in the The mechanism goes as follows [2]: CP violation in discussion of neutrino masses, one would expect that the same Yukawa interaction of the right-handed neu- better understanding of neutrino mass physics would trinos, which go into giving nonzero neutrino masses clarify one of the deepest mysteries of cosmology both qualitatively as well as quantitatively. This question has been the subject of many investigations in E-mail address: [email protected] recent years [4–11] in the context of different neu- (R.N. Mohapatra). trino mass models and many interesting pieces of in-

0370-2693/$ – see front matter  2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physletb.2005.03.082 232 R.N. Mohapatra et al. / Physics Letters B 615 (2005) 231–239 formation about issues such as the spectrum of right- (B) µ–τ symmetry for leptons is broken only at handed neutrinos, upper limit on the neutrino masses high scale in the mass matrix of the right-handed neu- etc have been obtained. In a recent paper, [12],two trinos. of the authors showed that if one assumes that the The Letter is organized as follows: in Section 2,we lepton sector of minimal see-saw models has a lep- outline the general framework for our discussion; in tonic µ–τ interchange symmetry [14,15], then one Section 3, we rederive the result of Ref. [12] for the can under certain plausible assumptions indeed predict case of exact µ–τ symmetry; in Section 4, we derive the magnitude of the matter–antimatter asymmetry in the connection between l and oscillation parameters 2 terms of low energy oscillation parameter, m and for the case of approximate µ–τ symmetry. Section 4 a high scale CP phase. The choice of µ–τ symmetry is devoted to the case of two right-handed neutrinos, was dictated by the fact that it is the simplest sym- where we present the allowed range of θ13 dictated metry of neutrino mass matrix that explains the max- by leptogenesis argument. In Section 5, we describe imal atmospheric mixing as indicated by data. Using a class of simple gauge models where these conditions 2 present experimental value for m, one obtains the are satisfied. right magnitude for the baryon asymmetry of the Uni- verse. The results of the paper [12] were derived in the 2. Introductory remarks on lepton asymmetry in limit that µ–τ interchange symmetry is exact. If how- type I see-saw models ever a nonzero value for the neutrino mixing angle θ13 is detected in future experiments, this would imply that We start with an extension of the minimal super- this symmetry is only approximate. Also, since in the symmetric Standard Model (MSSM) for the generic Standard Model νµ and ντ are members of the SU(2)L the type I see-saw model for neutrino masses. The doublets Lµ ≡ (νµ,µ) and Lτ ≡ (ντ ,τ), any symme- effective low energy superpotential for this model is try between νµ and ντ must be a symmetry between given by Lµ and Lτ at the fundamental Lagrangian level. The observed difference between the muon and tau masses cT cT MR cT c W = e YLHd + N YνLHu + N N . (1) would therefore also imply that the µ–τ symmetry has 2 c c to be an approximate symmetry. In view of this, it Here L, e , ν are leptonic superfields; Hu,d are the is important to examine to what extent the results of Higgs fields of MSSM. Yν and MR are general ma- Ref. [12] carry over to the case when the symmetry trices where we choose a basis where Y is diagonal. is approximate. We find two interesting results under We do not display the quark part of the superpoten- some very general assumptions: (i) a simple formula tial which is same as in the MSSM. After electroweak relating the lepton asymmetry and neutrino oscillation symmetry breaking, this leads to the type I see-saw observables for the case of three right-handed neutri- formula for neutrino masses given by = 2 + 2 2 nos, i.e., l (am bmAθ13) and (ii) a relation ∝ 2 v2 tan2 β of the form l θ13 for the case of two right-handed M =− T −1 wk ν Yν f Yν . (2) neutrinos. Measurement of θ13 will have important im- vR plications for both the models; in particular, we show The constraints of µ–τ symmetry will manifest them- that in a class of models with two right-handed neutri- selves in the form of the Y and M . It has been nos with approximate µ–τ symmetry breaking, there ν R pointed out that if we go to a basis where the right- is a lower limit on θ , which is between 0.1 to 0.15 de- 13 handed neutrino mass matrix is diagonal, we can solve pending on the values of the CP phase. These values for Y in terms of the neutrino masses and mixing an- are in the range which will be probed in experiments ν gles as follows [17]: in near future [16]. The basic assumption under which the two results 1/2 1/2 Y v = iMd R(z ) Md U †, (3) are derived are the following: ν R ij ν (A) type I see-saw formula is responsible for neu- where R is a complex matrix with the property that trino masses; RRT = 1. The unitary matrix U is the lepton mixing R.N. Mohapatra et al. / Physics Letters B 615 (2005) 231–239 233 matrix defined by other under the µ–τ symmetry. Both cases are in agreement with the observed neutrino mass differ- M = ∗Md † ν U ν U . (4) ences and mixings. The complex orthogonal matrices R can be parameter- It follows from Eq. (9) that ized as: 3M Im[m2R2 + m2R2 + m2R2 ] =− 1 1 11 2 12 3 13 . (10) = l 2| M † |2 R(z12,z23,z13) R(z23)R(z13)R(z12) (5) 8π v R(zij ) νR (zij ) 11 with Since the matrix R is an orthogonal matrix, we have the relation cos z12 sin z12 0 R(z ) = − sin z cos z 0 (6) 2 + 2 + 2 = 12 12 12 R11 R12 R13 1. (11) 001 Using this equation in Eq. (10), we get and similarly for the other matrices. zij are complex 2 2 2 2 angles. 3M Im[mR + m R ] =− 1 12 A 13 . (12) Let us now turn to lepton asymmetry: the formula l 2 | |2 8π v j ( R1j mj ) for primordial lepton asymmetry in this case, caused by right-handed neutrino decay is This relation connects the lepton asymmetry to both the solar and the atmospheric mass difference square [ ˜ ˜ †]2 1 Im YνYν M [5]. To make a prediction for the lepton asymmetry, we = 1j F 1 , l † (7) need to the lengths of the complex quantities R .The 8π (Y˜ Y˜ ) Mj 1j j ν ν 11 out of equilibrium condition does provide a constraint | | where Y˜ is defined in a basis where right-handed neu- on R1j as follows: ν trinos are mass eigenstates and their masses are de- | |2 −3 2 R1j mj 10 eV. (13) noted by M where F(x)=−1 [ 2x −ln(1 + x2)] 1,2,3 x x2−1 j=1,2,3 [18]. In the case where that the right-handed neutrinos Naively interpreted, this would have meant a strong have a hierarchical mass pattern, i.e., M1 M2,3,we get F(x)−3x. In this approximation, we can write constraint on the degenerate neutrino spectrum. How- the lepton asymmetry in a simple form [19] ever, as has been shown in Ref. [5] the preferred range | |2 −3 for j=1,2,3( R1j mj ) is from 10 to 0.1 eV with [ M† T ] 3 M1 Im Yν νYν 11 no strict upper bound, although an upper bound of on l =− , (8) 2 1/2 2 ˜ ˜ † ( m ) of 0.1 eV can be deduced from washout 8π v (YνYν )11 i processes. It is clear from Eq. (13) that if neutrinos are where using the expression for Yν given above, we can quasidegenerate based on this argument, we conclude rewrite l as: that a degenerate mass spectrum with m0 0.1eV d 1/2 d 2 d 1/2 will most likely be in conflict with observations, if 3 Im[M R(zij )M R(zij )M ]11 =− R ν R . type I see-saw is responsible for neutrino masses. It l 8π 2| M † |2 v R(zij ) νR (zij ) 11 must however be noted that a more appealing and nat- (9) ural scenario for degenerate neutrino masses is type II We will now apply this discussion to calculate the see-saw formula [20], in which case the above con- lepton asymmetry in the general case without any siderations do not apply. Therefore, it is not possible symmetries. In the following sections, we follow it up to conclude based on the leptogenesis argument alone with a discussion of two cases: (i) the cases of exact that a quasi-degenerate neutrino spectrum is inconsis- µ–τ symmetry and (ii) the case where this symmetry tent. is only approximate. Since the formula in Eq. (9) as- In a hierarchical neutrino mass picture, Eq. (13) 2 2 sumes that there are three right-handed neutrinos, we implies that |R13| 0.02 and |R12| 0.1. If we as- will focus on this case in the next two sections. In a sume that the upper limit in the Eq. (13) is saturated, subsequent section, we consider the case of two right- then we get the atmospheric neutrino mass difference handed neutrinos (Nµ,Nτ ), which transform into each square in Eq. (12) to give the dominant contribution. 234 R.N. Mohapatra et al. / Physics Letters B 615 (2005) 231–239

˜ We will see below that if one assumes an exact µ–τ where U = U (π/4) U 0 . Using this, we can cast 23 01 l symmetry for the neutrino mass matrix, the situation in the form: becomes different and it is the solar mass difference 2 2 3 M Im(cos z12)m square that dominates. = 1 . l 2 2 2 (17) 8π v (| cos z12| m1 +|sin z12| m2) This could also have been seen from Eq. (12) by real- 3. Three right-handed neutrinos and exact µ–τ izing that for the case of exact µ–τ symmetry, we have symmetry z13 = 0 and z23 = π/4. The above result reproduces the direct proportion- In this section, we consider the case of three right- ality between l and solar mass difference square handed neutrino with an exact µ–τ symmetryinthe found in Ref. [12]. To simplify this expression further, Dirac mass matrix as well as the right-handed neutrino let us note that out of equilibrium condition for the de- mass matrix. In this case, the right-handed neutrino cay of the lightest right-handed neutrino leads to the mass matrix MR and the Dirac–Yukawa coupling Yν condition: can be written, respectively, as: M2 M2 1 m | z |2 + m | z |2 1 , M11 M12 M12 2 1 cos 12 2 sin 12 14 (18) vwk MP MR = M12 M22 M23 , M M M which implies that 12 23 22 h11 h12 h12 2 2 −3 m1| cos z12| + m2| sin z12| 2 × 10 eV. (19) Yν = h21 h22 h23 , (14) h21 h23 h22 Since solar neutrino data require that in a hierarchical neutrino mass picture m 0.9 × 10−2 eV, in where M and h are all complex. An important 2 ij ij Eq. (19),wemusthave| sin z |2 ∼ 0.2. If we para- property of these two matrices is that they can be cast 12 meterize cos2 z = ρeiη, we recover the conclusions into a block diagonal form by the same transformation 12 of Ref. [12]. This provides a different way to arrive at matrix U (π/4) ≡ 10 on the ν’s and N’s. Let 23 0 U(π/4) the conclusions of Ref. [12]. ˜ us denote the block diagonal forms by a tilde, i.e., Yν ˜ ˜ and MR. We then go to a basis where the MR is subsequently diagonalized by the most general 2×2 unitary 4. Lepton asymmetry and µ–τ symmetry matrix as follows: breaking T × T × V (2 2)U23(π/4)MRU23(π/4)V (2 2) In this section, we consider the effect of breaking of = d MR, (15) µ–τ symmetry on lepton asymmetry. Within the see- saw framework, this breaking can arise either from the where V(2 × 2) = V 0 where V is the most general 01 Dirac mass matrix for the neutrinos or from the right- × = iα 2 2 unitary matrix given by V e P(β)R(θ)P(γ). handed neutrino sector or both. We focus on the case, × × The 3 3 case therefore reduces to a 2 2 prob- when the symmetry is broken in the right handed sec- lem. The third mass eigenstate in both the light and tor only. Such a situation is easy to realize in see-saw the heavy sectors play no role in the leptogenesis as models where the theory obeys exact µ–τ symmetry at well as generation of solar mixing angle [12].Note high scale (above the see-saw scale) prior to B–L sym- = also that we have θ13 0. The see-saw formula in the metry breaking as we show in a subsequent section. 1–2 subsector has exactly the same form except that We will also show that in this case there is a simple all matrices in the left- and right-hand side of Eq. (9) generalization of the lepton asymmetry formula that × are 2 2 matrices. The formula for the Dirac–Yukawa we derived in the exact µ–τ symmetric case [12].1 coupling in this case can be inverted to the form: ˜ Yν(2 × 2) 1 Leptogenesis in a speciﬁc µ–τ symmetric model where the = = d 1/2 × Md 1/2 × ˜ † Dirac Yukawa coupling has the form Yν diag(a,b,b) has been iMR (2 2)R(z12) ν (2 2)U , (16) discussed in Ref. [13]. Our discussion applies more generally. R.N. Mohapatra et al. / Physics Letters B 615 (2005) 231–239 235

In this case the neutrino Yukawa matrix is given in to first order by the mass eigenstates basis of the right-handed neutrinos by m1 m2 −iδ z13 = R21 − R22 θ13e cθ . (26) m3 m3 ˜ = + + + Yν V1/3V1/2V2/3Yν, (20) This proves that the matrix element R13 that goes into where Yν is the neutrino Dirac matrix in the flavor ba- the leptogenesis formula is directly proportional to the + × sis; The notation Vi/j denotes a unitary 2 2matrix physically observable parameter θ13. This enables us = 2 2 2 in the (i, j) subspace. In the above equation, V2/3 to write = am + bm θ . A consequence of ˜ l A 13 V2/3(π/4). Now if we substitute for Yν the expression this is that if the coefficient of proportionality is cho- in Eq. (3) and use maximal mixing for the atmospheric sen to be of order one, then as experimental upper limit neutrino we obtain goes down, unlike the generic type I see-saw case in ˜ Section 2, the solar mass difference square starts to Y × 0 1/2 + + 2 2 = V M R R m1/2U U . dominate for the LMA solution to the solar neutrino 0 y˜ 1/3 R 1/2 1/3 ν 1/2 1/3 3 problem. (21) Since the µ–τ symmetry breaking is assumed to be small and from reactor neutrino experiments θ13 1 5. Lepton asymmetry for two right-handed we will expand the mixing matrices in the 1–3 sub- neutrinos space to first order in mixing parameter: In this section, we consider the case of two right- (V,R,U)1/3 1 + (,z,θ)13E, (22) handed neutrinos which transform into one another where under µ–τ symmetry. The leptogenesis in this model 001 with exact µ–τ symmetry was discussed in [12] and E = 000. (23) was shown that it vanishes. In this model therefore, −100 a vanishing or very tiny θ13 would not provide a viable model for leptogenesis. Turning this argument around, To first order in 13, z13 and θ13 we have enough leptogenesis should provide a lower limit on 1/2 + 1/2 1/2 + the value of θ13. z13M R1/2EmνU + 13EM R1/2m U R 1/2 R ν 1/2 To set the stage for our discussion, let us first review − 1/2 1/2 + = θ13MR R1/2mν U1/2E 0. (24) the argument for the exact µ–τ symmetry case [12]. The symmetry under which (N ↔ N ) and L ↔ L It is straightforward to show that the perturbation pa- µ τ µ τ whereas the m = m constrains the general structure rameters should satisfy the following equations µ τ of Yν and MR as follows: + 13MR3 m3 z13MR1 m3R11 = M22 M23 − −iδ − MR , θ13e MR1 cθ (m1R11 m2R12) 0, M23 M22 M (m R s − m R c ) 13 R2 2 12 θ 1 11 θ = h11 h22 h23 − Yν . (27) − − iδ h11 h23 h22 z13MR3 m1cθ θ13e MR3 m3 0, + + 13MR2 (m1R11sθ m2R12cθ ) z13MR3 m1sθ 0, In order to calculate the lepton asymmetry using Eq. (7), we first diagonalize the right-handed neu- z13MR2 m3R21 ˜ − trino mass matrix and change the Yν to Yν . Since − θ e iδM c (m R − m R ) 0. (25) 13 R2 θ 1 21 2 22 MR is a symmetric complex 2 × 2 matrix, it can be diagonalized by a transformation matrix U(π/4) ≡ Where Rij are the matrix elements of R1/2 and cθ and √1 11 T , i.e., U(π/4)M U (π/4) = diag(M1,M2) sθ are the sine and cosine of the solar neutrino mixing 2 −11 R angle. Hence one can see that the parameter z13 is pro- where M1,2 are complex numbers. In this basis we ˜ portional to the θ13 neutrino mixing angle and is given have Yν = U(π/4)Yν . We can therefore rewrite the 236 R.N. Mohapatra et al. / Physics Letters B 615 (2005) 231–239 formula for n as M ∝ Im U(π/4)Y Y†U T (π/4) 2 F 1 . l ν ν 12 M j 2 (28) † AB Now note that YνYν has the form BA which can be diagonalized by the matrix U(π/4). Therefore it follows that = 0. Let us now introduce µ–τ symmetry breaking. If we introduce a small amount of µ–τ breaking in the right-handed neutrino sector as follows: we keep the Yν symmetric but choose the right-handed neutrino Fig. 1. Plot of ηB vs θ13 for the case of two right-handed neutrinos mass matrix as: with approximate µ–τ symmetry and CP phases δ = π/4andπ/3. The values of θ13 are predicted to be 0.1 and 0.15, respectively. The obs = +0.4 × −10 M22 M23 horizontal line corresponds to ηB (6.5−0.3) 10 [21]. MR = . (29) M23 M22(1 + β) After the right-handed neutrino mass matrix is diago- asymmetry. If we take this upper bound, then we get nalized, the 3 × 2 Y takes the form (for θ 1 and ν 13 an absolute lower bound on θ 0.015–0.008. Also in the basis where the light neutrino masses are diago- 13 we note that for values of M < 7 × 1011 GeV, the nal): 1 baryon asymmetry becomes lower than the observed ABwθ value. 13 . (30) xθ13 yθ13 D

Here B,D,x,y,w are of order one and θ13 ∝ β. To first order in the small mixing θ13, the complex 6. A model for µ–τ symmetry for neutrinos parameters A,B,D satisfy the constraint In this section, we present a simple extension of the A ∼ θ ,Bv2 m M , 13 2 1 minimal supersymmetric standard model (MSSM) by 2 Dv m3M2. (31) adding to it specific high scale physics that at low ener- Using these order of magnitude values, we now find gies can exhibit µ–τ symmetry in the neutrino sector that as well as real Dirac masses for neutrinos. First we recall that MSSM needs to be extended 3 M sin η[m2θ 2 ξ] 1 3 13 , by the addition of a set of right-handed neutrinos (ei- l 2 (32) 8π v m2 ther two or three) to implement the see-saw mecha- where ξ is a function of order one. It is clear that very nism for neutrino masses [1]. We will accordingly add small values for θ13 will lead to unacceptably small l. three right-handed neutrinos (Ne,Nµ,Nτ ) to MSSM. In Fig. 1, we have plotted ηB against θ13 for values of We then assume that at high scale, the theory has µ–τ the parameters in the model that fit the oscillation data S2 symmetry under which N± ≡ (Nµ ± Nτ ) are even and find a lower bound on θ13 0.1–0.15 for two dif- and odd combinations; similarly, we have for leptonic ferent values of the CP phases (Fig. 1). In this figure, doublet superfields L± ≡ (Lµ ± Lτ ) and leptonic sin- 11 c ≡ c ± c we have chosen, M1 7 × 10 GeV. For higher val- glet ones ± (µ τ ); two pairs of Higgs doublets ues of M1 the allowed range θ13 moves to the lower (φu,± and φd,±), and a singlet superfields S±. Other − c range and goes down like M 1/2.Itmusthowever superfields of MSSM such as Ne,Le,e as well as 1 quarks are even under the µ–τS symmetry. Now sup- be noted that in Ref. [23] an upper bound on M1 of 2 about 1014 GeV has been derived from the constraint pose that we write the superpotential involving the S −5 fields as follows: m˜ 1 10 eV that follows from the requirement that there must be a large enough density of the light- = + est right-handed neutrinos to lead to sufficient lepton WS λ1φu,−φd,+S− λ2φu,−φd− S+, (33) R.N. Mohapatra et al. / Physics Letters B 615 (2005) 231–239 237 then when we give high scale vevs to S±=M±, then The Higgs sector of the low energy superpotential below the high scale there are only the usual MSSM is determined from this theory after left–right gauge Higgs pair Hu ≡ φu,+ and Hd ≡ (cφd,+ + sφd,−) that group is broken down to the Standard Model gauge survive whereas the other pair becomes superheavy group by the vevs of χc. The phenomenon of doublet– and decouple from the low energy Lagrangian. The doublet spitting leaves only two Higgs doublets out of effective coupling at the MSSM level is then given the four in Φ± and is determined by a generic super- by: potential of type

c c c W = heLeHd e + h1LeHd + + h2LeHd m− = c + ¯ ¯ c WDD λij k χiΦj χk λij k χiφj χk c c c + h3L+Hd e + h4L−Hd e + h5L+Hd + i,j,k c c + − + − + + c c h6L Hd m− h7L Hd + f1LeHu, Ne + M1 χ±χ¯± + χ±χ¯± , (36) + f L H +N+ + f L+H +N 2 e u, 3 u, e where i, j, k go over ‘+’ and ‘−’ for even and odd + f4L+Hu,+N+ + f5L−Hu,+N−. (34) and only even terms are allowed by µ–τ invariance, e.g., λ+++,λ+−−,... are nonzero. Now suppose that Note that the µ–τ symmetry is present in the Dirac χc =0but χc = 0 and ¯χc = 0. These vevs neutrino mass matrix whereas it is not in the charged + − ± break the left–right group to the Standard Model gauge lepton sector as would be required to. group. It is then easy to see that below the χc We show below that it is possible to have a high scale, there are only one Higgs pair where Hu = φu,+ scale supersymmetric theory which would lead to real = and Hd i=+,−, , aiφd,i. Here we have denoted Dirac–Yukawa couplings (fi ) if we require the high 3 4 the Φ ≡ (φ ,φ ) and φ = χ±. The upshot of scale theory to be left–right symmetric. To show how u d d,3,4 all these discussions is that the right-handed neutrino this comes about, consider the gauge group to be Yukawa couplings are µ–τ even and therefore have the SU(2)L ×SU(2)R ×U(1)B–L with quarks and leptons form: assigned to left- and right-handed doublets as usual c − −   [22], i.e., Q(2, 1, 1/3), Q (1, 2, 1/3); L(2, 1, 1) h11 he+ 0 c and L (1, 2, +1); Higgs fields Φ(2, 2, 0); χ(2, 1, +1); =  ∗  c Yν he+ h++ 0 . (37) χ(¯ 2, 1, −1); χ (1, 2, −1) and χ¯c(1, 2, −1). The new 00h−− point specific to our model is that we have two sets of the Higgs fields with the above quantum numbers, It is easy to see that redefining the fields appropriately, one even and the other odd under the µ–τS2 permuta- we can make Yν real. So the only source of complex c c tion symmetry, i.e., Φ±, χ±, χ¯±, χ± and χ¯± (plus for phase in this model is in the RH neutrino mass matrix, fields even under S2 and ‘−’ for fields odd under S2). which in this model are generated by higher dimen- Furthermore, we will impose the parity symmetry un- sional couplings of the form LcLcχ¯ cχ¯ c as we discuss ∗ ∗ ∗ ∗ der which Q ↔ Qc , L ↔ Lc , (χ, χ¯ ↔ χc , χ¯c ), now. Φ ↔ Φ†. The most general nonrenormalizable interactions The Yukawa couplings of this theory invariant un- that can give rise to right-handed neutrino masses are der the gauge group as well as parity are given by the of the form: superpotential: = 1 c ¯ c 2 + c ¯ c 2 = T c WNR Leχ+ Leχ− W h11Le Φ+Le M T c T c T c + h++L Φ+L h−−L Φ+L h +L Φ+L c c 2 c c 2 c c 2 + + − − e e + + L+χ¯+ L−χ¯− + L−χ¯+ + ∗ T c + T c he+L+Φ+Le he−Le Φ−L− + Lc χ¯ c 2 Lc χ¯ c Lc χ¯ c . (38) + ∗ T c + T c + − + − − + he−L−Φ−Le h+−L+Φ−L− c ∗ T c Note that since both χ¯± acquire vevs, the last term in + h+−L−Φ−L (35) , the above expression will give rise to µ–τ breaking where h11,h++,h−− are real. in the RH neutrino sector while preserving it in the 238 R.N. Mohapatra et al. / Physics Letters B 615 (2005) 231–239

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Exclusive Higgs and dijet production by double pomeron exchange. The CDF upper limits

Adam Bzdak

M. Smoluchowski Institute of Physics, Jagiellonian University, Reymonta 4, 30-059 Kraków, Poland Received 1 February 2005; received in revised form 11 April 2005; accepted 12 April 2005 Available online 20 April 2005 Editor: P.V. Landshoff

Abstract We use as a starting point the original, central inclusive Bialas–Landshoff model for Higgs and dijet production by double pomeron exchange in pp (pp)¯ collisions. Next, we propose the simple extension of this model to the exclusive processes. We ﬁnd the extended model to be consistent with the CDF Run I, II upper limits for double diffractive exclusive dijet production. The predictions for the exclusive Higgs boson production cross sections at the Tevatron and the LHC energies are also presented.  2005 Elsevier B.V. All rights reserved.

PACS: 14.80.Bn; 13.87.Ce; 12.40.Nn

1. Introduction reﬂects our present limited understanding of the nature of the diffractive (pomeron) reactions. The discovery of the Higgs boson is one of the main The best way to reduce these uncertainties is to goals of searches at the present and next hadronic col- study other double pomeron exchange processes and liders, the Tevatron and the LHC. compare them with existing data. A particularly en- One appealing production mode, the double pome- lightening process is the DPE production of two jets ron exchange (DPE) one, see Fig. 1, was proposed (dijets). Such a process was originally discussed at the some time ago in Refs. [1,2]. In the following papers Born level in [12]. Later the dijet production was stud- this subject was discussed from different perspectives ied in [5,13] and in [8–11,14–19]. [3–11]. Despite some progress the serious uncertain- One generally considers two types of DPE events ties are still present that do not allow to get fully re- when colliding hadrons remain intact, namely, exclu- liable predictions needed for future experiments. This sive and central inclusive one (or central inelastic). In the exclusive DPE event the central object H is produced alone, separated from the outgoing hadrons by E-mail address: [email protected] (A. Bzdak). rapidity gaps:

2 where |Mqq| is the production amplitude squared for colliding quarks:1

2 s − − |M |2 = C δ2 2α(t1)δ2 2α(t2) qq (u )2(u )2 1 2 1 2 2 × exp 2β(t1 + t2) R . (4) Fig. 1. Production of Higgs boson H , dijet jj, by double pomeron Transverse momenta of the produced gluons are de- exchange. The colliding hadrons remain intact. noted by u1 and u2. The constants C and R will be defined later. α(t) = 1 + + αt is the pomeron Regge ≈ . α = . −2 t t pp¯ → p + gap + H + gap +¯p. (1) trajectory with 0 08, 0 25 GeV ( 1, 2 are defined in Fig. 1). F(t)= exp(λt) is the nucleon form- −2 In the central inclusive DPE event there is an addi- factor with λ = 2GeV . δ1, δ2 are defined as δ1,2 ≡ tional radiation X accompanying the central object H : 1 − k1,2/p1,2 (k1, k2, p1, p2, are defined in Fig. 1). −2 The factor exp(2β(t1 + t2)) with β = 1GeV [23] pp¯ → p + gap + HX+ gap +¯p. (2) takes into account the effect of the momentum transfer dependence of the non-perturbative gluon propagator Recently, using the Bialas–Landshoff [2] model givenby(p2 is the Lorentz square of the momentum for central inclusive double diffractive production the carried by the non-perturbative gluon): cross-section for gluon jet production was calculated 2 2 2 [18,19]. In this model in some approximation pomeron D p = D0 exp p /τ . (5) exchange corresponds to the exchange of a pair of non- perturbative gluons which takes place between a pair The constants C and R are defined as: of colliding quarks [20]. The obtained results together 2 2 1 6 g /4π with those for quark–antiquark jets calculated some C = D G2τ τ 2 , (6) (27π)2 0 G2/4π time ago [15] give the full cross-section for dijet production in double pomeron exchange reactions. The R = 9 dQ 2 Q 2 exp −3Q 2 = 1. (7) model was found [19] to give correct order of magni- T T T tude for the measured [21] central inclusive dijet cross Here G and g are the non-perturbative and per- sections. turbative quark–gluon couplings, respectively.2 τ is In this Letter we propose the simple extension of the range of the non-perturbative gluon propagator (5) this model to the exclusive processes. We find the ex- and D0 its magnitude at vanishing momentum trans- tended model to be consistent with the CDF Run I, fer. From data on the elastic scattering of hadrons one II upper limits [21,22] for double diffractive exclusive 2 −1 infers D0G τ = 30 GeV and τ = 1GeV. dijet production. We also present the predictions for The constant R reflects the structure of the loop in- the exclusive Higgs boson production cross sections at tegral. QT is the transverse momentum carried by each the Tevatron and the LHC energies. of the three non-perturbative gluons. R was shown explicitly in Eq. (4) for the reason which will become clear in the next section. 2. Central inclusive dijet production Taking into account (4) we obtain the following result for the differential cross-section [19]: The matrix element for two gluon jet production in the Bialas–Landshoff model is given [18] by the 1 This formula is only valid in the limit of δ1,2 1 and for small s-channel discontinuity of the diagrams shown in momentum transfer between initial and final quarks. Fig. 2. The square of the matrix element (averaged and 2 One should note that the non-perturbative quark–gluon cou- summed over spins and polarizations) is of the form: pling G does not depend on the scale of the process. 242 A. Bzdak / Physics Letters B 615 (2005) 240–246

Fig. 2. Three diagrams contributing to the amplitude of the process of gluon pair production by double pomeron exchange. The dashed lines represent the exchange of the non-perturbative gluons.

dσ However, we believe that this is not a serious objec- 2 d(E )d( y)dy tion to our model. In the Bialas–Landshoff approach T 2 the produced object (Higgs, dijet, etc.) is coupled to 2 −4 s = R CE(ET) the non-perturbative gluons via the perturbative cou- 4E2 cosh2( y/2) T pling g. It is not clear and the question of consistency π 3/4α 2 could be addressed. Finally, let us note that estimates × . (8) λ+β 2ET cosh( y/2) 2 2 − ln √ − y in the present Letter are based on the basis of the pure α s forward direction. It was first mentioned in [14] that 8 Here CE = 81C/(16(2π) ). ET =|u1|=|u2| is the such approach may lead to incorrect results. transverse energy of one of the produced gluons. y = Integrating (8) over the CDF Run I kinematical y1 − y2, y = (y1 + y2)/2 where y1,2 are the rapidities range [21] for the central inclusive production of dijets of the produced gluons. For completeness it is neces- of ET > 7 GeV we obtain [19] the result to be about 2 = 2 sary to say that the rapidities y1,2 are connected with 70 nb (with G /4π 1 and no Sgap), to be compared ± δ1, δ2 and ET in the following way: with the CDF measurement of 43 nb (43 26 nb). We √ thus scale our cross section by a factor of 43/70 ≈ 0.6, δ s = E exp(y ) + E exp(y ), 1√ T 1 T 2 that is: δ s = E (−y ) + E (−y ). 2 T exp 1 T exp 2 (9) √ 2 = Sgap( s 2TeV) The result (8) does not take gap survival effect = . . 2 2 2 0 6 (10) (Sgap) into account, i.e., the probability of the gaps not (G /4π) to be populated by secondaries produced in the soft rescattering. From [5,24] we expect that for the Teva- This completes the summary of [18] and [19]. 2 tron energy it is about 0.05–0.1. The factor Sgap is not a universal number but it depends on the initial energy and the particular final state. Theoretical predictions of 2 3. Exclusive dijet production—Sudakov factor the survival factor, Sgap, can be found in Ref. [25]. The main uncertainty in the expression (8) is the value of G2/4π (see (6)). It is expected to be [26] As was already mentioned the calculation pre- about 1 but in fact it should be considered only as an sented in the previous section, based on the original order of magnitude estimate. Bialas–Landshoff model, is a central inclusive one, Let us now make clear the rather ad hoc nature i.e., the radiation is present in the central region of the of many of the assumptions inherent in the Bialas– rapidity. Landshoff approach [2]. The predictions of this model In order to describe the exclusive processes one depend only weakly on energy (∼ s2 ). This is a con- has to forbid this radiation. To do it we include the sequence of the Regge-like dependence implied by Sudakov survival factor T(QT,µ) inside the loop in- Eq. (4). There are some controversies if such assump- tegral (7) over QT. The Sudakov factor T(QT,µ) is tion is justified. In our calculations we assume the ex- the survival probability that a gluon with transverse ponential form of the non-perturbative gluon propaga- momentum QT remains untouched in the evolution up tor (5). As was already stated in [2] there is no reason to the hard scale µ = Mgg/2, where Mgg is the mass to believe that the true form of D is as simple as this. of the produced gluons. The function T(QT,µ)can be A. Bzdak / Physics Letters B 615 (2005) 240–246 243 y calculated as [5]: µ = E cosh . (15) T 2 µ2 2 2 αs(kT) dkT Naturally a question of internal consistency arises. T(QT,µ)= exp − 2π k2 Namely, the Sudakov factor uses perturbative gluons 2 T QT whilst in our calculations the Born amplitude (4) uses − 1 ∆ non-perturbative gluons. It is not clear what the non- perturbative gluon is and the extension of the original × zPgg(z) + Pqg(z) dz . q Bialas–Landshoff model to the exclusive processes is 0 (11) not straightforward. We hope that taking the Sudakov Here ∆ =|kT|/(µ +|kT|), Pgg(z) and Pqg(z) (we take factor in the loop integral into account we obtain an q = u, d, s, u,¯ d,¯ s¯) are the GLAP spitting functions. approximate insight into exclusive processes. It should 3 αs is the strong coupling constant. be emphasized that at present our calculation is a hy- Taking into account the leading-order contributions brid of perturbative and non-perturbative ideas. [27] to the GLAP splitting functions: At the end of this section let us notice that the Su- dakov factor (11) does not depend on the sum of the z 1 − z = + P (z) = 6 + + z(1 − z) dijet rapidities y (y1 y2)/2. This together with the gg 1 − z z observation that 11 nf + δ(1 − z) − , 2 3 2 + 2 = y (λ β)/α 144, P (z) = z2 + (1 − z)2 /2, (12) qg 2 2 = 4ET cosh ( y/2)/s δ1δ2 1, we obtain: 1−∆ leads to the conclusion that the differential cross 11 section for DPE exclusive dijets production very zP (z) dz =− + 12∆ − 9∆2 + 4∆3 gg 2 weekly depends on the sum of the dijet rapidity y. 0 This feature agrees with the observation found in − 4 E 6.5 − 3∆ − Ref. [13]. Moreover, the observed power law T 6ln∆, ˜2 ∼ −2.2 2 (with R ET ) is close to the observation of − 1−∆ Ref. [7] (∼ E 7.3). 1 ∆ ∆2 ∆3 T P (z) dz = − + − . (13) qg 3 2 2 3 0 Now to describe the exclusive processes we use the 4. CDF Run I, II upper limits formula (8) with R2 = 1 replaced by R˜2(µ) where ˜ 4 R(µ) is defined as: CDF Collaboration has presented results on upper limits on exclusive DPE dijet production cross sec- µ2 tions. ˜ = 2 2 − 2 √ R(µ) 9 dQT QT exp 3QT T(QT,µ). (14) At Run I ( s = 1.8TeV)[21] the upper bound for Λ2 exclusive dijets production was measured to be 3.7 nb ≡ µ = M / E for the kinematic range of 0.035 <δ2 δp¯ < 0.095 The hard scale gg 2 can be expressed by T − and y in the following way: and jets of ET > 7 GeV confined within 4.2 10 GeV [ET > 25 GeV] 4 Notice that µ>1.5 GeV is required so that was measured to be 970 ± 65(stat) ± 272(syst) pb 2 9 µ dQ2 Q2 exp(−3Q2 ) = 1. (34 ± 5(stat) ± 10(syst) pb). The kinematics is follow- Λ2 T T T 244 A. Bzdak / Physics Letters B 615 (2005) 240–246

Table 1 Comparison of the CDF upper limits for DPE exclusive dijet production with the results obtained in the presented model Transverse CDF Model energy upper limits 2 2 2 = Sgap/(G /4π) 0.6 Fig. 3. The diagram contributing to the amplitude of the process of E > 7GeV 3.7 [nb] 1 [nb] T Higgs boson production by double pomeron exchange. The Higgs E > 10 GeV 970 ±337 [pb] 300 [pb] T coupling is taken to be through a t-quark loop. The dashed lines E > 25 GeV 34 ±15 [pb] 3 [pb] T represent the exchange of the non-perturbative gluons.

− − 2 2 2 2 2α(t1) 2 2α(t2) 2 ing:5 0.03 <δ ≡ δ ¯ < 0.1, jets are conﬁned within |Mpp | = BN s δ δ F(t1)F (t2) 2 p 1 2 −2.5

Table 2 about two orders of magnitude smaller for the LHC en- Our result for DPE√ exclusive Higgs production cross section for ergy. It reflects the possible large uncertainties of the = the Tevatron energy, s 2 TeV. Our prediction is about 10 times presented approach and the general fact that the per- smaller than the prediction based on the KMR model turbative QCD predictions, on the contrary to the non- KMR model Our model perturbative two-gluon-exchange-type models, show a 2 = 2 2 2 = Sgap 0.05 Sgap/(G /4π) 0.6 strong increase of the cross sections with increasing σ [fb] 0.06 0.005 energy. Hopefully a study of the dijets production as a function of the energy will clearly be able to discrim- Table 3 inate between the perturbative QCD determinations and the non-perturbative model approaches. Our result for√ DPE exclusive Higgs production cross section for the LHC energy ( s = 14 TeV). A distinct difference, ∼ 102, with the KMR model prediction is observed KMR model Our model Acknowledgements 2 = 2 2 2 = Sgap 0.02 Sgap/(G /4π) 0.25 σ [fb] 2 0.015 It is a pleasure to thank Dr. Leszek Motyka, Prof. Andrzej Bialas and Prof. Robert Peschanski for useful discussions. This investigation was supported by the 2 2 ln(sδ /MH ) what leads to the final result: Polish State Committee for Scientific Research (KBN) under grant 2 P03B 043 24. BN2 s 2 σ = 5 3 2 2 4 π α MH References R˜2(M /2) sδ2 × H √ ln . (20) 2 2 [1] A. Schafer, O. Nachtmann, R. Schopf, Phys. Lett. B 249 (1990) ((λ + β)/α − ln[MH / s ]) M H 331. Now we are ready to give our predictions for DPE [2] A. Bialas, P.V. Landshoff, Phys. Lett. B 256 (1991) 540. exclusive Higgs production at the Tevatron and the [3] J.R. Cudell, O.F. Hernandez, Nucl. Phys. B 471 (1996) 471. LHC energies. We also compare our results with those [4] D. Kharzeev, E. Levin, Phys. Rev. D 63 (2001) 073004. [5] V.A. Khoze, A.D. Martin, M.G. Ryskin, Eur. Phys. J. C 14 obtained in a model developed by Khoze, Martin and (2000) 525. Ryskin (KMR model). [6] V.A. Khoze, A.D. Martin, M.G. Ryskin, Eur. Phys. J. C 19 √ In Table 2 the prediction for the Tevatron energy, (2001) 477; s = 2 TeV, is shown. The mass of the Higgs is taken V.A. Khoze, A.D. Martin, M.G. Ryskin, Eur. Phys. J. C 20 (2001) 599, Erratum. to be 120 GeV and αs(MH ) is about 0.1. We also in- → [7] A.B. Kaidalov, V.A. Khoze, A.D. Martin, M.G. Ryskin, Eur. clude the αs virtual correction [5,30] to the gg H Phys. J. C 33 (2004) 261. vertex factor, so-called K-factor to be about 1.5. As [8] B. Cox, J. Forshaw, B. Heinemann, Phys. Lett. B 540 (2002) was discussed earlier we take δ = 0.1 and assume 263. 2 2 2 [9] M. Boonekamp, R. Peschanski, C. Royon, Phys. Rev. Lett. 87 Sgap/(G /4π) to be 0.6. (2001) 251806. Before√ we present the prediction for the LHC = [10] M. Boonekamp, A. De Roeck, R. Peschanski, C. Royon, Acta energy, s 14 TeV, we have to take into ac- Phys. Pol. B 33 (2002) 3485; count the s dependence of the gap survival√ fac- M. Boonekamp, R. Peschanski, C. Royon, Nucl. Phys. B 669 2 2 = tor Sgap. Following√ [25] we expect that Sgap( s (2003) 277; 2TeV)/S2 ( s = 14 TeV) ≈ 0.4 what allows us to M. Boonekamp, R. Peschanski, C. Royon, Phys. Lett. B 598 gap (2004) 243. 2 2 2 = assume Sgap/(G /4π) 0.25. The obtained result [11] N. Timneanu, R. Enberg, G. Ingelman, Acta Phys. Pol. B 33 for DPE exclusive Higgs (MH = 120 GeV) produc- (2002) 3479; tion at the LHC energy is presented in Table 3. R. Enberg, G. Ingelman, A. Kissavos, N. Timneanu, Phys. Rev. As can be seen from Table 2 and Table 3 the re- Lett. 89 (2002) 081801; R. Enberg, G. Ingelman, N. Timneanu, Phys. Rev. D 67 (2003) sults for DPE exclusive Higgs production are about 011301. one order of magnitude smaller than those obtained [12] A. Berera, J. Collins, Nucl. Phys. B 474 (1996) 183; in the KMR model [5,6] for the Tevatron energy and A. Berera, Phys. Rev. D 62 (2000) 014015. 246 A. Bzdak / Physics Letters B 615 (2005) 240–246

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Higgs mass in the gauge-Higgs uniﬁcation

Naoyuki Haba a, Kazunori Takenaga b, Toshifumi Yamashita c

a Institute of Theoretical Physics, University of Tokushima, Tokushima 770-8502, Japan b Department of Physics, Osaka University, Toyonaka, Osaka 560-0043, Japan c Department of Physics, Kyoto University, Kyoto 606-8502, Japan Received 14 February 2005; received in revised form 27 March 2005; accepted 1 April 2005 Available online 20 April 2005 Editor: T. Yanagida

Abstract The gauge-Higgs unification theory identifies the zero mode of the extra-dimensional component of the gauge field as the usual Higgs doublet. Since this degree of freedom is the Wilson line phase, the Higgs does not have the mass term nor quartic coupling at the tree level. Through quantum corrections, the Higgs can take a vacuum expectation value, and its mass is induced. The radiatively induced mass tends to be small, although it can be lifted to O(100) GeV by introducing the O(10) numbers of bulk fields. Perturbation theory becomes unreliable when a large number of bulk fields are introduced. We reanalyze the Higgs mass based on useful expansion formulae for the effective potential and find that even a small number of bulk field can have the suitable heavy Higgs mass. We show that a small (large) number of bulk fields are enough (needed) when the SUSY breaking mass is large (small). We also study the case of introducing the soft SUSY breaking scalar masses in addition to the Scherk–Schwarz SUSY breaking and obtain the heavy Higgs mass due to the effect of the scalar mass.  2005 Elsevier B.V. All rights reserved.

1. Introduction

There are much progress in the higher-dimensional gauge theories. One of the most fascinating motivations for the higher-dimensional gauge theory is that gauge and Higgs fields can be unified [1–13]. The higher-dimensional components of gauge fields become scalar fields bellow the compactification scale, and these scalar fields are identified with the Higgs fields in the gauge-Higgs unification theory. In fact, the adjoint Higgs fields can emerge through the S1 compactification from 5D theory, while the Higgs doublet fields can appear through the orbifold 1 compactification such as S /Z2.

E-mail addresses: [email protected], [email protected] (N. Haba), [email protected] (K. Takenaga), [email protected] (T. Yamashita).

In order to obtain the Higgs doublets from the gauge fields in higher dimensions, the gauge group must be lager than the Standard Model (SM) gauge group. The gauge symmetries are reduced by the orbifolding boundary conditions of the extra dimensions and can be broken further by the Hosotani mechanism [2]. The Higgs fields have only finite masses of order the compactification scale because the masses of the Higgs fields are forbidden by the higher-dimensional gauge invariance. In the previous works [11,12], we have studied the possibility of the dynamical electro-weak symmetry breaking in two gauge-Higgs unified models, SU(3)c × SU(3)W and SU(6) models. We calculated the one-loop effective potential of Higgs doublets and analyze the vacuum structure of the models, and a similar analysis of the 6D gauge- Higgs unification model is studied in Ref. [13]. We found that the introduction of the appropriate numbers and representation of extra bulk fields are required for the desirable symmetry breaking. Since the Higgs is essentially the Wilson line degree of freedom, the mass term nor quartic coupling does not exist in the Higgs potential at the tree level. Through quantum corrections, the Higgs can develop a vacuum expectation value, which means the dynamical electro-weak symmetry breaking is realized and accordingly its mass is induced. The induced Higgs mass tends to be small, less than the weak scale, reflecting the nature of the Coleman–Weinberg mechanism [14]. It is possible to lift the magnitude of the Higgs mass to O(100) GeV by introducing the O(10) numbers of bulk fields in Refs. [11,12]. The perturbation expansion is given by g2/16π 2 × [bulk fields degrees of freedom] (g: gauge coupling), so that the analysis of the one-loop effective potential cannot be reliable when there are a large number of bulk fields. Furthermore, it seems artificial to introduce a large number of extra fields. In this Letter we reanalyze and study the Higgs mass analytically, not only numerically in the gauge-Higgs unification. Based on the expansion formulae for the effective potential, we show that a small (large) number of bulk fields are needed when the SUSY breaking mass is large (small). We find that even a small number of bulk field can make the Higgs mass suitably heavy. The expression for the Higgs mass is obtained by using the formula, which makes it clear that what mainly controls the magnitude of the Higgs mass. The analyses made in this Letter by using the expansion formula are applicable to the bulk field with an arbitrary representation under an arbitrary gauge group. We also study the case that the soft SUSY breaking scalar masses exist in addition to the Scherk–Schwarz (SS) SUSY breaking [15–19]. The soft scalar masse also plays the role to lift the Higgs mass. And we show that also in this case a small (O(1)) number of extra bulk fields can realize the suitable electro-weak symmetry breaking and the Higgs mass of O(100) GeV. This Letter is organized as follows. In Section 2, we briefly overview the previous works. In Section 3 we will present useful expansion formulae for the effective potential, and by using them, we study the Higgs mass in the gauge-Higgs unified models. In Section 4, we also study the Higgs mass for the case of existing soft SUSY breaking scalar masses. Section 5 devotes summary and discussion.

2. Gauge-Higgs uniﬁcation

In this section we give a brief review of the dynamical electro-weak symmetry breaking in the SU(3)c ×SU(3)W and SU(6) models based on Refs. [11,12]. In the gauge-Higgs unification models, the 5th-dimensional coordinate 1 is compactified on an S /Z2 orbifold. The Higgs doublets are identified as the zero modes of the extra-dimensional component of the 5D gauge field, A5. We check whether the dynamical electro-weak symmetry breaking is possible or not by calculating one-loop effective potential of the Higgs doublets. Denoting y as the coordinate of the 5th dimension, parity operator, P (P ) are defined according to the Z2 transformation, y →−y (πR−y → πR+y). In the SU(3)c ×SU(3)W model [4–6],wetakeP = P = diag(1, −1, −1) 1 (P = P = I) in the base of SU(3)W (SU(3)c). Then, there appears Higgs doublet as the zero mode of A5,

1 Taking account of the scalar degrees of freedom in the gauge super-multiplet, we can easily show that there appear two Higgs doublets in the SUSY theory. N. Haba et al. / Physics Letters B 615 (2005) 247–256 249 √ H = 2πRA5. (1) √ 2 The 4D gauge coupling constant is defined as g4 = g/ 2πR. The vacuum expectation value (VEV) of A5 is parameterized by a/(2gR)E3, where E3 is the 3 × 3 matrix having 1 at (1, 3) and (3, 1) elements, while the other elements being zero [11,12]. The relation between the VEV and electro-weak scale is given by √ 4 = a0 = ∼ 2πR A5 v 246 GeV. (2) g4R 4 = a a a 4 = 1 Here the component gauge field A5 is defined by A5 a A5T through the generators T , where T 2 E3. The compactification scale must be above the weak scale, and when we take it as a few TeV, for examples, a0 should be a parameter of O(10−1∼−2). Let us study SUSY theory with SS SUSY breaking. We define + 1 J ( )[a,β,n]≡ 1 − cos(2πnβ) cos(πna), n5 − 1 J ( )[a,β,n]≡ 1 − cos(2πnβ) cos πn(a − 1) , n5 where β(0 β 0.5) parameterizes the magnitude of the SS SUSY breaking. Then, the soft mass parameters become O(β/R) [11,12]. The contribution of the gauge multiplet to the effective potential is written as

∞ gauge =− (+)[ ]+ (+)[ ] Veff 2C J 2a,β,n 2J a,β,n , (3) n=1 where C ≡ 3/(64π 7R5).TheVEVofσ , which forms the real part of scalar component of N = 1 chiral multiplet at low-energies, becomes zero by calculation of the effective potential for σ [20]. The minimum of the effective potential (3) is located at a0 = 1 (mod 2), which means that the suitable electro-weak scale VEV, (0 <) a0 1 and electro-weak symmetry breaking are not realized. Thus, for the desirable dynamical electro-weak symmetry (±) (±) breaking, one needs to introduce the extra bulk fields, which are Nfnd and Nadj species of hypermultiplets of fundamental and adjoint representations, respectively. Here the index, (±), denotes the sign of the intrinsic parity of PP defined in Refs. [11,12]. The effective potential from the bulk fields is given by

∞ matter = (+) (+)[ ]+ (+)[ ] + (−) (−)[ ]+ (−)[ ] Veff 2C Nadj J 2a,β,n 2J a,β,n Nadj J 2a,β,n 2J a,β,n n=1 + (+) (+)[ ]+ (−) (−)[ ] Nfnd J a,β,n Nfnd J a,β,n . (4) (+) = (−) = (−) = (+) = = −1 Ref. [11] shows one example, Nadj Nadj 2, Nfnd 4, Nfnd 0 with β 0.1 and R of order a few TeV, in which the suitable electro-weak symmetry breaking is realized by the small VEV, a0 = 0.047. The Higgs mass ≡ ¯ = gauge + matter is calculate by the second derivative of the effective potential, Veff CVeff Veff Veff with respect to a at the minimum, a = a0, √ 2 ¯ 1/2 2 3 ∂ Veff vg4 mH ∼ × , (5) 4π 3 ∂a2 a a=a0 0

2 We should take g4 1 for the wall-localized kinetic terms being the main part of the MSSM kinetic terms [11]. Hence, we take g4 = O(1) in this Letter. 250 N. Haba et al. / Physics Letters B 615 (2005) 247–256 wherewehaveused(2). In this case Higgs mass is calculated as3

0.025g 2 m2 ∼ 4 ∼ 118g2 GeV 2, (6) H R 4 where g4 = O(1), as explained above. The Higgs mass is likely to be smaller than the weak scale, 246 GeV (Eq. (2)) since it is zero at the tree level and is induced through the radiative corrections (Coleman–Weinberg mechanism). As for the SU(6) model [5,6], we take the parities, P = diag(1, 1, 1, 1, −1, −1) and P = diag(1, −1, −1, −1, −1, −1), which induces Higgs doublet in A5 as the zero mode. The VEV of A5 is written as a/(2gR)E6, where E6 is the 6 × 6 matrix having 1 at (1, 6) and (6, 1) elements while the other elements being zero [11,12]. The gauge part of the effective potential is given by ∞ gauge =− (+)[ ]+ (+)[ ]+ (−)[ ] Veff 2C J 2a,β,n 2J a,β,n 6J a,β,n . (7) n=1

As in the SU(3)c × SU(3)W model, the suitable symmetry breaking cannot be realized only by the gauge sector. This situation can be changed by introducing the extra bulk ﬁelds, which induce the effective potential, ∞ matter = (+) (+)[ ]+ (+)[ ]+ (−)[ ] Veff 2C Nadj J 2a,β,n 2J a,β,n 6J a,β,n n=1 + (−) (−)[ ]+ (−)[ ]+ (+)[ ] Nadj J 2a,β,n 2J a,β,n 6J a,β,n + (+) (+)[ ]+ (−) (−)[ ] Nfnd J a,β,n Nfnd J a,β,n . (8) (+) = (−) = We show one example of the suitable symmetry breaking in Ref. [11], which is the case of Nadj 2, Nfnd 10, (−) = (+) = = −1 = Nadj Nfnd 0 with β 0.1 and R of order a few TeV. In this case, the minimum exists at a0 0.047, and the Higgs mass squared is calculated as

0.024g 2 m2 ∼ 4 ∼ 120g2 GeV 2. (9) H R 4 In the above two examples O(10) numbers of bulk fields are required for the suitable symmetry breaking and Higgs mass. Naively, this situation seems inevitable in the gauge-Higgs unification theory since Higgs doublets are originally Wilson line phases and do not have the quartic couplings nor mass terms in the tree level. In Section 3 we obtain the effective mass term and quartic coupling by expanding the cosine functions with respect to a in the effective potential and study the condition for the suitable symmetry breaking and Higgs mass. We will check whether a large numbers of extra fields are really needed or not.

3. Higgs mass

The Higgs mass is deﬁned by the 2nd derivative of the effective potential. Here we concentrate on the mass term and quartic coupling in the radiatively induced effective potential. We comment on higher order terms in the last section. The effective potentials in the previous section are written by the linear combinations of J (±)[a,β,n] and J (±)[2a,β,n]. Effective potentials are generally the linear combination of J (±)[ma,β,n], (m: integer) (see, for examples, Refs. [11,12,21]). Here we study the contribution from the fundamental representation bulk ﬁelds for

3 In Ref. [11], we calculated it by using approximation formulae, and got slightly different value. N. Haba et al. / Physics Letters B 615 (2005) 247–256 251 simplicity. Although the gauge sector and adjoint representation bulk ﬁelds also induce J (±)[2a,β,n] terms (higher representations can induce J (±)[ma,β,n] in general), these contributions can be incorporated straightforwardly. By using approximations for small x, we obtain, up to O(x8), that

∞ cos(nx) ζ (3) x4 25 x6 x8 ∼ ζ (5) − R x2 − ln x2 + x4 + + , (10) n5 R 2 48 288 8640 4838400 n=1 ∞ cos(n(x − π)) 15 3 x4 x6 x8 ∼− ζ (5) + ζ (3)x2 − ln 2 + + . (11) n5 16 R 8 R 24 2880 322560 n=1 Let us note that these formulae are also applicable to the bulk ﬁeld with the higher representations under the gauge group. Then, we can show that

∞ 2 2 + a β a a J ( ) , ,n ∼ 25a2 − 432β2 − 6a2 ln + 144β2 ln 4β2 , (12) π π 288 4β2 n=1

∞ 2 − a β β J ( ) , ,n ∼− a4 − 48a2 ln 2 , (13) π π 48 n=1 for a β.4 Then, the coefﬁcients of a2 and a4 in Eq. (12) are roughly given by

β2 25 − 6ln(a2/4β2) − 432 − 144 ln 4β2 (< 0) and 0 (> 0), (14) 288 288 respectively. On the other hand, coefﬁcients of a2 and a4 in Eq. (13) are

β2 ln 2 (> 0) and −β2/48 (< 0), (15) respectively. For realizing the suitable heavy Higgs mass, the quartic coupling should be large and positive. On the other hand, the VEV (W and Z boson masses) should be maintained small (a0 1) comparing to the compactification scale. For this purpose, large negative contribution in the first term in Eq. (14) must be almost cancelled by (−) = O 2 introducing Nfnd (ln 4β ) numbers bulk fields acting on the first term in Eq. (15). This means that the less (more) bulk fields are needed when β becomes large (small). Eqs. (14) and (15) shows that even in the case of 4 this cancellation, the coefficient of a is still positive and large enough when a0 β. Thus, the heaviness of Higgs − 2 2 mass is mainly controlled by the factor ln(a0/β ) in the effective quartic coupling, which implies that the smaller 2 2 (larger) a0/β becomes, the larger (smaller) the Higgs mass becomes. A typical numerical example for realizing this in the SU(3)c × SU(3)W model is given in Table 1.

Table 1 (+) (−) (+) (−) 2 Nadj Nadj Nfnd Nfnd βa0 mH /g4 (GeV) 22020.10 0.0891 95 22020.13 0.0574 117 22020.14 0.0379 130

4 In the usual scenario, a<βshould be satisﬁed since the SUSY breaking mass, O(β/R) must be larger than the electro-weak scale, O(a/R). 252 N. Haba et al. / Physics Letters B 615 (2005) 247–256

The observation given above becomes clearer if we apply (12) and (13) to the effective potential, Veff and take the 4 order up to O(x ). Then, the Higgs mass for the SU(3)c × SU(3)W model is calculated as √ m 3 a2 H v B 0 + , 2 4 ln 2 const (16) g4 4π 4β where

−1 + + B ≡ 18 N ( ) − 1 + N ( ) , (17) 24 adj fnd and the constant term depends on β and the number of flavors. Eq. (17) shows that a few adjoint bulk fields are enough and essential for the large quartic coupling. The contribution from the adjoint bulk field overcome the loop factor ∼ 1/4π to enhance the magnitude of the Higgs mass. Let us note that the dependence of the Higgs mass on the supersymmetry breaking parameter is logarithmic, as expected. 2 If we tune the values of β to βc 0.14865 ... at which the coefficient of a vanishes at one-loop level, the smaller VEV a0 is realized within the validity of perturbation theory, and the scale R should be smaller due to Eq. (2). In this sense two examples below are different theories from each other, since they should have different 2 2 initial setup of the value, R. The magnitude of the Higgs mass is enhanced because of the large ln(a0/β ) as shown × (+) (−) (+) (−) = 5 in Table 2 for the SU(3)c SU(3)W model with (Nadj ,Nadj ,Nfnd ,Nfnd ) (2, 2, 0, 2). 2 Two-loop contributions become dominant in the coefficient of a if we tune β close to βc. The effect of the two- loop, however, is almost absorbed into the values of β by adjusting β at one-loop level as long as the perturbation is valid. If one chooses β such that the coefficient of a2 taken account of higher-loops almost vanishes, one expects very small values of a0, so that the magnitude of the Higgs mass becomes larger than the values obtained in the above table. Let us comment on the heavy Higgs mass in non-SUSY gauge models. We can see from Eqs. (10) and (11) that it is possible to cancel the a2 terms between the matter with even parity and the one with odd parity, keeping the positive and large quartic coupling, by an appropriate choice of the matter content. In this case, we have the similar situation with the SUSY case studied above and expect the desirable size of the Higgs mass. In fact, the non-SUSY model with the appropriate matter content, which realizes the suitable dynamical electro-weak symmetry breaking, is presented in Ref. [11].

4. SUSY gauge-Higgs with bulk mass

In this section we show another example for realizing the dynamical electro-weak symmetry breaking with the small number of extra bulk ﬁelds. We introduce explicit soft SUSY breaking scalar mass in addition to the SS parameter for the bulk superﬁelds.6 In this Letter we do not introduce the soft gaugino masses because the mass terms are odd under the Z2 operation.

Table 2 (+) (−) (+) (−) 2 Nadj Nadj Nfnd Nfnd βa0 mH /g4 (GeV) 22020.1486 0.0023 191 22020.14865 0.0009 208

5 Note that too small a0 induces a very large log factor that spoils the validity of perturbation theory. 6 The effective potentials and vacuum structures with soft scalar masses on S1 have been studied in Refs. [22,23]. N. Haba et al. / Physics Letters B 615 (2005) 247–256 253

Let us study the SU(3)c × SU(3)W model at ﬁrst. The contribution of the gauge multiplet to the effective potential is the same as Eq. (3). We introduce the soft SUSY breaking mass, m for the bulk hypermultiplets and deﬁne a dimensionless parameter, z ≡ mR (< 1). We denote

2 + 1 (2πzn) − I ( )[a,β,z,n]≡ 1 − 1 + 2πzn+ e 2πzn cos(2πnβ) cos(πna), (18) n5 3

2 − 1 (2πzn) − I ( )[a,β,z,n]≡ 1 − 1 + 2πzn+ e 2πzn cos(2πnβ) cos πn(a − 1) , (19) n5 3 in which I (±)[a,β,z,n] is reduced to J (±)[a,β,n] in the limit of z → 0(m → 0). The contribution of the matter hypermultiplet to the effective potential is given by

∞ matter = (+) (+) (+) + (+) (+) Veff 2C Nadj I 2a,β,zadj ,n 2I a,β,zadj ,n n=1 + (−) (−) (−) + (−) (−) + (+) (+) (+) Nadj I 2a,β,zadj ,n 2I a,β,zadj ,n Nfnd I a,β,zfnd ,n + (−) (−) (−) Nfnd I a,β,zfnd ,n , (20) (±) (±) (±) where zrep stands for the explicit soft mass defined by zrep ≡ mrep R(<1) for each representation field. Eq. (20) becomes Eq. (4) in the limit of the vanishing soft scalar mass, m → 0. We find some examples of extra matter contents and SUSY breaking parameters, for which the suitable VEV and Higgs mass are realized, and we summarize them in Table 3. 2 The Higgs mass mH /g4 is measured in GeV unit. This table shows that even small number of extra bulk fields can realize the suitable dynamical electro-weak symmetry breaking with the heavy Higgs mass. The effect of the bulk masses increases not only the degrees of freedom of parameter space, but also induces a similar effect of large β, which is necessary for the symmetry breaking with a small number bulk fields, as explained in the last section. We show an example in which one can see the enhancement of the magnitude of the Higgs mass due to the existence 2 of the bulk mass in Table 4, where the Higgs mass mH /g4 is measured in GeV unit. Next we study the case of SU(6) model. The contribution of the gauge multiplet to the effective potential is the same as Eq. (7). On the other hand, the contribution of the matter hypermultiplet to the effective potential is given

Table 3 (+) (−) (+) (−) (+) (−) (+) (−) 2 Nadj Nadj Nfnd Nfnd βzadj zadj zfnd zfnd a0 mH /g4 (1)23040.05 0.01 0.01 – 0.045 0.0040 164 (2)24260.05000.050.05 0.0037 176 (3)24060.025 0.025 0.025 – 0.025 0.0066 129 (4)21020.10.10.1– 1 0.0097 150 (5)11020.0111–10.0196 125 (6)22020.1400–00.0379 130

Table 4 (+) (−) (+) (−) (+) (−) (+) (−) 2 Nadj Nadj Nfnd Nfnd βzadj zadj zfnd zfnd a0 mH /g4 21020.100–00.2362 42 21020.10.10.1– 1 0.0097 150 254 N. Haba et al. / Physics Letters B 615 (2005) 247–256

Table 5 (+) (−) (+) (−) (+) (−) (+) (−) 2 Nadj Nadj Nfnd Nfnd βzadj zadj zfnd zfnd a0 mH /g4 (7)200100.10.05 – – 0.05 0.0207 139 (8)20060.15 0.1– – 0.10.0268 139 (9)200160.040––0.03 0.0021 173 (10) 2 0 0 4 0.07 0.5– – 0.50.0366 138 (11) 2 0 0 2 0.320––00.0594 135 by ∞ matter = (+) (+) (+) + (+) (+) + (−) (+) Veff 2C Nadj I 2a,β,zadj ,n 2I a,β,zadj ,n 6I a,β,zadj ,n = n 1 + (−) (−) (−) + (−) (−) + (+) (−) Nadj I 2a,β,zadj ,n 2I a,β,zadj ,n 6I a,β,zadj ,n + (+) (+) (+) + (−) (−) (−) Nfnd I a,β,zfnd ,n Nfnd I a,β,zfnd ,n , (21) which becomes Eq. (8) in the zero limit of explicit soft scalar masses. Some examples for realizing the suitable dynamical electro-weak symmetry breaking are shown in Table 5.

5. Summary and discussions

We have studied the Higgs mass in the gauge-Higgs unification theory. Since the Higgs doublet corresponds to the Wilson line phases, it does not have the mass term nor quartic coupling at the tree level. Through the quantum corrections, the Higgs can take a VEV, and its mass is induced. The radiatively induced mass, however, tends to be small, so we lift it to O(100) GeV by introducing the O(10) numbers of bulk fields in the previous works. The perturbation theory cannot be reliable when there are a large number of bulk fields. In this Letter we have reanalyzed the Higgs mass and have found that even a small number of bulk field can have the suitable heavy Higgs mass, accompanying the desirable electro-weak symmetry breaking. The expansion formulae for the effective potential are useful to discuss and study analytically the Higgs mass. And we have shown that a small (large) number of bulk fields are enough (needed) when the SUSY breaking mass is large (small). The Higgs mass has the logarithmic dependence on the supersymmetry breaking parameter of the Scherk–Schwarz mechanism. The fine tuning of β yields smaller VEV, a0, and accordingly enhances the magnitude of the Higgs mass. The analyses in the Letter can be applied to the bulk field with an arbitrary representation under an arbitrary gauge group. We have also studied the case of introducing the soft SUSY breaking scalar masses in addition to the SS SUSY breaking. In this case the suitable electro-weak symmetry breaking and the O(100) GeV Higgs mass can also be realized by O(1) numbers of bulk fields. Finally, we would like to discuss the higher order operators of Higgs self interactions. We see that the effective n n n potential contains a interactions by the expansion of the cosine function, which implies a = (g4RH) from n Eqs. (1) and (2). When g4R is of order a few TeV, higher order operators, H (n 6) have the dimensionful suppression of order a few TeV. This means that the contributions from the higher order operators are not so significant. In connection with new physics expected in the scenario of the gauge-Higgs unification, it may be interesting to comment on the effective 3-point self coupling of H in the models. The coupling is important for the search of the new physics in the future linear colliders [24]. The coupling of the effective λH 3 interaction is given by 3g3 3 λ ≡ 4 ∂ (V /C) | , and the deviation from the tree level SM coupling, λ = 3m2/v, is estimated by λ = (λ − 32π6R ∂a3 a0 SM h − × (+) = (−) = λSM)/λSM [24].Thevalueofλ becomes 17.4% for the example of SU(3)c SU(3)W model (Nadj Nadj 2, N. Haba et al. / Physics Letters B 615 (2005) 247–256 255

Table 6 (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) λ (%) −8.6 −8.3 −14.0 −10.2 −3.1 −13.7 −12.0 −12.0 −7.6 −11.2 −12.7

(−) = (+) = = − (+) = (−) = (−) = (+) = Nfnd 4, Nfnd 0 with β 0.1) and 16.6% for SU(6) model (Nadj 2, Nfnd 10, Nadj Nfnd 0 with β = 0.1). As for the examples of Section 4, we show them in Table 6. The effective 3-point self couplings tend to be small comparing to that of the SM. We should notice again that the Higgs field in our model has VEV in A5 not σ .TheVEVofA5 should be distinguished from that of σ in the dynamically induced effective potential in the gauge-Higgs unification theory [20]. Since H is the field of the D-flat direction, which is massless at tree level, it corresponds to the lighter Higgs scalar in the MSSM, h0. Since h0 becomes the SM-like Higgs in the large soft SUSY breaking masses, we have compared the effective 3-point self coupling to the SM one in the above estimation of λ. Other Higgs eigenstates, charged Higgs, heavier neutral scalar, and pseudo-scalar, are corresponding to the 6,7 5 4 5 fields of directions, σ , σ , and σ , respectively. (We denote the generators in the SU(3)c × SU(3)W , T having −i/2(i/2) in (1, 3) ((3, 1)) element and T 6 having 1/2in(2, 3) and (3, 2), and so on.) The masses of them can be calculated by the effective potential for these directions, which might be O(100) GeV for the pseudo-scalar, MZ +O(100) GeV for the heavier neutral scalar, and MW +O(100) GeV for the charged scalar. Here O(100) GeV shows the effect from the radiative corrections as we have calculated for the lighter neutral scalar H(h0). In the tree level, the gauge-Higgs unification models with the SS SUSY breaking have µ =−˜m, m2 =−˜m2, B = 0atthe hu,hd compactification scale, where m˜ = β/R [6].TheD-flat direction is really flat, and the vacuum is not determined at one point. However, as we have shown above, the vacuum is selected uniquely through the quantum corrections, in which Higgs squared masses and B term are induced effectively, and all physical Higgs particles become massive of O(100) GeV. The detail analyses will be shown in the forthcoming Letter [25].

Acknowledgements

We would like to thank M. Tanabashi and Y. Okada for useful discussions which become one of the motivations of this work. We would like to thank N. Okada for a lot of useful and helpful discussions. K.T. would thank the colleagues in Osaka University and gives special thanks to the professor Y. Hosotani for valuable discussion. K.T. is supported by the 21st Century COE Program at Osaka University. T.Y. would like to thank the Japan Society for the Promotion of Science for ﬁnancial support. N.H. is supported in part by Scientiﬁc Grants from the Ministry of Education and Science, Grant Nos. 14740164, 16028214, and 16540258.

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Higgs boson mass and electroweak–gravity hierarchy from dynamical gauge–Higgs uniﬁcation in the warped spacetime

Yutaka Hosotani, Mitsuru Mabe

Department of Physics, Osaka University, Toyonaka, Osaka 560-0043, Japan Received 4 March 2005; accepted 11 April 2005 Available online 28 April 2005 Editor: T. Yanagida

Abstract Dynamical electroweak symmetry breaking by the Hosotani mechanism in the Randall–Sundrum warped spacetime is examined, relations among the W-boson mass (mW), the Kaluza–Klein mass scale (MKK), and the Higgs boson mass (mH)being 1/2 2 derived. It is shown that MKK/mW ∼ (2πkR) (π/θW) and mH/mW ∼ 0.058kR(π/θW),wherek , R,andθW are the curvature and size of the extra-dimensional space and the Wilson line phase determined dynamically. For typical values kR = 12 19 and θW = (0.2–0.4)π, one ﬁnds that MKK = 1.7–3.5TeV,k = (1.3–2.6) × 10 GeV, and mH = 140–280 GeV.  2005 Elsevier B.V. All rights reserved.

Although the standard model of the electroweak interactions has been successful to account for all the experimental data so far observed, there remain a few major issues to be settled. First of all, Higgs particles are yet to be discovered. The Higgs sector of the standard model is for the most part unconstrained unlike the gauge sector where the gauge principle regulates the interactions among matter. Secondly, the origin of the scale of the electroweak interactions characterized by the W-boson mass mW ∼ 80 GeV or the vacuum expectation value of the Higgs field v ∼ 246 GeV becomes mysterious once one tries to unify the electroweak interactions with the strong interactions in the framework of grand unified theory, or with gravity, where the energy scale is given by 15 17 19 MGUT ∼ 10 –10 GeV or MPl ∼ 10 GeV, respectively. The natural explanation of such hierarchy in the energy scales is desirable. In this Letter we show that the Higgs sector of the electroweak interactions can be integrated in the gauge sector, and the electroweak energy scale is naturally placed with the gravity scale within the framework of dynamical gauge–Higgs unification in the Randall–Sundrum warped spacetime. The scheme of dynamical gauge–Higgs unification was put forward long time ago in the context of higher- dimensional non-Abelian gauge theory with non-simply connected extra-dimensional space [1,2]. In non-simply

E-mail addresses: [email protected] (Y. Hosotani), [email protected] (M. Mabe).

0370-2693/$ – see front matter  2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physletb.2005.04.039 258 Y. Hosotani, M. Mabe / Physics Letters B 615 (2005) 257–265 connected space there appear non-Abelian Aharonov–Bohm phases, or Wilson line phases, which can dynamically induce gauge symmetry breaking even within configurations of vanishing field strengths. The extra-dimensional components of gauge potentials play a role of Higgs fields in four dimensions. The Higgs fields are unified with the gauge fields and the gauge symmetry is dynamically broken at the quantum level. It was originally designed that Higgs fields in the adjoint representation in SU(5) grand unified theory are unified with the gauge fields. The attempt to identify scalar fields as parts of gauge fields was made earlier by utilizing symmetry reduction. Witten observed that gauge theory in four-dimensional Minkowski spacetime with spherical symmetry reduces to a system of gauge fields and scalar fields in two-dimensional curved spacetime [3]. This idea was extended to six- dimensional gauge theory by Fairlie [4] and by Forgacs and Manton [5] to accommodate the electroweak theory in four dimensions. It was recognized there that to yield SU(2)L × U(1)Y symmetry of electroweak interactions in four dimensions one need start with a larger gauge group such as SU(3), SO(5) or G2. The reduction of the symmetry to SU(2)L × U(1)Y was made by an ad hoc ansatz for field configurations in the extra-dimensional space. For instance, Manton assumed spherically symmetric configurations in the extra-dimensional space S2.As was pointed out later [6], such a configuration can be realized by a monopole configuration on S2.1 However, classical non-vanishing field strengths in the background would lead to the instability of the system. In this regard gauge theory defined on non-simply connected spacetime has big advantage in the sense that even with vanishing field strengths Wilson line phases become dynamical and can induce symmetry breaking at the quantum level by the Hosotani mechanism. Recently significant progress has been achieved along this line by considering gauge theory on orbifolds which are obtained by modding out non-simply connected space by discrete symmetry such as Zn [7–21]. With the orbifold symmetry breaking induced from boundary conditions at fixed points of the orbifold, a part of light modes in the Kaluza–Klein tower expansion of fields are eliminated from the spectrum at low energies so that chiral fermions in four dimensions naturally emerge [7]. Further, in SU(5) grand unified theory (GUT) on orbifolds the triplet–doublet mass splitting problem of the Higgs fields [10] and the gauge hierarchy problem [8] can be naturally solved. The orbifold symmetry breaking, however, accompanies indeterminacy in theory. It poses the arbitrariness problem of boundary conditions [15]. One needs to show how and why a particular set of boundary conditions is chosen naturally or dynamically, which is achieved, though partially, in the scheme of dynamical gauge–Higgs unification. Quantum dynamics of Wilson line phases in GUT on orbifolds was first examined in Ref. [14] where it was shown that the physical symmetry is determined by the matter content. Several attempts to implement dynamical gauge–Higgs unification in the electroweak theory have been made since then. The most intriguing among those is the U(3) × U(3) model of Antoniadis, Benakli and Quiros [9]. The effective potential of the Wilson line phases in this model has been recently evaluated to show that the electroweak symmetry breaking dynamically takes place with minimal addition of heavy fermions [20]. The model is restrictive enough to predict the Kaluza–Klein mass scale (MKK) and the√ Higgs boson mass (mH) with the W-boson mass (mW) as an input. It turned out that MKK ∼ 10mW and mH ∼ αwmW, which contradicts with the observation. We argue that this is not a feature of the specific model examined, but is a general feature of orbifold√ models in which extra-dimensional space is flat. Unless tuning of matter content is enforced, the relation mH ∼ αwmW is unavoidable in flat space as shown below. To circumvent this difficulty, it is necessary to have curved extra- dimensional space. 1 Randall and Sundrum introduced warped spacetime with an extra-dimensional space having topology of S /Z2 which is five-dimensional anti-de Sitter spacetime with boundaries of two flat four-dimensional branes [22].It was argued there that the standard model of electroweak interactions is placed on one of the branes such that the electroweak scale becomes natural compared with the Planck scale chracterizing gravity. Since then many

1 8 2 × The monopole configuration for AM of the SU(3) gauge fields on S realizes the envisaged symmetry reduction to SU(2) U(1) in Ref. [5]. Y. Hosotani, M. Mabe / Physics Letters B 615 (2005) 257–265 259 variations of the Randall–Sundrum model have been investigated. The standard model can be placed in the bulk five-dimensional spacetime, not being restricted on one of the branes [23]. However, fine-tuning of the Higgs potential remains necessary. More promising is to consider dynamical gauge–Higgs unification in the Randall–Sundrum background where gauge theory is defined in the bulk five-dimensional spacetime without five-dimensional scalar fields. The first step in this direction has been made by Oda and Weiler who evaluated the 1-loop effective potential for Wilson line phases in the SU(N) gauge theory [24]. We will show in the present Letter that the electroweak symmetry breaking can be naturally implemented in dynamical gauge–Higgs unification on the Randall–Sundrum background to avoid the aforementioned difficulty concerning MKK and mH. We show that in this scheme the Higgs mass mH should be between 140 GeV and 280 GeV, and the Kaluza–Klein mass scale MKK must be between 1.7 TeV and 3.5 TeV. It is exciting that the predicted ranges of mH and MKK fall in the region where experiments at LHC can explore in the near future.2 We consider gauge theory in the Randall–Sundrum warped spacetime whose metric is given by 2 −2σ(y) µ ν 2 ds = e ηµν dx dx + dy (1)

(µ, ν = 0, 1, 2, 3).Hereσ(y) = k|y| for |y| πR, σ(y + 2πR) = σ(y) and ηµν = diag(−1, 1, 1, 1). Points (xµ, −y) and (xµ,y + 2πR) are identiﬁed with (xµ,y). The resultant spacetime is an anti-de Sitter spacetime (0

2 1 µ ν 1 2 ds = ηµν dx dx + dw . (2) w 4k2w2 Boundary conditions for the gauge potentials in the original coordinate system (xµ,y) are given in the form − = − + † = = ∈ 2 = = (Aµ,Ay)(x, yj y) Pj (Aµ, Ay)(x, yj y)Pj , where y0 0, y1 πR, Pj U(3) and Pj 1(j 0, 1) 1 µ [14,18,20]. They follow from the S /Z2 nature of the spacetime. In the new coordinate system (x ,w), the boundary conditions are summarized as A A ∂ A −∂ A µ = µ † w µ = w µ † (x, wj ) Pj (x, wj )Pj , (x, wj ) Pj (x, wj )Pj , (3) Aw −Aw ∂wAw ∂wAw 2πkR where w0 = 1 and w1 = e . Similarly, for a fermion in the fundamental representation 5 5 ψ(x,wj ) = ηj Pj γ ψ(x,wj ), ∂wψ(x,wj ) =−ηj Pj γ ∂wψ(x,wj ), (4) where ηj =±1. We take −1 P0 = P1 = −1 (5) 1

2 Cosmological consequences of the Hosotani mechanism in curved spacetime has been previously investigated in Ref. [25]. The Hosotani mechanism in the Randall–Sundrum warped spacetime has been applied to the electroweak symmetry breaking in Ref. [26]. 260 Y. Hosotani, M. Mabe / Physics Letters B 615 (2005) 257–265 to ensure the electroweak symmetry. The advantage of the w coordinate over the y coordinate lies in the fact that zero modes of Aw(x, w) become independent of w.Inthey coordinate Ay(x, y) has cusp singularities at y = 0 and y = πR. To observe it explicitly, we specify the gauge-fixing term in the action. A general procedure in curved spacetime has been given in Ref. [2]. It is convenient to adopt the prescription for gauge-fixing given in Ref. [24]. As is justified a posteriori, the effective c = c potential is evaluated√ in the background field method with a constant background AM δMwAw. The gauge fixing 4 term d xdw −gLg.f. is chosen to be L =− 2 c µ + 2 c 2 g.f. w Tr DµA 4k wDwAw , (6) c ≡ + [ c ] c µ ≡ µν c = where DM AN ∂M AN ig AM ,AN and DµA η DµAν . In the path integral formula we write AM c + q q AM AM and expand the action in AM . The bilinear part of the action including the ghost part is given by w 1 =− 4 1 q µ + 2 c c qν + q µ + 2 c c q Ieff d x dw Tr Aν ∂ ∂µ 4k wDwDw A 2k Tr Aw ∂ ∂µ 4k DwwDw Aw 2kw w0 1 µ 2 c c − Tr η¯ ∂ ∂µ + 4k wD D η . (7) 2kw2 w w µ Partial integration necessary in deriving (7) is justified as Tr Aµ∂wA and Tr Aw∂wAw vanish at w = w0,w1 with the boundary conditions (3). = 8 1 a a a 0 Let us denote AM a=0 2 λ AM with the standard Gell-Mann matrices λ (λ represents the U(1) part). a = b = = With (3) and (5), Aµ (a 0, 1, 2, 3, 8) and Aw (b 4, 5, 6, 7) satisfy Neumann boundary conditions at w a = b = w0,w1, whereas Aµ (a 4, 5, 6, 7) and Aw (b 0, 1, 2, 3, 8) satisfy Dirichlet boundary conditions. Zero modes a = b = independent of w areallowedforAµ (a 0, 1, 2, 3, 8) and Aw (b 4, 5, 6, 7). It is found from (7) that they c = × a indeed constitute massless particles in four dimensions when Aw 0. Gauge fields of SU(2)L U(1)Y are in Aµ = b = (a 0, 1, 2, 3, 8), whereas doublet Higgs fields are in Aw (b 4, 5, 6, 7). We note that in the y coordinate system = 2ky Ay 2ke Aw so that the zero modes are not constant in y, which gives rise to unphysical cusp singularities at y = 0,πR. Mode expansion for Aµ(x, w) is inferred from (7) to be

d2 Aa (x, w) = Aa (x)f (w), −4k2w f (w) = λ f (w), µ µ,n n dw2 n n n n w1 1 dw f (w)f (w) = δ . (8) 2kw n m nm w0

For Aw(x, w) one ﬁnds d d Aa (x, w) = Aa (x)h (w), −4k2 w h (w) = λˆ h (w), w w,n n dw dw n n n n w1

dw2khn(w)hm(w) = δnm. (9)

w0 ˆ a a = Given boundary conditions, (λn,fn(w)) and (λn,hn(w)) are determined. Aµ (Aw) has a zero mode√ λ0 0 ˆ (λ0 = 0) only√ with Neumann boundary conditions at w = wj . For the zero modes f0(w) = 1/ πR and h0(w) = 1/ 2k(w1 − w0) [27]. Y. Hosotani, M. Mabe / Physics Letters B 615 (2005) 257–265 261

ˆ Except for the zero modes, both λn and λn are positive. Apart from the normalization factors eigen-functions √ √ ˆ are given by fn(w) = wZ1( λnw/k) and hn(w) = Z0( λnw/k) where Zν(z) is a linear combination of Bessel ˆ functions Jν(z) and Yν(z) of order ν. (λn,fn) with the Neumann boundary conditions and (λn,hn) with the Dirich- let boundary conditions are determined by √ √ J0(βn w0 ) J (βn w ) √ = 0 √ 1 , (10) Y0(βn w0 ) Y0(βn w1 ) ˆ whereas (λn,fn) with the Dirichlet boundary conditions and (λn,hn) with the Neumann boundary conditions are determined by √ √ J1(βn w0 ) J (βn w ) √ = 1 √ 1 . (11) Y1(βn w0 ) Y1(βn w1 ) √ √ √ − √ Here β = λ /k or λˆ /k.Forβ 1, β = πn/( w − w ).Forw 1/2 β 1, β = (n − 1 )π/ w n √n n n n 1 0 1 n n√ 4 1 + 1 ∼ or (n 4 )π/ w1 for the case (10) or (11), respectively. The ﬁrst excited state is given by β1 w1 2.6or3.8. Hence, the Kaluza–Klein mass scale is given by πk −1 → = √ √ = R for k 0, MKK − (12) w1 − w0 πke πkR for eπkR 1. 1 ± 2 With Pj in (5), the W boson and the weak Higgs doublet Φ are contained in the zero modes of (Aµ iAµ)(x, w) b = and Aw(x, w) (b 4, 5, 6, 7): √1 1 + 2 ⇒ √1 1 + 2 = √1 Aµ iAµ (x, w) Aµ,0 iAµ,0 (x)f0(w) Wµ(x), 2 2 πR 4 − 5 4 − 5 1 Aw iAw 1 Aw,0 iAw,0 Φ(x) √ (x, w) ⇒ √ (x) h0(w) = √ . (13) A6 − iA7 6 − 7 2k(w − w ) 2 w w 2 Aw,0 iAw,0 1 0 There is no potential term for Φ at the classical level, but nontrivial effective potential is generated at the quantum level. As in the model discussed in Ref. [20], the effective potential is supposed to have a global minimum at Φ = 0, inducing dynamical electroweak symmetry breaking. Making use of the residual SU(2) × U(1) invariance, we need to evaluate the effective potential for the conﬁguration  

= c = =   Aw Aw αΛ, Λ 1 . (14) 1 √ √ 0 Note that v = 2 Φ =2 2k(w1 − w0)α. 4 1 1 The Randall–Sundrum warped spacetime has topology of R × (S /Z2).AsS is not simply connected, there arise Aharonov–Bohm phases, or Wilson line phases, which become physical degrees of freedom [1,2]. The Wilson { }· line phases are deﬁned by eigenvalues of P exp ig C dwAw U, where the path C is a closed non-contractible 1 1 loop along S and U = P1P0. In the present case U = I so that all gauge potentials are periodic on S . It follows that α in (14) is related to the Wilson line phase by

θW = 2gα(w1 − w0). (15)

It will be shown below that θW and θW + 2π are gauge equivalent. The SU(2)L gauge coupling constant in four −1/2 dimensions, g4, is easily found by inserting Aµ(x, w) ∼ (πR) Aµ,0(x) into Fµν : g g4 = √ . (16) πR 262 Y. Hosotani, M. Mabe / Physics Letters B 615 (2005) 257–265

Nonvanishing θW or v gives the W boson a mass mW. In our scheme the mass term for W arises from the term − q c c qν = 1 dw2k Tr Aν DwDwA in (7). The resultant relation is the standard one, mW 2 g4v. Thus one finds 1 θW 1/2 MKK for k → 0, = g4v = πk θW = 2 π mW 1 θ πkR (17) 2 2R(w1 − w0) π √ W M for e 1, 2πkR π KK where MKK is given in (12). The precise value of θW depends on the details of the model. If the effective potential is minimized at θW = 0, then the electroweak symmetry breaking does not occur. If it occurs, θW takes a value typically around 0.2π to 0.4π, unless artificial tuning of matter content is made. As an example, in the model discussed in Ref. [20] in flat space, θW ∼ 0.25π, which, with mW = 80.4 GeV inserted, yielded too small MKK ∼ 640 GeV. In the present case, with the value of θW given, kR determines MKK and k. Recall that the four- and five- 2 ∼ 3 ∼ dimensional Planck constants Mpl and M5d are related by Mplk M5d. To have a natural relation M5d Mpl, kR must be in the range 11

Having established the phase nature of θW, we estimate Veff(θW). Veff(θW) in the models in flat orbifolds has been evaluated well [12,14,16,18,20]. Veff(θW) in the Randall–Sundrum spacetime in the SU(N) gauge theory has c been evaluated by Oda and Weiler [24]. With the background Aw or θW, the spectrum λn of each field degree of freedom depends on θW as well as on the boundary conditions of the field. Its contribution to four-dimensional Veff(θW) at the one loop level is summarized as i d4p V (θ ) =∓ ln −p2 + λ (θ ) , (21) eff W 2 ( π)4 n W 2 n where ‘−’(‘+’) sign is for a boson (fermion). The spectrum λn for θW = 0 is determined as described in the ∼ 2 2 discussions from Eq. (7),toEq.(11). It is found there that λn MKKn for large n. Hence one can write, after making a Wick rotation, as 1 d4q V (θ ) =± M4 E ln q2 + ρ (θ ) + const, (22) eff W 2 KK ( π)4 E n W 2 n = 2 1 where ρn(θW) λn/MKK. It is known that on an orbifold with topology of S /Z2, fields form a Z2 doublet pair to have an interaction with θW [14]. The resultant spectrum for a Z2 doublet is cast in the form where the sum in (22) extends over from n =−∞to n =+∞. Further, ρn(θW + 2π) = ρn+(θW) (: an integer), and 2 ρn(θW) ∼[n + γ(θW)] for large |n|, where γ(θW + 2π) = γ(θW) + . For instance, in the U(3) × U(3) model 2 in flat space, ρn(θW) =[n + θW/2π + (const)] with = 0, ±1, ±2 [20]. The important feature is that as θW is shifted to θW + 2π by a large gauge transformation, each eigen mode is shifted to the next KK mode in general, but the spectrum as a whole remains the same. Recall the formula ∞ ∞ 1 d4q 3 cos 2nπx E ln q2 + (n + x)2 =− h(x) + const,h(x)= . (23) 2 (2π)4 E 64π 6 n5 n=−∞ n=1 The x-dependent part is finite. In the present case we have (±)h[γ(θW)]. The total effective potential takes the form 3 V (θ ) = N M4 f(θ ), (24) eff W eff 128π 6 KK W where f(θW + 2π)= f(θW) and its amplitude is normalized to be an unity. Once the matter content of the model is specified, the coefficient Neff is determined. In the minimal model or its minimal extension, Neff = O(1) as supported by examples. = min When Veff(θW) has a global minimum at a nontrivial θW θW , dynamical electroweak symmetry breaking min = min ∼ takes place. It typically happens at θW (0.2–0.3)π [20]. It is possible to have a very small θW 0.01π by fine-tuning of the matter content as shown in Ref. [19], which, however, is eliminated in the present consideration min for the artificial nature. The mass mH of the neutral Higgs boson is found by expanding Veff(θW) around θW and −1 † 1/2 using θW = g[(w1 − w0)k Φ Φ] . One finds 3α R(w − w ) m2 = N f θ min w 1 0 M4 , (25) H eff W 64π 4 k KK = 2 min ∼ where αw g4/4π. In a generic model f (θW ) 1. Making use of (12) and (17), one finds   3αw 1/2 = 3αw 1/2 π →  c 3 MKK c 3 min mW for k 0, 32π 8π θW mH = √ (26)  3αw 1/2 = 3αw 1/2 π πkR c 2 kRMKK c 32π kR min mW for e 1, 64π θW 264 Y. Hosotani, M. Mabe / Physics Letters B 615 (2005) 257–265

=[ min ]1/2 min where c Nefff (θW ) . The values of θW and c depend on details of the model. In the models analyzed min in Ref. [20], (θW ,c)ranges from (0.269π,2.13) to (0.224π,1.63), which justifies our estimate. Hereafter we set = = = = min c 1.9, understanding 20% uncertainty. Inserting αw 0.032 and kR 12, we obtain that mH 0.70(π/θW )mW = → = min = and MKK 12.4mH. In flat space (in the k 0 limit), mH 0.037(π/θW )mW and MKK 53.9mH, which yielded too small mH. There appears a large enhancement factor kR in the relation connecting mH and mW in the min = Randall–Sundrum warped spacetime. For a typical value θW (0.2–0.4)π, the mass of the Higgs boson and the Kaluza–Klein mass scale are given by mH = (140–280) GeV and MKK = (1.7–3.5) TeV, respectively. The relations (17) and (26) reveal many remarkable facts. First of all, only the parameter kR in the Randall– Sundrum spacetime appears in the relations connecting mW, mH and MKK. Secondly, if one supposes that k = O(Mpl), then kR = 12 ± 1 to have the observed value for mW. The electroweak-gravity hierarchy is accounted for min by a moderate value for kR. Thirdly, another quantity θW involved in those relations is dynamically determined, once the matter content of the model is specified. In case the electroweak symmetry breaking takes place, it typically min takes (0.2–0.4)π. mH and MKK are predicted up to the factor θW . Fourthly and most remarkably, the predicted value for mH, 140–280 GeV, is exactly in the range which can be explored in the experiments at LHC and other planned facilities in the near future. In conjunction with it, we recall that in the minimal supersymmetric standard model the Higgs boson mass is predicted in the range 100

Acknowledgements

This work was supported in part by Scientiﬁc Grants from the Ministry of Education and Science, Grant No. 13135215 and Grant No. 15340078 (Y.H.).

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Instability of tachyon supertube in type IIA Gödel spacetime

Wung-Hong Huang

Department of Physics, National Cheng Kung University, Tainan, Taiwan Received 17 January 2005; received in revised form 10 March 2005; accepted 13 April 2005 Available online 25 April 2005 Editor: M. Cveticˇ

Abstract We study the tachyon supertube probes in a type IIA supergravity background which is a stringy-like Gödel spacetime and contains closed timelike curve. In the case of small value of f , which is a parameter of the background, we use the Minahan– Zwiebach tachyon action to obtain a single regular tube solution and argue that the tube is a BPS D2-brane. However, we find that the fluctuation around the tube configuration has a negative-energy mode. This means that the tachyon supertube, despite being a BPS configuration, develops an instability in the pathological spacetime with closed timelike curve which violates the causality.  2005 Elsevier B.V. All rights reserved.

1. Introduction ogy protection conjecture, with a quantum mechanism enforcing it by superselecting the causality violat- Gödel had found a homogeneous and simply con- ing field configurations from the quantum mechanical nected universe with closed timelike curves (CTCs) phase space [3]. [1]. This particular solution is not an isolated pathol- In a previous paper [4] we have calculated the ogy and in fact the CTCs are a generic feature of energy–momentum tensor of a scalar field propagat- gravitational theories, in particular in higher dimen- ing in a one-parameter family of solutions that in- sions with or without the supersymmetry [2]. CTCs cludes the four-dimensional generalized Gödel space- are not linked to the presence of strong gravitational time [5]. As the parameter is varied, we had shown fields. They are global in nature and will violates the that the energy–momentum tensor becomes divergent causality. Physicists usually label the spacetimes with precisely at the onset of CTCs. This gives a support CTCs as pathological, unphysical and forget about to the Hawking’s chronology protection conjecture. In them. However, Hawking had formulated the chronol- [6] Gauntlett et al. calculated the holographic energy– 5 momentum tensor in a deformation of AdS5 × S to investigate the problem in the string theory. Their re- E-mail address: [email protected] (W.-H. Huang). sults, however, showed that the holographic energy–

0370-2693/$ – see front matter  2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physletb.2005.04.031 W.-H. Huang / Physics Letters B 615 (2005) 266–272 267 momentum tensor remains finite even when the CTCs the single regular tube solution with circular cross- appear.1 section in the stringy-like Gödel spacetime. We will In recent many physicists hope that the string the- see that the energy of the single tubular configura- ory could provide a mechanism to rule out the patho- tion comes entirely from the D0 and strings at critical logical solutions or eliminate the CTCs [6–13].For Born–Infeld (BI) electric field. Thus the solution is example, Herdeiro showed that the causality bound supersymmetric [14] and the tachyon supertube solu- corresponds to a unitarity bound in the CFT [7], and tion is a BPS configuration. In Section 3 we calculate therefore that the over rotating spacetimes, which will the fluctuation spectrum around the tube solution and form CTCs, are not genuine solutions of string the- show that there is a negative-energy mode. This means ory. Boyda et al. found that holography can serve as that the tachyon supertube, despite being a BPS con- a chronology protection agency [8]. Astefanesei et al. figuration, develops an instability in the pathological argued that the conjectured AdS/CFT correspondence spacetime with closed timelike curve which violates may teach us something about the physics in space- the causality. Our result provides an alternative proof times containing closed timelike curve [9]. Caldarelli of Drukker’s result [11] while using the tachyon field et al. showed that a violation of the Pauli exclusion theory. We make a conclusion in the last section. principle in the phase space of the fermions is thus intimately related to causality violation in the dual geometries [10]. 2. Tube solution in a stringy-like Gödel spacetime The philosophy behind this is that, in string theory there are dynamical extended objects which may pro- We adopt a specific solution of type IIA supergrav- vide us with probes suited particularly well to study ity with rotation in a single plane. This solution is a non-local issues like closed timelike curves. stringy-like Gödel spacetime preserving one quarter One of the interesting objects in the string theory of the maximal number of supersymmetries [18].It is the supertube found by Mateos and Townsend [14]. has a nontrivial metric in three dimensions, which we It is a tubular bound state of D0-branes, fundamental parameterize by the time coordinate t, and polar coor- strings (F1) and D2-branes, which is supported against dinates r and θ in the plane. There is also a NS–NS collapse by the angular momentum generated by the flux as well as RR 2-form and 4-form fluxes. The flux Born–Infeld (BI) electric and magnetic fields. Drukker fields are used to render the spacetime supersymmet- et al. studied the dynamics of cylindrical D2-brane in ric. The metric and fluxes of this stringy-like Gödel type IIA Gödel universe and claimed that the super- spacetime are given by tube develops an instability in the CTC region despite 2 =− + 2 2 + 2 + 2 being a BPS configuration [11]. ds dt fr dθ dz dr 2 2 i j The investigations in [11] use the supertubes as + r dθ + δij dx dx , (2.1) 2 2 probes, in which the supertubes are described as BPS BNS = fr dz∧ dθ, C3 = fr dθ ∧ dt ∧ dz, D2-branes in DBI action. As well-known that, in the C =−fr2 dθ. (2.2) spirit of the Sen’s conjecture [15], the BPS branes 1 could also be viewed as tachyon kinks of non-BPS The spacetime has CTC when r>f−1 [18].How- branes in higher dimension, it is therefore interesting ever, as the stringy-like Gödel universe is homoge- to adopt the tachyon tube [16,17] as a probe to inves- neous there are CTCs through every point in space. tigate the problem. This is the work of this Letter. In this section we will adopt the MZ tachyon ac- In Section 2, we first describe a type II supergrav- tion [19] to analyze the supertube in the stringy-like ity background which behaves as the Gödel spacetime Gödel spacetime (2.1). The Minahan–Zwiebach (MZ) [18] and contains closed timelike curve. Then we use tachyon action is a derivative truncation of the BSFT the Minahan–Zwiebach tachyon action [19] to find action of the non-BPS branes [20], which embodies the tachyon dynamics for unstable D-branes in (super)string theories and was first proposed as a sim- 1 The inconsistence between [4] and [6] has not yet been clari- plified action to capture the desirable properties of fied. string theories. The action had been successfully used 268 W.-H. Huang / Physics Letters B 615 (2005) 266–272 to study the phenomena of kink condensation and vor- which, however, could not be solved exactly and we tex condensation in the unstable non-BPS branes [21]. shall adopt some approximations. The results support the Sen’s conjecture of the ‘Brane In the case of f 1 we have found the solution Descent Relations’ of tachyon condensation. It had − B f(1 Ec) 2 also been used to investigate the problems of the re- Tc(r) ≈ √ ln(r/r0) + r , (2.7) combination of intersecting branes [22]. 2 4 The Minahan–Zwiebach tachyon action of the non- in which the critical value of electric field Ec is defined BPS D3 brane, including the Wess–Zumino terms, is by described by [18,19] √ √ = − + 2 =−T + 2 + 1F 2 Ec 2 1 fB. (2.8) S 3 V(T) 1 (∂µT) µν 2 4 The above solution is consistent with our previous pa- + ∧ + ∧ F V(T)dT (C3 C1 ). (2.3) per [17] in the case of flat space, i.e., f = 0. The value of r in above is an arbitrary integration constant. As The field strength F includes the BI gauge field 0 the value of |T(r)c| becomes zero near r = r0 the ra- strength FBI that are turned on in the brane and those dius of tachyon tube will depend on the value of r0. induced by the NS field strength BNS in (2.2), i.e., The tube radius and r0 are determined by the BI EM F = FBI + BNS, field, i.e., the charges of D0 and strings on the brane. It is noted that, as that in the previous cases [17],the F = Edt∧ dz+ Bdz∧ dθ (2.4) BI above tube solution is irrelevant to the function form in which we allow for a time-independent electric field of the tachyon potential V(T). E and magnetic field B. The supertube we considered To proceed we define the electric displacement de- is a cylindrical D2-brane which is extended in the z di- fined by Π = ∂L/∂E and thus from (2.5) rection as well as the angular direction θ at fixed radius r about the origin. It is supported against collapse by Π =−2πV(T) E f 2R2 − 1 r the angular momentum generated by the Born–Infeld − f B − fr2 r − T fBr2 . (2.9) (BI) electric field E and magnetic fields B [14]. Under these conditions the Lagrangian is The associated energy density defined by H = ΠE − L becomes 2 1 2 2 2 L =−2πV (T )r 1 + T + E f R − 1 2 1 H = 2πV(T) 1 + T 2 − E2 f 2R2 − 1 − 2 2 2 − − 2 + (B fr ) fE B fr 2 2 2r2 (B − fr ) + r − fr2T . (2.10) 2r2 − fr(1 − E)T , (2.5) In Fig. 1 we plot the typical behaviors of function in which the tachyon field is a function of radius as we H(r) which shows that there is a peak at finite radius consider only the case of circular tube. The associated for the solution (2.7). field equation is We can now use the regular tachyon solution (2.7) to evaluate the D0 charge q0 and F -string charge q1, T 2V(T) T (x) + − f(1 − E) which are defined by [16,17] r 1 1 2 − V (T ) 1 − T + fr(1 − E)T q1 = dϕΠ, q0 = dϕB, (2.11) 2π 2π 1 respectively. It is seen that energy density U could be + E2 f 2R2 − 1 − fE B − fr2 2 expressed as (B − fr2)2 + = 0, (2.6) U = E q + q . (2.12) 2r2 c 1 0 W.-H. Huang / Physics Letters B 615 (2005) 266–272 269

3. Fluctuation and instability of tachyon supertube

Let us now consider the ﬂuctuation t around the tubular solution

T(r)= T(r)c + t(r). (3.1)

Substituting the tubular solution T(r)c in (2.7) into the action (2.5) and considering only the quadratic terms of fluctuation field t(r) we obtain √ S =−2π dr 1 + ( 2 − fr) rtˆ 2 Fig. 1. The behaviors H(r)in (2.10) in the case of B = 2, f = 0.01 − 2 and V(T)= e T . There is a peak at finite radius r which specifies B √ the size of the circular cross-section tube of solution (2.7). + 4B − fr 4( 2 − 1) 8r √ + − + 2 As the energy density of tachyon tube is just given by ( 2 2) 4 B r Ln(r/r0) √ √ the sum of charges it carries this solution represents a + 2B ( 2 − 2)f r2 − B(1 + fr− 2fr) BPS tube, as dictated by supersymmetry [14,16,17]. It is also interesting to mention the physical mean- 2 ˆ2 × Ln(r/r0) t ing of the critical value of electric field E . Substitut- c ing the tachyon solution (2.7) into the Lagrangian we ∂2tˆ 2B2 − B4w2 =−2π dw tˆ + tˆ2 + δHtˆ2 see that ∂w2 4 √ √ B 2 − 2 1 − 2 (3.2) Lc ≈−B − fr+ Bf V(Tc). r 2 2 in which

(2.13) w √ √ = e − 2 + − + 2 2 w = δH f 4( 2 1)B 2( 2 2 )B w e Thus increasing E to its ‘critical’ value E Ec would 8 √ reduce the D2-brane tension to zero if the magnetic + 4w (1 − 2 ) B2 + B4 field were zero [14]. This implies that the tachyon √ 3 w tube has no energy associated to the tubular D2-brane − ( 2 − 2) 4 + B e , (3.3) tension; its energy comes entirely from the electric and we have used the partial integration, field redefin- and magnetic fields, which can be interpreted as ‘dis- ition solved’ strings and D0-branes, respectively. The en- ˆ −1/2 −T 2 ergy from the D2-brane tension has been canceled by t ≡ V(Tc) t with V(Tc) = e c , (3.4) the binding energy released as the strings and D0- branes are dissolved by the D2-brane. The phenomena and new variable √ that the tubular D2-brane tension has been canceled w = ln(r/r0) − ln 1 + ( 2 − 1)f r/r0 . (3.5) was crucial to have a supersymmetric tube configuration [14]. Without the δH term (i.e., f = 0 and space be- ˆ We conclude this section a comment. The critical comes flat) we see that the fluctuation t obeys a electric in DBI action is Ec = 1 [14] which is different Schrödinger equation of a harmonic oscillator, thus from that in the MA tachyon action. The inconsistence the mass squared for the fluctuation is equally spaced may be traced to the fact that the MZ tachyon action and specified by an integer n, is just a derivative truncation of the BSFT action of m2 = 2nB2,n 0. (3.6) the non-BPS branes [17]. However, we hope that the t truncated action could capture desirable properties of Thus there is no tachyonic fluctuation, the mass tower the brane theories. starts from a massless state Ψ(w)0 and has the equal 270 W.-H. Huang / Physics Letters B 615 (2005) 266–272 spacing. This result is consistent with the identification Lagrangian of a static straight tube of circular cross- of the tachyon tube as a tubular BPS D2-brane. This is section with radius R becomes [11] the result obtained in [17]. Now we can use the perturbation method of quan- L =− R2 + ∆−1B˜ 2 − E˜ 2R2∆ − fR2 + fR2E, tum mechanism to perform the calculation including (4.2) the δH term effect. We then find that the energy of the where ground state Ψ(w)0 becomes ∆ ≡ 1 − f 2R2, B˜ ≡ B − fR2, = 2 ˜ ≡ − ˜ −1 δEΨ(w)0 2π dwδH Ψ(w)0 E E f B∆ . (4.3) 1 √ The momentum conjugate to E takes the form = πe4B − − 2 (1 2 )B(1 B) L ˜ 2 2 ∂ ER ∆ 2 √ Π ≡ = + fR . (4.4) 3 ∂E 2 −1 ˜ 2 ˜ 2 2 + e 4B (2 − 2 )(2 − B) R + ∆ B − E R ∆ √ Solving the above equation we have the relation ( 2 − 1)π ≈− f |B|3 if |B|1, (3.7) 4 Π˜ R2∆2 + B˜ 2 E˜ = s , which is a negative value. (Note that to have CTCs R∆ R2∆2 + Π˜ 2 the parameter f must be positive.) We thus conclude Π˜ = Π − fR2,s= sign(Π/∆).˜ (4.5) that the tachyon supertube, despite being a BPS configuration, develops an instability in the pathological The corresponding Hamiltonian density per unit length spacetime with closed timelike curve. is H ≡ − L (R) ΠE 4. Discussions 1 R2∆2 + B˜ 2 = R2∆2 + sΠ˜ 2 R∆ R2∆2 + Π˜ 2 In normal background of spacetime a BPS state f ˜ ˜ 2 satisfies a Bogomolnyi bound, which implies that the + BΠ + fR . (4.6) ∆ energy is minimized for a given charge, and state is Then, the tubular bound state of the F -string and absolutely stable. However, in the pathological space- D0-brane may be formed if the energy H(R) has a time with closed timelike curve, a BPS state will be minimum at finite radius R. For example in Fig. 1 unstable, as shown in [11] using DBI action and in this we plot the radius-dependent energy for the case of Letter by using the tachyonic BDI action. f = 0.1, B = Π = 5. The tube with radius R = 5is In fact, the instability of the BPS supertube in type found. The functional form of energy H(R) in (4.6) IIA Gödel spacetime could be easily seen from the can be analyzed if B = Π. In this case we have a sim- analyses of DBI action, as briefly described in below. ple relation The world-volume theory on the supertube is just that of a D2-brane in curved background, which includes B2 + R2 − 2fBR2 H(R) = , (4.7) the Dirac–Born–Infeld and Wess–Zumino terms R − fR2 − and it has a solution tube with radius R = B. The as- S =− e φ − det(G + F ) − (C + C ∧ F ), c 3 1 sociated energy density is (4.1) H(Rc) = 2B ∼ q0 + q1. (4.8) where G is the pullback of the metric and C1 and C3 the pullbacks of the RR potentials. The field strength As the tube energy density is just given by the sum F includes the BI gauge field strength FBI that are of charges it carries (note that in the case of B = Π turned on in the brane and those induced by the NS the charges is proportional to B), this solution rep- field strength BNS in (2.2). Under these conditions the resents a BPS tube, as dictated by supersymmetry W.-H. Huang / Physics Letters B 615 (2005) 266–272 271

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Condition for superradiance in higher-dimensional rotating black holes

Eylee Jung, Sung Hoon Kim, D.K. Park

Department of Physics, Kyungnam University, Masan 631-701, South Korea Received 24 March 2005; received in revised form 11 April 2005; accepted 12 April 2005 Available online 19 April 2005 Editor: N. Glover

Abstract It is shown that the superradiance modes always exist in the radiation by the (4 + n)-dimensional rotating black holes. Using a Bekenstein argument the condition for the superradiance modes is shown to be 0 <ω

Recent brane-world scenarios such as large extra ground have been studied analytically [8] and numer- dimensions [1,2] or compactiﬁed extra dimensions ically [9]. It was shown that the presence of the extra with warped factor [3] predict a TeV-scale gravity. dimensions in general decreases the absorptivity and The emergence for the TeV-scale gravity in the higher- increases the emission rate on the brane. The decrease dimensional theories opens the possibility to make the of the absorptivity may be due√ to the decrease of the black hole factories in the future high-energy collid- effective radius [10] rc ≡ σ∞/π, where σ∞ is a ers [4–7]. In this context it is of interest to examine high-energy limit of the total absorption cross section. the various properties of the higher-dimensional black Although it may explain why the absorptivity is sup- holes. pressed in the high-energy regime, it does not seem to The absorption and emission for the different parti- provide a satisfactory physical reason for the suppres- cles by the (4 + n)-dimensional Schwarzschild back- sion of the absorptivity in the full range of the particle energy. The enhancement of the emission rate may be caused by the increase of the Hawking temperature in the presence of the extra dimensions. This means E-mail addresses: [email protected] (E. Jung), [email protected] (S.H. Kim), [email protected] that the Planck factor is more crucial than the grey- (D.K. Park). body factor in the Hawking radiation. For the case of

0370-2693/$ – see front matter  2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physletb.2005.04.022 274 E. Jung et al. / Physics Letters B 615 (2005) 273–276 the minimally coupled massless scalar the low-energy the superradiance modes [18,19] is predicted analyt- absorption cross section (LACS) always equals to the ically and numerically in the presence of the extra horizon area [11]. Thus, for the brane-localized scalar dimensions. The existence of the superradiance is very 2 the LACS is always 4πrH while for the bulk scalar it important for the experimental signature in the fu- n+2 ture colliders because it may change [15,20,21] the is equal to Ωn+2rH , where standard claim that black holes radiate mainly on the 2π (n+3)/2 brane. In this context it is important to derive a crite- Ω + = n 2 [(n + 3)/2] rion for the existence of the superradiance. In this short note we will derive this criterion using a Bekenstein’s + is the area of a unit (n 2)-sphere. The ratio of argument [22]. the LACS for the Dirac ﬁeld to that for the scalar The gravitational background around a (4 + n)- was shown to be 2(n−3)/(n+1) for the brane-localized −(n+3)/(n+1) dimensional, rotating, uncharged black hole having case [8] and 2 for the bulk case [12]. single angular momentum parameter about an axis in Therefore, the ratio factor 1/8, which was obtained the brane is given by [23] by Unruh long ago [13], was recovered when n = 0. The dependence on the dimensionality in these ratio µ 2aµsin2 θ factors may be used to prove the existence of the ex- ds2 =− 1 − dt2 − dt dφ Σrn−1 Σrn−1 tra dimensions in the future black hole experiments. Σ The relative bulk-to-brane energy emissivity was also + dr2 + Σdθ2 calculated in Ref. [9] numerically, which conﬁrmed ∆ the main result of Ref. [10], i.e., black holes radiate a2µ sin2 θ + r2 + a2 + sin2 θdφ2 mainly on the brane. Σrn−1 For the higher-dimensional charged black holes the 2 2 full absorption and emission spectra have been com- + r cos θdΩn, (3) puted numerically in Ref. [14]. It has been shown that contrary to the effect of the extra dimension the where presence of the nonzero inner horizon parameter r− µ ∆ = r2 + a2 − , generally enhances the absorptivity and suppresses the rn−1 emission rate. It has been shown also that the relative Σ = r2 + a2 cos2 θ, (4) bulk-to-brane emissivity decreases with increasing the inner horizon parameter r−. The LACS for the min- and dΩn is a line-element on a unit n-sphere. imally coupled massless scalar always equals to the It is worthwhile noting that the (4+n)-dimensional horizon area. For the Dirac fermion the LACS be- rotating black holes can have 1 + n/2 angular momen- comes [12] tum parameters for even n and (3 + n)/2 parameters + n+1 n 2 for odd n maximally [23]. Although our following ar- − n+3 r− n+1 BL = n+1 − BL gument can be applicable to this general case, it seems σF 2 1 σS (1) r+ to be complicated in the calculation. Thus, we would for the bulk case and like to consider the simpler case by reducing the angular momentum parameters. That is why we choose n+1 2 n−3 r− n+1 a single angular momentum parameter in Eq. (3).The BR = n+1 − BR σF 2 1 σS (2) detailed calculation for the spacetime background hav- r+ ing multiple angular momentum parameters will be BL for the brane-localized case. In Eqs. (1) and (2) σS reported elsewhere. BR The horizon radius r is determined from ∆ = 0, and σS are the LACSs for the bulk and brane- H localized scalars, respectively. i.e., The absorption and emission problems in the µ higher-dimensional rotating black holes were recently r2 + a2 − = . H n−1 0 (5) discussed in Refs. [15–17], where the existence of rH E. Jung et al. / Physics Letters B 615 (2005) 273–276 275

The horizon area A˜,massM, angular momentum J Inserting Eqs. (12) and (13) into Eq. (8),Eq.(8) be- and Hawking temperature TH are given by comes in the following n + ˜ ˜ ΩnrH (n 2)Ωn+2 dJ ∂A A = A, M = µ, dA˜ = 1 − Ω dM, (14) n + 1 16π dM ∂M 2 J = Ma, where + n 2 a 2 8π(n− 1)M Ω = (15) T = r + , (6) r2 + a2 H H + n H A (n 2)Ωn+2rH is a rotational frequency of the black hole. Bekenstein = (N+1)/2 [ + ] where ΩN 2π / (N 1)/2 is an area of showed in Ref. [22] that for scalar, electromagnetic = 2 + 2 unit N-sphere and A 4π(rH a ). It is easy to show and gravitational waves dJ/dM becomes that the various quantities in Eq. (6) are related to each T r other in the form dJ =− φ = m r , (16) µ (n − 1)A dM Tt ω AT = 2r + (n − 1) = 2r + . (7) H H n H where m and ω are azimuthal quantum number and rH 4πrH energy of the incident wave, respectively, and T is a Now we assume M and J are independent variables. µν stress-energy tensor. Thus, Eq. (14) becomes Then elementary mathematics gives m ∂A˜ ∂A˜ ∂A˜ dA˜ = 1 − Ω dM. (17) dA˜ = dM + dJ. (8) ω ∂M ∂M ∂J ˜ Firstly, let us calculate ∂A/∂M, which is given by Since ∂A/∂M is always positive from Eq. (12) and dA>˜ 0 because A/˜ 4 is a black hole entropy, Eq. (17) ∂A ∂r ∂a = 8π r H + a . (9) gives a condition ∂M H ∂M ∂M 0 <ω

Canonical sectors of ﬁve-dimensional Chern–Simons theories

Olivera Miškovic´ a,b, Ricardo Troncoso a, Jorge Zanelli a

a Centro de Estudios Cientíﬁcos (CECS), Casilla 1469, Valdivia, Chile b Departamento de Física, P. Universidad Católica de Chile, Casilla 306, Santiago 22, Chile Received 7 April 2005; accepted 20 April 2005 Available online 28 April 2005 Editor: M. Cveticˇ

Abstract The dynamics of five-dimensional Chern–Simons theories is analyzed. These theories are characterized by intricate self couplings which give rise to dynamical features not present in standard theories. As a consequence, Dirac’s canonical formalism cannot be directly applied due to the presence of degeneracies of the symplectic form and irregularities of the constraints on some surfaces of phase space, obscuring the dynamical content of these theories. Here we identify conditions that define sectors where the canonical formalism can be applied for a class of non-Abelian Chern–Simons theories, including supergravity. A family of solutions satisfying the canonical requirements is explicitly found. The splitting between first and second class constraints is performed around these backgrounds, allowing the construction of the charge algebra, including its central extension.  2005 Elsevier B.V. All rights reserved.

1. Introduction and for higher odd dimensions in [6–8]. For vanishing cosmological constant supergravity theories sharing The best known gauge theories whose dynamical this geometric structure have also been constructed field is a connection on a fiber bundle are described by in [9–11]. However, this elegant geometrical setting Yang–Mills and Chern–Simons (CS) actions. Three- leads to a rich and quite complex dynamics. Indeed, dimensional CS theories are topological and also pro- for the purely gravitational sector, the Lagrangian in = vide descriptions for gravitation and supergravity [1, D 5 dimensions, apart from the Einstein–Hilbert 2]. There also exist gravity theories in higher odd di- Lagrangian, also contains the Gauss–Bonnet term mensions described in terms of CS actions [3,4].For which is quadratic in the curvature, while for D 7 negative cosmological constant, in five dimensions the additional terms with higher powers of the curvature locally supersymmetric extension was found in [5], and explicitly involving torsion are also required. CS theories for D 5 have been studied in different contexts (see, e.g., [12–14]), and are not necessar- E-mail address: olivera@chopin.fis.puc.cl (O. Miškovic).´ ily topological theories but can contain propagating

µ degrees of freedom [15]. Their dynamical structure GK dx where K = 1,...,∆, and ∆ is the dimension depends on the location in phase space, and can drasti- of the Lie group. The five-dimensional Chern–Simons cally change from purely topological sectors to others form is such that its exterior derivative is an invariant with different numbers of local degrees of freedom. 6-form, Sectors where the number of degrees of freedom is not = 3 = K ∧ L ∧ M maximal are degenerate and on them additional local dL k F kgKLMF F F , (1) symmetries emerge [16]. K where F = dA + A ∧ A = F GK is the field strength Furthermore, the symmetry generators in CS the- 2-form and k is a dimensionless constant. Here ··· ories may be functionally dependent in some regions stands for a symmetrized1 trilinear invariant form, of phase space. Sectors where this happens are called which defines the third rank invariant tensor gKLM = irregular [17,18]. Around irregular configurations the 2 GK , GL, GM . The action is standard Dirac procedure, required to identify the physical observables (propagating degrees of freedom, 1 1 I[A]= L(A) = k AF2 − A3F + A5 , (2) conserved charges, etc.), is not directly applicable. 2 10 Furthermore, the naive linearization of the theory fails M M to provide a good approximation to the full theory where M is a five-dimensional manifold. The field around irregular backgrounds [19,20]. equations read Degeneracy and irregularity are two independent 2 conditions which may occur simultaneously in CS the- F GK = 0. (3) ories, and it is not yet fully understood how to deal with them. Irregular sectors are also found in the Ple- 2.1. Warming up with the linearized approximation banski theory [21]. Although these features are rarely found in field theories, they naturally arise in fluid dy- Non-Abelian CS theories are characterized by in- namics, as in the description of vortices through the tricate self couplings which give rise to dynamical Burgers equation [22], or in transonic wave propaga- features not present in standard theories. These non- tion in compressible fluids described by the Chaplygin trivial properties can be captured in the linearized the- and Tricomi equations [23]. ory in five dimensions. The action in the linear approx- ¯ The presence of degenerate and irregular sectors imation around a background solution A has the form ¯ ¯ obscures the dynamical content of these theories, as I[A + u]=I[A]+Ieff[u], where the effective action Dirac’s canonical formalism cannot be directly applied is3 to them. In Section 2, for simplicity we consider a non- [ ]= ¯ ∇¯ Abelian CS theory in five dimensions, which captures Ieff u 3k uF u the dynamical behavior without loss of generality. In M Section 3, we identify conditions that define canonical 1 = d5x uK Ω¯ ij ∇¯ uL − uK C − h , (4) sectors, that is, those where the canonical formalism 2 i KL 0 j 0 K can be applied, and a family of solutions satisfying ≡ ij kl K ¯ L ∇¯ M the canonical requirements is explicitly found. In Sec- with the “potential” h(u) 3kε gKLMui F0j kul tion 4, the splitting between first and second class con- and the covariant derivative ∇¯ u = du +[A¯ , u].The straints is performed around these backgrounds, allow- constraint CK is given by ing the construction of the charge algebra, including = ¯ ij ∇¯ L its central extension. Section 5 contains the discussion CK ΩKL iuj , (5) and outlook.

1 In case of a superalgebra, the standard (anti)symmetrized form is assumed. 2. Dynamics 2 Hereafter, wedge products between forms will be omitted. 3 The five-dimensional manifold is assumed to be topologically Chern–Simons Lagrangians describe gauge theo- R ⊗ Σ, and the coordinates are chosen as xµ = (x0,xi ),wherexi , = K × = ries for a Lie-algebra-valued connection A Aµ with i 1,...,4 correspond to the space-like section Σ. O. Mišković et al. / Physics Letters B 615 (2005) 277–284 279 and the kinetic term is defined by the symplectic form 2.2. Nonlinear dynamics

The field equations (3) can be written as ¯ ij ≡ ij ¯ =− ij kl ¯ M Ω Ω (F) 3kgKLMε Fkl , (6) µνλρσ L M = KL KL ε gKLMFµνFλρ 0. (7) Therefore, the field equations (7) split into the dynam- which explicitly depends on the curvature. K ical equations, Thus, the time evolution of the perturbations ui de- ¯ K ij L = pends on the background field strength, Fkl , and hence ΩKLF0j 0, (8) the dynamics is crucially sensitive to the particular and the constraints, background around which it is explored. Overlook- ing this issue may lead to the paradoxical situation 1 C = Ωij F L ≈ 0. (9) that the linearized theory around some backgrounds K 4 KL ij may seem to have more degrees of freedom than the ij fully nonlinear theory [19]. This is due to the fact that The symplectic matrix ΩKL(F ), defined in Eq. (6), × i j around those backgrounds, the linear approximation is a 4∆ 4∆ array with indices K and L with at ij L = i ≈ eliminates some constraints from the action. least four zero modes (since ΩKLFjk δkCK 0), Since the rank of Ω¯ is not fixed, the number of corresponding to the spatial diffeomorphisms. The ex- dynamical degrees of freedom can change throughout istence of these four zero modes implies that the rank phase space: of Ω cannot exceed (4∆ − 4) [15]. Generic configurations have maximal rank of Ω¯ As in the linearized approximation, the symplectic K and form an open set in phase space. The theory matrix is a function of the field strength Fij , its rank around this kind of configurations has maximal num- is not necessarily constant throughout phase space. ber of degrees of freedom [15]. A configuration is said to be generic if Ω has maxi- Degenerate configurations are the ones for which mum rank, 4∆ − 4. Configurations of lower rank are the rank of Ω¯ is not maximal so that they have ad- degenerate; they have additional gauge symmetries ditional gauge symmetries, and thus fewer degrees of and, consequently, fewer degrees of freedom. For ex- freedom. ample, any “vacuum” solution F K = 0 is maximally On the other hand, not only the rank of the sym- degenerate since the symplectic form vanishes on it plectic form can vary, but the linear independence of and hence no local excitations can propagate around the constraints CK is not guaranteed either, and can these configurations. Since the rank cannot change un- fail on some backgrounds. der small deformations, generic sectors form open sets If the constraints CK are independent, the sector is in field space, whereas degenerate configurations de- regular. This is the case in all standard theories. How- fine sets of measure zero. ever, in CS theories, the constraints CK can become Regular configurations are those on which the con- dependent on some backgrounds, and these sectors are straints CK = 0 are functionally independent, that is, called irregular. they define a smooth surface with a unique tangent In an irregular configuration, there is always a lin- space on an open set around the configuration.4 This ear combination of C’s that identically vanishes. Con- means that the variations sequently, in the linear approximation the number of 1 ij L degrees of freedom seems to increase in irregular sec- δCK = Ω δF , (10) 2 KL ij tors. However, this is an illusion induced by using the linear approximation which is no longer valid in this at a regular configuration, are ∆ linearly independent case. Moreover, the Dirac approach is not directly ap- vectors in phase space. Consequently, regularity is sat- plicable in irregular sectors [24]. isfied if the Jacobian of Eq. (10) has maximal rank, ∆, Indeed, careful analysis of the full non-linear theory shows that the number of degrees of freedom can- 4 Regular configurations also form open sets in field space, while not increase. This is discussed in the next section. irregular ones form sets of measure zero. 280 O. Mišković et al. / Physics Letters B 615 (2005) 277–284

ij × = where now ΩKL has to be regarded as a ∆ 6∆ ma- For this kind of configurations, the rank (ΩKL) ij (Ω ¯ ¯ ) = 4∆−4, and therefore, they provide generic trix with indices (K) and L . KL Note that genericity does not imply regularity, and backgrounds.5 vice-versa. This is because in one case the components Among the configurations (12), the regular ones are Ω are regarded as the entries of a square matrix and, those for which the variations of the constraints (10) in the other, as the entries of a rectangular one. are linearly independent, and this depends on the value Dirac’s Hamiltonian formalism requires that, in an of α, open set, the symplectic matrix be of constant rank and 1 ij L¯ 1 ij z the constraints be functionally independent. Hence, we δC ¯ = Ω ¯ ¯ δF + Ω ¯ δF , K 2 KL ij 2 Kz ij call canonical sectors of phase space those that are 1 ¯ 3 simultaneously generic and regular, because that is δC = Ωij δF K − kαεij kl F z δF z . (13) z 2 Kz¯ ij 2 ij kl where the canonical formalism applies without mod- ij ifications. Around degenerate or irregular configura- If the (∆ − 1) × 6(∆ − 1) block Ω ¯ ¯ is non- − KL tions, the dynamical content of the theory requires ex- degenerate, δCK¯ represent ∆ 1 linearly independent ¯ tending Dirac’s formalism as in [16,18]. vectors expressed as a linear combination of δF K .The problem of regularity then, reduces to the question of linear independence of the vector δCz relative to the 3. Canonical sectors δCK ’s. If α = 0, the block Ωzz is non-vanishing, so that δCz in (13) always contains the term proportional to The identification of the canonical sectors for a z δF , giving a vector linearly independent from δCK¯ . Chern–Simons theory in general is a non-trivial task. Therefore, one concludes that: However, a little bit of information about the group and the invariant tensor allows, in some cases, to iden- For α = 0, the dynamics of the sectors defined by tify these sectors. Let us split the generators as GK = open sets around configurations of the form (12) is al- { } GK¯ , Z . If the group admits a third rank invariant ten- ways canonical. sor gKLM such that g ¯ ¯ := g ¯ ¯ is invertible, and KLz KL However, for α = 0, the variations of Cz do not de- g ¯ = 0, then the search for canonical sectors if much ¯ Kzz pend on F z but on the remaining components F K .In simpler. These conditions are trivially fulfilled by a K¯ = non-Abelian group of the form G = G˜ ⊗ U(1), and particular, note that for a configuration with F 0, ij z the block Ω =−3kεij kl g ¯ ¯ F is invertible and also apply to a larger class of theories including su- K¯ L¯ KL kl pergravity, for which the relevant group is super AdS5, hence, this configuration is generic. However, this con- = SU(2, 2|N). figuration is irregular because δCz 0. One therefore Thus, the symplectic matrix reads concludes that: For α = 0, the theory contains additional irregular ij ij Ω ¯ ¯ Ω ¯ sectors which do not exist if α = 0. Thus, a necessary ij = KL Kz ΩKL condition to ensure regularity for configurations of the Ωij Ωij Lz¯ zz form (12),forα = 0, is that at least one component of ¯ M¯ + z M¯ F K does not vanish. gK¯ L¯ M¯ F gK¯ L¯ F gK¯ M¯ F =−3kεij kl kl kl kl , M¯ z In a canonical sector the counting of degrees of gL¯ M¯ Fkl αFkl freedom can be safely done following the standard (11) := where α gzzz. ij ij 5 This can be explicitly seen from (Ω ) = (Ω ) + Consider the following class of configurations, KL K¯ L¯ ij ij (Σ ), where the matrix Σ vanishes for CM ≈ 0, as it is given ij i il −1 K¯ L¯ jknm z −1 by Σ = (δ Cz − Ω ¯ (Ω ) C ¯ )ε Fnm,andΩ is the ij z k zK lk L Ω = non-degenerate, det F = 0. (12) ij K¯ L¯ ij inverse of the invertible block Ω only. K¯ L¯ O. Mišković et al. / Physics Letters B 615 (2005) 277–284 281 procedure [24], and in this case the number is N = and they satisfy ∆ − 2 [15]. i j = ij A simple example of a solution of the constraints in φK ,φL ΩKL. (19) the canonical sector is one whose only non-vanishing Since the symplectic matrix Ωij has at least four null K¯ KL components of F is eigenvectors, some φ’s are first class and the explicit ¯ separation cannot be performed in general. However, F K dx1 ∧ dx2 = 0, (14) 12 for the class of canonical configurations described in for at least one K¯ and the previous chapter, it is possible to separate them as i z = z = First class: GK =−CK +∇iφ ≈ 0, F34 0, with det Fij 0. (15) K ¯ ¯ H = F z φj − φk Ω−1 KLΩlj i ij z K¯ kl Lz¯ = K j ≈ 4. Constraints and charge algebra Fij φK 0, i Second class: φ ¯ ≈ 0, (20) The canonical sectors satisfy all conditions neces- K ∇ i = i + M L i sary for the Dirac formalism to apply. That is, the where iφK ∂iφK fKL Ai φM , is the covariant first and second class constraints can be clearly de- derivative, and the constraints GK satisfy the algebra fined and the counting and identifying of degrees of {G ,G }=f M G , G ,φi = f M φi . freedom can be explicitly done [25]. The explicit sepa- K L KL M K L KL M ration between first and second class constraints might (21) be extremely difficult or impossible to carry out. This The constraints GK and Hi are generators of obstacle prevents, among other things, the canonical gauge transformations and improved spatial diffeo- construction of the conserved charges. morphisms, respectively.6 The second class constraints The advantage of the class of canonical sectors can be eliminated by introducing Dirac brackets, described above, is that this splitting can be per- ¯ ¯ ∗ 4 i −1 KL formed explicitly and, as a consequence, the conserved {X, Y } ≡{X, Y }− d x X, φ ¯ (x) Ω (x) charges and their algebra are obtained following the K ij Σ Regge–Teitelboim approach [26]. × φj (x), Y , (22) L¯ 4.1. Hamiltonian structure and the reduced phase space is parametrized by ¯ {AK ,Az,πi}. The Hamiltonian formalism applied to CS systems i i z was performed in [15] and can be easily extended to 4.2. Conserved charges the supersymmetric case [19]. The action in Hamil- tonian form reads The separation (20) allows the construction of the conserved charges and the algebra following the I[A]= d5x Li A˙K − AK C , (16) K i 0 K Regge–Teitelboim approach [26]. The symmetry generators are where the constraints CK are defined in (10), GQ[λ]=G[λ]+Q[λ], (23) i ij kl L M 1 M L N S L ≡ kε gKLM F A − fNS A A A , K jk l 4 j k l where (17) G[λ]= d4xλK G , (24) M K and fNS are the structure constants of the Lie group. Additional constraints arise from the definition of the Σ momenta, 6 An improved diffeomorphism, δ AK = F K ξ ν , differs from i i i ξ µ µν φ = π − L ≈ 0, (18) K =− µ K K K K the Lie derivative by the gauge transformation with λ ξ Aµ . 282 O. Mišković et al. / Physics Letters B 615 (2005) 277–284 and Q[λ] is a boundary term such that GQ[λ] has ories with α = 0 which accept simple generic con- ¯ well-defined functional derivatives. According to the figurations of the form (12), given by F K = 0 with Brown–Henneaux theorem, in general the charge al- det(F z) = 0. These configurations are generic but ir- gebra is a central extension of the gauge algebra [27], regular (and therefore not canonical), since δCz in ∗ Eq. (13) identically vanishes. If one naively chooses Q[λ],Q[η] = Q[[λ,η]] + C[λ,η], (25) a configuration of this type as a background in the ex- K K L M where [λ,η] = fLM λ η . pression for the charges (26), one obtains that the U(1) Thus, being in a canonical sector allows writing the charge identically vanishes. This example simply re- charges as (see Appendix A),7 flects the fact that, for irregular configurations, U(1) generator becomes functionally dependent, so that the K ¯ L M Q[λ]=−3k gKLMλ F A , (26) naive application of Dirac’s formalism within irregular sectors would lead to ill-defined expression for the ∂Σ charges. ¯ where F is the background field strength and the para- Canonical sectors represent typical initial configu- meters λK (x) approach covariantly constant fields at rations around which one can prepare the system to let the boundary, and the central charge is it evolve. In its evolution, the system may reach degenerate or irregular configurations. Although either K ¯ L M C[λ,η]=3k gKLMλ F dη . (27) degenerate or irregular configurations are easily iden- ∂Σ tified, it is not yet fully understood how to deal with the dynamics around them in general, and it is clear that The charge algebra can be recognized as the WZW4 extension of the full gauge group [28]. In an irregular one must proceed with caution. However, the analy- sector the charges are not well defined and the naive sis of degenerate mechanical systems provide simple application of the Dirac formalism would at best lead models that describe irreversible processes in which a to a charge algebra associated to a subgroup of G. system may evolve into a configuration with fewer degrees of freedom [16]. From these results one infers that, under certain conditions, a CS system may fall 5. Discussion into a degenerate state from which it can never escape, losing degrees of freedom in an irreversible manner Configurations in the canonical sectors satisfy all [16]; or it can also pass through irregular states un- necessary conditions to safely apply the Dirac formal- harmed [18]. ism to five-dimensional CS theories. The identification The possibility that the fate of an initial configu- of these sectors in CS theories considered here, al- ration in a canonical sector of a higher-dimensional lows the explicit separation of first and second class CS system could be to end in a degenerate state, leads constraints. Consequently, the conserved charges and to an interesting effect: in Ref. [11],anewkindof their algebra are constructed following the Regge– eleven-dimensional supergravity was constructed as a Teitelboim approach. CS system for the M-algebra. It was observed that the theory admits a class of vacuum solutions of the form As a direct application of these results in the con- − S10 d × Y + Y + R text of supergravity, canonical BPS states saturating d 1, where d 1 is a warped product of d a Bogomol’nyi bound for the conserved charges (26) with a -dimensional spacetime. Remarkably, it was can be constructed [29]. One should expect that these found that a non-trivial propagator for the graviton d = results extend to higher odd dimensions. Indeed, con- exists only for 4 and positive cosmological conserved charges have been found in the purely gravita- stant, and that perturbations of the metric around this tional case following a different approach [30]. solution reproduce linearized general relativity around Overlooking regularity obstructs the construction four-dimensional de Sitter spacetime. of well-defined canonical generators. Consider the- Since this solution is a degenerate state one may regard it as the final stage of a wide class of initial canonical configurations that underwent a sort of dy- 7 Hereafter, the forms are defined at the spatial section Σ. namical dimensional reduction. O. Mišković et al. / Physics Letters B 615 (2005) 277–284 283

Acknowledgements where the term proportional to vanishes on the constraint surface. We gratefully acknowledge insightful discussions Note that the second term dλz AA¯ identically with M. Henneaux. This work was partially funded by vanishes when it is evaluated on the background A = ¯ ¯K ¯L FONDECYT grants 1020629, 1040921, 1051056, and A, since gzKLA A ≡ 0. 3040026. The generous support to CECS by Empresas Since the asymptotic behavior of the fields ap- CMPC is also acknowledged. CECS is a Millennium proaches a background configuration as A ∼ A¯ + A, Science Institute and is funded in part by grants from the parameters of the asymptotic symmetries are of the Fundación Andes and the Tinker Foundation. form λ ∼ λ¯ + λ. In particular, λ¯ z is a constant, and then dλzAA¯ ∼ d λz A¯ (A) , (A.7) Appendix A is a subleading contribution which vanishes as it ap- On the reduced phase space φi = 0, the generators proaches the boundary. Therefore, the charges are K¯ are given by (26) =− =− + i [ ]=− ¯ GK¯ CK¯ ,Gz Cz ∂iφz, (A.1) Q λ 6k λFA . (A.8) and the smeared generators become ∂Σ G[λ]=3k λF2 + λz d , (A.2) References Σ Σ [1] A. Achúcarro, P.K. Townsend, Phys. Lett. B 180 (1986) 89. = − { }− 1 3 [2] E. Witten, Nucl. Phys. B 311 (1988) 46. where the 3-form π k( A, F 2 A ) is dual of i [3] A. Chamseddine, Phys. Lett. B 233 (1989) 291. the constraints, i.e., jkl = φ εij kl . z [4] R. Troncoso, J. Zanelli, Class. Quantum Grav. 17 (2000) 4451. The variation of the generators (A.2) yields [5] A. Chamseddine, Nucl. Phys. B 346 (1990) 213. [6] R. Troncoso, J. Zanelli, Phys. Rev. D 58 (1998) R101703. δG[λ]=6k λF∇δA+ λz dδ . (A.3) [7] R. Troncoso, J. Zanelli, Int. J. Theor. Phys. 38 (1999) 1181. [8] R. Troncoso, J. Zanelli, Chern–Simons supergravities with off- Σ Σ shell local superalgebras, in: C. Teitelboim, J. Zanelli (Eds.), Integrating by parts this expression in order to obtain a Black Holes and Structure of the Universe, World Scientific, well defined functional derivative, the leftover bound- Singapore, 1999, hep-th/9902003. ary term is the variation of the charge [9] M. Bañados, R. Troncoso, J. Zanelli, Phys. Rev. D 54 (1996) 2605. [10] M. Hassaine, R. Olea, R. Troncoso, Phys. Lett. B 599 (2004) z δQ[λ]=−6k λFδA−2k dλ AδA 111. [11] M. Hassaine, R. Troncoso, J. Zanelli, Phys. Lett. B 596 (2004) ∂Σ ∂Σ 132; − λzδ . (A.4) M. Hassaine, R. Troncoso, J. Zanelli, Proc. Sci. WC2004 (2005) 006. ∂Σ [12] R. Floreanini, R. Percacci, Phys. Lett. B 224 (1989) 291. This expression can be integrated out provided the [13] R. Floreanini, R. Percacci, R. Rajaraman, Phys. Lett. B 231 (1989) 119. connection is fixed at the boundary, as [14] V.P. Nair, J. Schiff, Phys. Lett. B 246 (1990) 423; ¯ V.P. Nair, J. Schiff, Nucl. Phys. B 371 (1992) 329. A → A at ∂Σ, (A.5) [15] M. Bañados, L.J. Garay, M. Henneaux, Phys. Rev. D 53 (1996) ¯ 593; where A is a background configuration in the canoni- M. Bañados, L.J. Garay, M. Henneaux, Nucl. Phys. B 476 cal sector. Then, the charges are (1996) 61. [16] J. Saavedra, R. Troncoso, J. Zanelli, J. Math. Phys. 42 (2001) Q[λ]=−6k λFA¯ −2k dλz AA¯ , (A.6) 4383; J. Saavedra, R. Troncoso, J. Zanelli, Rev. Mex. Fis. 48 (2002) ∂Σ ∂Σ 387. 284 O. Mišković et al. / Physics Letters B 615 (2005) 277–284

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Tensionless strings, correspondence with SO(D, D) sigma-model

G. Savvidy

Theory Division, CERN, CH-1211 Geneva 23, Switzerland Demokritos National Research Center, GR-15310 Athens, Greece Received 23 February 2005; received in revised form 31 March 2005; accepted 1 April 2005 Available online 12 April 2005 Editor: M. Cveticˇ

Abstract The string theory with perimeter action is tensionless by its geometrical nature and has a pure massless spectrum of higher spin gauge particles. I demonstrate that the linear transformation of the world-sheet fields defines a map to the SO(D, D) sigma- model equipped by additional Abelian constraint, which breaks SO(D, D) to a diagonal SO(1,D− 1). The effective tension is equal to the square of the dimensional coupling constant of the perimeter action. This correspondence allows to view the perimeter action as a “square root” of the Nambu–Goto area action. The aforementioned correspondence between tensionless strings and SO(D, D) sigma-model allows to introduce vertex operators in full analogy with the standard string theory and to confirm the form of the vertex operators introduced earlier, the value of the intercept a = 1 and the critical dimension D = 13.  2005 Elsevier B.V. All rights reserved.

It is generally expected that high energy limit or, requires therefore a nonperturbative treatment of the what is equivalent, the tensionless limit α →∞of problem [4–8].1 2 = string theory should have massless spectrum MN The tensionless model with perimeter action sug- (N − 1)/α → 0 and should recover genuine symme- gested in [17–20] does not appear as an α →∞limit tries of the theory [1–3]. Of course this observation of the standard string theory, as one could probably ignores the importance of the high genus G diagrams, think, but has a tensionless character by its geomet- the contribution of which AG exp{−α s/(G + 1)} rical nature [17]. Therefore it remains mainly unclear is exponentially large compared to the tree level di- at the moment how these two models are connected. agram [1–3]. The ratio of the corresponding scatter- However the perimeter model shares many properties 2 ing amplitudes behaves as AG+1/AG exp{α s/G } with the area strings in the sense that it has world-sheet and makes any perturbative statement unreliable and conformal invariance and contains the correspond-

E-mail addresses: [email protected], 1 The different aspects and models of tensionless theories can be [email protected] (G. Savvidy). found in [9–16].

0370-2693/$ – see front matter  2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physletb.2005.04.001 286 G. Savvidy / Physics Letters B 615 (2005) 285–290 ing Virasoro algebra, which is extended by additional less string theory the intercept a = 1 and therefore Abelian generators. This makes mathematics used in only N = 1 state realizes fixed helicity representa- the perimeter model very close to the standard string tions, W = 0, whereas the ground state N = 0 and theory and allows to compute its massless spectrum, the rest of the excited states N 2 realize continu- critical dimension Dc = 13 [18–20] and to construct ous spin representations, W = 0, of the massless little appropriate vertex operators [21,22]. group SO(11). Comparing literally the spectrum of these two mod- The aforementioned correspondence allows to in- els one can see that instead of usual exponential grow- troduce the vertex operators in full analogy with the ing of states, in the perimeter case we have only linear standard string theory and to confirm the form of growing. In this respect the number of states in the the vertex operators introduced earlier in [21,22].The perimeter model is much less compared with the stan- n-point scattering amplitude of fixed helicity states dard string theory and is larger compared with the field W1 = 0 (N = 1) in terms of 13-dimensional momenta 2 theory models of the Yang–Mills type. From this point ki and polarizations ei is of view it is therefore much closer to the quantum field A ; ; theory rather than to the standard string theory. At the (k1,e 1 ... kn,en) ie1π1+···+ienπn same time its formulation and the symmetry structure = dπ1 ···dπn e is more string-theoretical. Perhaps there should be a strong nonperturbative rearrangement of the spectrum n × 2 ··· in the limit α →∞before the spectrum of the area d ζi Uk1,π1 (ζ1) Ukn,πn (ζn) , (1) and the perimeter strings can become close to each i other. where Uki ,πi (32) are fixed helicity vertex operators Our aim here is to give a partial answer to these (ki · πi) = 0, i = 1,...,n. This scattering amplitude questions. As we shall see the linear transforma- exhibits important gauge invariance with respect to the tion of the world-sheet fields defines a map to the gauge transformations [19]: SO(D, D) σ -model equipped by an additional Abelian e → e + k Λ (k ,...,k ), (2) constraint, which breaks SO(D, D) to a diagonal i i i i 1 n SO(1,D− 1). The effective string tension is equal to where Λi(k1,...,kn) are gauge parameters. This in- the square of the dimensional coupling constant m of variance is valid only for the states which are described the perimeter action by the fixed helicity vertex operator Uk,π (32),for 2 which W1 ∼ (k · π) = 0. 1 m2 = . 2πα π The perimeter string model was suggested in [17] and describes random surfaces embedded in D-dimen- This relation allows to view the perimeter action as sional space–time with the following action a “square root” of the Nambu–Goto area action m = √ / α m 2 1 2 . The mass-shell quantization condition of the S = mL = d2ζ h ∆(h)X , SO(13, 13)σ-model π µ where h is the world-sheet metric, ∆(h) = 1/ α M2 =−α K2 = (N − ), √ √ αβ N 1 αβ h∂α hh ∂β is Laplace operator and m has di- which defines the value of the first Casimir operator mension of mass. There is no Nambu–Goto area term K2 of the Poincaré algebra in 26 dimensions, is trans- in this action. The action has dimension of length lated through the dictionary into the quantization con- L and the dimensional coupling constant is m.Mul- = 2 dition for the square W wD−3 of the Pauli–Lubanski tiplying and dividing the Lagrangian by the square form wD−3 of the Poincaré algebra in 13 dimensions 2 root (∆(h)Xµ) one can represent it in the σ -model (k · π)2 W = = (N − a)2 = (N − 1)2, N m2 2 W defines fixed helicity states, when W = 0 (N = 1),and ◦ | = × N 1 because, as we shall see (26), K K in 26 dim. 2m continuous spin representations (CSR), when WN = 0 (N = 1) [18, (k · π)|in 13 dim. This demonstrates that in the tension- 19,27–29]. G. Savvidy / Physics Letters B 615 (2005) 285–290 287

ab ´ form [18]: where h Tab = 0. The central charge c = 2D of the √ Virasoro algebra remains untouched and demonstrates 1 2 αβ µ ν S =− d ζηµν hh ∂αΠ ∂β X , (3) the absence of additional contributions to the central π charge due to the primary constraint (6) (see also [24] where the operator Π µ is for alternative calculation). ∆(h)Xµ Correspondence with the SO(D, D) σ -model.Let Π µ = m . (∆(h)Xµ)2 us introduce new variables as follows: 1 µ = √1 µ + µ We shall consider the model B, in which two field Π Φ1 Φ2 , µ m2 variables X and hαβ are independent. The classical 2 1 equation is Xµ = √ Φµ − Φµ . (10) 2 1 2 (I) ∆(h)Π µ = 0 (4) Then the action (3) will take the form and world-sheet energy–momentum tensor m2 √ S =− d2ζη hhαβ = µ µ − cd µ µ = µν (II)Tαβ ∂{αΠ ∂β}X hαβ h ∂cΠ ∂d X 0. 2π × µ ν − µ ν (5) ∂αΦ1 ∂β Φ1 ∂αΦ2 ∂β Φ2 . (11) The operator Π is a space-like vector, If one considers the 2D-dimensional target space with ≡ µ µ − 2 = the combined coordinates (III)Θ Π Π m 0. (6) M µ1 µ2 a Φ = Φ ,Φ ,M= 1,...,2D, The energy–momentum tensor is conserved ∇ Tab = 1 2 ab 0 and is traceless h Tab = 0, thus we have two- and fully symmetric Lorentzian signature space–time dimensional world-sheet conformal field theory with metric with D pluses and D minuses the central charge c = 2D [18]. We have equa- ηµ1ν1 tion of motion (4) together with the primary con- ηMN = −ηµ2ν2 straints (5) and (6) and secondary constraints Θ1,0 = Π∂+Π, Θ0,1 = Π∂−Π, Θ1,1 = ∂+Π∂−Π of confor- −, +,...,+ = , (12) mal weights (1, 0), (0, 1) and (1, 1) [18]. The equiva- +, −,...,− lent form of the action (3) is [23] then the action (11) will have formal interpretation in 1 √ terms of σ -model being defined on a 2D-dimensional S´ =− d2ζ hhαβ π target space with the symmetry group SO(D, D) × µ ν + 2 − 2 2 √ ηµν∂αΠ ∂β X ωαβ Π m , (7) m 2 αβ M N S =− d ζηMN hh ∂αΦ ∂β Φ . (13) where the Π µ field is now an independent variable and 2π the ωαβ are Lagrange multipliers. The system of equa- From this expression of the action we can deduce that tions which follows from S´ the effective string tension Teff is equal to the square of the mass m µ = µ − αβ µ = ∆(h)Π 0, ∆(h)X 2h ωαβ Π 0, 1 m2 µ 2 = . (14) Π Πµ = m (8) 2πα π is equivalent to the original equations (4) and (6) and The last relations allow to view the tensionless string ´ the corresponding new energy–momentum tensor Tαβ theory, which is defined by the perimeter action (3), acquires an additional term which depends only on the as a “square root of the Nambu–Goto area action” field Π m = 1/2α . This interpretation has deep geometrical

origin because in some sense the perimeter L, which ´ 1 γδ 2 Tαβ = Tαβ + ωαβ − hαβ h ωγδ Π − 1 , (9) was deﬁned for the two-dimensional surfaces in (3), 2 can be considered as a square root of the surface area. 288 G. Savvidy / Physics Letters B 615 (2005) 285–290

This intuitive interpretation can be made more pre- following commutator relations cise if one recalls Zenodor–Minkowski isoperimetric µ ˆν µν µ ν µν 2 x , k = iη , e , πˆ = iη , inequality [25,26], which tells that L 4πS, with the µ ν = µν equality taking place only for a sphere. αn ,βl nη δn+l,0 (19) The crucial constraint (6) will take the form m2 × 2 and [α, α]=[β,β]=0 (the indexes are not shown). (Φ1 + Φ2) = 2 and it breaks SO(D, D) group of fully symmetric space–time MD,D down to the diag- It is also convenient to introduce the zero momentum operators αµ = kˆµ, βµ =ˆπ µ. The appearance of the onal group SO(1,D − 1) of the standard space–time 0 0 M1,D−1 with one time coordinate additional zero mode means that the wave function is a function of the coordinate variables xµ and eµ: SO(D, D) → diag SO(1,D− 1), ΨPhys = Ψ(x,e). as one can see from the component form of the above The coordinate variable xµ belongs to a Minkowski constraint3 µ ∈ 13 µ space x M and e belongs to a hyperboloid − 0 + 0 2 + + 2 =− 2 + 2 = 2 eµ ∈ H 13 which is defined by the relation e2 =−e2 + Φ1 Φ2 (Φ1 Φ2) Φ+ Φ+ 2/m . 0 (15) e 2 = 1 (6), (15) The X and Π fields (10) are actually light-cone co- µ µ M26 → M13 ⊗ H 13. (20) ordinates on MD,D and one can heuristically say that our strings are massless because they propagate on the It was suggested therefore in [18] that eµ should be light cone of MD,D. The Abelian constraint (15) can interpreted as a polarization vector, because from the also be considered as a “compactification” to a hyper- constraint (6), (15) it follows that [18] boloid manifold H D. 2 = · = 2 = The energy–momentum tensor (5) will take the k 0,ek 0,e1. form It is important to get a better idea about the algebra (19). The transformation (10) naturally leads to the os- T = ∂ Φ ∂ Φ − ∂ Φ ∂ Φ αβ α 1 β 1 α 2 β 2 cillators 1 − h hγδ(∂ Φ ∂ Φ − ∂ Φ ∂ Φ ). (16) αβ γ 1 δ 1 γ 2 δ 2 µ = √1 µ + µ µ = √1 µ − µ 2 An αn βn ,Bn αn βn 2 2 It is therefore clear that we shall have 2Dc = 26 and shall recover the previous result [18] and brings the algebra (19) to the form µ ν µν = A ; A =+η nδ + , Dc 13. (17) n m n m µ; ν =− µν Bn Bm η nδn+m, Operator algebra and vertexes. For the open strings Aµ; Bν = 0. (21) the solution of this two-dimensional world-sheet CFT n m is [18]: This is a standard algebra of the oscillators with the following signature 1 1 − Xµ = xµ + πˆ µτ + i βµe inτ cos nσ, m n n µν = − + + ∈ − n=0 η ( , ,..., ) SO(1,D 1), µν 1 − −η = (+, −,...,−) ∈ SO(D − 1, 1). Π µ = meµ + kˆµτ + i αµe inτ cos nσ, (18) n n n=0 In terms of the above oscillators the “target space” coordinates (10) ΦM = (Φµ1 ,Φµ2 ) have the form: ˆ 1 2 where kµ =−i∂/∂x and πˆ µ =−i∂/∂e are mo- µ µ √ mentum operators and α , β are oscillators with the µ 1 1 1 n n 2Φ = xµ + eµ + kˆµ + πˆ µ τ 1 m m2 m i 1 3 The SO(D, D) signature allows the D light-cone coordinates + µ −inτ An e cos nσ, ± = 0 ± 0 ± = ± m n Φ Φ1 Φ2 , Φ Φ1 Φ2. n=0 G. Savvidy / Physics Letters B 615 (2005) 285–290 289

√ µ =− µ + 1 µ + 1 ˆµ − 1 ˆ µ Indeed the general form of the vertex operators sug- 2Φ2 x e k π τ m m2 m gested in [21] is given by the formula

i 1 − ˜ ˜ + Bµe inτ cos nσ. (22) µ1µ1...,...µj µj n Uk,π (ζ ) m = n n 0 ˜ ˜ n1 µ1 n1 µ˜ 1 nj µj nj µ˜ j =:∂ X ∂¯ X ...... ∂ Π ∂¯ Π The above SO(D, D) σ -model interpretation of the ζ ζ ζ ζ · + · tensionless string theory allows to introduce vertex × eik X(ζ) iπ Π(ζ):, (27) operators in the full analogy with the standard string the conformal spin should be equal to zero, there- theory case. Indeed the vertex operator for the ground fore n1 +···+nj =ñ1 +···+n ˜j = N.Usingthe state has the form: world-sheet energy–momentum operator [18] T(ζ)= ◦ −:∂ X · ∂ Π: one can compute the anomalous dimen- V =:eiK Φ : ζ ζ K sion of the open strings vertex operators [21,22]: M = µ1 µ2 M = µ1 µ2 ◦ = where K (k1 ,k2 ), Φ (Φ1 ,Φ2 ), K Φ (k · π) M N ∆ = + N. (28) ηMNK Φ , and has conformal dimension equal to m the square of the momentum KM It must be equal to 1 in order to describe emission of K ◦ K physical states, therefore in tensionless string theory ∆ = α K2 = . (23) 2 the value of the intercept is equal to one, a = 1, be- 2m · cause (L − a)ψ = ( (k π) + N − a)ψ = 0 [19].The Therefore substituting the expressions for the field 0 m M = µ1 µ2 corresponding poles Φ (Φ1 ,Φ2 ) in terms of the original world-sheet µ µ (k · π) fields X and Π (10) we shall get = 1 − N (29) − + m =: ik1Φ1 ik2Φ2 :=: ikX iπΠ/m: VK e e , (24) are translated to the mass-shell condition on the σ - 2 = − where the momenta k and π are: model side: α K 1 N. Let us discuss what these relations mean. The 1 1 physical meaning of the invariant k · π is given k = √ (k1 + k2), π = √ (k1 − k2). (25) = 2 2 2m by W wD−3-square of the Pauli–Lubanski form µ ,...,µ − 1 D 3 ∼ µ1,...,µD−3,νλρ This is an interesting relation because it demon- wD−3 ε kνMλρ of the Poincaré strates how the 2D-dimensional momentum KM of algebra on M13, that is [18,19] the SO(D, D) σ -model splits into two parts which (k · π)2 form the physical momentum variable k of the W = . µ 2 (30) tensionless strings propagating in a 13-dimensional m Minkowski space–time xµ ∈ M1,D−1 and the momen- From (29), (30) we conclude that on the level N the tum π µ, which is conjugate to the polarization vector value of the square of the Pauli–Lubanski form is eµ ∈ H 1,D−1 equal to 2 WN = (1 − N) . (31) MSO(13,13) → MSO(1,12) ⊗ HSO(1,12). As it is well known it defines fixed helicity states, We can now translate the conformal dimension ∆ of when W = 0 and continuous spin representations, the ground state vertex operator VK into the language when W = 0 [18,19,27–29]. Therefore only N = 1 of our momenta k and π state realizes the fixed helicity representations, whereas ◦ 2 − 2 + 2 − 2 the ground state N = 0 and the rest of the excited K K k1 k2 (k mπ) (k mπ) ∆ = = = − states N 2 realize continuous spin representations 2m2 2m2 4m2 4m2 of the massless little group SO(11). The corresponding (k · π) = . (26) vertex operator Uk,π (N = 1) in open strings case is m = ◦ ˙ iK◦Φ = · ˙ + · ˙ ik·X+iπ·Π This clearly confirms the form of the vertex operator Uk,π ζ Φe (ξ Π ω X)e , and its conformal dimension obtained earlier in [21]. (32) 290 G. Savvidy / Physics Letters B 615 (2005) 285–290 where in 26 dimensions K ◦ ζ = 0, K ◦ K = 0 [18] G.K. Savvidy, Phys. Lett. B 552 (2003) 72. or, being translated through our dictionary (26) into [19] G.K. Savvidy, Int. J. Mod. Phys. A 19 (2004) 3171. 13 dimensions, it will take the form: k · ξ(k,π) + [20] I. Antoniadis, G. Savvidy, hep-th/0402077. · = · = [21] L. Alvarez-Gaume, I. Antoniadis, L. Brink, K. Narain, π ω(k,π) 0, k π 0. The Uk,π operators are G. Savvidy, Tensionless strings, vertex operators and scattering of the essential importance, because for them W ∼ amplitudes, Preprint CERN-PH-TH/2004-095 and NRCPS- (k · π)2 = 0, and they create fixed helicity massless HE-2004-13. gauge particles [30–41]. [22] G. Savvidy, hep-th/0409047. [23] A. Nichols, R. Manvelyan, G.K. Savvidy, Mod. Phys. Lett. A 19 (2004) 363. [24] J. Mourad, hep-th/0410009. Acknowledgements [25] H. Minkowski, Math. Ann. B 57 (1903) 447. [26] W. Blaschke, Griechische und Anschauliche Geometrie, Old- In conclusion I would like to thank Luis Alvarez- enbourg, München, 1953; Gaume, Ignatios Antoniadis, Ioannis Bakas, Lars Zenodor, Πρι ισoπριµτρων σχηµατων, 150 B.C. [27] E. Wigner, in: A. Salam (Ed.), Theoretical Physics, Interna- Brink and Kumar Narain for stimulating discussions tional Atomic Energy, Vienna, 1963, p. 59; and CERN Theory Division for hospitality. E. Wigner, Ann. Math. 40 (1939) 149. [28] L. 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Elementary particles, holography and the BMS group

Claudio Dappiaggi

Dipartimento di Fisica Nucleare e Teorica, Università degli Studi di Pavia, INFN, Sezione di Pavia, via A. Bassi 6, I-27100 Pavia, Italy Received 27 January 2005; received in revised form 14 March 2005; accepted 13 April 2005 Available online 22 April 2005 Editor: M. Cveticˇ

Abstract In the context of asymptotically ﬂat space–times, it has been suggested to label elementary particles as unitary irreducible representations of the BMS group. We analyse this idea in the spirit of the holographic principle advocating the use of this deﬁnition.  2005 Elsevier B.V. All rights reserved.

PACS: 04.62.+v; 04.20.Ha; 11.30.Cp; 04.60.-m

The concept of elementary particle plays a cen- P = SL(2, C) T 4, the semidirect product between tral role in the physical interpretation of quantum field the four-dimensional translations T 4 and SL(2, C),the theory; nonetheless we still lack a concrete and univer- group of conformal motions of the 2-sphere S2. Since sally accepted definition whenever gravity is included an elementary system may admit under internal pro- and, thus, a non-trivial background space–time is con- bing a more complex structure, an elementary parti- sidered. The aim of this Letter is to advocate that, in cle is defined as an elementary system whose states the framework of four-dimensional asymptotically flat cannot be physically connected to states of another space–times, a solution to this deficiency exists if the system. Albeit natural, the above definition is unsa- overall problem is set in the context of finding an holo- tisfactory for several reasons [3,4]: (1) in a Poincaré graphic description for a quantum field theory in such invariant theory, the mass operator admits only a con- class of space–times [1]. tinuous spectrum whereas observations show only a As a starting point, let us remember that, thought discrete spectrum of (rest) masses which cannot be de- as [2,3], a system is “elementary” when its Hilbert scribed by any finite-dimensional Lie group [5],(2) space carries a single irreducible representation (ir- by means of the Wigner approach, it is possible to rep.) of (the double cover of ) the full Poincaré group construct all the kinematical and the dynamical data of a Poincaré free field theory, but massless particles may be labelled either by discrete spins with a finite E-mail address: [email protected] (C. Dappiaggi). number of polarizations (unfaithful representations)

0370-2693/$ – see front matter  2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physletb.2005.04.028 292 C. Dappiaggi / Physics Letters B 615 (2005) 291–296 either by continuous spins with an infinite number of irreducible representation of SL(2, C) L2(S2).Al- polarizations (faithful representations); only the for- though this approach experienced an initial success, no mer have been experimentally observed though there further significant progress was achieved in this field is no theoretical reason to prefer any of the above after McCarthy analysis of BMS theory of induced two choices, (3) more importantly, the definition of representations. In [6], it was pointed out that, besides elementary particles assumes the flatness of the back- unitary irrep. related to the observed Poincaré mas- ground discarding any gravitational effect. In a gene- sive and massless fields, a plethora of other elementary ral relativity framework, even in presence of a weak particles existed, so far lacking any experimental evi- gravitational field, this is not a reasonable request: dence. As we have anticipated, considering McCarthy Poincaré invariance is assumed on the basis that the seminal work as a starting point, we will nonetheless underlying manifold is maximally symmetric i.e., it advocate the effectiveness of the whole approach; in is Minkowski whereas, according to Einstein’s theory, particular the unwanted pathologies disappear if we the degree of symmetry of any other bulk space–time interpret the BMS field theory as a boundary theory is smaller. encoding holographically the information from any A candidate solution for the above pathologies can asymptotically flat space–time. be formulated in the context of asymptotically flat Let us briefly comment that holography has been (AF) space–times where a natural and universal coun- introduced in order to solve the apparent paradox in- terpart for the P group exists [4]. In detail, all AF formation of black holes by means of a second theory manifolds share a common boundary structure at past living in a codimension one submanifold (usually the and future null infinity. In a Bondi reference frame boundary) with a density of data not exceeding the (u = t − r, r, θ, ϕ), these submanifolds, ±, topolo- Planck density. An explicit realization of these con- gically equivalent to S2 × R, can be endowed with a cepts consists on constructing a field theory on the degenerate metric boundary of the chosen space–time invariant under the action of the asymptotic symmetry group; the bulk 2 = 2 + 2 + 2 2 ds 0 du dθ sin θdϕ , data are reconstructed starting directly from those as- whose group of diffeomorphisms is the so-called sociated to the boundary, explaining how they generate Bondi–Metzner–Sachs group (BMS) which, up to a their dynamic and how they can reproduce classical stereographic projection sending (θ, ϕ) in (z, z)¯ ,is space–time. A concrete example is known in an AdS manifold as the AdS/CFT correspondence [9] and, → = ¯ + ¯ u u K(z,z) u α(z,z) , (1) only recently, a similar investigation has begun in the az + b context of AF space–times [10]. In this latter scenario, z → z = ,ad− bc = 1 ∧ a,b,c,d ∈ C, ± cz + d the aim is to develop a field theory on invariant (2) under a BMS transformation and, consequently, Mc- where K(z,z)¯ = (1 +|z|2)−1(|az + c|2 +|bz + d|2) Carthy analysis can be naturally interpreted as the ini- and where α(z,z)¯ is an arbitrary real scalar function tial framework where the boundary kinematic data are over S2 [6]. This transformation identifies the BMS as studied and classified. In particular, adopting notations the semidirect product SL(2, C) N, where N is the and nomenclatures as in [6],aBMS covariant wave set of α-functions endowed with a suitable topology function(al) is defined as a map usually, but not necessary, chosen as N = L2(S2), 2 2 n i.e., the collection of square integrable maps over the ψ : L (S ) → C , (3) 2-sphere [7,8]. The universality of the boundary struc- transforming under a BMS unitary representation, i.e., ture and the dual role of the BMS group as diffeomor- in a momentum frame and for any g = (Λ, p(θ, ϕ)) ∈ phism group of ± and as asymptotic symmetry group BMS, of any AF bulk metric naturally suggests to replace the Poincaré group with the BMS as the fundamen- λ i p,α 2 2 −1 D (g)ψ (p ) = e L (S ) Uλ(Λ)ψ Λ p , (4) tal group of symmetry; thus, an elementary particle in an AF space–time is defined by means of an ele- being Uλ(Λ) a unitary representation of SL(2, C) and ip,α mentary system whose Hilbert space carries a unitary e L2(S2) the character associated to p(θ,ϕ).Inor- C. Dappiaggi / Physics Letters B 615 (2005) 291–296 293 der to describe an elementary particle (or equivalently an orthoprojection equation where ρ(p) is a suitable a free field), the associated wave function should trans- non a priori invertible covariant operator which can- form under a unitary and irreducible representation; cels the redundant component of ψ(p) in Cn, i.e., the latter can be constructed from the unitary repre- the image of f is (isomorphic to) Cm. The second sentation of the little (isotropy) groups L ⊂ BMS and is an orbit constraint that reduces the support of (3) 2 2 consequently it is possible to introduce an induced from L (S ) to the coset space OL; although an ex- wave function: plicit expression is available for all little groups [10], we switch for sake of clarity to a specific example: SL(2, C) L = SU(2) where the orbit equation is ψ˜ : O ∼ → Cm,m 0, Γ , the double cover of SO(2), with m2 = 0 m = and a plethora of other non-connected isotropy sub- π(p)µ π plmYlm(θ, ϕ) groups, the most notables being the series of finite l=0 l=−m = alternate, cyclic and dihedral groups An, Cn, Dn with (p00,...,p11). n>2. The connected little groups provide exactly Let us emphasize that, while (7) is the BMS-equivalent the unitary irrep. giving rise to the observed Poincaré of the Klein–Gordon equation which holds for any spins; as a direct consequence, the arbitrariness in the Poincaré-covariant elementary particle, (6) is a com- choice of the irrep. associated to massless particles pact expression for the wave equations of any BMS disappears since the faithful one-dimensional repre- free field, i.e., they are the BMS counterpart for usual sentation proper of the BMS Γ little group is fully formulas such as the Dirac and the Proca wave equa- equivalent to its Poincaré counterpart induced from the tions. Thus, following this line of reasoning, the pair ⊂ two-dimensional Euclidean group E(2) P . {Dλ(Λ), ρ(p)} (from (4) and (6)) completely charac- The main handicap emerging from the analysis of terise the dynamic of a free field; each (BMS) elemen- the kinematic data is the total absence of an inter- tary particle is distinguished from another only by the pretation for the additional “non-Poincaré” degrees of values of the squared mass and of the spin. freedom. The paradigm we propose is the following: Since, according to the holographic principle, the if a BMS field theory encodes the data from all AF boundary theory should encode the bulk degrees of manifolds, an elementary particle, living in a fixed freedom, a comparison, between the classical dynamic background, such as, for example, Minkowski space– of a theory living on ± and of one living on a time, is described only by means of those boundary flat background, should be performed at a level of degrees of freedom allowing a proper reconstruction phase spaces. The subtlety lies in the intrinsic infinite- of the chosen bulk manifold. dimensional nature of the BMS field theory which In order to support such conjecture, the first step prevents a canonical approach to the construction of is to compare the dynamic of bulk and boundary free the phase space since the usual splitting of a four- fields. In the context of Wigner approach, the latter can 4 dimensional manifold M as 3 × R is meaningless be fully characterized as a set of constraints restric- in the boundary framework. Thus we introduce the ting the covariant wave function to the induced one; covariant phase space, the set of covariant wave func- in particular, in a BMS setting, starting from (3), these tion(al)s satisfying the equations of motion and, con- constraints are twofolds: the first is sequently, representing the dynamically allowed configurations; in the specific example of a BMS SU(2) ρ(p)ψ(p) = ψ(p), (6) field it is 294 C. Dappiaggi / Physics Letters B 615 (2005) 291–296 (cov) = 2 2 → R ΓBMS ψ : L (S ) , Within this framework, the set of cut functions appears to play a role similar to the Fefferman–Graham π(p)νπ(p) − m2 ψ(p)= 0, ν construction for an asymptotically AdS space–time p − π(p) ψ(p)= 0, (see, for example, [13,14]); this latter tool allows for

ρ(p)ψ(p) = ψ(p) . (8) an algebraic reconstruction of bulk data starting from boundary ones whereas the counterpart of this ap- The Poincaré counterpart of this expression for an proach in an asymptotically flat space–time produces SU(2) field is: a set of differential equations [15]. Conversely, the null surface formulation of general relativity and, more (cov) Γ = ψ : T 4 → R, pµp − m2 ψ(p)= 0, in detail, Z (z, z)¯ allows in an asymptotically flat P µ xa ρ pµ ψ pµ = ψ pµ . (9) space–time a reconstruction of bulk geometry (in particular, the metric) starting only from data living on It is straightforward to realize that, due to the or- + (or −) solving a set of algebraic equations. Fur- bit constraint [p − π(p)]ψ(p) = 0, (8) is in 1:1 thermore, also bulk fields on M4 can be seen as “de- correspondence with (9); furthermore, an identical pendent” only upon boundary data since, starting only claim holds between the covariant phase space of a from the cut functions, it is possible to construct the Poincaré E(2) massless field and a BMS Γ mass- following tetrad Θi living at null infinity: less particle with vanishing pure supertranslational u = Z(x ,z,z),¯ component [11]. In an “holographic” language, this a result grants us that the boundary theory fully en- ω = 1 +|z|2 ∂Z(x ,z,z),¯ a codes the bulk classical degrees of freedom (at least 2 ¯ ω¯ = 1 +|z| ∂Z(xa,z,z),¯ in Minkowski); conversely, from an “elementary parti- 2 2 ¯ cle” point of view, the results from [4] are considerably R = 1 +|z| ∂∂Z(xa,z,z),¯ improved since, not only the kinematic but also the dy- which can be (in principle) inverted as x = x [Θi, namic of massive and massless Poincaré elementary a a z, z¯]. Thus each local bulk field φλ : M4 → Cλ can be particles is fully reproduced in a BMS invariant theory. now rewritten as a functional of boundary data, i.e., As a final step, we need to provide evidences that φλ(x ) = φλ[Θi,z,z¯]. all other BMS little group do not encode any infor- a In particular, if we now consider the specific exam- mation allowing a full reconstruction of the physics ple of a Minkowski background and if we work in a and the geometry of a Minkowski space–time. A so- momentum frame, the cut function is unique [16]: lution to this obstacle lies in the so-called null surface formulation of general relativity. In this approach to = = a Zpa (θ, ϕ) p(θ,ϕ) pal (θ, ϕ), (10) Einstein’s theory, the main variable is a scalar function a ={ } Z : M4 × S2 → R (cut function) solution of the light where l Y00(θ, ϕ), . . . , Y11(θ, ϕ) . At a classical 4 level, (10) grants us that the momenta encoding the cone equation in M [12]; Z(xa,θ,ϕ) allows to uni- vocally reconstruct all the conformal data of the bulk information from a flat manifold automatically satisfy manifold and, in particular, up to a conformal rescaling a vanishing pure supertranslational constraint the metric itself. From an holographic perspective, the Zp − π(Zp ) = 0, i.e., p − π(p) = 0. (11) appealing aspect of the overall procedure arises reali- a a 4 zing that, helding fixed the bulk point xa ∈ M , the cut Thus a BMS elementary particle can be related to function is a real scalar map on ±, thus a boundary a Poincaré invariant counterpart living in Minkowski only if the equation of motion for the associated co- data. Moreover, since Zxa is smooth and single-valued in a suitable neighbourhood of ±, it can be naturally variant wave functional (8) includes (11).Forafixed identified as a BMS supertranslation. Thus, in a BMS little group L, the orbit equation imposes to the clas- field theory, the collection of data encoding the free sical free field an evolution on a finite-dimensional fields dynamic on a fixed background, can be extracted manifold embedded in L2(S2); the latter is generated C by the action of the coset group SL(2, ) on a fixed point from the degrees of freedom Zxa reconstructing a L specific manifold in the null surface formalism. p¯ ∈ L2(S2) such that Lp¯ =¯p. A decomposition in C. Dappiaggi / Physics Letters B 615 (2005) 291–296 295 spherical harmonics proves that the most general ex- partition function is pression for p¯ is [6,10] − Z = d[φ]eiS(φ) = const · det[B] 1/2, (15) l p¯ = m + p Y (θ, ϕ), (12) C lm lm ∞ l>1 m=−l = µν + 2 + 1 − B η Deµ Deν m (Qei Dei )Dei , l 2ζi i=1 p¯ = p0 + p0Y11(θ, ϕ) + plmYlm(θ, ϕ), (16) l>1 m=−l where ζ is an arbitrary real non-vanishing number (13) i and where Qei is the infinite-dimensional multiplica- respectively, for a massive and a massless field. From tion operator along the direction ei . The propagator the above two formulas, it is straightforward to see that G(x1 − x2) can be calculated as (11) is equivalent to the constraint plm = 0 for any l>1. A detailed analysis [10,11], proved that this re- BG(x1 − x2) = iδ(x1 − x2). (17) quest may be satisfied only by the connected isotropy Up to a Fourier transform, (17) satisfies: subgroups of the BMS group, i.e., SU(2), if we con- ∞ sider (12), and Γ if we consider (13); thus, we may µν − 2 + − 2 conclude that, at least at a classical level, only these η pµpν m piDei pi two BMS subgroups encode the information from a i=1 Minkowski elementary particle, discarding any physi- × G p(θ,ϕ) = i, (18) cal role for the other “pathological” little groups. where pµ and pi are the projections of p(θ,ϕ),re- In order to better clarify the role of the BMS group 2 2 spectively, along the directions eµ and ei ∈ L (S ). in the definition of elementary particles, we need to A physical analysis of (18) has been performed in [17], comment on the quantum aspects of the boundary the- but, in this Letter, we wish simply to emphasize the ory. In the above framework, we focused our atten- relation of the above formula with the flat counterpart tion mainly on the dynamically allowed configurations i.e., if we take into account (10) as the set of possible whereas, if we wish to calculate quantum data, such as values of p(θ,ϕ), (18) reduces to correlation functions, by means of path-integral tech- i niques, we should refer to all the kinematically al- G(p) = , µν 2 lowed configurations. The latter are a priori different η pµpν − m in a Poincaré and in a BMS field theory and it is which is the 2-point function in a Minkowski back- natural to wonder if the conjectured correspondence ground. Thus, this result suggests us that the conjec- holds also at this level. Thus we need to switch to ture to holographically describe Poincaré elementary a Lagrangian formalism; if we consider for sake of particles by means of the BMS group should hold also simplicity a BMS scalar field φ(x), the equation of at a quantum level. motion (7) can be derived minimizing the following To conclude this Letter, we wish to emphasize some action [17], remarks on the overall approach:

= µν − 2 • In a general picture, elementary particles may also S(φ) dµ φ(x) η Deµ Deν m φ(x) be characterized by an additional set of quan- L2(S2) tum number {σ } associated to internal degrees of ∞ freedom usually described by means of a (gauge) + γ (x)D φ(x) , (14) i ei Lie group G. Nonetheless, the indices {σ } act as = i 1 absolute superselection rules, i.e., external inter- being eµ an element of the set {Y00(θ,ϕ),..., actions can only modify the momentum and the } { } Y11(θ, ϕ) , ei one of the set Ylm(θ, ϕ) l>1, Dei the spin projection along a ﬁxed direction. The sug- inﬁnite-dimensional directional derivative along ei gestion in [18] to relate these degrees of freedom and γ (x) a Lagrange multiplier. The corresponding to the irrep. of I = BMS , does not seem to hold i T 4 296 C. Dappiaggi / Physics Letters B 615 (2005) 291–296

in an holographic framework; on the contrary, the boundary theory in [17], the leading role played faithful irrep. of I label the so-called IR-sectors by cut functions in the bulk reconstruction starting of gravity [11]. In a few words, the presence of from boundary data, appears to hold even in pre- different infrared sectors of the gravitational field sence of interactions. A tricky issue arises if one is a measure of the arbitrariness induced by the wishes to consider boundary gauge theory since BMS group in the choice of a specific Minkowski the usual construction coupling gauge fields and space–time describing the underlying geometry of elementary particles, well explained in [20], can- a bulk field approaching ±. This specific de- not be blindly applied in the infinite-dimensional gree of freedom is related to pure supertransla- context proper of a BMS setting; thus this issue is tions and, consequently, to the I group; it repre- still under analysis and development. sents a direct consequence of the obstruction to reduce the BMS to the Poincaré group, thus it has no reference with internal labels of an elementary Acknowledgements particle. • The absence of a physical interpretation for the The author is in debt with Mauro Carfora and Gio- non-connected little groups disappears in a generic vanni Arcioni for useful discussions and comments. (non-stationary) background. Conversely, they may carry information from specific bulk data and an example is provided by the discrete isotropy subgroups, related to gravitational instantons (see References [19] and references therein). • Further question concerns the application of the [1] G. ’t Hooft, gr-qc/9310026. [2] S. Weinberg, The Quantum Theory of Fields, vol. 1: Founda- hypothesis proposed in this Letter in a scenario tions, Cambridge Univ. Press, Cambridge, 1995. with a non-vanishing cosmological constant and [3] A.O. Barut, R. Raczka, Theory of Group Representations and in particular in the AdSd /CFTd−1 (d>3) corre- Applications, Word Scientific, Singapore, 1986. spondence. Let us briefly comment that, though [4] P.J.M. McCarthy, Phys. Rev. Lett. 29 (1972) 817. completely different from its asymptotically flat [5] L. O’Reifeartaigh, Phys. Rev. Lett. 14 (1965) 575. [6] P.J. McCarthy, Proc. R. Soc. London A 330 (517) (1972); counterpart (formulated only in 4D), in an AdS P.J. McCarthy, Proc. R. Soc. London A 333 (317) (1973); manifold holography relies to a certain extent on P.J. McCarthy, Proc. R. Soc. London A 335 (301) (1973). the equivalence between the bulk and the bound- [7] L. Girardello, G. Parravicini, Phys. Rev. Lett. 32 (1974) 565. ary (finite-dimensional) symmetry group; thus, a [8] M. Crampin, P.J.M. McCarthy, Phys. Rev. Lett. 33 (1974) 547. priori, there is no specific reason that do not al- [9] J.M. Maldacena, Adv. Theor. Math. Phys 2 (1998) 231. [10] G. Arcioni, C. Dappiaggi, Nucl. Phys. B 674 (2003) 553. lows to repeat the reasoning of this Letter in such [11] G. Arcioni, C. Dappiaggi, Class. Quantum Grav. 21 (2004) framework though a detailed analysis is not yet 5665. available. [12] S. Frittelli, C. Kozameh, E.T. Newman, J. Math. Phys. 36 • The role of interactions both in bulk and in the (1995) 4904. boundary has been discarded in this Letter since [13] K. Skenderis, Class. Quantum Grav. 19 (2002) 5849. [14] S. de Haro, S.N. Solodukhin, K. Skenderis, Commun. Math. our aim has been to develop an alternative defi- Phys. 217 (2001) 595. nition of elementary particles which are related [15] S. de Haro, K. Skenderis, S.N. Solodukhin, Class. Quantum to free fields. Nonetheless, in the spirit of fin- Grav. 18 (2001) 3171. ding an holographic correspondence in asympto- [16] C.N. Kozameh, E.T. Newman, J. Math. Phys. 24 (1983) 2481. tically flat space–times, it is imperative to under- [17] C. Dappiaggi, JHEP 0411 (2004) 011. [18] A. Komar, Phys. Rev. Lett. 15 (1965) 76. stand the role of interactions between BMS fields [19] E. Melas, J. Math. Phys. 45 (2004) 996. and whether they may “break the holographic ma- [20] A. Weinstein, Lett. Math. Phys. 2 (1978) 417. chinery”. According to the initial analysis of the Physics Letters B 615 (2005) 297 www.elsevier.com/locate/physletb Erratum Erratum to: “Meissner masses in the gCFL phase of QCD” [Phys. Lett. B 605 (2005) 362]

R. Casalbuoni a,b, R. Gatto c, M. Mannarelli d, G. Nardulli e,f, M. Ruggieri e,f

a Dipartimento di Fisica, Università di Firenze, I-50125 Firenze, Italy b INFN, Sezione di Firenze, I-50125 Firenze, Italy c Département de Physique Théorique, Université de Genève, CH-1211 Genève 4, Switzerland d Cyclotron Institute and Physics Department, Texas A&M University, College Station, TX 77843-3366, USA e Università di Bari, I-70126 Bari, Italy f INFN, Sezione di Bari, I-70126 Bari, Italy Received 13 April 2005 Available online 20 April 2005

In the abstract substitute a = 1, 2 with a = 1, 2, 3 and 8. After Eq. (29) substitute “On the other hand we note that gluons of colors a = 3 and a = 8 are continuous 2 = positive functions of Ms /µb both in the CFL and gCFL phase.” with “Also the gluons a 3, 8 show this behavior.” In the caption of Fig. 1 substitute “The short-dashed line denotes the gluons with color a = 3 and the long- = 2 − 2 2 dashed line a 8.” with “The short-dashed line denotes the function (m1(x) m3(x))/2mM , and the long-dashed + 2 − 2 2 one denotes 2 (m1(x) m8(x))/2mM .” In the conclusions substitute “gluons of color indices 1 and 2” with “gluons of color indices 1, 2, 3 and 8”.

Acknowledgements

We would like to thank Dr. Fukushima for his very useful comments.

DOI of original article: 10.1016/j.physletb.2004.11.045. E-mail address: casalbuoni@ﬁ.infn.it (R. Casalbuoni).

Cumulative author index to volumes 611–615

Abbaneo, D., 611, 66; 614,7 Amoroso, A., 612, 154 Abdalla, E., 611,21 Amsler, C., 615, 153 Abdalla, M.C.B., 613, 213 Anderhub, H., 613, 118; 615,19 Abe, K., 613, 20, 20; 614, 27, 27; 615, 39, 39 Anderson, B.D., 612, 181 Ablikim, M., 614,37 Anderson, J.D., 613,11 Abuki, H., 615, 102 Andreev, V.P., 613, 118; 615,19 Achard, P., 613, 118; 615,19 Andronic, A., 612, 173 Ackerman, L., 611,53 Anselmo, F., 613, 118; 615,19 Adachi, I., 614,27 Antonelli, A., 611, 66; 614,7 Adams, J., 612, 181 Antonelli, M., 611, 66; 614,7 Adler, C., 612, 181 Anzivino, G., 615,31 Adriani, O., 613, 118; 615,19 Aoki, S., 613, 105; 614, 155 Ageev, E.S., 612, 154 Arcidiacono, R., 615,31 Aggarwal, M.M., 612, 181 Areﬁev, A., 613, 118; 615,19 Aguilar-Benitez, M., 613, 118; 615,19 Arhrib, A., 612, 263 Aguilar-Saavedra, J.A., 613, 170 Arkhipkin, D., 612, 181 Aguirre, R., 611, 248 Arleo, F., 614,44 Ahammed, Z., 612, 181 Armstrong, S.R., 611, 66; 614,7 Aichelin, J., 612, 201 Artamonov, A., 613, 105; 614, 155 Aihara, H., 613, 20; 614, 27; 615,39 Asahi, K., 615, 186 Ajaltouni, Z.J., 614, 165 Asano, Y., 613, 20; 614, 27; 615,39 Akaishi, Y., 613, 140 Assunção, M., 615, 167 Akatsu, M., 614, 27; 615,39 Aulchenko, V., 615,39 Akhmetshin, R.R., 613,29 Aulchenko, V.M., 613,29 Alberg, M., 611, 111 Aushev, T., 613, 20; 614, 27; 615,39 Alcaraz, J., 613, 118; 615,19 Avelino, P.P., 611,15 Alemanni, G., 613, 118; 615,19 Avenier, M., 615, 153 ALEPH Collaboration, 611, 66; 614,7 Averett, T., 613, 148 Alexakhin, V.Yu., 612, 154 Averichev, G.S., 612, 181 Alexandrov, Yu., 612, 154 Awunor, O., 611, 66; 614,7 Alexandru, A., 612,21 Azemoon, T., 613, 118; 615,19 Alexeev, G.D., 612, 154 Aziz, T., 613, 118; 615,19 Alkofer, R., 611, 279 Azzurri, P., 611, 66; 614,7 Allaby, J., 613, 118; 615,19 Aloisio, A., 613, 118; 615,19 Baaquie, B.E., 615, 134 ALPHA Collaboration, 612, 313 Baba, H., 614, 174 Alviggi, M.G., 613, 118; 615,19 Babichev, E., 614,1 Amonett, J., 612, 181 Badaud, F., 611, 66; 614,7

0370-2693/2005 Published by Elsevier B.V. doi:10.1016/S0370-2693(05)00622-2 Cumulative author index to volumes 611–615 (2005) 298–321 299

Badełek, B., 612, 154 Belaga, V.V., 612, 181 Badyal, S.K., 612, 181 Belle Collaboration, 613, 20; 614, 27; 615,39 Bagliesi, G., 611, 66; 614,7 Bellucci, L., 613, 118; 615,19 Bagnaia, P., 613, 118; 615,19 Bellucci, S., 612, 283 Bahinipati, S., 613, 20; 615,39 Bellwied, R., 612, 181 Bai, J.Z., 614,37 Benabderrahmane, L., 612, 173 Bajo, A., 613, 118; 615,19 Bencivenni, G., 611, 66; 614,7 Bakich, A.M., 613, 20; 614, 27; 615,39 Bender, C.M., 613,97 Baksay, G., 613, 118; 615,19 Benzoni, G., 615, 160 Baksay, L., 613, 118; 615,19 Berbeco, R., 613, 118; 615,19 Balantekin, A.B., 613,61 Berdugo, J., 613, 118; 615,19 Baldew, S.V., 613, 118; 615,19 Berek, G., 612, 173 Balestra, F., 612, 154 Berezinsky, V., 612, 147 Balewski, J., 612, 181 Berger, J., 612, 181 Ball, J., 612, 154 Berges, P., 613, 118; 615,19 Ban, Y., 614, 27, 37; 615,39 Berglund, P., 612, 154 Bandos, I.A., 615, 127 Berkelman, K., 611, 66; 614,7 Banerjee, S., 611, 27; 613, 118; 614, 27; 615, 19, 39 Bernabéu, J., 613, 162 Banerjee, Sw., 613, 118; 615,19 Bernet, C., 612, 154 Banzarov, V.Sh., 613,29 Bertini, R., 612, 154 Barannikova, O., 612, 181 Bertucci, B., 613, 118; 615,19 Barate, R., 611, 66; 614,7 BES Collaboration, 614,37 Baratt, A., 613,29 Betev, B.L., 613, 118; 615,19 Barbuto, E., 613, 105; 614, 155 Beuselinck, R., 611, 66; 614,7 Barczyk, A., 613, 118; 615,19 Bevan, A., 615,31 Barea, J., 613, 134 Bezverkhny, B.I., 612, 181 Barger, V., 613, 61; 614,67 Bhardwaj, S., 612, 181 Barillère, R., 613, 118; 615,19 Bhati, A.K., 612, 181 Barklow, T., 611, 66; 614,7 Bhattacharyya, A., 611,27 Barkov, L.M., 613,29 Bian, J.G., 614,37 Barnby, L.S., 612, 181 Biasini, M., 613, 118; 615,19 Barr, G., 615,31 Bichsel, H., 612, 181 Barret, V., 612, 173 Bidder, S.J., 612,75 Bartalini, P., 613, 118; 615,19 Biglietti, M., 613, 118; 615,19 Bashtovoy, N.S., 613,29 Biino, C., 615,31 Basile, M., 613, 118; 615,19 Biland, A., 613, 118; 615,19 Basrak, Z., 612, 173 Billmeier, A., 612, 181 Bastid, N., 612, 173 Binétruy, P., 611,39 Batalova, N., 613, 118; 615,19 Birsa, R., 612, 154 Battiston, R., 613, 118; 615,19 Bisplinghoff, J., 612, 154 Battistoni, G., 615,14 Bissegger, M., 613,57 Baudot, J., 612, 181 Biswas, A., 613, 208 Bauer, C.W., 611,53 Bitenc, U., 613, 20; 614, 27; 615,39 Baum, G., 612, 154 Bizjak, I., 613, 20; 614, 27; 615,39 Bäumer, C., 612, 165 Bizzeti, A., 615,31 Bay, A., 613, 118; 614, 27; 615,19 Bjerrum-Bohr, N.E.J., 612,75 Becattini, F., 613, 118; 615,19 Blair, G.A., 611, 66; 614,7 Becherini, Y., 615,14 Blaising, J.J., 613, 118; 615,19 Becker, H.G., 615,31 Blaizot, J.-P., 615, 221 Becker, U., 613, 118; 615,19 Bland, L.C., 612, 181 Bedfer, Y., 612, 154 Blasi, N., 615, 160 Bednarczyk, P., 615, 160 Bleicher, M., 612, 201 Bedny, I., 613, 20; 614, 27; 615,39 Bloch-Devaux, B., 611, 66; 614,7 Behner, F., 613, 118; 615,19 Blondel, A., 611, 66; 614,7 Bekele, S., 612, 181 Blumenfeld, Y., 613, 128 300 Cumulative author index to volumes 611–615 (2005) 298–321

Blumenschein, U., 611, 66; 614,7 Buontempo, S., 613, 105; 614, 155 Blümlein, J., 614,53 Burger, J.D., 613, 118; 615,19 Blyth, C.O., 612, 181 Burger, W.J., 613, 118; 615,19 Blyth, S., 613, 20; 614, 27; 615,39 Burtin, E., 612, 154 Blyth, S.C., 613, 118; 615,19 Bussa, M.P., 612, 154 Bobbink, G.J., 613, 118; 615,19 Busto, J., 615, 153 Boccali, T., 611, 66; 614,7 Butter, D., 612, 304 Bocquet, G., 615,31 Bytchkov, V.N., 612, 154 Böhm, A., 613, 118; 615,19 Bzdak, A., 615, 240 Böhrer, A., 611, 66; 614,7 Boldizsar, L., 613, 118; 615,19 Cachazo, F., 611, 167 Bondar, A., 613, 20; 614, 27; 615,39 Cadman, R.V., 612, 181 Bondar, A.E., 612, 215; 613,29 Cai, X., 614,37 Bondarev, D.V., 613,29 Cai, X.D., 613, 118; 615,19 Bonissent, A., 611, 66; 614,7 Cai, X.Z., 612, 181 Bonner, B.E., 612, 181 Caines, H., 612, 181 Booth, C.N., 611, 66; 614,7 Calderón de la Barca Sánchez, M., 612, 181 Bordalo, P., 612, 154 Callot, O., 611, 66; 614,7 Borean, C., 611, 66; 614,7 Calvetti, M., 615,31 Borgia, B., 613, 118; 615,19 Camera, F., 615, 160 Bossi, F., 611, 66; 614,7 Cameron, W., 611, 66; 614,7 Botje, M., 612, 181 Canfora, F., 614, 131 Bottai, S., 613, 118; 615,19 Capell, M., 613, 118; 615,19 Boucham, A., 612, 181 Caplar,ˇ R., 612, 173 Boucrot, J., 611, 66; 614,7 Capon, G., 611, 66; 614,7 Bouhova-Thacker, E., 611, 66; 614,7 Cara Romeo, G., 613, 118; 615,19 Boumediene, D., 611, 66; 614,7 Carlino, G., 613, 118; 615,19 Bourilkov, D., 613, 118; 615,19 Carroll, J., 612, 181 Bourquin, M., 613, 118; 615,19 Cartacci, A., 613, 118; 615,19 Bowdery, C.K., 611, 66; 614,7 Cartiglia, N., 615,31 Bozek, A., 613, 20; 614, 27; 615,39 Cartwright, S., 611, 66; 614,7 Bozza, C., 613, 105; 614, 155 Casado, M.P., 611, 66; 614,7 Braccini, S., 613, 118; 615,19 Casalbuoni, R., 615, 297 Bracco, A., 615, 160 Casali, R., 615,31 Bracko,ˇ M., 613, 20; 614, 27; 615,39 Casaus, J., 613, 118; 615,19 Bradamante, F., 612, 154 Castillo, J., 612, 181 Bragin, A.V., 613,29 Castoldi, M., 615, 160 Branco, G.C., 614, 187 Catanesi, M.G., 613, 105; 614, 155 Brandenburg, A., 615,68 Cattaneo, M., 611, 66; 614,7 Brandin, A., 612, 181 Cavallari, F., 613, 118; 615,19 Brandt, S., 611, 66; 614,7 Cavallo, N., 613, 118; 615,19 Branson, J.G., 613, 118; 615,19 Cavanaugh, R., 611, 66; 614,7 Bravar, A., 612, 154, 181 Cavero-Pelaez, I., 613,97 Bravo, S., 611, 66; 614,7 Cebra, D., 612, 181 Bressan, A., 612, 154 Cecchi, C., 613, 118; 615,19 Brient, J.-C., 611, 66; 614,7 Cecchini, S., 615,14 Brihaye, Y., 615,1 Ceccucci, A., 615,31 Britto, R., 611, 167 Cenci, P., 615,31 Brochu, F., 613, 118; 615,19 Cerini, L., 612, 154 Brodzicka, J., 613, 20; 614, 27; 615,39 Cerna, C., 615, 153 Broggini, C., 615, 153 Cerrada, M., 613, 118; 615,19 Browder, T.E., 613, 20; 614,27 Cerri, C., 615,31 Brunelière, R., 611, 66; 614,7 Cerutti, F., 611, 66; 614,7 Bruski, N., 613, 105; 614, 155 Chaloupka, P., 612, 181 Buchmüller, O., 611, 66; 614,7 Chamizo, M., 613, 118; 615,19 Cumulative author index to volumes 611–615 (2005) 298–321 301

Chan, C.-T., 611, 193 Cirilli, M., 615,31 Chandrasekhar, B., 614, 207 Ciulli, V., 611, 66; 614,7 Chang, J.F., 614,37 Clare, I., 613, 118; 615,19 Chang, M.-C., 613,20 Clare, R., 613, 118; 615,19 Chang, P., 613,20 Clarke, D.P., 611, 66; 614,7 Chang, Y.H., 613, 118; 615,19 Clerbaux, B., 611, 66; 614,7 Chao, Y., 614,27 Cleymans, J., 615,50 Chapiro, A., 612, 154 Clifft, R.W., 611, 66; 614,7 Chattopadhyay, S., 612, 181 Cocco, A.G., 613, 105; 614, 155 Chbihi, A., 613, 128 Cofﬁn, J.P., 612, 181 Chemarin, M., 613, 118; 615,19 Cogan, J., 615,31 Chen, A., 613, 20, 118; 614, 27; 615, 19, 39 Coignet, G., 613, 118; 615,19 Chen, C.-M., 611, 156 Colaleo, A., 611, 66; 614,7 Chen, G., 613, 118; 615,19 Colantoni, M., 612, 154 Chen, G.M., 613, 118; 615,19 Colas, P., 611, 66; 614,7 Chen, H.F., 612, 181; 613, 118; 614, 37; 615,19 Colavita, A.A., 612, 154 Chen, H.S., 613, 118; 614, 37; 615,19 Cole, S., 614,27 Chen, H.X., 614,37 Colino, N., 613, 118; 615,19 Chen, J., 614, 37, 37 Collazuol, G., 615,31 Chen, J.-P., 613, 148 Combley, F., 611, 66; 614,7 Chen, J.C., 614,37 COMPASS Collaboration, 612, 154 Chen, K.-F., 614,27 Contalbrigo, M., 615,31 Chen, M.L., 614,37 Conte, E., 614, 165 Chen, S.-L., 612,29 Cordier, E., 612, 173 Chen, W.T., 613, 20; 614,27 Cormier, T.M., 612, 181 Chen, Y., 612, 21, 181 Costa, S., 612, 154 Chen, Y.B., 614,37 Costantini, F., 615,31 Cheon, B.G., 613, 20; 614, 27; 615,39 Costantini, S., 613, 118; 615,19 Chernenko, S.P., 612, 181 Cowan, G., 611, 66; 614,7 Cherney, M., 612, 181 Coyle, P., 611, 66; 614,7 Chernyak, V.L., 612, 215 Cozzi, M., 615,14 Cheshkov, C., 615,31 Cramer, J.G., 612, 181 Cheze, J.B., 615,31 Cranmer, K., 611, 66; 614,7 Chi, S.P., 614,37 Crawford, H.J., 612, 181 Chiarella, V., 611, 66; 614,7 Creanza, D., 611, 66; 614,7 Chiefari, G., 613, 118; 615,19 Crespo, J.M., 611, 66; 614,7 Chikanian, A., 612, 181 Crespo, M.L., 612, 154 Chikawa, M., 613, 105; 614, 155 Crochet, P., 612, 173 Chimento, L.P., 615, 146 Csatlós, M., 615, 175 Chistov, R., 613, 20; 614, 27; 615,39 Cucciarelli, S., 613, 118; 615,19 Chmeissani, M., 611, 66; 614,7 Cugnon, J., 614,44 Choi, S., 613, 148 Cuhadar, T., 615,31 Choi, S.-K., 614,27 Cui, X.Z., 614,37 Choi, Y., 613, 20; 614, 27; 615,39 Cundy, D., 615,31 Choi, Y.K., 613, 20; 614, 27; 615,39 Curien, D., 615, 160 Chollet, J.C., 615,31 Curtil, C., 611, 66; 614,7 Chomaz, P., 613, 128 Czosnyka, P., 615,55 Chong, Z.-W., 614,96 Czosnyka, T., 615,55 CHORUS Collaboration, 613, 105; 614, 155 Czy˙z, H., 611, 116 Christie, W., 612, 181 Chu, Y.P., 614,37 D’Agostini, G., 615,31 Chuvikov, A., 613, 20; 614, 27; 615,39 Dai, H.L., 614,37 Cicuttin, A., 612, 154 Dai, Y.S., 614,37 Cifarelli, L., 613, 118; 615,19 Dalla Torre, S., 612, 154 Cindolo, F., 613, 118; 615,19 Dalpiaz, P., 615,31 302 Cumulative author index to volumes 611–615 (2005) 298–321

Dalseno, J., 613, 20; 614, 27; 615,39 Derevschikov, A.A., 612, 181 D’Ambrosio, N., 613, 105; 614, 155 De Rosa, G., 613, 105; 614, 155 Danilov, M., 614, 27; 615,39 De Salvo, A., 613, 118; 615,19 Dappiaggi, C., 615, 291 De Schauenburg, B., 612, 173 Darabi, F., 615, 141 De Séréville, N., 615, 167 Daraktchieva, Z., 615, 153 Desesquelles, P., 613, 128 Darwish, E.M., 615,61 Deshpande, N.G., 615, 111 Das, D., 612, 181 Dessagne, S., 611, 66; 614,7 Das, S., 612, 181 Deur, A., 613, 148 Dasgupta, S.S., 612, 154 Dey, T.K., 613, 208 Dash, M., 613, 20; 614, 27; 615,39 Dhamotharan, S., 611, 66; 614,7 David, A., 611, 66; 614,7 Dhara, L., 612, 154 Davids, B., 615, 167 D’Hose, N., 612, 154 Davidson, P.M., 611,81 Diaz Kavka, V., 612, 154 Davier, M., 611, 66; 614,7 Dibon, H., 615,31 Davies, A.D., 611,81 Di Capua, E., 613, 105; 614, 155 Davies, G., 611, 66; 614,7 Di Capua, F., 613, 105; 614, 155 Davis, A.-C., 611,39 Didenko, L., 612, 181 Davis, S.C., 611,39 Diemoz, M., 613, 118; 615,19 De Angelis, G., 615, 160 Dierckxsens, M., 613, 118; 615,19 De Asmundis, R., 613, 118; 615,19 Dietel, T., 612, 181 D’Eath, P.D., 613, 181 Dietl, H., 611, 66; 614,7 De Azcárraga, J.A., 615, 127 Dinkelbach, A.M., 612, 154 De Beer, M., 615,31 Dionisi, C., 613, 118; 615,19 De Bonis, I., 611, 66; 614,7 Dissertori, G., 611, 66; 614,7 Debreczeni, J., 613, 118; 615,19 Dittmaier, S., 612, 223 Debu, P., 615,31 Dittmar, M., 613, 118; 615,19 Decamp, D., 611, 66; 614,7 Doble, N., 615,31 Dedek, N., 612, 154 Dolgopolov, A.V., 612, 154 De Filippis, N., 611, 66; 614,7 Dombrádi, Zs., 614, 174 Déglon, P., 613, 118; 615,19 Dong, L.Y., 614,37 Degré, A., 613, 118; 615,19 Dong, S.J., 612,21 Dehmelt, K., 613, 118; 615,19 Dong, W.J., 612, 181 De Huu, M.A., 612, 165 Dong, X., 612, 181 Deiters, K., 613, 118; 615,19 Donnachie, A., 611, 255 De Jager, C.W., 613, 148 Donoghue, J.F., 612, 311 De Jong, M., 613, 105; 614, 155 Donskov, S.V., 612, 154 De Jong, P., 613, 118; 615,19 Dore, U., 613, 105; 614, 155 Dekhissi, H., 615,14 Doria, A., 613, 118; 615,19 De la Cruz, B., 613, 118; 615,19 Dornan, P.J., 611, 66; 614,7 Delaere, C., 611, 66; 614,7 Dorofeev, V.A., 612, 154 Del Aguila, F., 613, 170 Dosanjh, R.S., 615,31 Delbar, T., 613, 105; 614, 155 Doshita, N., 612, 154 De Lellis, G., 613, 105; 614, 155 Døssing, T., 615, 160 Della Morte, M., 612, 313 Doté, A., 613, 140 Della Volpe, D., 613, 118; 615,19 Dotsenko, V.S., 611, 189 Delmeire, E., 613, 118; 615,19 Dova, M.T., 613, 118; 615,19 De Masi, R., 612, 154 Draper, J.E., 612, 181 De Moura, M.M., 612, 181 Draper, T., 612,21 Denes, P., 613, 118; 615,19 Drevermann, H., 611, 66; 614,7 Deng, Z.Y., 614,37 Drutskoy, A., 614, 27; 615,39 Denisov, O.Yu., 612, 154 Du, F., 612, 181 Denner, A., 612, 223 Du, S.X., 614,37 DeNotaristefani, F., 613, 118; 615,19 Du, Z.Z., 614,37 De Palma, M., 611, 66; 614,7 Dubey, A.K., 612, 181 Cumulative author index to volumes 611–615 (2005) 298–321 303

Duchesneau, D., 613, 118; 615,19 Faivre, J., 612, 181 Duclos, J., 615,31 Falagan, M.A., 613, 118; 615,19 Duda, M., 613, 118; 615,19 Falaleev, V., 612, 154; 615,31 Duﬂot, L., 611, 66; 614,7 Falciano, S., 613, 118; 615,19 Duic, V., 612, 154 Fallot, M., 613, 128 Dunbar, D.C., 612,75 Falvard, A., 611, 66; 614,7 Dunin, V.B., 612, 181 Fang, J., 614,37 Dunlop, J.C., 612, 181 Fang, S.S., 614,37 Dünnweber, W., 612, 154 Fantechi, R., 615,31 Dupieux, P., 612, 173 Farley, A.N.St.J., 613, 181 Dürr, S., 612, 313 Fatemi, R., 612, 181 Durrer, R., 614, 125 Fauland, P., 612, 154 Dutta Majumdar, M.R., 612, 181 Favara, A., 613, 118; 615,19 Dželalija, M., 612, 173 Favart, D., 613, 105; 614, 155 Fay, J., 613, 118; 615,19 Ealet, A., 611, 66; 614,7 Fayard, L., 615,31 Echenard, B., 613, 118; 615,19 Fayolle, D., 611, 66; 614,7 Eckardt, V., 612, 181 Fedin, O., 613, 118; 615,19 Edgecock, T.R., 611, 66; 614,7 Fedotovitch, G.V., 613,29 Eﬁmov, L.G., 612, 181 Felcini, M., 613, 118; 615,19 Efremov, A.V., 612, 233 Feng, B., 611, 167 Ehlers, J., 612, 154 Ferguson, D.P.S., 611, 66; 614,7 Eidelman, S., 613, 20; 614, 27; 615,39 Ferguson, T., 613, 118; 615,19 Eidelman, S.I., 613,29 Fernandez, E., 611, 66; 614,7 Eiges, V., 614,27 Fernandez-Bosman, M., 611, 66; 614,7 Eisermann, Y., 615, 175 Ferrari, R., 611, 215 Elekes, Z., 614, 174 Ferrero, A., 612, 154 El Hage, A., 613, 118; 615,19 Ferrero, L., 612, 154 Eline, A., 613, 118; 615,19 Fesefeldt, H., 613, 118; 615,19 Ellis, G., 611, 66; 614,7 Fiandrini, E., 613, 118; 615,19 El Mamouni, H., 613, 118; 615,19 Field, J.H., 613, 118; 615,19 Emelianov, V., 612, 181 Filimonov, K., 612, 181 Emori, S., 615, 186 Filip, P., 612, 181 Enari, Y., 613, 20; 614, 27; 615,39 Filthaut, F., 613, 118; 615,19 Engelage, J., 612, 181 Finch, A.J., 611, 66; 614,7 Engler, A., 613, 118; 615,19 Finch, E., 612, 181 Epifanov, D., 613, 20; 615,39 Finger, M., 612, 154 Epifanov, D.A., 613,29 Finger Jr., M., 612, 154 Eppard, K., 615,31 Fiorillo, G., 613, 105; 614, 155 Eppard, M., 615,31 Fiorini, L., 615,31 Eppley, G., 612, 181 Fischer, C.S., 611, 279 Eppling, F.J., 613, 118; 615,19 Fischer, G., 615,31 Erazmus, B., 612, 181 Fischer, H., 612, 154 Esposito, L.S., 615,14 Fisher, P.H., 613, 118; 615,19 Estienne, M., 612, 181 Fisher, W., 613, 118; 615,19 Eversheim, P.D., 612, 154 Fisk, I., 613, 118; 615,19 Extermann, P., 613, 118; 615,19 Fisyak, Y., 612, 181 Eyrich, W., 612, 154 Fleurot, F., 615, 167 Flierl, D., 612, 181 Fabbro, B., 611, 66; 614,7 Foà, L., 611, 66; 614,7 Fabro, M., 612, 154 Focardi, E., 611, 66; 614,7 Fachini, P., 612, 181 Fodor, Z., 612, 173 Faessler, M., 612, 154 Foley, K.J., 612, 181 Faestermann, T., 615, 175 FOPI Collaboration, 612, 173 Faine, V., 612, 181 Forconi, G., 613, 118; 615,19 304 Cumulative author index to volumes 611–615 (2005) 298–321

Formica, A., 615,31 Gaudichet, L., 612, 181 Forty, R.W., 611, 66; 614,7 Gautheron, F., 612, 154 Foster, F., 611, 66; 614,7 Gavrichtchouk, O.P., 612, 154 Fouchez, D., 611, 66; 614,7 Gay, P., 611, 66; 614,7 Fox, H., 615,31 Gazizov, A.Z., 612, 147 Frabetti, P.L., 615,31 Ge, X.-H., 612,61 Fraga, E.S., 614, 181 Gehrmann, T., 612, 36, 49 Frank, A., 613, 134 Gehrmann-De Ridder, A., 612, 36, 49 Frank, M., 611, 66; 614,7 Gentile, S., 613, 118; 615,19 Frankland, J., 613, 128 Gerassimov, S., 612, 154 Franz, J., 612, 154 Germain, M., 612, 181 Frascaria, N., 613, 128 Gershon, T., 613, 20; 614, 27; 615,39 Fratina, S., 613, 20; 614, 27; 615,39 Gershon, T.J., 615,31 Frekers, D., 612, 165; 613, 105; 614, 155 Geurts, F., 612, 181 Freudenreich, K., 613, 118; 615,19 Geweniger, C., 611, 66; 614,7 Friedrich, J.M., 612, 154 Geyer, R., 612, 154 Frigerio, M., 612,29 Gharibyan, V., 611, 231 Frolov, V., 612, 154 Ghazikhanian, V., 612, 181 Fry, J.N., 612, 122 Ghete, V.M., 611, 66; 614,7 Fu, C.D., 614,37 Ghosh, P., 612, 181 Fu, H.Y., 614,37 Ghosh, S.K., 611,27 Fu, J., 612, 181 Giacomelli, G., 615,14 Fuchs, U., 612, 154 Giagu, S., 613, 118; 615,19 Fujii, K., 611, 223 Giammanco, A., 611, 66; 614,7 Fülöp, Zs., 614, 174 Giannakis, I., 611, 137 Furetta, C., 613, 118; 615,19 Giannini, G., 611, 66; 614,7 Gianoli, A., 615,31 Gabyshev, N., 613, 20; 614, 27; 615,39 Gianotti, F., 611, 66; 614,7 Gabyshev, N.I., 613,29 Giassi, A., 611, 66; 614,7 Gadelha, A.L., 613, 213 Gibelin, J., 614, 174 Gagliardi, C.A., 612, 181 Gilman, R., 613, 148 Gagunashvili, N., 612, 181 Giorgi, M., 612, 154 Gaillard, M.K., 612, 304 Giorgini, M., 615,14 Galaktionov, Yu., 613, 118; 615,19 Girone, M., 611, 66; 614,7 Ganguli, S.N., 613, 118; 615,19 Girtler, P., 611, 66; 614,7 Ganis, G., 611, 66; 614,7 Giudici, S., 615,31 Gans, J., 612, 181 Glover, E.W.N., 612, 36, 49 Ganti, M.S., 612, 181 Go, A., 614,27 Gao, C.J., 612, 127 Gobbo, B., 612, 154 Gao, C.S., 614,37 Goeke, K., 612, 233 Gao, H., 613, 148 Goertz, S., 612, 154 Gao, Y., 611, 66; 614,7 Gogohia, V., 611, 129 Gao, Y.N., 614,37 Gokhroo, G., 613, 20; 614, 27; 615,39 Garbrecht, B., 612, 311 Goldberg, J., 613, 105; 614, 155 Garcia-Abia, P., 613, 118; 615,19 Golec-Biernat, K., 613, 154 Garcia-Bellido, A., 611, 66; 614,7 Golob, B., 614,27 Garfagnini, R., 612, 154 Gong, M.Y., 614,37 Garibaldi, F., 613, 148 Gong, W.X., 614,37 Garmash, A., 613, 20; 614, 27; 615,39 Gong, Z.F., 613, 118; 615,19 Garrido, Ll., 611, 66; 614,7 Gonidec, A., 615,31 Gasparic, I., 612, 173 Gonzalez, J.E., 612, 181 Gataullin, M., 613, 118; 615,19 González, S., 611, 66; 614,7 Gates, G., 613, 148 Gorbachev, D.A., 613,29 Gatignon, L., 615,31 Gorbar, E.V., 611, 207 Gatto, R., 615, 297 Gorbunov, P., 613, 105; 614, 155 Cumulative author index to volumes 611–615 (2005) 298–321 305

Gorin, A.M., 612, 154 Hallman, T.J., 612, 181 Gorini, B., 615,31 Hamed, A., 612, 181 Govi, G., 615,31 Hanke, P., 611, 66; 614,7 Goy, C., 611, 66; 614,7 Hannappel, J., 612, 154 Grachov, O., 612, 181 Hannen, V.M., 612, 165 Graesser, M.L., 611, 53; 613,5 Hansen, J.B., 611, 66; 614,7 Grafström, P., 615,31 Hansen, J.D., 611, 66; 614,7 Grajek, O.A., 612, 154 Hansen, J.R., 611, 66; 614,7 Granier de Cassagnac, R., 615,31 Hansen, P.H., 611, 66; 614,7 Grassi, C., 615, 160 Hara, T., 613, 105; 614, 27, 155 Grasso, A., 612, 154 Harakeh, M.N., 612, 165; 615, 167, 175 Graw, G., 615, 175 Hardtke, D., 612, 181 Graziani, G., 615,31 Harris, F.A., 614,37 Grebeniuk, A.A., 613,29 Harris, J.W., 612, 181 Grebenyuk, O., 612, 181 Hartmann, O.N., 612, 173 Green, M.G., 611, 66; 614,7 Harvey, J., 611, 66; 614,7 Greenﬁeld, M.B., 615, 193 Hasegawa, T., 612, 154 Grégoire, G., 613, 105; 614, 155 Hasenfratz, P., 613,57 Grella, G., 613, 105; 614, 155 Haseyama, T., 615, 186 Grenier, G., 613, 118; 615,19 Hashimoto, M., 611, 207 Grigoriev, D.N., 613,29 Hatakeyama, A., 611, 239 Grigorieva, S.I., 612, 147 Hatanaka, K., 615, 193 Grimm, O., 613, 118; 615,19 Hatano, M., 615, 193 Grinstein, B., 615, 213 Hatzifotiadou, D., 613, 118; 615,19 Grishkin, Yu., 612, 173 Hay, B., 615,31 Grivaz, J.-F., 611, 66; 614,7 Hayasaka, K., 613, 20; 614, 27; 615,39 Gronstal, S., 612, 181 Hayashii, H., 613, 20; 614, 27; 615,39 Grosnick, D., 612, 181 Hayes, O.J., 611, 66; 614,7 Grube, B., 612, 154 Hazumi, M., 613, 20; 614,27 Gruenewald, M.W., 613, 118; 615,19 He, H., 611, 66; 614,7 Grünemaier, A., 612, 154 He, K.L., 614,37 Grupen, C., 611, 66; 614,7 He, L., 615,93 Grzelinska,´ A., 611, 116 He, M., 614,37 Gu, S.D., 614,37 He, X., 614,37 Guedon, M., 612, 181 Hebbeker, T., 613, 118; 615,19 Guertin, S.M., 612, 181 Hedicke, S., 612, 154 Guida, M., 613, 118; 615,19 Heinsius, F.H., 612, 154 Guillot, J., 615, 167 Heinz, M., 612, 181 Güler, M., 613, 105; 614, 155 Heitger, J., 612, 313 Gulyás, J., 615, 175 Heng, Y.K., 614,37 Guo, Y.N., 614,37 Henley, E.M., 611, 111 Guo, Y.Q., 614,37 Henry, T.W., 612, 181 Guo, Z.J., 614,37 Hepp, V., 611, 66; 614,7 Gupta, A., 612, 181 Heppelmann, S., 612, 181 Gupta, V.K., 613, 118; 615,19 Hermann, R., 612, 154 Gurtu, A., 613, 118; 615,19 Herrmann, N., 612, 173 Gutay, L.J., 613, 118; 615,19 Herskind, B., 615, 160 Gutierrez, T.D., 612, 181 Hertenberger, R., 615, 175 Hervé, A., 613, 118; 615,19 Haas, D., 613, 118; 615,19 Heß, C., 612, 154 Haba, J., 613, 20; 614,27 Hess, J., 611, 66; 614,7 Haba, N., 615, 247 Heusse, Ph., 611, 66; 614,7 Habs, D., 615, 175 Higuchi, T., 614,27 Hagemann, G.B., 615, 160 Hildenbrand, K.D., 612, 173 Hagino, K., 615,55 Hinterberger, F., 612, 154 306 Cumulative author index to volumes 611–615 (2005) 298–321

Hinz, L., 613, 20; 614,27 Ibe, M., 615, 120 Hippolyte, B., 612, 181 Ichikawa, Y., 614, 174 Hirayama, Y., 611, 239 Iconomidou-Fayard, L., 615,31 Hirsch, A., 612, 181 Ideguchi, E., 614, 174 Hirsch, J.G., 613, 134 Ignatov, F.V., 613,29 Hirschfelder, J., 613, 118; 615,19 Igo, G., 612, 181 Hjort, E., 612, 181 Iijima, T., 613, 20; 614, 27; 615,39 Hodgson, P.N., 611, 66; 614,7 Ijaduola, R.B., 612, 154 Hofer, H., 613, 118; 615,19 Ilgenfritz, E.-M., 611,27 Hoffmann, G.W., 612, 181 Ilgner, C., 612, 154 Hofmann, F., 612, 165 Imbergamo, E., 615,31 Hohlmann, M., 613, 118; 615,19 Imoto, A., 613, 20; 614, 27; 615,39 Hokuue, T., 613, 20; 614,27 Inami, K., 613, 20; 614, 27; 615,39 Holder, M., 615,31 Ioukaev, A.I., 612, 154 Hölldorfer, F., 611, 66; 614,7 Ishihara, A., 612, 181 Holstein, B.R., 612, 311 Ishikawa, A., 613, 20; 614, 27; 615,39 Holzner, G., 613, 118; 615,19 Ishimoto, S., 612, 154 Hong, B., 612, 173 Itakura, K., 615, 221 Horikawa, N., 612, 154 Itoh, R., 614, 27; 615,39 Horikawa, S., 612, 154 Ivanov, O., 612, 154 Horns, D., 611, 297 Ivanov, R.I., 611,34 Horsley, M., 612, 181 Ivanova, T.A., 612,65 Horváth, Á., 614, 174 Iwasa, N., 614, 174 Horváth, I., 612,21 Iwasaki, H., 614, 174 Horváthy, P.A., 615,87 Iwasaki, M., 613, 20; 615,39 Hoshi, Y., 613, 20; 614, 27; 615,39 Iwasaki, Y., 613, 20; 614, 27; 615,39 Hoshino, K., 613, 105; 614, 155 Iwata, T., 612, 154 Hosotani, Y., 615, 257 Izumi, H., 611, 239 Hou, S., 613, 20; 614, 27; 615,39 Hou, S.R., 613, 118; 615,19 Jacholkowska, A., 611, 66; 614,7 Hou, W.-S., 613, 20; 614, 27; 615,39 Jack, I., 611, 199 Hristov, P., 615,31 Jackson, K.P., 611, 239 Hristova, I.R., 613, 105; 614, 155 Jacobs, P., 612, 181 Hu, H., 611, 66; 614,7 Jacobs, W.W., 612, 181 Hu, H.M., 614,37 Jahn, R., 612, 154 Hu, T., 614,37 Jakobs, K., 611, 66; 614,7 Huang, G.S., 614,37 Janata, A., 612, 154 Huang, H.Z., 612, 181 Janik, M., 612, 181 Huang, L., 614,37 Janot, P., 611, 66; 614,7 Huang, S.L., 612, 181 Jarlskog, C., 615, 207 Huang, T., 611, 260 Jastrz˛ebski, J., 615,55 Huang, W.-H., 615, 266 Jeitler, M., 615,31 Huang, X., 611, 66; 614,7 Jenkins, A., 613,5 Huang, X.P., 614,37 Jézéquel, S., 611, 66; 614,7 Hughes, E., 612, 181 Ji, X.B., 614,37 Hughes, E.W., 613, 148 Jia, Q.Y., 614,37 Hughes, G., 611, 66; 614,7 Jiang, C.H., 614,37 Humanic, T.J., 612, 181 Jiang, H., 612, 181 Hunyadi, M., 615, 175 Jiang, J., 615, 111 Hutchcroft, D.E., 611, 66; 614,7 Jiang, X., 613, 148 Hüttmann, K., 611, 66; 614,7 Jiang, X.S., 614,37 Jin, B.N., 613, 118; 615,19 Iacopini, E., 615,31 Jin, D.P., 614,37 Iancu, E., 615, 221 Jin, S., 611, 66; 614,7,37 Iaselli, G., 611, 66; 614,7 Jin, Y., 614,37 Cumulative author index to volumes 611–615 (2005) 298–321 307

Jindal, P., 613, 118; 615,19 Keane, D., 612, 181 Johnson, I., 612, 181 Kecskemeti, J., 612, 173 Jones, D.R.T., 611, 199 Kekelidze, V., 615,31 Jones, L.W., 613, 118; 615,19 Kennedy, J., 611, 66; 614,7 Jones, P.G., 612, 181 Ketzer, B., 612, 154 Jones, R.W.L., 611, 66; 614,7 Khan, E., 613, 128 Joosten, R., 612, 154 Khan, H.R., 613, 20; 615,39 Josa-Mutuberría, I., 613, 118; 615,19 Khaustov, G.V., 612, 154 Jost, B., 611, 66; 614,7 Khazin, B.I., 613,29 Jouravlev, N.I., 612, 154 Khlebnikov, S., 615,55 Jousset, J., 611, 66; 614,7 Khodyrev, V.Yu., 612, 181 Judd, E.G., 612, 181 Khokhlov, Yu.A., 612, 154 Juget, F., 615, 153 Khomutov, N.V., 612, 154 Jung, E., 614, 78; 615, 273 Khotilovich, V., 611, 223 Khovansky, V., 613, 105; 614, 155 Kabana, S., 612, 181 Kichimi, H., 613, 20; 614, 27; 615,39 Kabuß, E., 612, 154 Kiener, J., 615, 167 Kachelrieß, M., 614,1 Kienzle-Focacci, M.N., 613, 118; 615,19 Kado, M., 611, 66; 614,7 Kijima, G., 615, 186 Kalinin, S., 613, 105; 614, 155 Kile, J., 611, 66; 614,7 Kalinnikov, V., 612, 154 Kim, H.J., 613, 20; 614, 27; 615,39 Kalinovsky, Y., 614,44 Kim, J.E., 612, 293 Kalmus, G.E., 615,31 Kim, J.H., 613, 20; 614, 27; 615,39 Kalmykov, Y., 612, 165 Kim, J.K., 613, 118; 615,19 Kalter, A., 615,31 Kim, S.H., 614, 78; 615, 273 Kameda, D., 615, 186 Kim, S.K., 613, 20; 615,39 Kamiya, J., 615, 193 Kim, S.M., 613, 20; 614, 27; 615,39 Kamon, T., 611, 223 Kim, Y.J., 612, 173 Kang, D., 612, 154 Kinoshita, K., 614,27 Kang, J.H., 613, 20; 614, 27; 615,39 Kirejczyk, M., 612, 173 Kang, J.S., 613, 20; 614, 27; 615,39 Kirkby, J., 613, 118; 615,19 Kanno, S., 614, 174 Kiryluk, J., 612, 181 Kanungo, R., 614, 174 Kisiel, A., 612, 181 Kao, C., 614,67 Kisielinski,´ M., 615,55 Kaplan, M., 612, 181 Kisselev, Yu., 612, 154 Kapusta, P., 614,27 Kitamura, Y., 615, 193 Karliner, M., 612, 197 Kitazawa, M., 615, 102 Karpov, S.V., 613,29 Kittel, W., 613, 118; 615,19 Karstens, F., 612, 154 Klay, J., 612, 181 Kastaun, W., 612, 154 Klein, F., 612, 154 Kataoka, S.U., 613, 20; 615,39 Klein, S.R., 612, 181 Katayama, N., 613, 20; 614, 27; 615,39 Kleinert, H., 611, 182 Kato, G., 615, 186 Kleinknecht, K., 611, 66; 615,31 Kato, H., 615, 193 Klimentov, A., 613, 118; 615,19 Kato, Y., 611, 223 Kluge, E.E., 611, 66; 614,7 Kaur, M., 613, 118; 615,19 Klyachko, A., 612, 181 Kaus, P., 611, 147 Kmiecik, M., 615, 160 Kawada, J., 613, 105; 614, 155 Kneringer, E., 611, 66; 614,7 Kawai, H., 611, 269; 613, 20; 614, 27; 615,39 Knowles, I., 615,31 Kawai, S., 614, 174 Ko, P., 611,87 Kawamura, T., 613, 105; 614, 155 Koang, D.H., 615, 153 Kawasaki, T., 613, 20; 614, 27; 615,39 Koblitz, S., 612, 154 Kayis-Topaksu, A., 613, 105; 614, 155 Koch, U., 615,31 Kayser, F., 611, 66; 614,7 Koczon, P., 612, 173 Kazanin, V.F., 613,29 Kodama, K., 613, 105; 614, 155 308 Cumulative author index to volumes 611–615 (2005) 298–321

Koetke, D.D., 612, 181 Kroumchtein, Z.V., 612, 154 Koivuniemi, J.H., 612, 154 Krueger, K., 612, 181 Kolev, D., 613, 105; 614, 155 Krüger, A., 613, 118; 615,19 Kollegger, T., 612, 181 Kubischta, W., 615,31 Kolosov, V.N., 612, 154 Kuhn, C., 612, 181 Komatsu, M., 613, 105; 614, 155 Kuhn, D., 611, 66; 614,7 Komissarov, E.V., 612, 154 Kühn, J.H., 611, 116 Kondo, K., 612, 154 Kuhn, R., 612, 154 Kondo, Y., 611, 93; 614, 174 Kulasiri, R., 615,39 König, A.C., 613, 118; 615,19 Kulikov, A.I., 612, 181 Königsmann, K., 612, 154 Kumar, A., 612, 181 Konoplyannikov, A.K., 612, 154 Kunde, G.J., 612, 181 Konorov, I., 612, 154 Kundrát, V., 611, 102 Konstandin, T., 612, 311 Kunihiro, T., 615, 102 Konstantinov, V.F., 612, 154 Kunin, A., 613, 118; 615,19 Koop, I.A., 613,29 Kunne, F., 612, 154 Kopal, M., 613, 118; 615,19 Kunz, C.L., 612, 181 Köpke, L., 615,31 Kunz, J., 614, 104 Koppenburg, P., 614,27 Kunz, M., 614, 125 Kopytine, M., 612, 181 Kuo, C.C., 613, 20; 614, 27; 615,39 Kordyasz, A., 615,55 Kurek, K., 612, 154 Korentchenko, A.S., 612, 154 Kuroki, T., 611, 269 Korolija, M., 612, 173 Kutuev, R.Kh., 612, 181 Korpar, S., 614, 27; 615,39 Kuzmin, A., 613, 20; 615,39 Korsch, W., 613, 148 Kuzmin, A.S., 613,29 Korzenev, A., 612, 154 Kuznetsov, A.A., 612, 181 Köse, U., 613, 105; 614, 155 Kwon, Y.-J., 613, 20; 614, 27; 615,39 Kotchenda, L., 612, 181 Kyriakis, A., 611, 66; 614,7 Kotte, R., 612, 173 Kotzinian, A.M., 612, 154 L3 Collaboration, 613, 118; 615,19 Koutchinski, N.A., 612, 154 Lacourt, A., 615,31 Koutsenko, V., 613, 118; 615,19 Ladron de Guevara, P., 613, 118; 615,19 Kovalenko, A.D., 612, 181 Ladygin, M.E., 612, 154 Kovalenko, A.V., 613,52 Lai, A., 615,31 Kowalczyk, M., 615,55 Lai, Y.F., 614,37 Kowalik, K., 612, 154 Laktineh, I., 613, 118; 615,19 Kraan, A.C., 611, 66; 614,7 Lamanna, G., 615,31 Kräber, M., 613, 118; 615,19 Lamanna, M., 612, 154 Kraemer, R.W., 613, 118; 615,19 Lamblin, J., 615, 153 Kramer, K., 613, 148 Lamont, M.A.C., 612, 181 Kramer, M., 612, 181 Lançon, E., 611, 66; 614,7 Kraniotis, G.V., 611, 156 Landgraf, J.M., 612, 181 Krasznahorkay, A., 615, 175 Landi, G., 613, 118; 615,19 Kravchuk, N.P., 612, 154 Langacker, P., 614,67 Kravchuk, V.L., 615, 167 Lange, J.S., 614,27 Kravtsov, P., 612, 181 Lange, S., 612, 181 Kravtsov, V.I., 612, 181 Lasiuk, B., 612, 181 Krein, G., 614, 181 Laue, F., 612, 181 Kress, T., 612, 173 Laurelli, P., 611, 66; 614,7 Krivokhizhin, G.V., 612, 154 Laurent, H., 615, 167 Krivonos, S., 612, 283 Lauret, J., 612, 181 Križan, P., 615,39 Laville, J.L., 613, 128 Krogulski, T., 615,55 Lazkoz, R., 615, 146 Krokovny, P., 613, 20; 614, 27; 615,39 Lazzeroni, C., 615,31 Krokovny, P.P., 613,29 Lebeau, M., 613, 118; 615,19 Cumulative author index to volumes 611–615 (2005) 298–321 309

Lebedev, A., 612, 173, 181; 613, 118; 615,19 Ligabue, F., 611, 66; 614,7 Leberig, M., 612, 154 Likhoded, S., 613, 118; 615,19 Lebrun, D., 615, 153 Limosani, A., 614,27 Lebrun, P., 613, 118; 615,19 Lin, C.H., 613, 118; 615,19 Lechtenfeld, O., 612,65 Lin, J., 611, 66; 614,7 Lecomte, P., 613, 118; 615,19 Lin, S.-W., 613, 20; 614, 27; 615,39 Lecoq, P., 613, 118; 615,19 Lin, W.T., 613, 118; 615,19 Le Coultre, P., 613, 118; 615,19 Linde, F.L., 613, 118; 615,19 Leder, G., 613, 20; 615,39 Lindenbaum, S.J., 612, 181 Lednický, R., 612, 181 Link, O., 615, 153 Lee, F.X., 612,21 Lipkin, H.J., 612, 197 Lee, H.-S., 614,67 Lisa, M.A., 612, 181 Lee, J., 611,87 Liska, T., 612, 154 Lee, J.-C., 611, 193 Lista, L., 613, 118; 615,19 Lee, S., 614, 113 Litke, A.M., 611, 66; 614,7 Lee, S.E., 613, 20; 614, 27; 615,39 Liu, C.X., 614,37 Lee, S.H., 613, 20; 614,27 Liu, D.-J., 611,8 Lee, T., 611,87 Liu, F., 612, 181; 614, 37, 37 Lees, J.-P., 611, 66; 614,7 Liu, H.M., 614,37 Lefebvre, A., 615, 167 Liu, J.B., 614,37 Le Goff, J.M., 612, 154; 613, 118; 615,19 Liu, J.P., 614,37 Lehto, M., 611, 66; 614,7 Liu, K.F., 612,21 Leibenguth, G., 611, 66; 614,7 Liu, L., 612, 181 Leifels, Y., 612, 173 Liu, Q.J., 612, 181 Leiste, R., 613, 118; 615,19 Liu, R.G., 614,37 Leitner, O., 614, 165 Liu, Z., 612, 181 Lemaire, M.-C., 611, 66; 614,7 Liu, Z.A., 613, 118; 614, 37; 615,19 Lemaitre, V., 611, 66; 614,7 Liu, Z.X., 614,37 Lenti, M., 615,31 Liventsev, D., 614, 27; 615,39 Leoni, S., 615, 160 Liyanage, N., 613, 148 Lesiak, T., 613, 20; 614, 27; 615,39 Ljubicic, T., 612, 181 LeVine, M.J., 612, 181 Llanes-Estrada, F.J., 611, 279 Levtchenko, M., 613, 118; 615,19 Llope, W.J., 612, 181 Levtchenko, P., 613, 118; 615,19 Lo Bianco, G., 615, 160 Levy, C.D.P., 611, 239 Locci, E., 611, 66; 614,7 Li, C., 612, 181; 613, 118; 615,19 Logashenko, I.B., 613,29 Li, F., 614,37 Lohmann, W., 613, 118; 615,19 Li, G., 614,37 Lokajícek,ˇ M., 611, 102 Li, H.H., 614,37 Long, H., 612, 181 Li, J., 613, 20; 614, 27, 37; 615,39 Longacre, R.S., 612, 181 Li, J.C., 614,37 Longo, E., 613, 118; 615,19 Li, Q., 612, 181 Lopes da Silva, P., 615,31 Li, Q.J., 614,37 Lopez, X., 612, 173 Li, R.B., 614,37 Lopez-Noriega, M., 612, 181 Li, R.Y., 614,37 Love, W.A., 612, 181 Li, S.M., 614,37 Loverre, P.F., 613, 105; 614, 155 Li, W.G., 614,37 Lu, F., 614,37 Li, X.-Z., 611,8 Lu, G.R., 614,37 Li, X.L., 614,37 Lü, H., 614,96 Li, X.Q., 614,37 Lu, J.G., 614,37 Li, X.S., 614,37 Lu, Y.S., 613, 118; 615,19 Liang, Y.F., 614,37 Lu, Z., 615, 200 Liao, H.B., 614,37 Lubrano, P., 615,31 Lichtenstadt, J., 612, 154 Luci, C., 613, 118; 615,19 Liddick, S.N., 611,81 Ludlam, T., 612, 181 310 Cumulative author index to volumes 611–615 (2005) 298–321

Ludovici, L., 613, 105; 614, 155 Mans, J., 613, 118; 615,19 Ludwig, I., 612, 154 Mantica, P.F., 611,81 Lukin, P.A., 613,29 Manuilov, I.V., 612, 154 Luminari, L., 613, 118; 615,19 Manvelyan, R., 613, 197 Luo, C.L., 614,37 Manweiler, R., 612, 181 Luo, X.L., 614,37 Manzoor, S., 615,14 Lustermann, W., 613, 118; 615,19 Mao, Z.P., 614,37 Lütjens, G., 611, 66; 614,7 Marchand, C., 612, 154 Lynch, J.G., 611, 66; 614,7 Marchetto, F., 615,31 Lynn, D., 612, 181 Marfatia, D., 613,61 Lysenko, A.P., 613,29 Margetis, S., 612, 181 Margiotta, A., 615,14 Ma, B.-Q., 615, 200 Marinelli, N., 611, 66; 614,7 Ma, E., 612,29 Markert, C., 612, 181 Ma, F.C., 614,37 Markou, C., 611, 66; 614,7 Ma, J., 612, 181 Markytan, M., 615,31 Ma, J.M., 614,37 Marotta, A., 613, 105; 614, 155 Ma, J.P., 613,39 Marouelli, P., 615,31 Ma, L.L., 614,37 Marras, D., 615,31 Ma, Q.M., 614,37 Marroncle, J., 612, 154 Ma, W.G., 613, 118; 615,19 Martin, A., 612, 154 Ma, X.Y., 614,37 Martin, F., 611, 66; 614,7 Ma, Y.G., 612, 181 Martin, J.P., 613, 118; 615,19 Mabe, M., 615, 257 Martin, L., 612, 181 Machefert, F., 611, 66; 614,7 Martin, V., 615,31 MacNaughton, J., 614,27 Martina, L., 615,87 Madigojine, D., 615,31 Martinez, M., 611, 66; 614,7 Maeda, Y., 615, 193 Martínez de la Ossa, A., 613, 170 Magestro, D., 612, 181 Martini, M., 615,31 Maggi, G., 611, 66; 614,7 Marx, J., 612, 181 Maggi, M., 611, 66; 614,7 Marzano, F., 613, 118; 615,19 Maggiora, A., 612, 154 Marzec, J., 612, 154 Maggiora, M., 612, 154 Máté, Z., 615, 175 Magnon, A., 612, 154 Mathur, N., 612,21 Mahajan, S., 612, 181 Matis, H.S., 612, 181 Mahapatra, D.P., 612, 181 Mato, P., 611, 66; 614,7 Maier, A., 615,31 Matsuda, T., 612, 154 Maier, H.J., 615, 175 Matsumoto, T., 613, 20; 614, 27; 615,39 Maina, E., 614, 216 Matsuo, M., 615, 160 Maj, A., 615, 160 Matulenko, Yu.A., 612, 181 Majka, R., 612, 181 Matyja, A., 613, 20, 118; 614, 27; 615,39 Majumder, G., 614, 27; 615,39 Maximov, A.N., 612, 154 Makhlioueva, I., 613, 105; 614, 155 Mayes, V.E., 611, 156 Malgeri, L., 613, 118; 615,19 Mayet, F., 614, 143 Malinin, A., 613, 118; 615,19 Mazumdar, K., 613, 118; 615,19 Mallot, G.K., 612, 154 Mazzucato, E., 615,31 Maña, C., 613, 118; 615,19 McClain, C.J., 612, 181 Mandl, F., 613, 20; 614, 27; 615,39 McNamara III, P.A., 611, 66; 614,7 Mandrioli, G., 615,14 McNeil, R.R., 613, 118; 615,19 Mangiarotti, A., 612, 173 McShane, T.S., 612, 181 Mangotra, L.K., 612, 181 Medcalf, T., 611, 66; 614,7 Mannarelli, M., 615, 297 Medved, K.S., 612, 154 Mannelli, I., 615,31 Meinhard, H., 613, 105; 614, 155 Männer, W., 611, 66; 614,7 Meissner, F., 612, 181 Mannocchi, G., 611, 66; 614,7 Mele, S., 613, 118; 615,19 Cumulative author index to volumes 611–615 (2005) 298–321 311

Meljanac, S., 613, 221 Mizuk, R., 614, 27; 615,39 Mellado, B., 611,60 Mnich, J., 613, 118; 615,19 Melnick, Yu., 612, 181 Mo, X.H., 614,37 Melnitchouk, W., 613, 148 Moch, S.-O., 614,53 Meloni, D., 613, 170 Mohanty, B., 612, 181 Melzer-Pellmann, I., 615,31 Mohanty, G.B., 613, 118; 615,19 Menichetti, E., 615,31 Mohapatra, D., 614, 27; 615,39 Menzel, S., 612, 233 Mohapatra, R.N., 615, 231 Merle, E., 611, 66; 614,7 Moinester, M.A., 612, 154 Merola, L., 613, 118; 615,19 Molke, H., 612, 313 Merschmeyer, M., 612, 173 Molnar, L., 612, 181 Mertzimekis, T.J., 611,81 Molokanova, N., 615,31 Meschanin, A., 612, 181 Moloney, G.R., 614,27 Meschini, M., 613, 118; 615,19 Monteil, S., 611, 66; 614,7 Meshkov, S., 611, 147 Moore, C.F., 612, 181 Messina, M., 613, 105; 614, 155 Mora-Corral, M.J., 612, 181 Messineo, A., 611, 66; 614,7 Moradi, S., 613,74 Mestvirishvili, A., 615,31 Moretti, S., 614, 216 Metlitski, M.A., 612, 137 Mori, T., 615,39 Metz, A., 612, 233 Morimatsu, O., 611,93 Metzger, W.J., 613, 118; 615,19 Morita, T., 611, 269 Meyer, W., 612, 154 Morozov, D.A., 612, 181 Meziani, Z.-E., 613, 148 Morozov, V., 612, 181 Michel, B., 611, 66; 614,7 Morris, C.L., 615, 193 Middleton, C., 613, 189 Moser, H.-G., 611, 66; 614,7 Mielech, A., 612, 154 Motobayashi, T., 614, 174 Migliozzi, P., 613, 105; 614, 155 Moulin, E., 614, 143 Mihul, A., 613, 118; 615,19 Moutoussi, A., 611, 66; 614,7 Mikhailov, K.Yu., 613,29 Muanza, G.S., 613, 118; 615,19 Mikhailov, Yu.V., 612, 154 Muciaccia, M.T., 613, 105; 614, 155 Mikulec, I., 615,31 Muijs, A.J.M., 613, 118; 615,19 Milcent, H., 613, 118; 615,19 Mukherji, S., 613, 208 Miller, M.L., 612, 181 Müller, A.-S., 611, 66; 614,7 Million, B., 615, 160 Munday, D.J., 615,31 Milosevich, Z., 612, 181 Munhoz, M.G., 612, 181 Milstein, A.I., 613,29 MUNU Collaboration, 615, 153 Milton, K.A., 613,97 Murtas, G.P., 611, 66; 614,7 Minaev, N.G., 612, 181 Musicar, B., 613, 118; 615,19 Minard, M.-N., 611, 66; 614,7 Musy, M., 613, 118; 615,19 Mirabelli, G., 613, 118; 615,19 Mutterer, M., 615,55 Miransky, V.A., 611, 207 Mironov, C., 612, 181 Nabi, J.-U., 612, 190 Mischke, A., 612, 181 Nagae, D., 615, 186 Mishra, D., 612, 181 Nagamine, T., 613, 20; 614, 27; 615,39 Misiejuk, A., 611, 66; 614,7 Nagasaka, Y., 613, 20; 614, 27; 615,39 Miškovic,´ O., 615, 277 Nagy, S., 613, 118; 615,19 Mitaroff, W., 613, 20; 614, 27; 615,39 Nähle, O., 612, 154 Mitchell, J., 612, 181 Nakamura, M., 613, 105; 614, 155 Mitra, I., 611, 289 Nakano, E., 613, 20; 614, 27; 615,39 Miyabayashi, K., 613,20 Nakano, T., 613, 105; 614, 155 Miyake, H., 613, 20; 614, 27; 615,39 Nakao, M., 613, 20; 614, 27; 615,39 Miyanishi, M., 613, 105; 614, 155 Nakazawa, H., 615,39 Miyata, H., 613, 20; 614, 27; 615,39 Nandi, B.K., 612, 181 Miyatake, H., 611, 239 Nanopoulos, D.V., 611, 156 Miyoshi, H., 615, 186 Napoli, D., 615, 160 312 Cumulative author index to volumes 611–615 (2005) 298–321

Napolitano, M., 613, 118; 615,19 Obraztsov, V.F., 612, 154 Nappi, A., 615,31 Ocariz, J., 615,31 Nardulli, G., 615, 297 Odyniec, G., 612, 181 Narita, K., 613, 105; 614, 155 Oeschler, H., 615,50 Nasri, S., 615, 231 Oﬁerzynski, R., 613, 118; 615,19 Nassalski, J., 612, 154; 615,31 Ogawa, A., 612, 181 Nasseri, F., 614, 140 Ogawa, S., 613, 20, 105; 614, 27, 155; 615,39 Natale, S., 613, 118; 615,19 Ohnishi, T., 614, 174 Natkaniec, Z., 613, 20; 614, 27; 615,39 Ohshima, T., 613, 20; 614, 27; 615,39 Navarro-Lérida, F., 614, 104 Okabe, T., 613, 20; 614, 27; 615,39 Nayak, G.C., 613,45 Okamura, H., 615, 193 Nayak, S.K., 612, 181 Okhapkin, V.S., 613,29 Nayak, T.K., 612, 181 Okorokov, V., 612, 181 Nedel, D.L., 613, 213 Okuno, S., 613, 20; 614, 27; 615,39 Needham, M.D., 615,31 Okusawa, T., 613, 105; 614, 155 Negus, P., 611, 66; 614,7 Olaiya, E., 615,31 Neliba, S., 612, 154 Oldeman, R.G.C., 613, 105; 614, 155 Nelson, J.M., 612, 181 Oldenburg, M., 612, 181 Nessi-Tedaldi, F., 613, 118; 615,19 Olsen, S.L., 613, 20; 614, 27, 37; 615,39 Nesterenko, I.N., 613,29 Olshevsky, A.G., 612, 154 Netrakanti, P.K., 612, 181 Olson, D., 612, 181 Neubert, M., 612,13 Önengüt, G., 613, 105; 614, 155 Neubert, W., 612, 173 Organtini, G., 613, 118; 615,19 Neuhofer, G., 615,31 O’Shea, V., 611, 66; 614,7 Newman, H., 613, 118; 615,19 Ostrick, M., 612, 154 Neyret, D.P., 612, 154 Ostrowicz, W., 613, 20; 614, 27; 615,39 Ngac, A., 611, 66; 614,7 Otboev, A.V., 613,29 Nie, J., 614,37 Ouyang, Q., 611, 66; 614,7 Nie, Z.D., 614,37 Ozawa, A., 614, 174 Nielsen, J., 611, 66; 614,7 Nieto, M.M., 613,11 Paccetti Correia, F., 613,83 Nikitin, V.A., 612, 181 Pacheco, A., 611, 66; 614,7 Nikolaenko, V.I., 612, 154 Padee, A., 612, 154 Nikulin, M.A., 613,29 Pagano, P., 612, 154 Nilsson, B.S., 611, 66; 614,7 Paic, G., 612, 181 Nisati, A., 613, 118; 615,19 Pakhlov, P., 613, 20; 615,39 Nishida, S., 613, 20; 614, 27; 615,39 Pakvasa, S., 613,61 Nishikawa, T., 611,93 Pal, I., 613, 118; 615,19 Nitoh, O., 613, 20; 614, 27; 615,39 Pal, S.K., 612, 181 Niu, K., 613, 105; 614, 155 Pal, S.S., 614, 201 Niwa, K., 613, 105; 614, 155 Paleni, A., 615, 160 Nogach, L.V., 612, 181 Palestini, S., 615,31 Nojiri, M.M., 611, 223 Palka, H., 613, 20; 614, 27; 615,39 Nonaka, N., 613, 105; 614, 155 Palla, F., 611, 66; 614,7 Norman, B., 612, 181 Pallin, D., 611, 66; 614,7 Norton, A., 615,31 Palomares, C., 613, 118; 615,19 Norton, P.R., 611, 66; 614,7 Pan, Y.B., 611, 66; 614,7 Notani, M., 614, 174 Panebianco, S., 612, 154 Novak, T., 613, 118; 615,19 Panebratsev, Y., 612, 181 Nowak, H., 613, 118; 615,19 Panitkin, S.Y., 612, 181 Nowell, J., 611, 66; 614,7 Panman, J., 613, 105; 614, 155 Nozaki, T., 614, 27; 615,39 Panzer-Steindel, B., 615,31 Nozdrin, A.A., 612, 154 Panzieri, D., 612, 154 Nurushev, S.B., 612, 181 Paolucci, P., 613, 118; 615,19 Nuzzo, S., 611, 66; 614,7 Papavassiliou, J., 613, 162 Cumulative author index to volumes 611–615 (2005) 298–321 313

Paramatti, R., 613, 118; 615,19 Piasecki, E., 615,55 Park, C.W., 613, 20; 614, 27; 615,39 Piasecki, K., 615,55 Park, D.K., 614, 78; 615, 273 Picariello, M., 611, 215 Park, H., 614, 27; 615,39 Piccini, M., 615,31 Park, J.-H., 611,87 Piccolo, D., 613, 118; 615,19 Park, K.S., 614,27 Picha, R., 612, 181 Parker, M.A., 615,31 Picón, M., 615, 127 Parrini, G., 611, 66; 614,7 Pierazzini, G., 615,31 Parslow, N., 613, 20; 614, 27; 615,39 Pierella, F., 613, 118; 615,19 Pascolo, J.M., 611, 66; 614,7 Pietrzyk, B., 611, 66; 614,7 Passalacqua, L., 611, 66; 614,7 Pignanelli, M., 615, 160 Passaleva, G., 613, 118; 615,19 Piilonen, L.E., 613, 20; 614, 27; 615,39 Passera, M., 613, 162 Pilkuhn, H., 614,62 Pastrone, N., 615,31 Ping, R.G., 611, 123 Patricelli, S., 613, 118; 615,19 Pioppi, M., 613, 118; 615,19 Patrizii, L., 615,14 Piragino, G., 612, 154 Paul, S., 612, 154 Pirjol, D., 615, 213 Paul, T., 613, 118; 615,19 Piroué, P.A., 613, 118; 615,19 Pauluzzi, M., 613, 118; 615,19 Pistolesi, E., 613, 118; 615,19 Paus, C., 613, 118; 615,19 Plagnol, E., 613, 128 Pauss, F., 613, 118; 615,19 Planinic, M., 612, 181 Pavlinov, A.I., 612, 181 Platchkov, S., 612, 154 Pavšic,ˇ M., 614,85 Platzer, K., 612, 154 Pawlak, T., 612, 181 Pluta, J., 612, 181 Payre, P., 611, 66; 614,7 Plyaskin, V., 613, 118; 615,19 Peak, L.S., 615,39 Pochodzalla, J., 612, 154 Pearson, M.R., 611, 66; 614,7 Pohl, M., 613, 118; 615,19 Pedace, M., 613, 118; 615,19 Pojidaev, V., 613, 118; 615,19 Peitzmann, T., 612, 181 Polikarpov, M.I., 613,52 Pelte, D., 612, 173 Pollacco, E.C., 613, 128 Peng, H.P., 614,37 Polunin, A.A., 613,29 Pensotti, S., 613, 118; 615,19 Polyakov, D., 611, 173 Pepe, M., 615,31 Polyakov, V.A., 612, 154 Pereira, H.D., 612, 154 Ponomarev, V.Yu., 612, 165 Perevedentsev, E.A., 613,29 Popa, V., 615,14 Perevoztchikov, V., 612, 181 Pope, C.N., 614,96 Perez, P., 611, 66; 614,7 Popov, A.A., 612, 154 Perkins, C., 612, 181 Popov, A.S., 613,29 Perkins, W.B., 612,75 Porile, N., 612, 181 Pernicka, M., 615,31 Porter, J., 612, 181 Perret, P., 611, 66; 614,7 Poskanzer, A.M., 612, 181 Perret-Gallix, D., 613, 118; 615,19 Potekhin, M., 612, 181 Peryt, W., 612, 181 Pothier, J., 613, 118; 615,19 Peshekhonov, D.V., 612, 154 Potrebenikov, Yu., 615,31 Peshekhonov, V.D., 612, 154 Potrebenikova, E., 612, 181 Pestotnik, R., 613, 20; 614, 27; 615,39 Potukuchi, B.V.K.S., 612, 181 Peters, A., 615,31 Prange, G., 611, 66; 614,7 Petersen, A.K., 614, 104 Pretz, J., 612, 154 Petit, F., 612, 105 Prindle, D., 612, 181 Petrache, C.M., 615, 160 Prodanov, E.M., 611,34 Petrov, V.A., 612, 181 Prokoﬁev, D., 613, 118; 615,19 Petrovici, M., 612, 173 Pruneau, C., 612, 181 Petrucci, F., 615,31 Puglierin, G., 615, 153 Peyaud, B., 615,31 Putschke, J., 612, 181 Phatak, S.C., 612, 181 Putz, J., 611, 66; 614,7 314 Cumulative author index to volumes 611–615 (2005) 298–321

Putzer, A., 611, 66; 614,7 Renk, B., 611, 66; 614,7;615,31 Rescigno, M., 613, 118; 615,19 Qi, N.D., 614,37 Retiere, F., 612, 181 Qian, C.D., 614,37 Reucroft, S., 613, 118; 615,19 Qin, H., 614,37 Reymann, J., 612, 154 Qiu, J.-W., 613,45 Rezaei-Aghdam, A., 615, 141 Qiu, J.F., 614,37 Richter, A., 612, 165 Quadri, A., 611, 215 Ridiger, A., 612, 181 Quayle, W., 611,60 Riemann, S., 613, 118; 615,19 Quintans, C., 612, 154 Riles, K., 613, 118; 615,19 Rith, K., 612, 154 Radu, E., 615,1 Ritter, H.G., 612, 181 Ragusa, F., 611, 66; 614,7 Roberts, J.B., 612, 181 Raha, S., 611,27 Robertson, N.A., 611, 66; 614,7 Rahal-Callot, G., 613, 118; 615,19 Roe, B.P., 613, 118; 615,19 Rahaman, M.A., 613, 118; 615,19 Rogachevski, O.V., 612, 181 Rahman, M.-U., 612, 190 Rolandi, L., 611, 66; 614,7 Rai, G., 612, 181 Rolf, J., 612, 313 Raics, P., 613, 118; 615,19 Romano, G., 613, 105; 614, 155 Raja, N., 613, 118; 615,19 Romero, J.L., 612, 181 Rakers, S., 612, 165 Romero, L., 613, 118; 615,19 Rakness, G., 612, 181 Rondio, E., 612, 154; 615,31 Ramelli, R., 613, 118; 615,19 Rong, G., 614,37 Rami, F., 612, 173 Root, N., 613,20 Ramos, S., 612, 154 Root, N.I., 613,29 Rancoita, P.G., 613, 118; 615,19 Rosa, G., 613, 105; 614, 155 Rander, J., 611, 66; 614,7 Rosca, A., 613, 118; 615,19 Ranieri, A., 611, 66; 614,7 Rose, A., 612, 181 Ranieri, R., 613, 118; 615,19 Rosemann, C., 613, 118; 615,19 Raniwala, R., 612, 181 Rosenbleck, C., 613, 118; 615,19 Raniwala, S., 612, 181 Rosier-Lees, S., 613, 118; 615,19 Ranjard, F., 611, 66; 614,7 Ross, D.A., 614, 216 Rapaport, J., 615, 193 Roth, M., 612, 223 Raso, G., 611, 66; 614,7 Roth, S., 613, 118; 615,19 Raspereza, A., 613, 118; 615,19 Rothberg, J., 611, 66; 614,7 Rastkar, A.R., 615, 141 Rougé, A., 611, 66; 614,7 Ratabole, R., 611, 289 Rouhani, S., 613,74 Ravel, O., 612, 181 Roussel-Chomaz, P., 613, 128 Ray, R.L., 612, 181 Rowley, N., 615,55 Razin, S.V., 612, 181 Roy, C., 612, 181 Razis, P., 613, 118; 615,19 Roynette, J.C., 613, 128 Rebelo, M.N., 614, 187 Rozanov, A., 613, 105; 614, 155 Rebourgeard, P.C., 612, 154 Rozanska, M., 614,27 Redin, S.I., 613,29 Rozhdestvensky, A.M., 612, 154 Redlich, K., 615,50 Ruan, L.J., 612, 181 Reichenbach, T., 612, 275 Ruban, A.A., 613,29 Reicherz, G., 612, 154 Rubio, J.A., 613, 118; 615,19 Reichhold, D., 612, 181 Rudolph, G., 611, 66; 614,7 Reid, J.G., 612, 181 Ruggieri, F., 611, 66; 614,7 Reisdorf, W., 612, 173 Ruggieri, M., 615, 297 Reitz, B., 612, 165 Ruggiero, G., 613, 118; 615, 19, 31 Ren, D., 613, 118; 615,19 Rühl, W., 613, 197 Ren, H.-C., 611, 137 Ruiz, H., 611, 66; 614,7 Ren, Z.Y., 614,37 Rutherford, S.A., 611, 66; 614,7 Renault, G., 612, 181 Rykaczewski, H., 613, 118; 615,19 Cumulative author index to volumes 611–615 (2005) 298–321 315

Ryskulov, N.M., 613,29 Schlatter, D., 611, 66; 614,7 Schmeling, S., 611, 66; 614,7 Sacco, R., 615,31 Schmidt, I., 612, 258 Sadovski, A.B., 612, 154 Schmidt, M.G., 613,83 Sagawa, H., 613, 20; 614, 27; 615,39 Schmidt, S.A., 615,31 Sahoo, R., 612, 181 Schmidt, T., 612, 154 Saito, K., 612,5 Schmitt, H., 612, 154 Saito, T., 615, 193 Schmitt, L., 612, 154 Saitta, B., 613, 105; 614, 155 Schmitz, N., 612, 181 Sakai, H., 615, 193 Schneider, O., 613, 20; 614, 27; 615,39 Sakai, Y., 613, 20; 614, 27; 615,39 Schönharting, V., 615,31 Sakellariadou, M., 614, 125 Schopper, H., 613, 118; 615,19 Sakemi, Y., 615, 193 Schotanus, D.J., 613, 118; 615,19 Sakharov, A., 613, 118; 615,19 Schroeder, L.S., 612, 181 Sakrejda, I., 612, 181 Schué, Y., 615,31 Sakurai, H., 614, 174 Schuller, F.P., 612,93 Salicio, J., 613, 118; 615,19 Schümann, J., 613, 20; 614, 27; 615,39 Saller, E., 612, 154 Schüttauf, A., 612, 173 Salur, S., 612, 181 Schwanda, C., 614,27 Samoylenko, V.D., 612, 154 Schweda, K., 612, 165, 181 Samsarov, A., 613, 221 Schweitzer, P., 612, 233 Sanchez, E., 613, 118; 615,19 Sciabà, A., 611, 66; 614,7 Sandacz, A., 612, 154 Sciacca, C., 613, 118; 615,19 Sander, H.-G., 611, 66; 614,7 Scotto Lavina, L., 613, 105; 614, 155 Sandweiss, J., 612, 181 Sedgbeer, J.K., 611, 66; 614,7 Sanguinetti, G., 611, 66; 614,7 Seger, J., 612, 181 Sans, M., 612, 154 Sekiguchi, K., 615, 193 Santacesaria, R., 613, 105; 614, 155 Selvaggi, G., 611, 66; 614,7 Santachiara, R., 611, 189 Semenov, S., 614,27 Santos, D., 614, 143 Senyo, K., 613, 20; 615,39 Sapeta, S., 613, 154 Seres, Z., 612, 173 Sapozhnikov, M.G., 612, 154 Serin, L., 611, 66; 614,7 Sarangi, T.R., 614,27 Servoli, L., 613, 118; 615,19 Saremi, S., 613, 118; 615,19 Setare, M.R., 612, 100 Sarkar, S., 613, 118; 615,19 Seth, K.K., 612,1 Sarrazin, M., 612, 105 Settles, R., 611, 66; 614,7 Sato, N., 613, 20; 615,39 Seuster, R., 614,27 Sato, O., 613, 105; 614, 155 Sevior, M.E., 614,27 Sato, Y., 613, 105; 614, 155 Seyboth, P., 612, 181 Satta, A., 613, 105; 614, 155 Sguazzoni, G., 611, 66; 614,7 Savin, I., 612, 181 Shahaliev, E., 612, 181 Savin, I.A., 612, 154 Shajesh, K.V., 613,97 Savrié, M., 615,31 Shamanov, V., 613, 105; 614, 155 Savvidy, G., 615, 285 Shamov, A.G., 613,29 Scarpaci, J.A., 613, 128 Shan, L.Y., 614,37 Schael, S., 611, 66; 614,7 Shang, L., 614,37 Schäfer, C., 613, 118; 615,19 Shao, M., 612, 181 Schaile, O., 615, 175 Shao, W., 612, 181 Schakel, A.M.J., 611, 182 Sharatchandra, H.S., 611, 289 Schambach, J., 612, 181 Sharma, M., 612, 181 Scharenberg, R.P., 612, 181 Shatunov, Yu.M., 613,29 Schegelsky, V., 613, 118; 615,19 Shcherbakov, A., 612, 283 Schiavon, P., 612, 154 Shen, D.L., 614,37 Schietinger, T., 613, 20; 614, 27; 615,39 Shen, X.Y., 614,37 Schill, C., 612, 154 Shen, Y.-G., 612, 61; 614, 195 316 Cumulative author index to volumes 611–615 (2005) 298–321

Sheng, H.Y., 614,37 Smolyankin, V., 612, 173 Shestermanov, K.E., 612, 181 Snellings, R., 612, 181 Shevchenko, A., 612, 165 Snopkov, I.G., 613,29 Shevchenko, O.Yu., 612, 154 Soffer, J., 612, 258 Shevchenko, S., 613, 118; 615,19 Solodov, E.P., 613,29 Shi, F., 614,37 Sommer, R., 612, 313 Shi, X., 614,37 Somov, A., 613, 20; 614, 27; 615,39 Shibuya, H., 613, 20, 105; 614, 27, 155; 615,39 Son, D., 613, 118; 615,19 Shimada, K., 615, 186 Song, J.S., 613, 105; 614, 155 Shimanskii, S.S., 612, 181 Soni, N., 613, 20; 614, 27; 615,39 Shimizu, Y., 615, 193 Sood, G., 612, 181 Shimoda, T., 611, 239 Sorensen, P., 612, 181 Shimoura, S., 614, 174 Sorrentino, M., 613, 105 Shindler, A., 612, 313 Sorrentino, S., 613, 105; 614, 155 Shishkin, A.A., 612, 154 Souga, C., 613, 118; 615,19 Shivarov, N., 613, 118; 615,19 Sowinski, J., 612, 181 Shoutko, V., 613, 118; 615,19 Sozzi, F., 612, 154 Shrivastava, A., 613, 128 Sozzi, M., 615,31 Shu, F.-W., 614, 195 Spada, F.R., 613, 105; 614, 155 Shumilov, E., 613, 118; 615,19 Spagnolo, P., 611, 66; 614,7 Shvorob, A., 613, 118; 615,19 Spillantini, P., 613, 118; 615,19 Shwartz, B., 613, 20; 615,39 Spinka, H.M., 612, 181 Shwartz, B.A., 613,29 Spurio, M., 615,14 Si, Z.G., 615,68 Srivastava, B., 612, 181 Sibidanov, A.L., 613,29 Srnka, A., 612, 154 Sidorov, V., 613,20 Stamen, R., 613, 20; 614, 27; 615,39 Sidorov, V.A., 613,29 Stanic,ˇ S., 613, 20; 614, 27; 615,39 Siebert, H.-W., 612, 154 Stanislaus, T.D.S., 612, 181 Siegel, E.R., 612, 122 STAR Collaboration, 612, 181 Sikora, B., 612, 173 Staric,ˇ M., 613, 20; 614, 27; 615,39 Silva-Marcos, J.I., 614, 187 Steffens, F.M., 612,5 Silvestris, L., 611, 66; 614,7 Stenzel, H., 611, 66; 614,7 Sim, K.S., 612, 173 Sterman, G., 613,45 Simion, V., 612, 173 Steuer, M., 613, 118; 615,19 Simon, F., 612, 181 Stichel, P.C., 615,87 Simopoulou, E., 611, 66; 614,7 Stickland, D.P., 613, 118; 615,19 Sin, S.-J., 614, 113 Stinzing, F., 612, 154 Singaraju, R.N., 612, 181 Stock, R., 612, 181 Singh, J.B., 614, 27; 615,39 Stockmeier, M.R., 612, 173 Sinha, B., 611,27 Stoicea, G., 612, 173 Sinha, L., 612, 154 Stolarski, M., 612, 154 Sioli, M., 615,14 Stolpovsky, A., 612, 181 Siopsis, G., 613, 189 Stoyanov, B., 613, 118; 615,19 Sirignano, C., 613, 105; 614, 155 Straessner, A., 613, 118; 615,19 Sirri, G., 615,14 Strikhanov, M., 612, 181 Sissakian, A.N., 612, 154 Stringfellow, B., 612, 181 Siwek-Wilczynska,´ K., 612, 173 Strolin, P., 613, 105; 614, 155 Skachkova, A., 612, 154 Strong, J.A., 611, 66; 614,7 Skoro, G., 612, 181 Struck, C., 612, 181 Skrinsky, A.N., 613,29 Stuchbery, A.E., 611,81 Slifer, K., 613, 148 Stutz, A., 615, 153 Slunecka, M., 612, 154 Su, R.-K., 611,21 Smirnov, G.I., 612, 154 Suaide, A.A.P., 612, 181 Smirnov, N., 612, 181 Suda, K., 615, 193 Smizanska, M., 611, 66; 614,7 Sudhakar, K., 613, 118; 615,19 Cumulative author index to volumes 611–615 (2005) 298–321 317

Sugarbaker, E., 612, 181 Tavartkiladze, Z., 613,83 Sugiyama, A., 614,27 Taylor, G., 611, 66; 614,7 Sugonyaev, V.P., 612, 154 Taylor, L., 613, 118; 615,19 Suire, C., 612, 181 Tchalishev, V.V., 612, 154 Sulc, M., 612, 154 Teixeira-Dias, P., 611, 66; 614,7 Sulej, R., 612, 154 Tellili, B., 613, 118; 615,19 Sultanov, G., 613, 118; 615,19 Tempesta, P., 611, 66; 614,7 Šumbera, M., 612, 181 Tenchini, R., 611, 66; 614,7 Sumisawa, K., 613, 20; 614, 27; 615,39 Teramoto, Y., 613, 20; 614, 27; 615,39 Sumiyoshi, T., 613, 20; 614, 27; 615,39 Tessarotto, F., 612, 154 Sun, H.S., 614,37 Teubert, F., 611, 66; 614,7 Sun, L.Z., 613, 118; 615,19 Teufel, A., 612, 154 Sun, S.S., 614,37 Teyssier, D., 613, 118; 615,19 Sun, Y.Z., 614,37 Tezuka, I., 613, 105; 614, 155 Sun, Z.J., 614,37 Thacker, H.B., 612,21 Surrow, B., 612, 181 Thein, D., 612, 181 Sushkov, S., 613, 118; 615,19 Thers, D., 612, 154 Suter, H., 613, 118; 615,19 Thirolf, P.G., 615, 175 Suzuki, S., 614,27 Thomas, J.H., 612, 181 Suzuki, S.Y., 613, 20; 614, 27; 615,39 Thompson, A.S., 611, 66; 614,7 Swain, J.D., 613, 118; 615,19 Thompson, J.A., 613,29 Swiderski,´ Ł., 615,55 Thompson, J.C., 611, 66; 614,7 Symons, T.J.M., 612, 181 Thompson, L.F., 611, 66; 614,7 Syritsyn, S.N., 613,52 Tian, X.C., 613, 20; 614, 27; 615,39 Szanto de Toledo, A., 612, 181 Tian, Y.R., 614,37 Szarwas, P., 612, 181 Tilquin, A., 611, 66; 614,7 Szillasi, Z., 613, 118; 615,19 Timmermans, C., 613, 118; 615,19 Szleper, M., 615,31 Timoshenko, S., 612, 181 Ting, S.C.C., 613, 118; 615,19 Tadsen, A., 615, 153 Ting, S.M., 613, 118; 615,19 Tai, A., 612, 181 Tioukov, V., 613, 105; 614, 155 Tajima, O., 613, 20; 614, 27; 615,39 Tittel, K., 611, 66; 614,7 Takabayashi, N., 612, 154 Tkatchev, L.G., 612, 154 Takahashi, J., 612, 181 Tobe, K., 615, 120 Takasaki, F., 613, 20; 614, 27; 615,39 Toeda, T., 612, 154 Takasugi, E., 611,27 Togano, Y., 614, 174 Takenaga, K., 615, 247 Togo, V., 615,14 Takeshita, E., 614, 174 Tokarev, M., 612, 181 Takeuchi, S., 614, 174 Toki, H., 611,27 Takook, M.V., 613,74 Tolun, P., 613, 105; 614, 155 Tamhankar, S., 612,21 Tomalin, I.R., 611, 66; 614,7 Tamii, A., 615, 193 Tomlin, B.E., 611,81 Tamura, N., 613, 20; 614, 27; 615,39 Tong, G.L., 614,37 Tanaka, M., 613, 20; 614, 27; 615,39 Tonjes, M.B., 612, 181 Tang, A.H., 612, 181 Tonwar, S.C., 613, 118; 615,19 Tang, X., 614,37 Toshito, T., 613, 105; 614, 155 Tang, X.W., 613, 118; 615,19 Tóth, J., 613, 118; 615,19 Tanihata, I., 614, 174 Trainor, T.A., 612, 181 Tao, N., 614,37 Trentalange, S., 612, 181 Tarjan, P., 613, 118; 615,19 Tretyak, V.I., 612, 154 Tatischeff, V., 615, 167 Triantafyllopoulos, D.N., 615, 221 Tatishvili, G., 615,31 Tribble, R.E., 612, 181 Taureg, H., 615,31 Tricomi, A., 611, 66; 614,7 Taurok, A., 615,31 Trocmé, B., 611, 66; 614,7 Tauscher, L., 613, 118; 615,19 Troncoso, R., 615, 277 318 Cumulative author index to volumes 611–615 (2005) 298–321

Trousov, S., 612, 154 Venturi, A., 611, 66; 614,7 Trzaska, W.H., 615,55 Verdini, P.G., 611, 66; 614,7 Tsai, O., 612, 181 Veszpremi, V., 613, 118; 615,19 Tsenov, R., 613, 105; 614, 155 Vesztergombi, G., 613, 118; 615,19 Tsuboyama, T., 613, 20; 615,39 Vetlitsky, I., 613, 118; 615,19 Tsukamoto, T., 613, 20; 614, 27; 615,39 Videau, H., 611, 66; 614,7 Tsukerman, I., 613, 105; 614, 155 Videau, I., 611, 66; 614,7 Tsukui, M., 615, 186 Viertel, G., 613, 118; 615,19 Tsushima, K., 612,5 Vigdor, S.E., 612, 181 Tuchming, B., 611, 66; 614,7 Vigezzi, E., 615, 160 Tully, C., 613, 118; 615,19 Vilain, P., 613, 105; 614, 155 Tung, K.L., 613, 118; 615,19 Vilasi, G., 614, 131 Tupper, G.B., 612, 293 Villa, S., 613, 20, 118; 615, 19, 39 Turlay, R., 615,31 Villegas, M., 611, 66; 614,7 Turyshev, S.G., 613,11 Viollier, R.D., 612, 293 Tyminski, Z., 612, 173 Virius, M., 612, 154 Vivargent, M., 613, 118; 615,19 Uchigashima, N., 615, 193 Viyogi, Y.P., 612, 181 Uehara, S., 613, 20; 614, 27; 615,39 Vlachos, S., 613, 118; 615,19 Ueno, H., 615, 186 Vlassov, N.V., 612, 154 Ueno, K., 614,27 Vodopianov, I., 613, 118; 615,19 Uglov, T., 613, 20; 614, 27; 615,39 Vogel, H., 613, 118; 615,19 Uiterwijk, J.W.E., 613, 105; 614, 155 Vogt, H., 613, 118; 615,19 Ulbricht, J., 613, 118; 615,19 Voloshin, S.A., 612, 181 Ullrich, T., 612, 181 Von Harrach, D., 612, 154 Ulrych, S., 612,89 Von Hodenberg, M., 612, 154 Unal, G., 615,31 Von Neumann-Cosel, P., 612, 165 Underwood, D.G., 612, 181 Von Wimmersperg-Toeller, J.H., 611, 66; 614,7 Uno, S., 613, 20; 614, 27; 615,39 Vorobiev, I., 613, 118; 615,19 Ushida, N., 613, 105; 614, 155 Vorobyov, A.A., 613, 118; 615,19 Vuilleumier, J.-L., 615, 153 Valassi, A., 611, 66; 614,7 Vznuzdaev, M., 612, 181 Valente, E., 613, 118; 615,19 Valishev, A.A., 613,29 Wachsmuth, H., 611, 66; 614,7 Vallage, B., 611, 66; 614,7;615,31 Wadhwa, M., 613, 118; 615,19 Van Buren, G., 612, 181 Waggoner, W., 612, 181 Van Dantzig, R., 613, 105; 614, 155 Wagner, M., 612, 154 Van den Berg, A.M., 612, 165; 615, 167 Wagner, P., 612, 173 Van der Aa, O., 611, 66; 614,7 Wahl, H., 615,31 VanderMolen, A.M., 612, 181 Wakasa, T., 615, 193 Van de Vyver, B., 613, 105; 614, 155 Walker, A., 615,31 Van de Walle, R.T., 613, 118; 615,19 Walker, J.W., 611, 156 Varanda, M., 612, 154 Wambach, J., 612, 165 Varela, O., 615, 127 Wan, S.-L., 615,79 Varma, R., 612, 181 Wang, B., 611,21 Varner, G., 613, 20; 614, 27; 615,39 Wang, C.C., 613, 20; 614, 27; 615,39 Varner, G.S., 614,37 Wang, C.H., 613, 20; 614, 27; 615,39 Varvell, K.E., 614,27 Wang, D.Y., 614,37 Vasilevski, I., 612, 181 Wang, F., 612, 181 Vasiliev, A.N., 612, 181 Wang, G., 612, 181, 181 Vasquez, R., 613, 118; 615,19 Wang, J.Z., 614,37 Vayaki, A., 611, 66; 614,7 Wang, K., 614,37 Veillet, J.-J., 611, 66; 614,7 Wang, L., 614,37 Velasco, M., 615,31 Wang, L.S., 614,37 Velázquez, V., 613, 134 Wang, M., 614,37 Cumulative author index to volumes 611–615 (2005) 298–321 319

Wang, M.-Z., 614,27 Wise, M.B., 611, 53; 613,5 Wang, P., 614,37 Wislicki,´ W., 612, 154 Wang, P.L., 614,37 Wislicki, W., 615,31 Wang, Q., 613, 39, 118; 615,19 Wisniewski,´ K., 612, 173 Wang, S.Z., 614,37 Wissink, S.W., 612, 181 Wang, T., 611, 66; 614,7 Witecki, M., 615,55 Wang, W.F., 614,37 Witt, R., 612, 181 Wang, X.L., 612, 181; 613, 118; 615,19 Wittgen, M., 615,31 Wang, Y., 612, 181 Wohlfarth, D., 612, 173 Wang, Y.F., 614,37 Wohlfarth, M.N.R., 612,93 Wang, Z., 614, 37, 37, 37 Wolf, G., 611, 66; 614,7 Wang, Z.-G., 615,79 Wood, J., 612, 181 Wang, Z.M., 612, 181; 613, 118; 615,19 Wörtche, H.J., 612, 165 Wang, Z.Y., 614,37 Worthy, L.A., 611, 199 Wanke, R., 615,31 Wotton, S.A., 615,31 Ward, H., 612, 181 Wronka, S., 615,31 Ward, J.J., 611, 66; 614,7 Wu, C., 614, 174 Wasserbaech, S., 611, 66; 614,7 Wu, J., 611, 66; 612, 181; 614,7 Watanabe, H., 615, 186 Wu, N., 614,37 Watanabe, M., 613, 20; 614, 27; 615,39 Wu, S.L., 611, 60, 66; 614,7 Watanabe, Y., 614, 27; 615,39 Wu, X., 611, 66; 614,7 Watson, J.W., 612, 181 Wu, X.-G., 611, 260 Webb, J.C., 612, 181 Wu, Y.M., 614,37 Webb, R., 612, 154 Wu, Z.C., 612, 115; 613,1 Weber, M., 613, 118; 615,19 Wunsch, M., 611, 66; 614,7 Wei, C.L., 614,37 Wynhoff, S., 613, 118; 615,19 Wei, D.H., 614,37 Weise, E., 612, 154 Xia, L., 613, 118; 615,19 Weitzel, Q., 612, 154 Xia, X.M., 614,37 Wells, R., 612, 181 Xiao, Z.-G., 612, 173 Westfall, G.D., 612, 181 Xie, X.X., 614,37 Wheaton, S., 615,50 Xie, Y., 611, 66; 614,7 White, R., 611, 66; 614,7 Xin, B., 614,37 White, T.O., 615,31 Xu, G.F., 614,37 Whitten Jr., C., 612, 181 Xu, H., 614,37 Widhalm, L., 615,31 Xu, N., 612, 181 Wiedenmann, W., 611, 66; 614,7 Xu, R., 611, 66; 614,7 Wieders, L.H., 612, 223 Xu, Y., 614,37 Wiedner, U., 612, 154 Xu, Z., 612, 181 Wieland, O., 615, 160 Xu, Z.Z., 612, 181; 613, 118; 615,19 Wieman, H., 612, 181 Xue, S., 611, 66; 614,7 Wiesmann, M., 612, 154 Xue, S.T., 614,37 Willis, A., 615, 167 Willson, R., 612, 181 Yabsley, B.D., 614, 27; 615,39 Wilquet, G., 613, 105; 614, 155 Yagi, M., 611, 239 Wilschut, H.W., 615, 167 Yako, K., 615, 193 Wilson, A.N., 611,81 Yamaguchi, A., 613, 20; 614, 27; 615,39 Wilson, J., 615, 160 Yamaguchi, Y., 614, 174 Windmolders, R., 612, 154 Yamamoto, E., 612, 181 Wingerter-Seez, I., 615,31 Yamamoto, J., 613, 118; 615,19 Winhart, A., 615,31 Yamashita, T., 615, 247 Winter, K., 613, 105; 614, 155 Yamashita, Y., 613, 20; 614, 27; 615,39 Winter, W., 613,67 Yamauchi, M., 613, 20; 614, 27; 615,39 Wirth, H.-F., 615, 175 Yamazaki, T., 613, 140 Wirth, S., 612, 154 Yan, M.L., 614,37 320 Cumulative author index to volumes 611–615 (2005) 298–321

Yanagida, T., 615, 120 Zaremba, K., 612, 154 Yanagisawa, Y., 614, 174 Zeitnitz, C., 611, 66; 614,7 Yang, B.Z., 613, 118; 615,19 Zeng, Y., 614, 37, 37 Yang, C.G., 613, 118; 615,19 Zerguerras, T., 613, 128 Yang, F., 614,37 Zhang, B.X., 614,37 Yang, H., 613, 20; 614,27 Zhang, B.Y., 614,37 Yang, H.J., 613, 118; 615,19 Zhang, C.C., 614,37 Yang, H.X., 614,37 Zhang, D.H., 614,37 Yang, J., 614,37 Zhang, H., 612, 181 Yang, J.-C., 613, 148 Zhang, H.Y., 614,37 Yang, J.-J., 612, 258 Zhang, J., 611, 66; 614, 7, 27, 37 Yang, M., 613, 118; 615,19 Zhang, J.-Z., 613,91 Yang, S.D., 614,37 Zhang, J.B., 612,21 Yang, W.-M., 615,79 Zhang, J.W., 614,37 Yang, Y.X., 614,37 Zhang, J.Y., 614,37 Yano, H., 611, 239 Zhang, L., 611, 66; 614,7 Ye, M., 614,37 Zhang, L.M., 613, 20; 614, 27; 615,39 Ye, M.H., 614,37 Zhang, L.S., 614,37 Ye, Y.X., 614,37 Zhang, Q.J., 614,37 Yeh, S.C., 613, 118; 615,19 Zhang, S.N., 612, 127 Yepes, P., 612, 181 Zhang, S.Q., 614,37 Yi, L.H., 614,37 Zhang, W.M., 612, 181 Yi, Z.Y., 614,37 Zhang, X., 611,1 Yim, K.K., 615, 134 Zhang, X.M., 614,37 Yin, Q.-J., 613,91 Zhang, X.Y., 614,37 Ying, J., 614, 27; 615,39 Zhang, Y., 614,37 Ynduráin, F.J., 612, 245 Zhang, Y.J., 614,37 Yoon, C.S., 613, 105; 614, 155 Zhang, Y.Y., 614,37 Yoshida, A., 614, 174 Zhang, Z., 612, 207 Yoshida, K., 611, 269; 614, 174 Zhang, Z.P., 612, 181; 613, 20, 118; 614, 27, 37; 615, 19, 39 Yoshimi, A., 615, 186 Zhang, Z.Q., 614,37 Yu, C.S., 614,37 Zhao, D.X., 614,37 Yu, G.W., 614,37 Zhao, J., 612, 154; 613, 118; 615,19 Yu, H.-b., 615, 231 Zhao, J.B., 614,37 Yuan, C.Z., 614,37 Zhao, J.W., 614,37 Yuan, J.M., 614,37 Zhao, M.G., 614,37 Yuan, Y., 613, 20; 614,37 Zhao, P.P., 614,37 Yudin, Yu.V., 613,29 Zhao, W., 611, 66; 614,7 Yue, Q., 614,37 Zhao, W.-Q., 612, 207 Yüksel, H., 613,61 Zhao, W.R., 614,37 Yurevich, V.I., 612, 181 Zhao, X.J., 614,37 Yusa, Y., 614, 27; 615,39 Zhao, Y.B., 614,37 Yushmanov, I., 612, 173 Zhao, Z.G., 614,37 Yuting, B., 612, 181 Zhaomin, Z.P., 612, 181 Zheng, H.Q., 614,37 Zacek, V., 615, 153 Zheng, J.P., 614,37 Zachariadou, K., 611, 66; 614,7 Zheng, L.S., 614,37 Zaitsev, A.S., 613,29 Zheng, Z.P., 614,37 Zakharov, V.I., 613,52 Zhilich, V., 613, 20; 614, 27; 615,39 Zalite, An., 613, 118; 615,19 Zhilin, A., 612, 173 Zalite, Yu., 613, 118; 615,19 Zhong, X.C., 614,37 Zanelli, J., 615, 277 Zhou, B.Q., 614,37 Zanetti, A.M., 612, 154 Zhou, G.M., 614,37 Zanevski, Y.V., 612, 181 Zhou, H.Q., 611, 123 Zang, S.L., 614,37 Zhou, L., 614,37 Cumulative author index to volumes 611–615 (2005) 298–321 321

Zhou, M.-Z., 611, 260 Zinchenko, A., 615,31 Zhou, N.F., 614,37 Ziolkowski, M., 615,31 Zhu, G.Y., 613, 118; 615,19 Zito, G., 611, 66; 614,7 Zhu, K.J., 614,37 Zizong, Z.P., 612, 181 Zhu, Q.M., 614,37 Zobernig, G., 611, 66; 614,7 Zhu, R.Y., 613, 118; 615,19 Zöller, M., 613, 118; 615,19 Zhu, Y., 614,37 Zołnierczuk,˙ P.A., 612, 181 Zhu, Y.C., 614,37 Žontar, D., 613, 20; 614, 27; 615,39 Zhu, Y.S., 614,37 Zou, B.S., 611, 123; 614,37 Zhu, Z.A., 614,37 Zoulkarneev, R., 612, 181 Zhuang, B.A., 614,37 Zoulkarneeva, J., 612, 181 Zhuang, H.L., 613, 118; 615,19 Zubarev, A.N., 612, 181 Zhuang, P., 615,93 Zucchelli, P., 613, 105; 614, 155 Zichichi, A., 613, 118; 615,19 Zucchiati, A., 615, 160 Ziegler, R., 612, 154 Zuker, A.P., 613, 134 Ziegler, T., 611, 66; 614,7 Zverev, S.G., 613,29 Zimmermann, B., 613, 118; 615,19 Zvyagin, A., 612, 154