Physics of Atomic Nuclei, Vol. 68, No. 5, 2005, pp. 723–747. Translated from Yadernaya Fizika, Vol. 68, No. 5, 2005, pp. 755–779. Original Russian Text Copyright c 2005 by Agasian.
Nonperturbative Phenomena in QCD at Finite Temperature
N. O. Agasian* Institute of Theoretical and Experimental Physics, Bol’shaya Cheremushkinskaya ul. 25, Moscow, 117259 Russia Received July 26, 2004
Abstract—The main objective of the present study is to analyze various nonperturbative phenomena in QCDboth at low, T
1. INTRODUCTION which was predicted theoretically in recent years (for an overview, see [3–5]), can be realized. The physics of strong interactions, QCD, has been flourishing over the past three decades. For many Quantum chromodynamics is a quantum the- years, investigations into the behavior of strongly in- ory of Yang–Mills gauge fields interacting with quark fermion fields. The asymptotic-freedom phe- teracting matter under various external effects have nomenon [6, 7] and a topologically nontrivial struc- been of great topical interest. In the real world, these ture of the vacuum of non-Abelian gauge theories [8– are primarily temperature and the baryon density. In- 12] were discovered in the 1970s. A further devel- terest in the behavior of matter under extreme con- opment of the theory revealed that it is precisely the ditions (high temperatures commensurate with the complicated nonperturbative structure of the vacuum characteristic QCDscale, T ∼ 200 MeV, and high −3 (nonperturbative fluctuations of vacuum fields) that baryon densities, n>n0 0.17 fm ,wheren0 is the is responsible for the phenomena of confinement and normal nuclear density) is motivated by the fact that spontaneous chiral-symmetry breaking and, thereby, an increase in energy was achieved in experiments for the formation of the physical hadron spectrum. aimed at studying heavy-ion collisions. In view of Thus, the development of new theoretical approaches this, it is expected that densities and temperatures at was required for describing nonperturbative phenom- which a phase transition to a new state of strongly ena in quantum field theories. interacting matter, quark–gluon plasma [1, 2], is pos- Investigations of the vacuum state at finite tem- 1) sible are reached in such experiments. perature and finite values of the chemical potential Extreme conditions existed at the initial stage of and external fields lead to new interesting phenom- expansion of the Universe. Within the time interval ena, including various phase transitions. Accordingly, − − there arises the need for developing a theoretical for- t ∼ 10 6−10 5 s after the Big Bang, the Universe malism for exploring the behavior of a quantum-field passed the stage of a strong phase transition in system and its vacuum state under external effects. which the system transformed into a hadronic phase dominated by the essentially nonperturbative phe- Quantum field theory at finite temperature and a nomena of confinement and spontaneous chiral- finite value of the chemical potential has been studied symmetry breaking. Extremely high baryon densities predominantly along three lines: (i) Perturbative calculations have been performed, (n ∼ 10n0) also exist in central regions of neu- tron stars, where the color-superconductivity phase, and various resummation schemes for perturbation- theory series have been constructed. Advances have *e-mail: [email protected] predominantly been made in developing the hard- 1)It should be emphasized from the outset that there is thermal-loop (HTL) and hard-dense-loop (HDL) ap- presently no consensus on the theoretical interpretation of proximations (for an overview, see [13, 14]). the experiments in question. The main reason is that it is not (ii) Various effective models that describe one quite clear whether the initial stage of a heavy-ion collision can be justifiably described in terms of a steady-state ther- nonperturbative phenomenon in actual QCDor an- modynamically equilibrium system in the phase of quark– other have been constructed. For example, various gluon plasma. approaches based on employing sigma models to
1063-7788/05/6805-0723$26.00 c 2005 Pleiades Publishing, Inc. 724 AGASIAN describe the chiral order parameter in QCDhave metries and constraints that these symmetries im- been developed; investigations within effective chiral pose on physical characteristics of the system are theory at T =0 have been performed; and various of particular importance in QCD, which is a the- generalizations of the Nambu–Jona-Lasinio model ory that involves confinement and where composite to the case of finite temperature, a finite value of the states (hadrons) appear as observables. Low-energy chemical potential, and finite external fields have been theorems, or Ward identities (scale and chiral ones), introduced. Other effective models have also been play a crucial role in the understanding of nonpertur- developed (for an overview, see [3, 5]). bative vacuum properties of QCD. Stricly speaking, (iii) The complex nonperturbative structure of the low-energy theorems were formulated almost simul- vacuum of non-Abelian gauge theories—in partic- taneously with the beginning of the application of ular, QCDat finite T and n—has been studied by quantum-field methods in particle physics (see, for means of numerical simulations with the aid of com- example, Low theorems [26]). In QCD, they were puters. Interesting and important results have been obtained in the early 1980s [27]. Low-energy theo- obtained along this line, and it is expected that an rems in QCD, which follow from general symmetry increase in the power of computers and the creation of properties and which are independent of the details new computational schemes would make it possible of the confinement mechanism, make it possible to to obtain deeper insights into the nonperturbative obtain information that sometimes cannot be deduced dynamics of QCD(for an overview, see [15 –17]). in any other way; they can also be used as “physi- It should be noted that the high-temperature cally reasonable” constraints in constructing effective phase (T>Tc), which involves restored chiral sym- theories and various models of the QCDvacuum. In metry and deconfined quarks and gluons, is described the present study, we develop a method that makes in a conventional way as a quark–gluon plasma it possible to generalize low-energy QCDtheorems by analogy with an electromagnetic plasma. In this to the case of finite temperature. The nonperturbative approach, the approximation of noninteracting quark QCDvacuum and condensates at T =0are studied and gluon gases is used for a zero-order approxima- by using this method. tion; further, the interaction is taken into account in As was indicated above, nonperturbative fluc- the form of a series in powers of the QCDrunning tuations of gluon fields play a crucial role in the coupling constant g(T ). In view of the asymptotic- vacuum of non-Abelian gauge theories and determine freedom phenomenon in QCD, the coupling constant many features of QCD. The instanton-liquid model, must decrease with temperature. In order to improve which was proposed in [28, 29], is an elaborate theory the convergence of a relevant series—for example, in that explains quite successfully some phenomena in calculating pressure, which is among the main ther- QCD. In this pattern, well-separated and not very modynamic quantities that characterize the system— strongly interacting instantons and anti-instantons use is made of the method for summing hard ther- (from here, the name “liquid” comes) are the main mal loops, which effectively reduces to introducing nonperturbative fields. The instanton density is ap- −4 the so-called magnetic gluon mass. The presence proximately N/V4 =1 fm . This model makes it of a nonperturbative magnetic mass, ∝g2(T )T ,in possible to solve a number of problems in QCD—in the theory is necessary for removing well-known particular, the phenomenon of spontaneous chiral- infrared singularities [18]. At the same time, the symmetry breaking naturally arises there and the QCDinteraction remains strong, g(5Tc) ∼ 1,evenat η -meson mass is explained within it. Neverthe- temperatures T ∼ 5Tc, so that the use of perturbation less, the model in question possesses a number of theory, at least as a standard means for taking into serious drawbacks—namely, it is not known how account small corrections in the interaction to the the instanton–anti-instanton ensemble is stabilized zero-order approximation, is not quite legitimate. (problem of an infrared inflation of instantons) and it Moreover, the magnetic-confinement phenomenon is impossible to explain the confinement phenomenon in non-Abelian gauge theories, which occurs at all within the instanton-liquid model. However, the temperatures, inevitably requires the presence of fluc- vacuum involves, in addition to semiclassical instan- tuating nonperturbative chromomagnetic fields [19– tons, other nonperturbative fields, which enables one, 22] (see also [23–25]). Thus, QCDthermodynamics among other things, to solve the infrared problem of above the critical temperature must be described in instantons. terms of a phase that involves a strong chromomag- The nonperturbative QCDvacuum can be param- netic condensate, against whose background there eterized by a set of nonlocal gauge-invariant vacuum arise excitations of hot quarks and gluons. expectation values of gluon-field strengths [30–32]. Relations that are obtained as corollaries of the It appears that, in this way, one can describe well a symmetry properties of the theory play an impor- large number of hadron-physics phenomena (see the tant role in quantum field theory. Searches for sym- review article of Di Giacomo et al. [33]). In order to
PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 5 2005 NONPERTURBATIVE PHENOMENA IN QCD725 explain qualitatively relevant effects, it is sufficient, in temperature is given. Analytic expressions are derived many cases, to take into account only a bilocal cor- for the temperature dependences of the quark and relation function (stochastic-vacuum model). More- gluon condensates at low temperatures, T
PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 5 2005 726 AGASIAN dynamics of the chromomagnetic sector in terms of where Fπ 93 MeV is the axial coupling constant static correlation functions. The relative contribution and Λq 470 MeV (Nf =2) [48]. It is well known of nonzero Matsubara modes to the spacelike string that, owing to the smallness of the π-meson mass tension is calculated. At T =2Tc, this contribution is with respect to the characteristic mass scale of about 5%. strong interactions, Mπ 1 GeV,the pion stands out In Section 5, the bilocal correlation function in among other strongly interacting particles. Therefore, gluodynamics at finite temperature is calculated the chiral limit, Mπ → 0, is a good approximation for within the instanton-gas model. It is shown that many QCDproblems arising at zero temperature. the correlation length in the instanton vacuum de- On the other hand, it is obvious that, in finite- creases with increasing temperature. Our results are temperature QCD, there arises a new mass scale compared with lattice data for the bilocal correlation that is determined by the critical (phase-transition) function at T =0 . The problem of the instanton den- temperature Tc, which separates two fundamentally sity and the possible structures of the nonperturbative different phases of strongly interacting matter— vacuum are discussed. the hadronic phase, which involves confinement The main results of the present study are formu- and spontaneous chiral-symmetry breaking, and lated in the Conclusion. the quark–gluon phase, where there is no color confinement and where chiral invariance is restored. Numerically, the critical temperature appears to be 2. NONPERTURBATIVE QCDVACUUM close to the pion mass, T ≈ M .2) In the hadronic AT FINITE TEMPERATURE c π phase, there is therefore no small parameter Mπ/T , At low temperatures, T PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 5 2005 NONPERTURBATIVE PHENOMENA IN QCD727 anonzerovalueofG2. However, this is not a spon- this dependence, we will use general renormalizabil- taneous symmetry breaking; therefore, this does not ity properties and renormalization-group relations for generate Goldstone particles. In gluodynamics, the the QCDpartition function. In the ensuing exposi- mass of the lightest excitation of the 0++ glueball—it tion, we will follow the approach developed in [57, 58]. can be identified with a dilaton—is directly related to The QCDpartition function normalized to pertur- 2 1/4 the gluon condensate, mD ∝ (G ) . Thus, we can bation theory at T =0with two sorts of quarks (the state that, in the gluon sector of QCD, thermal ex- generalization to the case of arbitrary Nf is trivial) citations of massive glueballs (M0++ ∼ 1.5 GeV) are canbewrittenintheform(β =1/T ) suppressed in proportion to exp{−Mgl/T },sothat 1 their contribution to the finite-temperature shift of the Z = [DA] [Dq¯][Dq] (2) ZPT 2 2 q=u,d gluon condensate is small, ∆G /G ∼0.1% at T = 200 MeV [50]. β × − 3 L Further, we note that, in the one-loop approxima- exp dx4 d x , tion of chiral perturbation theory, the pions are de- 0 V scribed as a gas of noninteracting massless particles. Obviously, this system is scale-invariant; therefore, where, as usual, we have imposed periodic bound- it does not contribute to the trace of the energy– ary conditions on boson (gauge) fields and antiperi- momentum tensor and, accordingly, to G2.Itwas odic boundary conditions on fermion fields. Here, the QCDLagrangian has the form demonstrated in [51] that the gluon condensate ap- 1 pears to be temperature dependent only at the three- L = (Ga )2 (3) loop level of chiral perturbation theory upon tak- 4g2 µν 0 ing into account the interaction between Goldstone λa bosons. + q¯ γ ∂ − i Aa + m q, µ µ 2 µ 0q In this section, we develop a method that makes it q=u,d possible to calculate the temperature dependences of where we have omitted the ghost and gauge-fixing quark and gluon condensates beyond chiral perturba- terms since they are immaterial for our further con- tion theory. On the basis of general renormalization- siderations. The free-energy density is given by group relations at T =0 and m =0 ,wederivean q 1 equation that relates the anomalous contribution to F (T,m0u,m0d)=− ln Z. the trace of the energy–momentum tensor (gluon βV condensate) to thermodynamic pressure in QCDin- From Eq. (2), we obtain the expression for the gluon cluding light quarks. Methodologically, this approach condensate in the form is a generalization of low-energy QCDtheorems to 2 ∂F the case of finite temperature. The proposed method G (T,m0u,m0d)=4 . (4) ∂(1/g2) is quite general and model-independent and can be 0 used to study various nonperturbative phenomena in The system described by the partition function (2) QCDat finite temperature and in external fields. In is characterized by the set of dimensional parame- particular, an investigation of nonperturbative QCD ters M, T ,andm (M) and by the dimensionless 0q in a magnetic field (at T =0and T =0) on the basis charge g2(M), where the bare quark masses m of this method makes it possible to predict a nontrivial 0 0q and the coupling constant g2 are determined at the behavior of the quark condensate (freezing of the or- 0 ultraviolet-cutoff mass M. On the other hand, one der parameter for the chiral phase transition [52, 53]) can go over to a renormalized free energy F and, and of the gluon condensate in response to variations R by using the renormalization-invariance properties of in the magnetic field [54–56]. FR, recast (4) into a form that involves derivatives with respect to physical parameters, T and the renor- 2.1. Relation between Thermodynamic Pressure malized masses mq. For this, we will use а dimen- and Condensates in QCD: Renormalization-Group sional transmutation, which is a well-known prop- Equations and Low-Energy Theorems at T =0 erty of QCD. The dimensional transmutation leads to the appearance of the dimensional nonperturbative At a nonzero quark mass (mq =0), scale invari- parameter ance is broken even at the classical level. This implies ∞ that, in the ideal gas approximation, thermal excita- dαs π Λ=M exp , (5) tions of mesons will change the gluon condensate β(αs) with increasing temperature. In order to determine αs(M) PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 5 2005 728 AGASIAN 2 ∂FR where αs = g0/4π and β(αs)=dαs(M)/d ln M is qq¯ (T )= . (12) the Gell-Mann–Low function. Furthermore, it is well ∂mq known that the quark mass m0 has an anomalous dimension γ and depends on the scale M.The m By using the above relations, one can obtain low- renormalization-group equation for the running mass energy QCDtheorems at finite temperature.4) For the m0(M) has the form d ln m0/d ln M = −γm,andwe sake of simplicity, we consider the chiral limit m =0. use the modified-subtraction (MS) scheme, where β q We introduce the field σ(τ = x4, x), and γm are independent of the quark mass [59]. The renormalization-invariant mass is then given by − b0 a 2 σ(τ,x)= (Gµν (τ,x)) , (13) α s(M) 32π2 γmq (αs) mq = m0q(M)exp dαs , (6) and the operator Dˆ, β(αs) ∂ Dˆ =4− T . (14) where the indefinite integral is taken at the point ∂T α (M). Since the physical (renormalized) free energy s 2 is a renormalization-invariant quantity, its anomalous By differentiating (4) with respect to 1/g0 n times and dimension is zero. Thus, the quantity FR has only taking into account Eqs. (7), (13), and (14), we obtain a normal (canonical) dimension, which is equal to Dˆ n+1F = Dˆ nσ(0, 0) (15) four. By using the renormalization invariance of the quantity Λ, we can represent FR in the most general 3 3 = dτnd xn ... dτ1d x1 form T mu md × F =Λ4f , , , (7) σ(τn, xn) ...σ(τ1, x1)σ(0, 0) c. R Λ Λ Λ The subscript c in (15) indicates that only connected where f is a function of the ratios T/Λ,....From diagrams are included in the vacuum expectation relations (5), (6), and (7), we obtain value. A similar argument applies to an arbitrary op- ∂FR ∂FR ∂Λ ∂FR ∂mq erator Oˆ(x) constructed from quark fields and gluon 2 = 2 + 2 , (8) ∂(1/g ) ∂Λ ∂(1/g ) ∂mq ∂(1/g ) fields or from both of them. We have 0 0 q 0 n ∂ ˆ ∂m γm (αs) T − d O (16) q = −4πα2m q . ∂T 2 s q (9) ∂(1/g0 ) β(αs) 3 3 With allowance for expression (4), we find for the = dτnd xn ... dτ1d x1 gluon condensate in QCDthat [57] 3) ×σ(τn, xn) ...σ(τ1, x1)Oˆ(0, 0)c, 16πα2 G2(T,m ,m )= s (10) u d where d is a canonical dimension of the operator Oˆ. β(αs) ˆ ∂ ∂ If the operator O has an anomalous dimension as × 4 − T − (1 + γ )m F . well, the corresponding γ function must be taken into ∂T mq q ∂m R q q account. In the one-loop approximation, we have β(αs) → The above relations (11) and (12) are general and − 2 → − bαs/2π and 1+γm 1,whereb =(11Nc 2Nf )/3. are applicable at any temperature T . They relate the Thus, we arrive at the following expressions for the nonperturbative condensates to the free energy or, condensates: which is the same, to the sign-reversed pressure. The 32π2 free energy of the system is determined by the spec- G2(T )=− (11) b trum of possible quantum states in a specific phase. Thus, finding the behavior of the condensates versus ∂ ∂ × 4 − T − m F ≡−DFˆ , temperature reduces to evaluating the free energy of ∂T q ∂m R R q q the phase being studied. Further, we will explore the hadronic state of strongly interacting matter—that is, 3)An anomalous contribution to the trace of the the confining phase. energy–momentum tensor in QCDis related to g 4) the gluon condensate by the equation θµµ = Low-energy theorems in gluodynamics at finite temperature 2 2 (β(αs)/(16παs )) G (T,mu,md). were considered in [60]. PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 5 2005 NONPERTURBATIVE PHENOMENA IN QCD729 2.2. Quark and Gluon Condensates at Low vacuum energy density. Following the same lines of Temperature reasoning, we obtain The effective pressure (we normalize the parti- 2 − ˆ −32π − ∂ tion function to perturbation theory at T =0), from Dεvac = 4 mq (25) b ∂mq which the condensates qq¯ (T ) and G2(T ) can be q extracted with the aid of relations (11) and (12), is b0 2 1 2 given by × − G + mqqq¯ = G . 128π2 4 q Peff(T )=−εvac + Ph(T ), (17) where ε = θ /4 is the nonperturbative vacuum- In order to find the dependence of the quark and vac µµ gluon condensates on T , we use the Gell-Mann– energy density at T =0and ≡− Oakes–Renner (GMOR) relation [61] (Σ qq¯ = b | | | ¯ | θ = − G2 + m qq¯ (18) uu¯ = dd ): µµ 32π2 q q=u,d 2 2 −1 ¯ Fπ Mπ = (mu + md) uu¯ + dd (26) is the trace of the energy–momentum tensor. In 2 =(m + m )Σ. Eq. (17), Ph is the thermodynamic pressure in the u d hadronic phase; for a gas of massive pions, we have We then have d3p ∂ Σ ∂ − − − 2 2 = , (27) Pπ = 3T 3 ln(1 exp( p + Mπ/T )). 2 2 (2π) ∂mq Fπ ∂Mπ (19) ∂ Σ ∂ ∂ m =(m + m ) = M 2 , The quark and gluon condensates are then given by q ∂m u d F 2 ∂M2 π ∂M2 q q π π π qq¯ (T )=−∂P /∂m , (20) eff q (28) 2 ˆ G (T )=DPeff, (21) 2 ˆ 32π − ∂ − 2 ∂ D = 4 T Mπ 2 . (29) where the operator Dˆ was defined by Eq. (11), b ∂T ∂Mπ 32π2 ∂ ∂ Bringing together all of the above formulas, we finally Dˆ = 4 − T − m . (22) represent the condensates as [57, 58] b ∂T q ∂m q q Σ(T ) − 1 ∂Ph(T ) =1 2 2 , (30) Let us consider the case of zero temperature, T = Σ Fπ ∂Mπ 0. By using the low-energy theorem for the derivative G2(T )=G2 (31) of the gluon condensate with respect to the quark mass [27], 32π2 ∂ ∂ + 4 − T − M 2 P (T ). ∂ b ∂T π ∂M2 h G2 = d4xG2(0)¯qq(x) (23) π ∂m q Let us consider the thermodynamics of the hadronic 96π2 phase at low temperatures, T M . It was indicated = − qq¯ + O(m ), π b q above that, in this region, the main contribution to thermodynamic quantities comes from thermal exci- where O(mq) is a term that is linear in the light-quark 5) tations of massive π mesons. The pion-gas density at mass, we arrive at the relation a temperature T is given by ∂εvac b ∂ 2 1 3 = − G + qq¯ (24) d p 1 ∂m 128π2 ∂m 4 nπ(T )=3 . (32) q q (2π)3 exp( p2 + M 2/T ) − 1 3 1 π = qq¯ + qq¯ = qq¯ . 4 4 For the mean distance between the particles of a gas, −1/3 We note that three-fourths of the quark condensate we then have Lπ nπ . The dilute-gas approxima- stems from the gluon part of the nonperturbative tion is applicable if the mean distance between the particles, Lπ, is much shorter than the mean free path 5)Equation (24) includes corrections proportional to calculated with allowance for collisions, Lπ λππ. mq∂ qq¯ /∂mq ∼ O(mq ), which are small for light u For a hadron gas, the mean free path is well known and d quarks. [62–64] at low temperatures, in which case the gas PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 5 2005 730 AGASIAN Lπ/λππ Σ(T)/Σ 1.5 1.0 1.0 0.6 0.5 0.2 0 40 80 120 160 0 50 100 150 T, MeV T, MeV Fig. 1. Ratio of the mean distance between particles, Lπ , Fig. 2. Quark condensate Σ(T )/Σ as a function of tem- and the free path λππ in the pion gas as a function of perature: (solid curve) results of the calculation by for- temperature. mula (34); (dash-dotted curve) three-loop result with- in chiral perturbation theory (1) in the chiral limit for Nf =2, and (dashed curve) numerical three-loop result consists almost completely of pions. An analysis of of chiral perturbation theory at a nonzero quark mass (M = 140 MeV) [48]. pion–pion scattering reveals that, in the tempera- π ture range 50 PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 5 2005 NONPERTURBATIVE PHENOMENA IN QCD731 For Mπ T ,wehave 〈G2〉(T)/〈G2〉 ∞ 1.000 ∆G2 24 M 2T 2 1 4π2 M 2T 2 = − π = − π . G2 b G2 n2 b G2 n=1 0.998 (38) 2 0.996 In the chiral limit, Mπ =0,wehave∆G (T )=0in accordance with the fact that a free gas of massless particles is conformally invariant. In the nonrelativis- 0.994 tic (low-temperature) limit, T Mπ, we obtain 2 3 0 50 100 150 ∆G 24 M T Mπ = − π K (39) T, MeV G2 b G2 1 T 2 2 5/2 1/2 5/2 3/2 Fig. 3. Gluon condensate G (T )/ G as a function 3 · 2 π M T − →− π e Mπ/T . of temperature: (solid curve) results obtained by for- b G2 mula (37) at G2 =0.5 GeV4 and (dash-dotted curve) three-loop results of chiral perturbation theory in the At the same time, we indicated above that, within chiral limit [see Eq. (41)]. chiral perturbation theory at zero quark mass, the temperature dependence of the gluon condensate 2.3. Low-Temperature Relations in QCD arises only at the three-loop level because of the interaction between massless pions [51]. In [47, 48], Within the approach proposed above, one can de- the pressure of Nf massless quarks was found in rive thermodynamic relations for the anomalous and the three-loop approximation of chiral perturbation normal (quark-mass term) contributions to the trace theory. The result is of the energy–momentum tensor in QCD[57, 58]. Nf =2 At low temperatures, the main contribution to pres- P (T )ChPT (40) sure comes from thermal excitations of π mesons. In π2 T 2 2 Λ general, the pressure of the gas formed by particles of = T 4 1+4 ln p + O(T 10), mass M can be written in the form 30 12F 2 T π π 4 Pπ = T ϕ(Mπ/T ), (42) where the numerical value of Λp 275 MeV [48] was where ϕ is a function of the ratio Mπ/T ; in the case obtained at mu = md =0 and fixed ms.Byusing Eqs. (21), (22), and (27), the following expression of a dilute gas, it is given by expression (33). The following relation then holds: (Leutwyler [51]) can be obtained for the gluon con- densate: ∂ 2 ∂ 2 ∂Pπ 4 − T − M Pπ = M . (43) G2(T ) 16π4 T 8 Λ 1 ∂T π ∂M2 π ∂M2 ChPT =1− ln p − . π π 2 4 2 G 135b Fπ G T 4 With the aid of Eqs. (20), (21), (24), (25), and (43), (41) one can obtain ∂P 32π2 ∂P Figure 3 shows the gluon condensate as a function ∆qq¯ = − π , ∆G2 = M 2 π , (44) ∂m b π ∂M2 of temperature at G2 =0.5 GeV4.6) One can see q π that, in the low-temperature region, thermal exci- where ∆qq¯ = qq¯ (T ) −qq¯ and ∆G2 = tations of π mesons change the gluon condensate G2(T ) −G2. Taking into account relation (28), only slightly. From the physical point of view, the we can recast (44) into the form smallness of the thermal shift ∆G2(T ) can be 2 2 32π associated with a great numerical value of G ∆G2 = − m ∆qq¯ . (45) b q at T =0 in relation to the characteristic param- q eters in the finite-temperature hadronic phase— 2 2∝ 2 2 2 −3 By differentiating expression (45) with respect to T , that is, ∆ G / G Mπ Tc / G 10 at Mπ = 2 4 we arrive at 0.14 GeV, Tc =0.17 GeV, and G =0.5 GeV . 2 2 ∂ G −32π ∂ qq¯ 6)This corresponds to the standard value of αs F 2 = = mq . (46) π µν ∂T b ∂T 0.012 GeV4 [66]. q PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 5 2005 732 AGASIAN 32π2 ∂P This equation can be rewritten in the form [58] = M 2 i . g q i 2 ∂θ ∂θ b ∂Mi µµ = µµ , (47) i=π,K,η ∂T ∂T For the temperature shift of the condensates, we then q g where θµµ = mqqq¯ and θµµ =(β(αs)/(16π × obtain [68] 2 2 αs)) G are, respectively, the quark and the gluon ∆Σ(T ) ∂P = g (52) contribution to the trace of the energy–momentum Σ ∂m tensor. u As was mentioned above, the π meson plays a spe- 1 ∂Pπ ∂PK 1 ∂Pη = 2 2 + 2 + 2 , cial role in QCDthermodynamics because its mass Fπ ∂Mπ ∂MK 3 ∂Mη is numerically close to the phase-transition temper- ∆Σ (T ) ∂P ature, while the masses of the remaining hadrons s = g (53) are severalfold larger than Tc.Theeffect of thermal Σs ∂ms excitations of massive hadrons on the features of 1 ∂PK 4 ∂Pη the quark and gluon condensates was investigated = 2 2 + 2 . in [67] within the conformally generalized nonlinear Fπ ∂MK 3 ∂Mη σ model. It was shown that, at low temperatures, We note that, since the π meson does not carry T<100 MeV, the contribution to qq¯ from massive strangeness, it does not participate in the evaporation states is below 5%.AtT = Mπ, this contribution is of the strange condensate (ss¯ ). The leading contri- approximately 10%. bution to ∆ss¯ (T ) is associated with the excitation Let us now consider leading corrections to the of the lightest particle carrying strangeness—that low-temperature relations (45)–(47). It is obvious is, with the K meson. The K-meson mass (MK = that these leading corrections are associated with the 493 MeV) is severalfold larger than the pion mass pion–pion interaction, since, in this case, its contri- Mπ. The contribution of the K meson to ∆Σs(T ) can bution to pressure is proportional to the square of easily be derived on the basis of the low-temperature − the pion density, ∝e 2Mπ /T . The inclusion of the s expression for the light-quark condensate ∆Σ(T ) quark also leads to contributions to pressure that are [see (36)] upon the obvious substitutions Mπ → MK − − proportional to e MK /T and e Mη/T and which are and Fπ → FK and the multiplication by a factor associated with thermal excitations of, respectively, K of 4/3. For the ratio of the condensates, we then and η mesons. The pressure in the hadron phase can have [68] then be represented in the form 1/2 2 ∆Σ (T )/Σ 4 M F s s K π (Mπ−MK )/T Ph(T )=Pg(T )+Pππ(T ), (48) = e , ∆Σ(T )/Σ 3 Mπ FK (54) P (T )= P (T ), (49) g i ∼ i=π,K,η and this ratio is about 0.13 at T 140 MeV.Employ- ing the same procedure as in deriving Eqs. (45) and 4 where Pi(T )=giT ϕ(Mi/T ) is the pressure of the taking into account Eqs. (50)–(53), we further have i-meson gas (i = π,K,η)andgi is the number of b degrees of freedom of the ith state with g =3, g = − ∆G2 = m ∆qq¯ +2P (55) π K 32π2 q ππ 4,andgη =1. The pressure associated with the pion– q=u,d,s pion interaction can be written in a general form as 1 ∂P + M 2 K . M 2 M π 2 4 π π 2 ∂MK Pππ = T 2 f , (50) Fπ T Let us introduce the functions ± g ± q − g ± q where f is a function of the ratio Mπ/T and the factor θ (T )= θµµ θµµ (T ) θµµ θµµ (0). (56) 2 2 Mπ /Fπ stems from the vertex corresponding to the + pion–pion interaction. Following the same lines of Here, θ (T ) is the finite-temperature part of the reasoning as in deriving relation (43), we can find, total trace of the energy–momentum tensor; that is, + tot − − with the aid of the Gell-Mann–Oakes–Renner for- θ (T )=∆θµµ (T )=ε 3P ,whereε = TdP/dT mula for three quark flavors, that P is the energy density. The function − DPˆ (T ) (51) θ (T ) g δ (T )= (57) θ θ+(T ) 2 32π − ∂ − ∂ = 4 T mq Pg(T ) 1 2 ∂PK − b ∂T ∂mq = 2Pππ + Mπ 2 (ε 3P ) q=u,d,s 2 ∂MK PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 5 2005 NONPERTURBATIVE PHENOMENA IN QCD733 2 can then be considered as a measure of the deviation decreases, ξm(T ) ∝ 1/(g (T )T ). This behavior of the from the low-temperature relation (45). Let us esti- magnetic correlation function explains the magnetic- mate this quantity numerically. For Pππ, we have [44] confinement phenomenon within the stochastic- 1 M 2 vacuum model. The resulting temperature depen- P = − π [g (M /T )]2 , (58) dence of the spacelike string tension agrees perfectly ππ 6 F 2 1 π π with lattice data [71–73] over the entire temperature d3p 1 range. g = , (59) 1 3 ω /T (2π) ωπ e π − 1 In view of all of the facts listed above, it is 2 2 necessary that a unified approach within Nf =2 ωπ = p + Mπ . QCDensure the following at the critical point Tc ∼ The pressure of the i-meson gas can be represented 175−190 MeV: an energy-density value of εc ∼ as 1−1.5 GeV/fm3; the vanishing of the quark con- d3p densate qq¯ (T ); and the evaporation of only half of P = −g T ln(1 − exp(− p2 + M 2/T )). i i (2π)3 i the gluon condensate (its chromoelectric component (60) responsible for string formation and confinement), this being required for the magnetic-confinement Under the assumption that Mπ = 140 MeV, MK = phenomenon to be preserved. 493 MeV, and Fπ =93MeV, numerical calculations yield In this section, we study the temperature proper- ties of the quark and gluon condensates within the δθ(T<140 MeV) < 0.04. (61) approach based on the description of the confining phase in terms of a hadron resonance gas and show that the aforementioned phenomena can be quanti- 3. HADRON RESONANCE GAS tatively explained within this approach upon appro- ANDNONPERTURBATIVE QCDVACUUM priately taking into account the temperature shift of Lattice calculations in finite-temperature QCD hadron masses [74]. show that deconfinement and the restoration of chi- We will consider QCDinvolving two light quark ral invariance occur at the same temperature and flavors. It was shown above that, if the pressure in N = that, in the case of two light quark flavors ( f the hadron phase, P (T ),isknown,thetemperature 2), the critical temperature falls within the interval h − dependence of the quark condensate can be deter- Tc 170 190 MeV [49, 69]. From an analysis of mined by using the Gell-Mann–Oakes–Renner re- QCDthermodynamics on a lattice and from experi- lation. The result is mental data on high-energy collisions, one can also qq¯ (T ) 1 ∂P (T ) deduce that the energy density of the system at the =1− h , (62) ∼ 2 2 point of the quark–hadron phase transition is ε(Tc) qq¯ Fπ ∂Mπ 1−1.5 GeV/fm3. where Fπ =93MeV is the axial pion decay constant. Recent numerical simulations on a lattice for The respective temperature dependence for the gluon SU(3) gauge theory featuring no quarks and for condensate, G2(T ) ≡(gGa )2(T ), was obtained N =2 QCDrevealed a strong suppression of the µν f above (see Subsections 2.1 and 2.2) in the form chromoelectric component and a weak growth of the chromomagnetic component of the gluon con- G2(T )=G2 densate with temperature above the critical temper- ature Tc [70]. The temperature dependence of the 32π2 ∂ ∂ gauge-invariant bilocal correlation function for the − − 2 + 4 T Mπ 2 Ph(T ), chromomagnetic-field strengths and the spacelike b ∂T ∂Mπ T string tension σs( ) was determined analytically where b =11N /3 − 2N /3=29/3 and G2∼ in [22] (see Section 4 below). As a consequence, it c f 0.5−1 GeV4. The expressions for qq¯ (T ) and G2(T ) was found that, in the region T<2Tc, the chromo- magnetic condensate grows slowly with temperature, in Nf =3 QCDwere obtained in [68]. Thus, we 2 2 can state that, if the pressure Ph as a function of H (T )=H coth(M/2T ),whereM =1/ξm 1.5 GeV is the inverse magnetic correlation length, temperature and the π-meson mass is known, one which is independent of temperature for T<2T .In can find the temperature dependence of the quark and c gluon condensates in the hadronic phase. the region T>2Tc, the amplitude of the chromomag- netic correlation function grows with temperature, In order to describe QCDthermodynamics in H2(T ) ∝ g8(T )T 4, while the correlation length the confining phase, we use the model of a hadron PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 5 2005 734 AGASIAN resonance gas.7) In this approach, the thermody- string tension, was proposed in [79]. For a specific namic properties of the system are determined by the choice of parameters, this formula describes well the total pressure of relativistic Bose and Fermi gases masses of all particles considered in [79]. describing thermal excitations of massive hadrons. One must also consider that the hadron masses The main motivation of employing the approach in change with increasing temperature. Within the question is that all significant degrees of freedom finite-temperature conformally generalized nonlinear of strongly interacting matter have been thereby σ model involving light and massive hadrons [67], it included in the consideration. Moreover, the use of was shown that the temperature shift of the hadron the hadron-resonance spectrum takes effectively into masses can be taken into account with the aid of the account interaction between stable particles. At the substitutions same time, the description of multiparticle production mh → mh(χT /χ0),Mπ → Mπ χT /χ0, (67) in heavy-ion collisions within the hadron-resonance- 2 2 1/4 gas model [76–78] leads to good agreement with χT /χ0 = G (T )/G , experimental data. Thus, the pressure in the confining phase can be represented in the form where χ is the dilaton field. That the dependence of the π-meson mass differs from those for other particles d3p −ωi/T 8) Ph = T giηi ln 1+ηie , (63) is a manifestation of its Goldstone nature. In the (2π)3 chiral limit, m → 0, the above relation for the hadron i q masses is a rigorous corollary of low-energy QCD 2 2 ωi = p + mi , theorems [27]. Formulas (31) and (62)–(67) determine the ther- +1, fermions, modynamic properties of the system in the hadronic ηi = (64) −1, bosons, phase and make it possible to calculate the quark and gluon condensates over the entire temperature range where gi is the spin–isospin degeneracy factor (for below the critical temperature Tc. example, gπ =3, gN =8, ...). The energy density We take into account all hadron states of mass − εh = T∂Ph/∂T Ph in the hadronic phase is given below 2.5 GeV for mesons and below 3.0 GeV by for baryons. In all, there are 2078 states (with al- d3p ω lowance for the degeneracy factors gi). It is obvious ε = g i . (65) that, at temperatures below the pion mass, T< h i (2π)3 exp(ω /T )+η i i i Mπ = 140 MeV, the main contribution to thermo- dynamic quantities comes from thermal excitations In order to study the condensates in the confining of π mesons, because the remaining states are phase quantitatively, it is necessary to know the pres- considerably heavier and are therefore exponentially sure Ph as a function of the light-quark mass (in the suppressed by a Boltzmann factor proportional to case of Nf =2) or, which is the same, as a function exp{−mh/T }.AtT>Mπ, however, a large number of the π-meson mass. Within the hadron-resonance- of heavy states begin exerting a significant effect on gas model, this is equivalent to knowing the depen- the thermodynamics of the system. In Fig. 4, the dence of the masses of all resonances on the π-meson pion contribution is depicted by the dashed curve. mass. This dependence was studied numerically in One can see that, up to the temperature of T = lattice calculations, and the five-parameter formula 120 MeV, the main contribution to the pressure Ph √ does indeed come from pions. At higher temperatures, m = N a x σ (66) i u 1 the main contribution to the pressure is due to all m + h , of the remaining hadron states. In Fig. 4, we also 1+a x + a x2 + a x3 + a x4 2 3 √ 4 5 present lattice data from [81] for the pressure Ph in x ≡ Mπ/ σ, Nf =2QCD. One can see that, in the temperature √ region T PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 5 2005 NONPERTURBATIVE PHENOMENA IN QCD735 4 〈 〉 〈 〉 Ph/T qq (T)/ qq 1.0 5 0.6 3 0.2 1 0.05 0.15 0.25 0.1 0.2 0.3 0.4 T, GeV T, GeV Fig. 6. Quark condensate qq¯ (T )/ qq¯ as a function 4 Fig. 4. Pressure Ph/T as a function of temperature: of temperature: (solid curve) zero-temperature hadron (solid curve) zero-temperature hadron spectrum; (dashed spectrum and (dashed curve) spectrum with allowance for curve) spectrum with allowance for the chromoelectric the chromoelectric shift, χT /χ0 =0.84. χ /χ =0.84 shift, T 0 ; (dash-dotted curve) results ob- tained with allowance for only pion excitations; and (dot- ted curve) lattice data borrowed from [81]. The vertical 〈G2〉(T)/〈G2〉 straight line corresponds to the critical temperature. 1.0 ε 3 h, GeV/fm 2.0 0.8 1.5 0.6 1.0 0.05 0.15 0.25 0.5 T, GeV Fig. 7. Gluon condensate G2 (T )/ G2 as a function 0.10 0.15 0.20 of temperature: (solid curve) zero-temperature hadron spectrum and (dashed curve) spectrum with allowance T, GeV 2 for the chromoelectric shift, χT /χ0 =0.84,at G = 4 Fig. 5. Energy density εh as a function of tempera- 0.87 GeV [82]. The shaded band corresponds to the ture: (solid curve) zero-temperature hadron spectrum and values of G2 =0.5−1.0 GeV4. (dashed curve) spectrum with allowance for the chromo- electric shift, χT /χ0 =0.84. 2 0.02G . With increasing temperature, T>Mπ, T 175 the gluon condensate decreases sharply; in a rather at MeV—that is, in the region of the phase- ∼ transition temperatures obtained in lattice calcula- narrow temperature interval, ∆T 50 MeV, there tions [16]. occurs a main change in the gluon condensate by 50% Figures 6 and 7 show the temperature depen- about . In accordance with the above, we present dences of, respectively, the quark and the gluon con- numerical results obtained with allowance for the temperature shift of the hadron masses by 16% densate. It is of importance that the temperature at 1/4 which the quark condensate vanishes is identical to (χT /χ0 =0.84 (0.5) ). We note that, even with- that at which half of the gluon condensate evaporates; out allowance for the decrease in the mass mh with with allowance for the chromoelectric shift of the temperature, the quark condensate and half of the hadron masses, this temperature is T 190 MeV. gluon condensate evaporate at the same temperature ∼ Strictly speaking, the temperature shift of the of T 215 MeV. gluon condensate must be found self-consistently (with the aid of the effective dilaton Lagrangian at 4. MAGNETIC CONFINEMENT T =0) with allowance for the shift of the hadron IN FINITE-TEMPERATURE SU(N) masses (see [67]). However, the results of numerical YANG–MILLS THEORY calculations show that, up to temperatures of T ∼ Mπ, the gluon condensate decreases very slowly; It is well known that, in SU(N) non-Abelian 2 at T = Mπ, the temperature shift is ∆G (T ) ≈ gauge theory, the deconfining phase transition from PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 5 2005 736 AGASIAN the low-temperature glueball phase to the hot-gluon- the behavior of these quantities in 4d gauge theory at matter phase occurs at a finite temperature. Upon high temperatures. In particular, the spacelike string the phase transition at the critical point Tc, thermo- tension σs(T ) at high temperatures, T Tc, agrees, dynamic characteristics of the system—such as the to a high accuracy, with the string tension σ3 calcu- energy density ε, the heat capacity, and the nonide- lated on a lattice in 3d gauge theory [72, 95]. 4 ality parameter (ε − 3P )/T —change their behavior In this section, we study the properties of the mag- substantially [15, 73, 83]. Moreover, it is known that, netic sector of 4dSU(N) Yang–Mills theory at finite in non-Abelian gauge theories, a rearrangement of temperature. It is shown that the finite-temperature the nonperturbative gluon vacuum occurs upon the behavior of physical quantities in the magnetic sector phase transition. For SU(3) gauge theory and for is qualitatively different in the low- and the high- QCDinvolving two quark flavors, the behavior of the temperature region. By these, we mean, for the sake of gauge-invariant two-point correlation function for definiteness, the region T<2Tc and the region T> the gluon-field strengths in the temperature region 2Tc, respectively. The factor of “2” in the expression around the phase-transition point was studied in [70] 2Tc should not be treated as an exact value, but, of by means of numerical simulations on a lattice. The course, it cannot be set to unity or, say, ten. (This is- data obtained in this way clearly demonstrated a sue is discussed below in Subsection 4.3.) In the low- strong suppression of the electric component of the temperature region, we will obtain analytic expres- correlation function and the invariability of its mag- sions for the magnetic correlation function, the mag- netic component. The magnetic correlation function netic condensate, and the spacelike string tension. and the magnetic gluon condensate, which is related We also calculate the relative contribution of nonzero to it, undergo virtually no changes upon the transition Matsubara modes to σs(T ) and study the tempera- from the confining (T PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 5 2005 NONPERTURBATIVE PHENOMENA IN QCD737 At finite temperature, the Euclidean spacetime contributions, and only the nonperturbative compo- O(4) symmetry is broken to the spatial O(3) sym- nents of the correlation functions contribute to the metry, and the bilocal correlation function can be de- string tension (see a detailed discussion of these is- scribed in terms of independent electric and magnetic sues in [33]). The magnetic gluon condensate is also correlation functions. Let us consider the magnetic determined in terms of the nonperturbative contribu- component of the correlation function. In general, we tions to the correlation function and is expressed in H H 2 have terms of the functions D and D1 at x =0.Forthe H 2 magnetic condensate, we have Dik (x)= g trHi(x)Φ(x, 0)Hk(0)Φ(0,x) (70) H DH (0) = g2trH2≡H2/2=3(DH (0) + DH (0)), xix ∂D ii i 1 = δ (DH + DH )+ δ − k x2 1 , (75) ik 1 ik x2 ∂x2 where H2≡(gHa)2. where Hi = εijkGjk/2 is the chromomagnetic-field i operator. Lattice data obtained in [70] show that, at least in ≤ By using the well-known method that was pro- the region of measured temperatures, T 1.5Tc,the H {−| | } posed in [31] and which is similar to the method for nonperturbative functions D = B0 exp x /ξm H {−| | } considering timelike Wilson loops, one can obtain the and D1 = B1 exp x /ξm are virtually indepen- area law for a spacelike W loop of dimension L ξm dent of temperature. The correlation length ξm does (ξm is the magnetic correlation length); that is, not change and is equal to its value at T =0,the contribution D1 being strongly suppressed, B1 ≈ W (C)spat ∼ exp{−σsSmin}. (71) 0.05B0 B0.OnlyB0 grows slowly with T , and this 2 The spacelike string tension σs can be expressed in leads to a slight increase in the condensate H (T ) terms of the magnetic correlation function as [19] at low temperatures. 1 σ = d2xg2trH (x)Φ(x, 0)H (0)Φ(0,x), s 2 n n 4.2. Low Temperatures (72) Let us consider the way in which one can de- where Hn is the magnetic-field component orthogo- rive analytic expressions for the magnetic correlation nal to the spacelike plane of the contour. function and, accordingly, for the spacelike string ten- Substituting (70) into (72) and considering that, sion and magnetic gluon condensate as functions of H T . It is well known that the introduction of tempera- taken together, the terms involving D1 form a total ture T for a quantum-field system in thermodynamic derivative, we recast σ into the form s equilibrium is equivalent to the compactification of 1 the Euclidean time coordinate x with radius β = σ = d2xDH (x). (73) 4 s 2 1/T and to imposing periodic boundary conditions (PBC) on boson fields (antiperiodic for fermions). H We note that only the term D appears in this ex- The temperature vacuum expectation values are pression for σs.AtT =0, the spacelike string tension defined in a standard way as obviously coincides with the physical (timelike) one 1 −Sβ[A] owing to O(4) Euclidean invariance—that is, σs = σ, ...β = [DA] ...e , (76) where σ = (420 MeV)2 is the standard value of the Zβ string tension in QCD. PBC H where the partition function is The magnetic correlation function Dik can be ab β written in a form including the matrix Uadj in the −Sβ [A] 3 adjoint representation. Since integration in (69) is Zβ = [DA]e ,Sβ = dx4 d xLYM. performed along the straight line connecting the PBC 0 points x and y,wehave (77) H 2 a ab b Dik (x, y)= g Hi (x)Uadj(x, y)Hk(y) , (74) It was indicated above that, in the low-temperature ab a b region, the gluelump mass M (inverse magnetic cor- Uadj(x, y)=2tr(t Φ(x, y)t Φ(y,x)). relation length 1/ξm) is independent of T .Thus,we H This representation is used in lattice calculations of can use the zero-temperature expression Dik (x,x4) the correlation length (gluelump mass). with periodic boundary conditions to obtain the H H finite-temperature correlation function DH = The functions D and D1 include both perturba- ik ∝ 4 ∝ {−| | } 2 a ab b tive ( 1/x ) and nonperturbative ( exp x /ξm ) g Hi UadjHk β constructed from the gauge fields PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 5 2005 738 AGASIAN +∞ Hi(x,x4) and the phase factor Uadj[A(x,x4)], which × exp (−M 2β2n2/4s). are taken at a specifictimepointx4.Wehave n=−∞ +∞ DH H ik (x,β)= Dik (x,nβ). (78) By using the summation property n=−∞ +∞ b2n2 exp − (83) This relation is generally used in the case of non- 4s interacting fields. For example, the finite-temperature n=−∞ √ ∞ Green’s function for a free scalar field is defined as the 2 πs + 4π2n2 sum over Matsubara frequencies of the propagator = exp − s , b b2 at T =0. At the same time, interaction changes the n=−∞ particle mass, rendering it dependent on T .Inthe case under consideration, we take into account only we represent (82) in the form ∞ a kinematical variation of the correlation function in +∞ 2 − − the low-T region and, accordingly, consider it as a f(x,β)= dse as b/s, (84) Mβ periodized zero-temperature correlation function. It n=−∞ is physically clear that expression (78) is valid in the 0 region β>2ξm—that is, in the case where the two where a =1+4π2n2/M 2β2 and b = M 2x2/4. nearest exponential functions do not overlap. Further, we use the value of 1/ξm =1.5 GeV; therefore, the The integral on the right-hand side of (84) reduces K condition T<ξm/2 = 750 MeV must hold. In the to the modified Bessel function 1. Finally, we have ∼ low-temperature region, T<2Tc 600 MeV, which f(x,T)=2MT|x| (85) is considered in this section, this condition is always +∞ 1 satisfied. Moreover, a comparison of the results of the × | | 2 2 K1( x M + ωn). ensuing calculations for physical quantities with the M 2 + ω2 corresponding lattice data corroborates the validity of n=−∞ n formula (78) for T<2Tc. By considering the asymptotic behavior of the func- If use is made of the Poisson summation formula tion f at zero, we derive the temperature dependence ∞ ∞ 2 1 + + of the chromomagnetic condensate: H (T )= eiωnx = δ(x − nβ), (79) H2f(|x|→0,T). Considering that K (x → 0) → β 1 n=−∞ n=−∞ 1/x,wehave ωn =2πn/β, +∞ H2(T ) 1 M =2MT =coth . one can see that formula (78) represents the ex- H2 M 2 + ω2 2T pansion of the magnetic correlation function D in a n=−∞ n Fourier series in Matsubara frequencies; that is, (86) H 1 ˜ H For T M, this reduces to Dik (x,nβ)= Dik (x,ωn), (80) β − − n n H2(T )/H2 =1+2e M/T + O(e 2M/T ). (87) ˜ where D(ω)= dτ exp(iωτ)D(τ) is the Fourier This is in perfect agreement with lattice data from [70] transform of D. on a weak growth of the chromomagnetic condensate Let us consider the function at low temperatures, T ≤ 1.5Tc. ∞ + From expression (73) for σ and formula (85) for − 2 2 2 s f(x,β)= exp ( M x + n β ), (81) the spacelike string tension, we obtain n=−∞ +∞ M ≡ 1/ξ . σ (T ) 4T 1 m s = , (88) σ M (1 + ω2 /M 2)2 In order to go over in (81) to summation over the n=−∞ n Matsubara frequencies ωn =2πnT,weusetheinte- gral representation for the exponential function and where we have used the normalization condition σ (0) = σ recast (81) into the form s and considered that ∞ ∞ 1 ds 2 3 f(x,β)=√ √ exp (−s − M 2x2/4s) (82) x dxK1(cx)=2/c . π s 0 0 PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 5 2005 NONPERTURBATIVE PHENOMENA IN QCD739 ∆ Upon calculating the sum on the right-hand side n(T) of (88), we obtain an expression for σs(T ) at low 0.6 temperatures: σs(T ) sinh(M/T)+M/T 0.4 = . (89) σ cosh(M/T) − 1 −M/T 0.2 We have σs(T M)/σ =1+2(M/T +1)e + O(e−2M/T ). ≡ 0 From (88), the relative contribution ∆n(T ) 0.5 1.5 2.5 n=0 σs (T )/σs(T ) from nonzero Matsubara frequencies T/Tc to σs(T ) can be obtained as a function of T .Isolating n=0 Fig. 8. Relative contribution ∆n(T ) ≡ σs (T )/σs(T ) the n =0term in (88), we obtain from nonzero Matsubara frequencies to the spacelike 4T cosh(M/T) − 1 string tension σs(T ) as a function of T/Tc for SU(3) ∆ (T )=1− . (90) gauge theory. n M sinh(M/T)+M/T n=0 n=0 ∆ (2T ) 0.05—that is, the contribution of non- It is obvious that σs (T )+σs (T )=σs(T ) and n c ∆n(0) = 1. static (nonzero) Matsubara frequencies to σs is ap- By using formula (88), one can obtain the contri- proximately 5%, which is less than the error of lat- bution of the nth individual nonzero Matsubara mode tice calculations of the spacelike string tension [71– in the form 73]. Therefore, the main contribution to the dynamics of the magnetic sector of gauge theory at tempera- σn(T ) M 4 1 s = . (91) tures T>2Tc comes from the zero (static) Matsub- n=0 4 2 2 2 σs (T ) ωn (1 + M /ωn) ara mode. Thus, we can see that, for T>M/2π,thecontribu- Further, it should be noted that, at any temper- ature T , SU(N) gauge theory is in the magnetic- tion of the nth nonzero mode to σ (T ) dies out fast s confinement phase; therefore, the magnetic correla- with increasing T against the contribution of the zero tion function at high temperatures does not change mode, n =0: its exponential form. At the same time, it is phys- σn(T ) M 4 1 M 6 1 ically clear that the two-point correlation function s = − 2 + .... n=0 4 6 is determined by the amplitude and the correlation σs (T ) 2πT n 2πT n (92) length, which depend on temperature. Therefore, the magnetic correlation function at high temperatures canbewrittenintheform 4.3. High Temperatures 1 DH (x,T)= δ H2(T ) (93) ik 6 ik By definition, M =1/ξm is the inverse magnetic correlation length. The correlation lengths at T =0 ∂DH × exp (−|x|/ξ (T )) + O x2 1 . were determined in various lattice calculations [82, m ∂x2 96, 97]. At the same time, M is the mass of the lowest 1+− magnetic gluelump. Gluelumps [98] are It was mentioned above that the O(...) term does not not physical objects, and their spectrum cannot be contribute to the chromomagnetic condensate or the measured experimentally. However, they play an spacelike string tension. From (72) and (93), we then important role in nonperturbative QCDsince their obtain masses determine field correlation functions. In par- π 2 2 σs(T )= H (T )ξm(T ). (94) ticular, the string tension at T =0is related to the 6 gluelump masses. They were calculated analytically It was shown in [71–73] that, at T>2Tc,thetem- within QCDsum rules [99] and the QCDstring perature dependence of the spacelike string tension, model [100] and were numerically determined in σ (T ), agrees, to a high accuracy, with the behavior of various lattice calculations [101]. By using the entire s the string tension σ3 in 3d Yang–Mills theory. Three- body of these results, the correlation length ξm was dimensional Yang–Mills theory is a superrenormal- determined to fall between 0.1 and 0.2 fm. ized theory, and all physical quantities are determined We will use the value of M =1.5 GeV for the by a single dimensional coupling constant g2 = g2T . T =0 3 inverse magnetic correlation length at .Fig- In [71–73], it was found that ure 8 shows the function ∆n(T ) versus T/Tc (Tc = 2 270 MeV for SU(3) gauge theory). It can be seen that σs(T )=cσg (T )T, (95) PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 5 2005 740 AGASIAN 3 15 σ 2 )/ 10 T ( s σ 5 1 048 012 T/Tc Fig. 9. Spacelike string tension σs(T )/σ for SU(2) gauge theory as a function of T/Tc: (solid curve) results of calculations by formula (89) and (dashed curve) results of calculations by formulas (95) and (96). The displayed lattice data (points) were borrowed from [71]. 2.5 7 2.0 5 σ )/ T 1.5 ( s σ 3 1.0 1 0.5 024 012 T/Tc Fig. 10. Spacelike string tension σs(T )/σ for SU(3) gauge theory as a function of T/Tc: (solid curve) results of calculations by formula (89) and (dashed curve) results of calculations by formulas (95) and (96). The displayed lattice data (points) were borrowed from [73]. where, for g2, use was made of the two-loop expres- From Eq. (94), one can determine H2(T ).At sion high temperatures, the inverse magnetic correlation length 1/ξm behaves as −2 T b1 T g (T )=2b0 ln + ln 2ln , (96) 2 Λ b Λ 1/ξm(T )=cmg (T )T, (97) σ 0 σ 11N 34 N 2 where c is a constant. By using expressions (94), b = ,b= . m 0 48π2 1 3 16π2 (95), and (97), we find for the temperature dependence of the nonperturbative chromomagnetic condensate The constants cσ and Λσ were evaluated on the basis that of a comparison with lattice data on σs(T ) by using 2 8 4 H (T )=cH g (T )T , (98) a two-parameter fit. We have cσ =0.369 ± 0.014 and Λσ =(0.076 ± 0.013)Tc for SU(2) gauge the- where ± ± ory [71] and cσ =0.566 0.013 and Λσ =(0.104 6 2 2 cH = cσcm. (99) 0.009)Tc for SU(3) gauge theory [73]. Figures 9 π and 10 show the graphs of σ (T )/σ for, respec- s One can determine the constant c from expres- tively, SU(2) and SU(3) gauge theories versus T/T H c sion (99). For example, we have c ≈ 0.37 [71] and SU(2) SU(3) σ (Tc = 290 MeV, Tc = 270 MeV). The solid cm ≈ 0.92 [102] in SU(2) Yang–Mills theory; there- and dashed curves represent the results of the cal- fore, the result is cH ≈ 0.22. It should be noted, how- culations by, respectively, formula (89) and formu- ever, that the quantity cH must be obtained from a las (95) and (96). The displayed lattice data were numerical lattice calculation of the thermal magnetic borrowed from [71, 73]. correlation function. PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 5 2005 NONPERTURBATIVE PHENOMENA IN QCD741 The result presented in (98) is natural from the The growth of the magnetic condensate with in- point of view of 3d gauge theory. It was indicated creasing temperature, H2(T ) ∝ T 4, is thermody- above that, at high temperatures, the thermal be- namically advantageous for a quantum-field system. havior of physical quantities within 4dSU(N) gauge The vacuum energy density is related through the theory coincides with their behavior in 3dSU(N) anomaly in the trace of the energy–momentum tensor gauge theory. Therefore, the nonperturbative chro- to the magnetic condensate by the equation εvac = 2 2 momagnetic condensate is determined by the dimen- θ00 = −(b/64π )H ,whereb =11N/3,andan 2 2 sional coupling constant g3,andwehaveH (T )= increase in H reduces the vacuum energy density. × 8 const g3. Strictly speaking, physical expectation values in superrenormalized 3d Yang–Mills theory 5. BILOCAL CORRELATION FUNCTION either are zero or are determined by the corresponding FOR GLUON-FIELDSTRENGTHS power of the dimensional coupling constant g3.The IN THE INSTANTON VACUUM AT FINITE vanishing of the magnetic condensate at high tem- TEMPERATURE peratures leads to σs =0, this contradicting lattice data on magnetic confinement. It follows that, at high In this section, we study the contribution of in- temperatures (in the scaling region for the space- stantons to the bilocal correlation function at finite temperature [103, 104]. We show that the correla- like string tension), the chromomagnetic correlation tion length for the caloron bilocal correlation function function has the form (that is, the bilocal correlation function for instan- H 1 8 4 2 tons at finite temperature) depends quite strongly D (x,T)= δ cH g (T )T exp(−cmg (T )T |x|). ik 6 ik on temperature in the lowest order in the caloron (100) density. This is not compatible with data from lattice In the high-temperature region, its amplitude grows calculations—it follows from these data that the cor- relation length for the chromomagnetic-field bilocal T 4 in proportion to with increasing temperature, correlation function is independent, within the errors, while the correlation length decreases in proportion of temperature over the entire temperature range from to 1/T . zero to Tc [70]. This discrepancy between the results By using the above results, one can estimate the makes it possible to set an upper bound on the possi- region of temperatures at which there occur a change ble instanton density in QCD. in the regime in the magnetic sector of SU(N) gauge We have derived above the analytic expression theory and the transition to the description of dynam- for the temperature dependence of the chromo- ics in terms of static correlation functions—that is, magnetic bilocal correlation function. It was found to 3d gauge theory. Physically, it is clear that, if the that the chromomagnetic condensate grows slowly compactification radius β (inverse temperature 1/T ) with increasing temperature up to 2Tc, while the becomes much less than the characteristic dimension correlation length ξm(T ) is independent of T in this l of the system (it is determined, for example, by the temperature range. At high temperatures, T>2Tc, magnetic-string thickness or by the characteristic di- the condensate behaves as H2(T ) ∝ g8(T )T 4,and mension of a magnetic gluelump, l =1/ σs(T )), the the magnetic correlation length decreases with in- correlation functions cease to feel the time coordinate creasing temperature according to the law ξm(T ) ∝ x4, becoming static ones. 1/(g2(T )T ). There exists a simple physical ex- Thus, we can estimate the temperature at which planation for the temperature independence of the ≤ the regime changes by employing the condition T = correlation length at T Tc. As was mentioned in σ (T ). For the sake of definiteness, we consider Section 4, the inverse magnetic correlation length s +− SU(3) gauge theory. In the region of low tempera- M =1/ξm is the mass of the lightest 1 magnetic M ≈ 1.5 tures, we find T− =1.42T from the numerical so- gluelump, GeV. Since the phase-transition c temperature is much smaller than the gluelump mass, lution to the equation T− = σs(T−) with σs(T ) Tc M, it is physically clear that the temperature determined by expression (89). At the same time, one dependence of the gluelump mass and, hence, of the can see that, at high temperatures, the solution to correlation length must be very weak over the low- the equation T+ = σs(T+) where σs(T ) is given temperature region (T PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 5 2005 742 AGASIAN is the order parameter for the deconfining phase 1 1 ∂Π 1 ∂2Π g (x)= + (103) transition; therefore, it is the correlation function for 1 r Π ∂r Π ∂τ2 † the Polyakov loops, L(x)L (0), that changes its 1 ∂Π 2 1 ∂Π 2 behavior significantly at the temperature T = T . − + , c Π2 ∂τ Π2 ∂r At finite temperature, O(4) Euclidean spacetime 2 ∂Π ∂Π 1 ∂2Π symmetry is broken to O(3) symmetry, and the bilocal − g2(x)= 2 + , correlation function is described by independent elec- Π ∂r ∂τ Π ∂r∂τ tric and magnetic correlation functions, which can 2 ∂Π 2 1 ∂2Π 1 1 ∂Π g (x)= − + . be expressed in terms of four independent functions 3 Π2 ∂r Π ∂r2 r Π ∂r DE(x2), DE(x2), DH (x2),andDH (x2) as 1 1 The chromoelectric field is equal to the chromomag- 2 † g tr Ei(x)Φ(x, y)Ej (y)Φ (x, y) (101) netic field by virtue of the self-duality of the caloron field, Fµν = F˜µν . ∂DE ∂DE = δ DE + DE + z2 1 + z z 1 , Further, we must calculate the parallel transporter ij 1 4 ∂z2 i j ∂z2 for the caloron. The result is † 1 g2tr H (x)Φ(x, y)H (y)Φ (x, y) i j µ Φ(x, y)=P exp igz dsAµ(y + sz) (104) ∂DH ∂DH H H 2 1 − 1 0 = δij D + D1 + z 2 zizj 2 , ∂z ∂z 1 − µ a a where Hi = εijkFjk/2 is the chromomagnetic-field = P exp iz η¯µν t ds∂ν ln Π(y + sz) operator, Ei = F4i is the chromoelectric-field oper- a a 0 ator, Fµν = Fµν t is the gluon-field-strength tensor, 1 and = P exp − ita ds η¯a z ∂ ln Π +η ¯a z ∂ ln Π 1 ij i j i4 i 4 µ 0 Φ(x, y)=P exp igz dsAµ(y + sz) (102) 0 a +¯η4jz4∂j ln Π , is the parallel transporter along the straight line con- necting the points x and y; here, z = x − y.Inthe z = x − y. case where these correlation functions are calculated E H for a self-dual field, it is obvious that D1 = D1 =0 With an eye to a comparison with data from lattice E H and D = D (see [105]). Therefore, it is sufficient to calculations, we consider the case of z4 =0.Wethen consider only the chromomagnetic correlation func- have tion. Φ(x, y) (105) E The very fact that, for instantons, we have D = 1 H D implies that the nonperturbative vacuum can- = P exp −ita ds η¯a z ∂ ln Π +η ¯a z ∂ ln Π . not be a simple superposition of instanton–anti- ij i j i4 i 4 instanton field configurations because, as is known 0 from lattice calculations, the chromomagnetic and In general, we must evaluate the path-ordered chromoelectric condensates manifest different tem- integral. However, there exist special cases where perature dependences. In particular, the chromoelec- the operators in the exponent at different values of tric condensate decreases sharply at the tempera- s commute, so that the evaluation of the integral ture of the deconfining phase transition, while the reduces to the calculation of an ordinary exponen- chromomagnetic condensate undergoes virtually no tial. It is obvious that this occurs at τ =0, β/2, changes [70]. and β since ∂4 ln Π(r, τ =0;β/2; β)=0. The single- The one-instanton solution at finite temperature, instanton field is yet another case where this is so the so-called caloron, can be represented in the well- inst inst since ∂4 ln Π = τ(1/r)(∂ ln Π /∂r). known form [106] In order to calculate the contribution of a dilute a a − a abc b c caloron gas to the two-point correlation function in Fµν = ∂µAν ∂νAµ + gε AµAν, x x x the lowest order in the density, it is necessary to gHa = −δ g (x)+ε j g (x)+ a i g (x), calculate the contribution of a single caloron and to i ai 1 aij r 2 r2 3 a a average the result over its position or, which is the Ei = Hi , same, to average it over y at a fixed value of z = x − y. PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 5 2005 NONPERTURBATIVE PHENOMENA IN QCD743 Averaging over y reduces to evaluating the integral dinst(τ, z2) β 30 4 3 d y = d y dy4. 0 Averaging over the three-dimensional vector y does 20 not involve any difficulties. As to the integral with respect to y4, it cannot be calculated since, by virtue of the aforesaid, the correlation function can be cal- 10 culated numerically only at τ =0, β/2,andβ.Nev- ertheless, we will see from a comparison with the single-instanton case that, at low temperatures, it is sufficient to perform calculations at τ =0and that, 0 at high temperatures, the correlation length at τ =0 00.4 0.8 1.2 z, fm coincides with that at τ = β/2, so that averaging over τ is trivial. Fig. 11. Function dinst(τ,z2) at τ =0, 0.15, 0.3, 0.45, Letusdeterminethefunctionsd and d .Wehave 0.6 fm for ρ =0.3 fm (the upper curve corresponds to the 1 value of τ =0). 2 † g tr Hi(x)Φ(x, y)Hj(y)Φ (x, y) (106) y ξτ, fm 2 ∂d1 − ∂d1 = δij d + d1 + z 2 zizj 2 , ∂z ∂z 0.3 2 where z4 =0and the functions d and d1 depend on z 2 2 and y4 ≡ τ: d = d(τ,z ) and d1 = d1(τ,z ).Averag- ing over y4,wehave 0.2 β 2 H 2 dτd(τ,z )=D (z ), (107) 0.1 0 β 2 H 2 dτd1(τ,z )=D1 (z )=0, 0.2 0.6 1.0 β 0 , fm DH (z2 =0)=4π2/3. Fig. 12. Correlation length ξτ for the functions d(τ = 2 2 The last equality follows from the relation 0,z ) and d(τ = β/2,z ): (solid curve) ξτ=0(β) and (dashed curve) ξτ=β/2(β); ρ =0.3 fm. β g2 d3y dy F a 2 =32π2 4 µν 0,z2 =0)and DH (z2)/DH (z2 =0), which are nor- 0 malized to unity, are nearly coincident. The point is that an instanton is a well-localized field configura- (by virtue of the self-duality of the field, this integral is proportional to the topological charge ∼ d4yFF˜). tion, its field decreasing fast away from its center. This is illustrated in Fig. 11, where the function dinst(τ,z2) Let us define the correlation length ξτ as the coef- calculated for a single instanton is depicted at sev- ficient in the exponent of the exponential function that eral values of τ. Since the amplitude of the function provides the best fit to the function d(τ,z2);thatis, d(τ,z2) decreases fast with increasing τ,theprofile −| | d(τ,z2) e z /ξτ . (108) of the function DH (z2) is actually determined by the function d(τ =0,z2). This is also true for a caloron As was discussed above, we will calculate only in the region β ρ, because, in this case, instantons d(τ =0,z2) and d(τ = β/2,z2). However, this in- in a chain along the time axis are well separated. At formation is sufficient for determining the correlation smaller values of β—that is, at β ≤ 2ρ—the func- length for the function DH (z2). Indeed, averaging tions d(τ =0,z2) and d(τ = β/2,z2) have virtually over τ shows that the functions d(τ =0,z2)/d(τ = the same correlation length (see Fig. 12). Thus, av- PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 5 2005 744 AGASIAN ξ 1.0 τ=0, fm 0.8 0.26 0.6 0.4 0.22 0.2 0 00.4 0.8 1.2 z, fm 0.18 0.1 0.3 0.5 2 2 Fig. 13. Ratio d(τ =0,z )/d(τ =0,z =0) at β = T, GeV 0.05, 0.1, 0.2, 0.3, 0.5, 0.86, ∞ fm−1 for ρ =0.3 fm. The maximum correlation length (upper curve) corresponds Fig. 14. Correlation length ξτ=0(T ) as a function of to zero temperature (β = ∞)—that is, to the instanton temperature for ρ =0.3 fm. contribution. conclude that an instanton gas does not correctly de- eraging over τ is trivial and merely leads to a change scribe the true vacuum of gluodynamics. There exist in the amplitude, which is determined by the condition a few possible explanations of lattice data. −4 DH (z2 =0)=4π2/3. (i) The instanton density is much less than 1 fm , and the main contribution to the correlation function Taking into account the aforesaid, we will con- comes from other nonperturbative fields,whichen- sider only the function d(τ =0,z2), whose correla- sure confinement. This pattern is in accord with the tion length is close to the correlation length for the stochastic vacuum model. function DH (z2) at all values of β.Atτ =0, β/2,and (ii) The instanton density is so high that higher β, the chromomagnetic correlation function has the order corrections in density change the result quali- form tatively, leading to a temperature-independent corre- lation length. 2 † g tr Hi(x)Φ(x, y)Hj(y)Φ (x, y) (109) (iii) The configuration of the instanton ensem- y ble is rearranged in such a way that instantons and anti-instantons form molecules, and the contribution = g2 d3y Ha(x)U ab(x, y)Hb(y) , i j of these instanton–anti-instanton molecules to the † bilocal correlation function differs significantly from U ab(x, y)=tr taΦ(x, y)tbΦ (x, y) the contribution of the instanton gas. This binding into pairs can have some bearing on the deconfin- 1 1 = δab cos(2φ) − εabcnc sin(2φ)+nanb sin2(φ), ing phase transition. For example, this occurs in the 2 2 three-dimensional model involving a Higgs field in 1 the adjoint representation, for which it was shown 1 a a 1 ∂ ln Π(y + sz) φ = n η¯ijziyj ds . that, in the deconfining phase, instantons and anti- 2 r ∂r instantons are bound into molecules [107–109]. The 0 formation of instanton–anti-instanton molecules in The correlation function calculated by formula (109) the QCDvacuum and its relation to the phase tran- at various values of β isshowninFig.13.Thecorre- sition restoring chiral symmetry are discussed in the lation length as a function of temperature is given in review article of Schafer and Shuryak [110]. Fig. 14. (iv) In recent years, a totally different scenario has been discussed for the caloron vacuum in the confin- The bilocal correlation function at finite tempera- ing phase (T PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 5 2005 NONPERTURBATIVE PHENOMENA IN QCD745 3 In principle, one can consider yet another possi- hadron resonance gas is εh(Tc) ∼ 1.5 GeV/fm . bility where the instanton-size distribution also de- Upon taking into account the chromoelectric shift of pends on temperature, and this dependence is such the hadron masses, the critical temperature becomes that the resulting correlation length is independent Tc 190 MeV. of T . However, there are lattice data [115] accord- ing to which the instanton-size distribution does not The magnetic-confinement phenomenon has been change within the errors over the entire temperature studied in 4dSU(N) Yang–Mills theory at finite range from zero to Tc. temperature. The temperature dependence of the gauge-invariant bilocal correlation function for the chromomagnetic-field strength and the spacelike 6. CONCLUSION string tension σs(T ) hasbeenderivedinananalytic We have considered a number of the most charac- form. It has been shown that, in the region T<2Tc, teristic nonperturbative phenomena in finite-tempe- the chromomagnetic condensate grows slowly with rature QCD. We have not touched upon problems temperature, H2(T )=H2 coth (M/2T ),where such as chiral and deconfining phase transitions, pos- M =1/ξm 1.5 GeV is the inverse magnetic corre- sible bound states in the high-temperature phase, and lation length, which is independent of temperature the behavior of condensates of higher dimensions (in in this region. At temperatures T>2Tc,theam- particular, a mixed quark–gluon condensate). These plitude of the magnetic correlation function grows and other problems require a dedicated detailed con- with increasing temperature according to the law sideration. In conclusion, we will briefly formulate the H2(T ) ∝ g8(T )T 4, while the correlation length basic results of this study. decreases, ξ (T ) ∝ 1/(g2(T )T ). This behavior of the On the basis of an analysis of the QCDpartition m function and its renormalization-group properties, we magnetic correlation function explains the magnetic- have obtained an equation that relates the anomalous confinement phenomenon in 4dSU(N) Yang–Mills contribution to the trace of the energy–momentum theory at finite temperature. The resulting temper- tensor to the QCDthermodynamic pressure. 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Narodetskii Institute of Theoretical and Experimental Physics, Bol’shaya Cheremushkinskaya ul. 25, Moscow, 117259 Russia Received May 31, 2004; in final form, September 29, 2004 Abstract—On the occasion of the celebration of the 70th birthday of Prof. Yu.A. Simonov, we contribute a brief review of the status of exotic baryons, which are most likely pentaquark states. We summarize the experimental status of exotic baryons, discuss the baryon antidecuplet to which exotic baryons possibly belong, and review theoretical expectations for the masses and widths of recently discovered Θ and Ξ3/2 baryons which have come from studies of QCD sum rules, lattice QCD, and quark models. We also pay special attention to the dynamical calculation of pentaquark masses in a framework of the QCD string approach originally elaborated by Simonov for baryons and using the Jaffe–Wilczek [ud]2q¯ approximation for the pentaquark states. c 2005 Pleiades Publishing, Inc. 1. INTRODUCTION baryon states provides an opportunity to refine our quantitative understanding of nonperturbative QCD Recently, the LEPS [1] and DIANA [2] Collabora- at low energy. tions reported the observation of a very narrow peak in + 0 In October 2003, the NA49 Collaboration at the K n and K p invariant mass distribution, whose CERN SPS [14] announced evidence for an addi- existence has been confirmed by several experimental tional narrow Ξ−π− resonance with I =3/2, a mass groups in various reaction channels [3–11]. These of 1862 ± 2 MeV, and a width below the detector experimental results were motivated by a pioneering resolution of about 18 MeV. NA49 also reported paper on the chiral soliton model (χSM) [12]. The re- evidence for a Ξ0(1860) decaying into Ξ(1320)π. ported mass determinations for Θ are very consistent, ∗− ± Recently, a narrow resonance Θc(3099) in D p and falling in the range 1540 10 MeV, with the width ∗+ smaller than the experimental resolution of 20 MeV D p¯ invariant mass combinations was observed for the photon- and neutrino-induced reactions and in inelastic electron–proton collisions at center- 300 320 of 9 MeV for the ITEP K+Xe → K0pXe experiment. of-mass energies of and GeV by the H1 Collaboration at HERA [15] with a mass of 3099 ± K+d A recent analysis of total cross section data [13] 6 MeV and decay width of 12 ± 3 MeV. It can be found a structure corresponding to the resonance at interpreted as a pentaquark baryon with minimal the mass 1.559 ± 0.003 GeV with the width 0.9 ± quark content (ududc¯),thusbeingthefirst exotic 0.3 MeV. baryon with anticharm quark, implying the existence From the soliton point of view, Θ is nothing ex- of other exotic baryons with heavy quarks. A partial otic compared with other baryons—it appears natu- summary of experimental results is given in Table 1. rally in the rigid rotator approach to the three-flavor The complete list of both positive and negative ex- Skyrme model.1) However, in the sense of the quark perimental results existing as of May 2004 is given, model, the Θ+(1540) baryon with positive amount of e.g., in [16]. While a dozen experiments have reported strangeness is manifestly exotic—its minimal con- evidence for the phenomenon, another dozen have not figuration cannot be satisfied by three quarks. The seen the states. In particular, the e+e−-collider data positive strangeness requires an s¯ and qqqq (where (Belle, BaBar, ALEPH, DELPHI) give null results. q refers to the lightest quarks u, d) are required for The Fermilab high-energy proton experiments do not + the net baryon number, thus making a pentaquark observe Ξ(1860), Θ ,orΘc (CDF); do not confirm ∗− − uudds¯ state as the minimal “valence” configuration. Θc in either D p or D p channels (FOCUS); and The experimental evidence for the manifestly exotic → 0 do not observe Θ pKS (HyperCP, E690) [17]. ∗ The results from KN scattering known prior This article was submitted by the author in English. ∗ 1)In this framework, the lowest lying exotic baryon is an S = 2003 and referring to the phenomenology of Z +1 member of the antidecuplet 10F , with other exotic repre- (the old name of KN resonances with S =1)can sentations, such as the 27F , 35F ,and35F ,beingheavier. be summarized as follows. The 1982 version of the 1063-7788/05/6805-0748$26.00 c 2005 Pleiades Publishing, Inc. PENTAQUARKS: FACTS AND PUZZLES 749 Table 1. Summary of experimental data on Θ and Ξ3/2 hyperons Collaboration Reaction Mass, MeV Γ,MeV Significance, σ LEPS [1] γn → K+K−n (12C) 1540 ± 10 ≤25 4.6 DIANA [2] K+Xe → K0pXe 1539 ± 2 ≤9 4.5 CLAS [3] γd → K+K−pn 1642 ± 5 ≤21 5.3 → + 0 ± ≤ SAPHIR [5] γp K KSn 1640 4 20 4.8 FNAL [6] Bubble chambers 1533 ± 5 ≤20 6.7 + HERMES [7] γd, Θ → pKS → pππ 1528 ± 3 ≥4–6 ∼6 NA49 [14] pp → Ξ−π−X 1862 ± 2 ≤18 4.6 H1 [15] e+e− 3099 ± 6 12 ± 3 “Review of Particle Properties” lists several Z0 and we provide an overview of the effective Hamiltonian Z1 structures in the energy region 1800–2100 MeV: approach to QCD pioneered by Yu.A. Simonov and 2) Z0(1780)(P 01), Z0(1865)(D03), Z1(1900)(P 13), pay special attention to the dynamical calculation of 3) pentaquark masses in the framework of this approach. Z1(2150),andZ1(2500) with widths of 100 MeV or more. At least some of these have been understood as “pseudoresonance” structures [18] arising from the 2. BARYON ANTIDECUPLET coupling to inelastic K∗N and K∆ channels. An earlier search [19] for a Z∗ in the reaction π−p → The possibility of low-lying q4s¯ states in the P K−Z∗ at π− momenta of 6 and 8 GeV/c found wave fitting in the antidecuplet flavor representation − enhancements in the missing mass of recoiling K 10F of the quark model was mentioned long ago by at 1.5 and 1.9 GeV/c2 being explained as kinematical Golovich [23]. Later on, it was discovered that the reflection of the f2(1275),a2(1320),andρ3(1690) 10F multiplet appears as the third (after the octet with resonances. The absence of fresh news has left this spin 1/2 and decuplet with spin 3/2) rotational state interpretation in limbo. A similar possibility [20] for in the SU(3) version of the χSM [24]. To get a feeling the enhancement observed in the K+n effective mass for the antidecuplet (to which the newly discovered distribution at the mass of Θ seems to be unlikely, + − Θ and Ξ3/2 presumably belong), we first recall the but should be carefully checked in the forthcoming Gell-Mann–Okubo mass formula for a SU(3)F mul- experiments. tiplet Existence of a narrow Θ+ state around 1540 MeV M = M + αY + βD3, (1) should also be seen in available K+d data. The lack 0 3 + + of a prominent Θ signature in the total K d cross where M0 is the center of gravity for a given multiplet, section seems to exclude the width of Θ beyond the 3 − 2 − few-MeV level [21]. Similar but even more restrictive D3 = I(I +1) Y /4 C/6, (2) + conclusions were drawn from the K N phase-shift and analysis [22]. 2 C =2(p + q)+ (p2 + pq + q2) (3) In this paper, we explore the phenomenology of, 3 and models for, the exotic baryons, the lowest of is expressed in terms of the labels (p, q) which de- which is Θ+(1540). The next section includes a note the SU(3) representation. Schematically, q is pedagogical discussion of the baryon antidecuplet F the number of boxes in the lower line of the Young to which exotic baryons possibly belong. Then, in tableau depicting an SU(3) representation and p is Section 3, we review the theoretical expectations for F the number of extra boxes in the upper line. For ex- the masses and widths of exotic baryons. Finally, ample, (p, q)=(1, 1), (3, 0),and(0, 3) for JP =1/2+ 2) P + P + We use the standard notation LI2J . octet, J =3/2 decuplet, and J =1/2 antide- 3)The last two have no spin or parity assignment. cuplet, respectively. In Eq. (1), α and β are constants PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 5 2005 750 NARODETSKII nK+ or pK0 The (Y , I3) diagram for the antidecuplet of baryons is Θ+(1540) uudds– shown in Fig. 1. Also shown is the quark content of each state together with the predicted masses for N10 2 2 and Σ3/2. The corners of this diagram [ud] s,¯ [ds] u¯, 2 N(1647) and [us] d¯are manifestly exotic. The nonzero strange quark mass also leads to the mixing of octet and antidecuplet states. The exotic Σ(1655) + members of the antidecuplet Θ and Ξ3/2 cannot mix with octet baryons, though in principle they can mix with higher exotic multiplets like 27F with J =1/2 Ξ 3/2(1862) and 3/2. Two other particles common to the octet and the antidecuplet are N and Σ, and only they can mix, – Ξ–π– or Σ– K– Ξ0π+ or Σ+ K0 – |N = |N,10 −c |N,8 , (6) ddssu– uussd 10 10 | | − | Σ10 = Σ, 10 c10 Σ, 8 . (7) Fig. 1. The (I3,Y) diagram for the baryonic antidecuplet. Estimation in the χSM yields for c10 a not negligible The corners of diagram are manifestly exotic. The Θ and quantity: c ∼ 0.1 [12]. The general consideration Ξ N Σ 10 3/2 masses are used as input; the 10 and 10 masses of octet–antidecuplet mixing has been given in [25] are predicted using Eq. (1). with the conclusion that a good octet candidate for mixing with a 10F may consist of N(1410), Λ(1600), generally different for different multiplets.4) Splitting Σ(1660), and Ξ(1690), although in any case the mix- of otherwise degenerate baryon masses inside each ing cannot be too large. SU(3)F multiplet is due to nonzero strange quark mass. It follows from Eq. (1) that, for the antidecuplet, 3. THE PENTAQUARK WIDTH there are not three but only two independent param- eters: M and δm = α − 3 β .5) Therefore, if The most striking experimental result is a very 10 10 10 2 10 small width of Θ+.IntheχSM, the key element in 10F does not mix with other multiplets, one gets the equidistant masses of antidecuplet baryons: the narrow ΓΘ+ prediction is the strong cancellation between the coupling constants G0 and G1 which MΘ = M10 +2δm10, (4) enter the decay operator: M = M + δm , N10 10 10 1 2 M = M ,M = M − δm , Γ∆ ∼ G0 + G1 − G2 , (8) Σ10 10 Ξ10 10 10 2 − 3 2 where δm10 = α10 2 β10. This suggests that, if any 1 1 Γ ∼ G − G − G . two of the masses in the antidecuplet are known, the Θ 0 2 1 2 2 other masses can be predicted by relations (4). For example, using MΘ = 1540 MeV and identifying the This cancellation seems, at first sight, unnatural be- recent measurement of MΞ3/2 = 1862 MeV by the cause the constants G1,2 are nonleading as far as NA49 experiment as MΞ10 , one obtains Nc counting is concerned. At leading order in the 1/Nc expansion, all the couplings of the 8F , 10F , MN10 = 1647 MeV,MΣ10 = 1755 MeV. (5) and 10F baryons to pseudoscalar mesons are pro- 4) portional to the dimensionless constant G0, in which α and β are related to the constants x, y, z ,andz in Eqs. (1), + (2) of [25] by case the width of Θ would be of the same order as 5 Γ∆. However, in antidecuplet decays, the G1 con- α = −y − x, β = x, 8 4 8 tribution gets an additional Nc enhancement from − 3 − 3 the SU(3)-flavor Clebsch–Gordan coefficients cal- α10 β10 = z ,α10 β10 = z. 2 2 culated in large Nc. A dedicated discussion of this 5) problem can be found in [26]. The nonrelativistic limit This is also true for the decuplet with δm10 = α10 + implying G1/G0 =4/5,G2/G0 =2/5 would predict (3/2)β10. However, the mass parameters are quite differ- ∼− ∼ Γ + Γ −− ent: δm10 107 MeV for the antidecuplet and δm10 a strong suppression of Θ and Ξ . Reliable pre- −150 MeV for the decuplet. dictions for the decay widths depend on modeling and PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 5 2005 PENTAQUARKS: FACTS AND PUZZLES 751 educated guesses, and hence are subject to additional results triggered vigorous theoretical activity and put uncertainties. According to [27], a renewed urge in the need to understand how baryon 3G2 properties are obtained from QCD. 10 1 3 ΓΘ+ = pK =20.6 MeV. (9) The Θ hyperon has hypercharge Y =2and third 8πM + M 5 Θ N component of isospin I3 =0. The apparent absence ++ + 6) Although this number is relatively small, it is con- of the I3 =+1, Θ in K p argues against I =1; siderably larger than recent experimental estimates therefore, the Θ is usually assumed to be an isosin- given in Table 1. glet, although other suggestions have been made in Note that, if the Θ and Ξ3/2 have the same spin– the literature [31]. The other quantum numbers are parity, then their widths can be related by assum- not established yet. ing SU(3)F symmetry for the matrix elements and All attempts at theoretical estimations of the pen- correcting for phase space. This should be reliable taquark masses can be subdivided into the follow- at the level of typical SU(3)F symmetry violation, ing four categories: (i) dynamical calculations using the SU(3) χSM, (ii) dynamical calculations using i.e., ∼30%. This estimate yields Ξ3/2 width of order 3.5 times the width of the Θ. the sum rules or lattice QCD, (iii) phenomenological From the hadronic point of view, the Θ+(1540) analyses of the SU(3)F mass relations, and (iv) phe- nomenological analysis of the hyperfine splitting in lies about 100 MeV above the K+n threshold at the center-of-mass momentum 270 MeV/c.There the Sakharov–Zeldovich quark model. are no other two-body hadronic channels coupled to The success of the prediction of [12] for the Θ+ K+n below the K∆ threshold at 1725 MeV. Since mass was somewhat fortuitous. Reference [12] iden- + 7) there are no qq¯ annihilation graphs in K n scatter- tified the N10 with N(1710) and assumed a value of ing, one expects that low-energy K+n scattering is the π–nucleon σ term Σ=45MeV. This latter value dominated by the ordinary exchange forces (see [28] was responsible for unsuccessful prediction of a very and references therein). The S-wave phase shifts are heavy mass ∼2070 MeV for the Ξ−−. More recently known to be repulsive at low energies, which implies Ellis et al. [27] critically discussed the calculation of that there are no 1/2− resonances. In the P -wave the Θ+ and Ξ−− masses and widths in the χSM and sector, the P 01 and P 13 waves are attractive, while explored the unbiased theoretical and phenomenolog- the P 03 and P 11 waves are repulsive. In the P 03 ical uncertainties. Overall, they found the ranges and P 11 states, attractive forces may generate a res- 1432 PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 5 2005 752 NARODETSKII Table 2. Flavor wave functions in the Jaffe–Wilczek model (Y,I,I3) 10F (Y,I,I3) 8F 2 (2, 0, 0) [ud] s¯ – – 1 1 2 1 2 1 1 1 2 2 1, , [ud][us]+s¯ + [ud] d¯ 1, , [ud][us]+s¯ − [ud] d¯ 2 2 3 3 2 2 3 3 1 1 2 1 2 1 1 1 2 2 1, , − [ud][ds]+s¯ + [ud] u¯ 1, , − [ud][ds]+s¯ − [ud] u¯ 2 2 3 3 2 2 3 3 2 1 1 2 (0, 1, 1) [ud][us] d¯+ [us]2s¯ (0, 1, 1) [ud][us] d¯− [us]2s¯ 3 + 3 3 + 3 1 1 2 (0, 1, 0) ([ud][ds] d¯+[ud][us] u¯ +[us][ds] s¯) (0, 1, 0) ([ud][ds] d¯+[ud][us] u¯) − [us][ds] s¯ 3 + + + 6 + + 3 + 2 1 1 2 (0, 1, −1) [ud][ds] u¯ + [ds]2s¯ (0, 1, −1) [ud][ds] u¯ − [ds]2s¯ 3 + 3 3 + 3 3 3 −1, , [us]2d¯ – – 2 2 3 1 2 1 2 1 1 1 2 2 −1, , [us][ds]+d¯+ [us] u¯ −1, , [us][ds]+d¯− [us] u¯ 2 2 3 3 2 2 3 3 3 1 2 1 2 1 1 1 2 2 −1, , − [ds][us]+u¯ + [ds] d¯ −1, , − [ds][us]+u¯ − [ds] d¯ 2 2 3 3 2 2 3 3 3 3 −1, , − [ds]2u¯ – – 2 2 1 – – (0, 0, 0) ([ud][ds] d¯− [ud][us] u¯) 2 + + of 1540 MeV. However, the most recent quenched P -wave excitation. A natural remedy would be to calculation [36] seems to reveal no evidence for a decrease the number of constituents. This leads one pentaquark state with quantum numbers I(JP )= to consider dynamical clustering into subsystems of 0(1/2±) near this mass region. diquarks like [ud]2s¯ and/or triquarks like [ud][uds¯] The constituent quark models (CQM), in which all which amplify the attractive color-magnetic forces. constituents are in a relative S wave, naturally predict There are two routes that emerge naturally. One − is that of [41], based on the Sakharov–Zeldovich the ground-state energy of a JP =1/2 pentaquark CQM [42], in which the hadron mass is just the sum to be lower than that of a JP =1/2+ one. Using the of the effective masses mi of its constituents plus arguments based on both Goldstone boson exchange hyperfine interaction between constituent quarks and color-magnetic ex- σ σ change, it was mentioned that the increase in hyper- M = m + i j vhyp. (11) i m m ij fine energy in going from negative- to positive-parity i i>j i j states can be quite enough to compensate the orbital excitation energy ∼200 MeV [37]. However, exist- The other is the JW model of [43] (see also [44]), ing dynamical calculations of pentaquark masses in where it has been proposed that the systematics of ex- CQM (see, e.g., [38, 39]) are subject to significant otic baryons can be explained by diquark correlations. + 4 uncertainties and cannot be considered as conclusive. In the last scenario in the Θ (1540) and other q q¯ Note that, if Θ is indeed ududs¯ with JP =1/2+,then baryons, the four quarks are bound into two scalar, the spin–orbit QCD forces necessarily imply that singlet isospin diquarks. Diquarks must couple to 3c there has to be the JP =3/2+ partner of Θ∗, which to join in a color singlet hadron. In total, there are 2 is probably only a few MeV heavier [40]. six flavor symmetric diquark pairs [ud] , [ud][ds]+, 2 2 [ds] , [ds][su]+, [su] ,and[su][ud]+ combining with the remaining antiquark, which give 18 pentaquark 5. JAFFE–WILCZEK MODEL states in 8F plus 10F . This is a general result: in the Pentaquark baryons are unexpectedly light. In- quark model one cannot get a 10F without an 8F .All deed, a naive quark model with quark mass ∼350 MeV these states are degenerate in the SU(3)F limit. The + predicts Θ at about 350 × 5 = 1750 MeV, plus flavor 10F and 8F wave functions of pentaquarks in ∼150 MeV for strangeness, plus ∼200 MeV for the the JW model are presented in Table 2. PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 5 2005 PENTAQUARKS: FACTS AND PUZZLES 753 The diquarks are postulated as building blocks for Table 3. Comparison of the EH approach predictions for multiquark states on equal footing with constituent spin-averaged masses (in GeV) of nucleons, hyperons, quarks. Note that the notion of diquarks is almost and pentaquark with lattice results and experimental data as old as that of quarks. In fact, diquarks were sug- (assuming Θ+ is an IP =0+ state) gested by Gell-Mann in his pioneering paper about quarks [45]. Later, it was recognized that the strong − − ∗ Θ+ (N, ∆) (N , ∆ ) (Λ, Σ, Σ ) chromomagnetic attraction and the attraction due to (IP =0+) pion exchange in the ud color triplet state with net spin 0 results in forces which may keep u and d Lattice 1.07 [53] 1.76 [54] 1.21 [53] 2.80 [33] quarks close together, thus forming an almost point- EH 1.14 [56] 1.63 [55] 1.24 [56] >2.07 [49] like diquark. Such an idea has a long history and Experiment 1.08 1.62 1.27 1.54 many phenomenological applications (for a review, see, e.g., [46]). Diquark correlations have played an important supporting role in QCD. A similar idea where H0 is the kinetic energy operator and V is the is a basis of color superconductivity in dense quark sum of the perturbative one-gluon-exchange poten- matter [47]. tials and the string potential, which is proportional to A crucial point of the JW approach is that, if both the total length of the strings, i.e., to the sum of the diquarks are identical scalars, Bose statistics would distances of antiquark or diquarks from the string- demand total symmetry under their exchange, while junction point. The dynamical masses µi (analogs the color wave function is antisymmetric. Since di- of the constituent ones) are expressed in terms of quarks have no spin, the only possibility is to make the the current quark masses mi from the condition of spatial wave function antisymmetric by putting one (0) diquark into a P -wave state, thus making the total the minimum of the hadron mass MH asafunction 8) parity positive. of µi : (0) ∂M (mi,µi) 6. EFFECTIVE HAMILTONIAN APPROACH H =0, (13) ∂µi TO QCD 3 2 Neither the Sakharov–Zeldovich model nor the (0) mi µi MH = + + E0(µi), quark constituent model has yet been derived from 2µi 2 QCD. Therefore, it is tempting to consider the ef- i=1 fective Hamiltonian (EH) approach in QCD, which, E0(µi) being eigenvalue of the operator√H0 + V . on one hand, can be derived from QCD and, on the Quarks acquire constituent masses µi ∼ σ due to other hand, leads to results for the qq¯ mesons and 3q the string interaction in (12). The EH in the form baryons which are equivalent to the quark model ones of (12) does not include chiral-symmetry-breaking with some important modifications. This approach effects. A possible interplay with these effects should has been suggested by Simonov (see, e.g., the review be carefully clarified in the future. paper [48] and references therein). The physical mass MH of a hadron is The EH approach contains the minimal number of input parameters, current (or pole) quark masses, the M = M (0) + C . (14) string tension σ, and the strong coupling constant H H i i αs, and does not contain fitting parameters such as the total subtraction constant in the Hamiltonian. It The (negative) constants Ci have the meaning of the should be useful and attractive to consider expanding constituent self-energies and are explicitly expressed this approach to include diquark degrees of freedom in terms of string tension σ [52]: with appropriate interactions. This program, based on 2σ assumption that chiral forces are responsible for the Ci = − ηi, (15) formation of ud diquarks in Θ, while the strings are πµi mainly responsible for binding constituents in Θ,was where ηq =1and ηs =0.88 is the correction factor partially done in [49]. In this and subsequent sections, due to nonvanishing current mass of the strange we briefly review application of the EH approach to the JW model of pentaquarks. 8)Technically, this is done using the auxiliary field (AF) The EH for the three constituents has the form approach to get rid of the square-root term in the La- 3 2 grangian [50, 51]. Applied to the QCD Lagrangian, this mi µi technique yields the EH for hadrons (mesons, baryons, pen- H = + + H0 + V, (12) taquarks) depending on auxiliary fields µ .Inpractice,these 2µi 2 i i=1 fields are finally treated as c numbers determined from (13). PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 5 2005 754 NARODETSKII − (a)(b) energy) corresponding to LP =0 . For a state with P + L =1 , lρ =1, lλ =0, the wave function in the ud ud ud ud lowest hyperspherical approximation K =1reads − ψ = R 5/2χ (R)u (Ω), (17) 1 1 8 u (Ω) = sin θ · Y (ρˆ), 1 π2 1m – – q q where R2 = ρ2 + λ2. Here, one unit of orbital mo- ρ Fig. 2. The Jaffe–Wilczek reduction of the uuddq¯ pen- mentum between the diquarks is with respect to the taquark (a)totheeffective [ud]2q¯ problem (b). variable, whereas the λ variable is in an S state. For a P + state with L =0 , lρ =1,lλ =1, the wave function in the lowest hyperspherical approximation K =2is quark. The self-energy corrections are due to con- −5/2 stituent spin interaction with the vacuum background ψ = R χ2(R)u2(Ω), (18) fields and equal zero for any scalar constituent. 4 u2(Ω) = √ sin θ cos θ(nρ · nλ). Accuracy of the method is illustrated in Table 3, in π3 which the results of the approach for baryon masses χ (R) K =1 2 ¨ are compared with the lattice ones and experiment. The functions K , , , satisfy the Schro- One observes that the accuracy of the EH method √dinger equations. Written in terms of the variable x = for the three-quark systems is ∼ 100 MeV or better. µR,whereµ is an arbitrary scale of mass dimension One can expect the same accuracy for the diquark– which drops off in the final expressions, these equa- diquark–antiquark system. We shall comment on this tions read point later. d2χ (x) a K +2 E + K − b x (19) dx2 0 x K (K +3/2)(K +5/2) 7. PENTAQUARKS IN THE EFFECTIVE − χ (x)=0, HAMILTONIAN METHOD 2x2 K In the quark model, five quarks are connected by with the boundary condition χ (x) ∼ O(x5/2+K ) seven strings (Fig. 2a). In the diquark approximation, K as x → 0 and the asymptotic behavior χK (x) ∼ the short legs on this figure shrink to points and the 1/3 →∞ five-quark system effectively reduces to the three- Ai((2bK ) x) as x . The constants aK and bK in (19) are given by body one (Fig. 2b), studied within the EH approach √ in [55, 57]. 2 aK = R µ VC(r1, r2, r3)uKdΩ, (20) Consider a pentaquark consisting of two identical diquarks with current mass m[ud] and antiquark with 1 b = √ V (r , r , r )u2 dΩ, current mass mq¯ (q = d, s). In the hyperspherical K R µ string 1 2 3 K formalism [58], the wave function ψ(ρ, λ) expressed in terms of the Jacobi coordinates ρ and λ can be where written in symbolic shorthand as 2 1 VC(r1, r2, r3)=− αs . (21) −5/2 3 rij ψ(ρ, λ)=R χK(R)Y[K](Ω), (16) i PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 5 2005 PENTAQUARKS: FACTS AND PUZZLES 755 can tune m[ud] to give the nucleon mass MN (in the Table 4. The pentaquark masses in the quark–diquark– quark–diquark approximation). Neglecting for sim- diquark approximations (shown are the masses of [ud]2q¯ plicity Coulomb-like interaction, one obtains states (q = d, s)forJ P =1/2+ pentaquarks) 2σ M = M (0) − , (22) N N µ[ud] µq M πµd 1 + AF 0.482 0.458 2.171 where [ud]2s¯ 2 2 SSE 0.463 0.468 2.070 m[ud] µ[ud] + µd (0) 1 + MN = + (23) 2 ¯ AF 0.476 0.415 2.091 2µ[ud] 2 [ud] d 2 2 1/3 SSE 0.469 0.379 1.934 σ µ + µd + γ [ud] . 2 µ[ud]µd of [ud]2s¯ and [ud]2d¯pentaquarks calculated using the In Eq. (23), µ[ud] and µd are defined from the mini- spinless Salpeter equation (SSE): mum condition (13), 3 (0) (0) 2 2 HS = p + m + V, ∂MN ∂MN i i = =0, (24) i=1 ∂µ[ud] ∂µd 3 2σ ηi where γ =2.338 is the first zero of the Airy function: M = M0 − , − π 2 2 Ai( γ)=0. Equating MN to the experimental value i=1 pi + mi of the N−∆ center of gravity (MN =1.085 GeV), we find that m[ud] varies in the interval 0 ≤ m[ud] ≤ where V isthesameasinEq.(12),M0 is the eigen- ≤ ≤ 2 2 1/2 2 2 1/2 300 MeV, when σ varies in the interval 0.15 σ value of HS, (p[ud] + m[ud]) , (pq¯ + mq¯) are 0.17 GeV2. The last value of σ is preferred by meson the average kinetic energies of diquarks and an an- spectroscopy, while the former one follows from the tiquark, and ηi are the same correction factors as lattice calculations of [60]. In what follows, we use 9) in (15). In Table 4 the quantities µ[ud] and µq de- 2 σ =0.15 GeV and explicitly include the Coulomb- note either the constituent masses calculated in the like interaction between quark and diquarks. 2 2 1/2 AF formalism using Eq. (15) or (p[ud] + m[ud]) , For the pedagogy, let us first assume m[ud] =0. (p2 + m2)1/2 found from the solution of SSE. It is ¯ q¯ q¯ This assumption leads to the lowest uuddd and uudds¯ seen from Table 4 that these quantities agree with pentaquarks. If the current diquark masses vanish, accuracy better than 5%. The diquark masses cal- then the [ud]2d¯pentaquark is dynamically analogous culated by the two methods differ by 100 MeV for − to the JP =1/2 nucleon resonance and the [ud]2s¯ ([ud]2s¯) and by 160 MeV for [ud]2d¯. The approxi- P − pentaquark is an analog of the J =1/2 Λ hyperon, mation of Vstring mentioned above introduces a cor- with one important exception. The masses of P -wave rection to the energy eigenvalues of ≤30 MeV, so baryons calculated using the EH method acquire the we conclude that the results obtained using the AF P − ∼ (negative) contribution 3Cq for J =1/2 nucleons formalism and the SSE agree within 5%, i.e., the P − and 2Cq + Cs for the J =1/2 Λ hyperons. These accuracy of the AF results for pentaquarks is the contributions are due to the interaction of constituent sameasfortheqq¯ system (see, e.g., [64]). spins with the vacuum chromomagnetic field. When the [ud] diquark mass is fitted on the However, the above discussion shows that the baryon spectra, the pentaquark mass is found around 2.2 GeV in the JW picture [65]. Increasing α up to self-energies C[ud] equal zero for the scalar diquarks. s This means that introducing any scalar constituent 0.6 (the value used in the Capstick–Isgur model [66]) 2 ∼ increases the pentaquark energy (relative to the N decreases the [ud] s¯ mass by 100 MeV. The hyper- and Λ P -wave resonances). The numerical calcu- fine interaction due to σ exchange between diquarks lation for m =0 yields the mass of [ud]2s¯ pen- [ud] 9)The numerical algorithm to solve the three-body SSE is taquark ∼2100 MeV (see Table 4 and subsequent based on an expansion of the wave function in terms of discussion). Similar calculations yield the mass of harmonic oscillator functions with different sizes [62]. In fact, 2 to apply this technique to the three-body SSE, we need to [ud] c¯ pentaquark ∼3250 MeV (for mc =1.4 GeV) 2¯ ∼ use an approximation of the three-body potential Vstring by and the mass of [ud] b pentaquark 6509 MeV (for a sum of the two- and one-body potentials (see [63]). This mb =4.8 GeV) [61]. For illustration of accuracy of the approximation, however, introduces a marginal correction to AF formalism, in Table 4 are also shown the masses the energy eigenvalues. PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 5 2005 756 NARODETSKII and a strange antiquark can lower the Θ+ energy neglects the confinement effects and concentrates ∼ 2 ∼ by 100 MeV for gσ/4π 1. Therefore, the lower on the pure chiral properties of baryons. The string bound for the pentaquark mass in the JW model is dynamics alone in its simplified form seems to pre- ∼1900 MeV, still 350 MeV higher than the observed dict overly high masses of pentaquarks. Therefore, Θ+(1540) state. the existence of Θ,ifconfirmed, provides a unique We therefore conclude that the string dynamics possibility to clarify the interplay between the quark alone in its simplified form predicts overly high and chiral degrees of freedom in light baryons. masses of pentaquarks. A drastic modification of the present results requires a completely new dynamics, ACKNOWLEDGMENTS either that of the chiral soliton type [12] or introducing other multiquark clusters, like a triquark uds¯ plus I am very grateful to Yu.A. Simonov for numerous adiquarkud in a relative P wave [41]. In the last discussions concerning the topic of this paper. model, pentaquark masses were computed using This work was supported by the Russian Foun- the Sakharov–Zeldovich mass formulas only. 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Ya.B.ZeldovichandA.D.Sakharov,Yad.Fiz.4, 395 68. T. R. Browder, I. R. Klebanov, and D. Marlow, Phys. (1966) [Sov. J. Nucl. Phys. 4, 283 (1966)]. Lett. B 587, 62 (2004); hep-ph/0401115. PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 5 2005 Physics of Atomic Nuclei, Vol. 68, No. 5, 2005, pp. 758–770. From Yadernaya Fizika, Vol. 68, No. 5, 2005, pp. 790–803. Original English Text Copyright c 2005 by Dremin. QCD and Models on Multiplicities in e+e− and pp¯ Interactions * I. M. Dremin** Lebedev Institute of Physics, Russian Academy of Sciences, Leninski ˘ı pr. 53, Moscow, 119991 Russia Received April 23, 2004; in final form, September 23, 2004 Abstract—A brief survey of theoretical approaches to describing multiplicity distributions in high-energy processes is given. It is argued that the multicomponent nature of these processes leads to some peculiar characteristics observed experimentally. Predictions for LHC energies are presented. It is shown that simi- larity of the energy dependence of average multiplicities in different reactions is not enough alone to suggest the universal mechanism of particle production in strongly interacting systems. Other characteristics of multiplicity distributions depend on the nature of colliding partners. c 2005 Pleiades Publishing, Inc. 1. INTRODUCTION are speculations about nonhomogeneous matter dis- tribution in impact parameters [15], not to speak of Multiplicity distributions are the most general quark–gluon plasma [16] behaving as a liquid [17], characteristics of any high-energy process of mul- etc.). tiparticle production. They depend on the nature of The theoretical description of the subprocesses colliding particles and on their energy. Nevertheless, differs drastically in e+e− and pp¯ processes. Jet evo- it has been found that their shapes possess some lution in e+e− is well described by perturbative QCD common qualitative features in all reactions studied. equations with the only adjustable parameter Λ . At comparatively low energies, these distributions are QCD This parameter is approximately known from other relatively narrow and have sub-Poissonian shapes. characteristics and is, therefore, bounded. The pro- With increasing energy, they widen and fit the Pois- duction of perturbative gluons and quark–antiquark son distribution. At even higher energies, the shapes pairs can be described in terms of dipole or “an- become super-Poissonian; i.e., their widths are larger tenna” radiation. Color interference plays a crucial than that for the Poisson distribution. The width role. Many predictions of the perturbative QCD ap- increases with energy and, moreover, some shoulder- proach have been confirmed by experimental data. like substructures appear. Concerning multiplicity distributions, their shape in Their origin is usually ascribed to multicomponent e+e− processes is known in QCD only implicitly from contents of the process. In a QCD description of studies of their moments. This is determined by the e+e− processes, these could be subjets formed inside fact that the QCD evolution equations are formu- quark and gluon jets (for reviews, see, e.g., [1, 2]). In lated in terms of the generating function. They can phenomenological approaches, the multiplicity distri- be rewritten as equations for the moments but not bution in a single subjet is sometimes approximated directly for probabilities of n-parton emission. Their by the negative binomial distribution (NBD) first solutions up to higher order perturbative QCD [18] proposed for hadronic reactions in [3]. For hadron- have predicted some completely new features of the initiated processes, these peculiarities are often as- moments. The moments also contain complete infor- cribed to multiple parton–parton collisions [4–8], mation about the distribution. However, the recon- which could lead, e.g., to two-, three-, ... ladder struction of the shape of the distribution from them is formation of the dual parton model (DPM) [9] or not a trivial task. Some attempts to solve the inverse problem [19, 20] and directly get the shape of the quark–gluon string model (QGSM) [10–12] and/or multiplicity distribution were successful only in the to different (soft, hard) types of interactions [13, lowest order perturbative QCD approximations with 14]. They become increasingly important as collision several additional assumptions. energy is increased. These subprocesses are related to the matter state during the collision (e.g., there The situation with hadronic processes is, in some respect, more complicated. The confinement property ∗This article was submitted by the author in English. is essential, and perturbative QCD methods cannot **e-mail: [email protected] be applied directly. Therefore, some models have been 1063-7788/05/6805-0758$26.00 c 2005 Pleiades Publishing, Inc. QCD AND MODELS ON MULTIPLICITIES 759 developed. Hadron interactions used to be consid- stage (e.g., these assumptions differ in the PYTHIA ered as proceeding via collisions of their constituent and the HERWIG cluster model). More important, partons. In preparton times, their role was played by the predictions at higher energies (in particular, for pions, and one-meson-exchange model [21] domi- multiplicity distributions) also differ in these models, nated. Pions were treated as hadron constituents. and it is necessary to try various approaches and Their high-energy interaction produced a ladder of confront them with experiment when the LHC enters one-pion t-channel exchanges with blobs of low- operation. Being interesting in themselves, these fits energy pion–pion interactions. This is the content provide the background of so-called minimum bias of the multiperipheral model. These blobs were first events for “triggered” experiments. interpreted as ρ mesons [22] and later called fire- balls [23], clusters [24], or clans [25] when higher 2. MOMENTS OF MULTIPLICITY mass objects were considered. Multiperipheral dy- DISTRIBUTIONS namics tells us that the length of a multiperipheral chain fluctuates and the number of these blobs is The shape of multiplicity distributions is so com- distributed according to the Poisson law. It was ar- plicated that it is difficult to get any analytical ex- gued that its convolution with the distribution of the pression for it from the solution of QCD equations. number of pions produced in each center can lead to It has been obtained only in the simplest perturbative the NBD of created particles. This supposition fits approximations [19, 20]. In particular, it has been experimental data on multiplicity distributions of pp demonstrated [20] that the recoil has a profound effect and pp¯ reactions at tens of GeV quite well. However, on the multiplicity distribution in QCD jets. It be- at higher energies, this fit by a single NBD becomes comes much narrower than according to the leading unsatisfactory. A shoulder appears at high multiplic- perturbative term, where energy conservation is not ities. Sums of NBD with different parameters were taken into account. However, no shoulders appear. used [14] to get agreement with experiment. Better On the contrary, the tail of the distribution is more fits are achieved at the expense of a larger number strongly suppressed. of adjustable parameters. These shortcomings can be An alternative and, in some sense, more accurate minimized if one assumes that each high-energy bi- approach is proposed by studies of moments of the nary parton collision is independent of some others si- distribution. One can get some QCD predictions for multaneously proceeding. With this supposition, the these moments [18] up to high orders in the perturba- whole process is described as a set of independent pair tive expansion. The moments of various ranks contain parton interactions (IPPI model, proposed in [26]). complete information about multiplicity distributions. The effective energy of a pair of partons does not Hence, the shape and energy evolution of multiplicity depend on how many other pairs interact and what distributions can be quantitatively described by the these interacting partons (quarks or gluons) are. The rank dependence and energy behavior of their mo- number of adjustable parameters does not increase ments. Moment analysis of multiplicity distributions compared to a single NBD if the probabilities of j can also be performed for models of hadronic pro- pairs’ interactions and the number of active pairs are cesses as well as for experimental data. Therefore, this known. approach is common to all processes and methods of analysis. Earlier, a somewhat different way to account for multiple parton collisions, which generalized the To introduce moments on the most general grounds, multiperipheral one-ladder model in the framework we write the generating function G(E,z) of the of the eikonal approximation, was proposed in DPM multiplicity distribution P (n, E): and QGSM (the latter one takes into account the ∞ Reggeization of exchanged particles). They differ G(E,z)= P (n, E)(1 + z)n. (1) from IPPI by probabilities of processes with different n=0 number of active parton pairs and by multiplicity dis- The multiplicity distribution is obtained from the gen- tributions of final particles. Also, the Lund model [27] erating function as with its parton cascades and the string hadronization n due to a linearly increasing QCD potential has been 1 d G(E,z) P (n)= n . (2) extremely successful in describing many features n! dz z=−1 of multiparticle production. Several Monte Carlo programs implement this model to provide some In what follows, we will use the so-called unnormal- ized factorial F and cumulant K moments defined hints to experimentalists at present-day experiments q q according to the formulas and give predictions at even higher energies (like PYTHIA, FRITIOF, etc.). Some assumptions have Fq = P (n)n(n − 1) ...(n − q +1) (3) to be used for the hadronization of partons at the final n PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 5 2005 760 DREMIN q d G(E,z) 3. QCD ON MOMENTS IN e+e− COLLISIONS = q , dz z=0 All moments can be calculated from the gener- ating function as explained above. The generating q d ln G(E,z) functions for quark and gluon jets satisfy definite K = . (4) q q equations in perturbative QCD (see [2, 19]). They are dz z=0 1 They determine, respectively, the total and genuine G 2 (i.e., irreducible to lower) order correlations among GG = dxKG (x)γ0 [GG(y +lnx) (9) the particles produced (for more details, see [2, 0 28]). The first factorial moment defines the average × GG(y +ln(1− x)) − GG(y)] multiplicity F = n, the second one is related to 1 1 the width (dispersion) of the multiplicity distribution F 2 F2 = n(n − 1), etc. Factorial and cumulant mo- + nf dxKG (x)γ0 [GF (y +lnx) ments are not independent. They are related by the 0 formula × GF (y +ln(1− x)) − GG(y)], q−1 F m K F 1 q = Cq−1 q−m m, (5) G 2 m=0 GF = dxKF (x)γ0 [GG(y +lnx) (10) where 0 − × GF (y +ln(1− x)) − GF (y)], m (q 1)! C − = (6) q 1 m!(q − m − 1)! where the labels G and F correspond to gluons and quarks, the energy scale of the process is defined by are the binomial coefficients. ≈ √y =ln(Q/Q0), Q = pθ is a virtuality of a jet, p Since both Fq and Kq strongly increase with their s/2 is its momentum, θ is an opening angle, and rank and energy, the ratio Q0 = const. Here, G (y)=dG/dy, nf is the number of active flavors, Hq = Kq/Fq, (7) 2Ncαs first introduced in [18], is especially useful due to γ2 = . (11) 0 π partial cancellation of these dependences. More im- portant is that some valuable predictions about its The running coupling constant in the two-loop ap- proximation is behavior can be obtained in perturbative QCD. Also, it will be shown below that H moments of the IPPI q 2π − β1 ln 2y −3 model depend on a smaller number of its adjustable αs(y)= 1 2 + O(y ), (12) β0y β y parameters than factorial and cumulant moments. 0 Thus, even though Fq, Kq,andHq are interrelated, where − they can provide knowledge about different facets of 11Nc 2nf β = , (13) the same multiplicity distribution. 0 3 2 − It is easy to find the ratio Hq from iterative formu- 17Nc nf (5Nc +3CF ) β1 = . las, 3 q−1 The kernels of the equations are Γ(q) FpFq−p H =1− H − , (8) 1 q Γ(p +1)Γ(q − p) q p F KG(x)= − (1 − x)[2 − x(1 − x)], (14) p=1 q G x once the factorial moments have been evaluated. F 1 2 − 2 KG (x)= [x +(1 x) ], (15) The factorial moments Fq are always positive by 4Nc definition [Eq. (3)]. The cumulant moments Kq can change sign. They are equal to zero for the Poisson G CF 1 − x KF (x)= 1+ , (16) distribution. Consequently, Hq =0in this case. Nc x 2 2 − Let us emphasize that Hq moments are very sensi- Nc =3 is the number of colors, and CF =(Nc tive to minute details of multiplicity distributions and 1)/2Nc =4/3 in QCD. can be used to distinguish between different models Here, one can get equations for any moment of and experimental data. the multiplicity distribution, for both quark and gluon PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 5 2005 QCD AND MODELS ON MULTIPLICITIES 761 jets. One should just equate the terms with the same recently in [34]. This will be demonstrated below in powers of u =1+z on both sides of the equations, Fig. 10. Let us mention that the oscillations shape where expressions (1) are substituted for both quarks depends crucially on the particular form of QCD and gluons. kernels (14)–(16), and their observation confirms the In particular, the nontrivial energy dependence of basic principles of QCD. mean multiplicity in quark and gluon jets has been However, one should be warned that the ampli- predicted. Within two lowest perturbative QCD ap- tudes of oscillations strongly depend on a multiplicity proximations, it has a common behavior distribution cutoff due to limited experimental statis- − 2 √ n = A y a1c exp(2c y), (17) tics (or by other reasoning) if it is done at rather low G,F G,F multiplicities [35]. Usually, there are no such cutoffs 1/2 where AG,F = const, c =(4Nc/β0) ,anda1 ≈ 0.3. in analytical expressions for Hq. One can control The main features of the solutions can be demon- the influence of cutoffs by shifting them appropri- strated in gluodynamics where only the first equation ately. The qualitative features of Hq behavior persist with nf =0 is considered. At asymptotically high nevertheless. In what follows, we consider very high energies, it can be reduced [19] to the differential energy processes where the cutoff due to experimental equation statistics is practically insignificant. Numerical esti- mates of the relation between the maximum multi- 2 − [ln G(y)] = γ0 [G(y) 1]. (18) plicity measured and effective ranks of the moments From the definitions of moments, one can easily guess are given in the Appendix. that this equation determines the asymptotic behavior of Hq, because ln G on the left-hand side gives rise to K F 4. NEGATIVE BINOMIAL DISTRIBUTION q and G on the right-hand side to q. The second AND IPPI MODEL derivative on the left-hand side would result in the factor q2. Thus, it can be shown [18] that asymptoti- Independently of progress in perturbative QCD cal (y →∞)valuesofHq moments are positive and calculations, the NBD fits of multiplicity distribu- − decrease as q 2. At present energies, they become tions were attempted for both e+e− and pp(pp¯) col- negative at some values of q and reveal a negative lisions [14, 36, 37]. The single NBD parametrization minimum at is 1 Γ(n + k) m n m −n−k qmin = +0.5+O(γ0), (19) P (n, E)= 1+ , h1γ0 NBD Γ(n +1)Γ(k) k k where h1 = b/8Nc =11/24, b =11Nc/3 − 2nf /3.At (20) 0 ≈ Z energy αs 0.12, and this minimum is located at where Γ denotes the gamma function. This distribu- ≈ about q 5. It moves to higher ranks with energy tion has two adjustable parameters m(E) and k(E), increase, because the coupling strength decreases. which depend on energy. However, the simple fitby Some hints to possible oscillations of Hq vs. q at formula (20) is valid until shoulders appear. In that higher ranks at LEP energies were predicted in [18]. case, this formula is often replaced [14] by the hybrid They were obtained with account of recoil effects NBD which simply sums up two or more expressions in higher order perturbative QCD and stressed the like (20). Each of them has its own parameters mj,kj . importance of both nonsingular terms in kernels These distributions are weighted with the energy- (14), (16) of the equations and energy conservation dependent probability factors wj which sum up to 1. in high-energy processes so often mentioned by Correspondingly, the number of adjustable parame- Andersson (see [27]). The approximate solution of ters drastically increases if the distributions are com- the gluodynamics equation for the generating func- pletely unrelated. tion [29] agrees with this conclusion and predicts the oscillating behavior at higher ranks. The same It was proposed recently [26] that hadron interac- conclusions were obtained from an exact solution to tions can be represented by a set of independent pair equations for quark and gluon jets in the framework parton interactions (IPPI model). This means that of fixed coupling QCD [30]. A recent exact numerical colliding hadrons are considered as “clouds” of par- solution of the gluodynamics equation in a wide tons which interact pairwise. It is assumed that each energy interval [31] coincides with the qualitative binary parton collision is described by the same NBD features of multiplicity distributions described above. distribution. The only justification for this supposition These oscillations were confirmed by experimental is given by previous fits of multiplicity distributions data for e+e− collisions and found also for hadron- at lower energies. Then the convolution of these dis- initiated processes first in [32], later in [33], and most tributions, subject to a condition that the sum of PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 5 2005 762 DREMIN Table 1. The values of wj according to (22) (left-hand side) For Hq moments, one gets and (27) (right-hand side) q−1 Γ(q) fpfq−p H =1− H − . (25) q Γ(p +1)Γ(q − p) q p f jmax 3 4 5 6 3 4 5 6 p=1 q w1 0.544 0.519 0.509 0.504 0.562 0.501 0.450 0.410 Note that, according to Eqs. (24) and (25), Hq are w2 0.295 0.269 0.259 0.254 0.278 0.255 0.236 0.219 functions of the parameter k only and do not depend on m in the IPPI model. This remarkable property of w3 0.161 0.140 0.131 0.128 0.160 0.153 0.152 0.147 Hq moments provides an opportunity to fit them with w4 0 0.072 0.067 0.065 0 0.091 0.100 0.104 one parameter. It is nontrivial because the oscillating H w5 0 0 0.034 0.033 0 0 0.062 0.073 shapes of q are quite complicated as shown below. Once the parameter k is found from fits of H , w 0 0 0 0.016 0 0 0 0.047 q 6 it is possible to get another parameter m rewriting Eq. (23) as follows: binary-collision multiplicities is the total multiplicity F 1/q n, leads [26] to a common distribution m = k q . (26) fq(k) jmax P (n; m, k)= wjPNBD(n; jm,jk). (21) This formula is a sensitive test for the whole approach, j=1 because it states that the definite ratio of q-dependent /q q This is the main equation of the IPPI model. The functions to the power 1 becomes -independent if NBD distribution on its right-hand side with prod- the model is correct. Moreover, this statement should k ucts jm and jk in its arguments is a consequence of be valid only for those values of which are deter- mined from H fits. Therefore, it can be considered as thepropertyoftheconvolutionofNBDstoproduce q new NBD. It can be easily shown if one considers a criterion of a proper choice of k and of the model the NBD generating function. One gets a sum of validity, in general. This criterion of constancy of m negative binomial distributions with shifted maxima happens to be extremely sensitive to the choice of k and larger widths for a larger number of collisions. No as shown below. new adjustable parameters appear in the distribution In [12], the energy dependence of the probabili- w for j pairs of colliding partons. The probabilities wj ties j was estimated according to the multiladder- are determined by collision dynamics and, in princi- exchange model [11] (its various modifications are ple, can be evaluated if some model is adopted (e.g., known as DPM or QGSM). They are given by the see [11, 12]). Independence of parton pair interactions following normalized expressions: implies that, at very high energies, wj is a product pj wj(ξj)= (27) of j probabilities w1 for one pair. Then, from the j max normalization condition pj j=1 jmax jmax j j−1 i wj = w1 =1, (22) 1 − Zj − Zj j=1 j=1 = 1 e , j max i! jZ p i=0 one can find w1 if jmax, which is determined by the j j maximum number of parton interactions at a given j=1 energy, is known. This is the only new parameter. It where depends on energy. Thus, three parameters are suffi- ∆ cient to describe multiplicity distributions at any en- 2 2Cγ s ξj =ln(s/s0j ),Zj = 2 2 ergy. Moreover, asymptotically, jmax →∞and w1 = R + αP ξj s0j 0.5 according to (22). (28) The factorial moments of the distribution (21) are with numerical parameters obtained from fits of ex- jmax Γ(jk + q) m q m q perimental data on total and elastic scattering cross F = w = f (k) (23) −2 2 −2 q j Γ(jk) k q k sections: γ =3.64 GeV , R =3.56 GeV , C = j=1 − −2 1.5, ∆=αP 1=0.08, αP =0.25 GeV ,ands0 = 2 with 1 GeV . It is seen that each wj depends on six ad- jmax justable parameters in these models. Γ(jk + q) f (k)= w . (24) Below, we will use both possibilities (22) and (27) q j Γ(jk) j=1 in our attempts to describe experimental data. The PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 5 2005 QCD AND MODELS ON MULTIPLICITIES 763 Hq m 0.3 14.5 14.0 0.2 13.5 0.1 13.0 0 12.5 210146 q Fig. 1. AcomparisonofHq moments derived from ex- 12.0 perimental data at 1.8 TeV (squares) with their values 2 61014q calculated with parameter k =4.4 (dash-dotted curve) and 3.7 (solid curve) [26]. Fig. 2. The q dependence of m for k =4.4 (squares), 3.7 (circles), and 7.5 (triangles) [26]. probabilities wj are different for them (see Table 1). In the IPPI model they decrease exponentially with the jmax =3and 4. Those at 300 and 1000 GeV are larger number of active partons, while in the ladder models for w1 by about 1% and smaller for w3 by about 3%. they are inversely proportional to this number with When energy increases, the processes with a larger additional suppression at large j due to the term in number of active pairs play a more important role in parentheses in (27). This is the result of the modified the modified ladder approach compared to the IPPI eikonal approximation. Let us stress that, when we model. Thus, the jmax cutoff is also more essential use expressions (27) for probabilities, this does not di- there. rectly imply comparison with DPM–QGSM because In principle, one can immediately try a two- in our case the NBD is chosen for the multiplicity parameter fit of experimental multiplicity distributions distribution in a single ladder, while it is a Poisson using Eq. (21) if wj are known according to Eqs. (22) distribution for resonances created in a ladder with or (27). However, the use of their moments is pre- a fluctuating length in DPM–QGSM. Thus, we will ferred. call it the modified ladder model. − We show the values wj for jmax =3 6 pairs calcu- 5. COMPARISON WITH EXPERIMENT lated according to Eq. (22) on the left-hand side of Ta- ble 1 and according to Eq. (27) on its right-hand side. Let us begin with hadronic collisions and then These values of jmax are chosen because previous compare them with other processes. analysis of experimental data [8] has shown that two We have compared [26] IPPI model conclusions pairsbecomeactiveatanenergyofabout120GeV with experimental (but extrapolated [8, 38] to the full and the number of binary collisions increases with phase space) multiplicity distributions of the E735 increasing energy. Thus, precisely these numbers will Collaboration [39] for pp¯ collisions at Tevatron ener- be used in comparison with experiment at higher en- gies 300, 546, 1000, and 1800 GeV. The multiplic- ergies. In particular, we shall choose jmax =3at 300 ity of charged particles was divided by 2 to get the and 546 GeV, 4 at 1000 and 1800 GeV, 5 at 14 TeV, multiplicity of particles with the same charge. Then and 6 at 100 TeV (see below). the above formulas for moments were used. Corre- One can clearly see the difference between the two spondingly, the parameters m and k refer to these distributions. The parameter j is chosen according approaches. The value of w1 is always larger than 0.5 max in the IPPI model, while it can become less than 0.5 to prescriptions discussed above. in the (modified) ladder model [9, 11] at high energies. Factorial and Hq moments were obtained from In the ladder model, wj depend explicitly on energy experimental data on P (n) according to Eqs. (3) and (not only on the jmax cutoff). We show their values at (8). Experimental Hq moments were fitted by Eq. (25) 546 and 1800 GeV in the right-hand side columns of to get the parameters k(E). We show in Fig. 1 how PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 5 2005 764 DREMIN Pn+ m 0.06 26 22 18 0.04 14 10 6 2 3 4 5 0.02 10 10 10 10 E, GeV Fig. 5. The energy dependence of m (squares) and its linear extrapolation (circles at 14 and 100 TeV) [26]. k 8 0 10 30 50 70 90 7 n+ 6 Fig. 3. The same-charge multiplicity distribution at 1.8 TeV, its fitatm =12.94,k=4.4. The dash-dotted 5 curve demonstrates what would happen if the NBD were replaced by a Poisson distribution [26]. 4 300 500 1000 3000 E, GeV Pn+ 0.04 Fig. 6. The values of k as calculated with wj satisfying relation (22) [26]. 0.03 given by Eq. (22). It is surprising that oscillations of Hq moments are well reproduced with one adjustable parameter k. The general tendency of this quite com- plicated oscillatory dependence is clearly seen. 0.02 With these values of the parameter k,wehave checked whether m is constant as a function of q as required by Eq. (26). The m(q) dependence is shown 0.01 in Fig. 2 for the same values of k and for the much larger value 7.5. The constancy of m is fulfilled with an accuracy better than 1% for k =4.4 up to q =16.The upper and lower curves in Fig. 2 demonstrate clearly 0 20 40 60 80 100 that this condition bounds substantially admissible n+ variations of k. Fig. 4. The decomposition of the fit in Fig. 3 to one, two, The same-charge multiplicity distribution at three, and four parton–parton collisions (corresponding 1.8 TeV has been fitted with parameters m =12.94 curves are ordered from left to right) [26]. and k =4.4 as shown in Fig. 3 (solid curve). To estimate the accuracy of the fit, we calculated 125 − 2 2 perfect these fits are at 1.8 TeV for k equal to 3.7 n=1(Ptheor(n) Pexp(n)) /∆ over all 125 exper- (solid curve) and 4.4 (dash-dotted curve). At this imental points. Here, Ptheor and Pexp are theoretical energy, we considered four active parton pairs with wj and experimental distributions and ∆ is the total PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 5 2005 QCD AND MODELS ON MULTIPLICITIES 765 experimental error. It includes both statistical and Pn+ systematic errors. Note that the latter ones are large 0.03 at low multiplicities in E735 data. This sum is equal to 50 for 125 degrees of freedom. No minimization of it was attempted. This is twice as good as the three- parameter fit by the generalized NBD considered in [40]. A Poisson distribution of particles in binary collisions is completely excluded. This is shown in Fig. 3 by the dash-dotted curve. In Fig. 4, we also show a breakdown of the total 0.02 distribution into contributions due to a different num- ber of interacting pairs, thus illustrating the mech- anism underlying the formation of the tails of the multiplicity distribution. The same procedure has been applied to data at energies 300, 546, and 1000 GeV. As stated above, we have assumed that three binary parton collisions are active at 300 and 546 GeV and four at 1000 GeV. We plot in Figs. 5 and 6 the energy dependence of 0.01 parameters m and k. The parameter m increases with energy logarithmically (Fig. 5). This is expected be- jmax cause the increase in M1 = j=1 wjj due to increas- ing number of active pairs at these energies leads to a somewhat faster than logarithmic increase in average multiplicity in accordance with experimental observations. The energy dependence of k is more complicated and rather irregular (Fig. 6). We tried to ascribe the latter to the fact that the 0 40 80 120 160 200 effective values of k, which we actually find from the n+ above-described fits, depend on the effective num- Fig. 7. The same-charge multiplicity distributions at 14 and 100 TeV obtained by extrapolation of parameters m ber of parton interactions, i.e., on wj variation at a and k with five active pairs at 14 TeV and six at 100 TeV threshold. The threshold effects can be important in [for the IPPI model, (solid curve) 14 TeV, k =4.4;(dash- this energy region. Then, the simple relation (22) is dotted curve) 14 TeV, k =8; and (dashed curve) 100 TeV, k . invalid. This influences the functions fq(k) (24) and, =44; for the modified ladder model, (dotted curve) 14 TeV, k =4.4] [26]. consequently, Hq calculated from Eq. (25). One can reduce the effective number of active pairs to about are marked not only by the change in w shown in 2.5 at 300 GeV and 3.5 at 1000 GeV if chooses the j Table 1 but also by the variation of the parameter k. following values of wj: 0.59, 0.34, 0.07 at 300 GeV and 0.54, 0.29, 0.14, 0.03 at 1000 GeV instead of The threshold effects become less important at those calculated according to (22) and shown in Ta- higher energies. We assume that there are five active ble1.Thisgivesrisetovaluesofk which are not pairs at 14 TeV and six at 100 TeV. Then we extrap- drastically different from previous ones. However, the olate to these energies. The parameter m becomes quality of fits becomes worse. Fits with two active equal to 19.2 at 14 TeV and 25.2 at 100 TeV if a pairs at 300 GeV and three pairs at 1000 GeV fail logarithmic dependence is adopted as shown in Fig. 5 completely. by the straight line. We choose two values of k equal Hence, we have to conclude that this effect results to 4.4 and 8 since we do not know which one is from dynamics of hadron interactions which is not yet responsible for thresholds. The predicted multiplic- understood and should be incorporated in the model. ity distributions with these parameters are plotted in The preliminary explanation of this effect could be Fig. 7. The oscillations of Hq still persist at these en- that, at the thresholds of a new pair formation, the ergies (see Fig. 8). The minima are, however, shifted previous active pairs produce more squeezed multi- to q =6at 14 TeV and 7 at 100 TeV as expected. plicity distributions due to smaller phase-space room The fit at 1.8 TeV with an approximation of wj available for them because of a newcomer. Therefore, according to the modified ladder model (27) with the single pair dispersion would decrease and the k the NBD for a binary parton collision is almost as values would increase. It would imply that thresholds successful as the fitwithvaluesofwj given by the PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 5 2005 766 DREMIN Hq 6. IS HADRONIC PRODUCTION SIMILAR 0.3 IN VARIOUS PROCESSES? This question was first raised by the statement of [43] that the average multiplicities in e+e− and 0.2 pp(pp¯) processes increase with energy in a similar way. Recently, the PHOBOS Collaboration [44] even claimed that the energy behavior of mean multiplic- 0.1 ities in all processes is similar. Therefore, it has been concluded that the dynamics of all hadronic processes is the same. Besides our general belief in QCD, we 0 cannot claim that other characteristics of multiple production processes initiated by different partners coincide. To answer the above question, we compare –0.1 characteristics of multiplicity distributions for pro- 210146 q cesses initiated by different partners. Fig. 8. The behavior of H predicted at 14 TeV [k =4.4 Average multiplicities. Total yields of charged q − (solid line); k =8(dash-dotted line)] [26]. particles in high-energy e+e , pp¯, pp, and central AA collisions become similar if special rescaling is done. IPPI model. However, some difference at 14 TeV (see The average charged-particle multiplicity in pp/pp¯ is + − Fig. 7) and especially at 100 TeV between these mod- similar [43] to that for e e collisions if the e√ffective els is predicted. To keep the same mean multiplicity energy seff is inserted in place of s,where seff is in both models at the same energy, we have chosen the pp/pp¯ center-of-mass energy minus√ the√ energy jmax of the leading particles. In practice, s = s/2 is different values of m according to n = m j=1 wjj eff and wj values shown in Table 1. Namely, their ra- chosen. This corresponds to the horizontal shift of empty squares to the diamond positions in Fig. 9, tios are mIPPI/mlad =0.988, 1.039, 1.145, 1.227 for j =3, 4, 5, 6, respectively. This shows that the borrowed from [44]. Then the diamonds lie very close max to the dashed curve, which shows QCD predictions maximum of the distribution moves to smaller mul- + − tiplicities and its width becomes larger in the mod- for multiplicities in e e collisions according to (17). ified ladder model compared to the IPPI model with For central nucleus–nucleus collisions, the particle increasing energy. yields have been scaled by the number of participating Surely, one should not overestimate the success of nucleon pairs Npart/2. Then the energy dependence the IPPI model in its present initial state. It has been of mean multiplicities is very similar for all these pro- cesses at energies exceeding 10 GeV up to 1 TeV,and applied just to multiplicity distributions. For more − detailed properties, say, rapidity distributions, one even at lower energies for e+e compared with pp/pp¯. would need a model for the corresponding features of This is well demonstrated in Fig. 9. the one-pair process. Moreover, the screening effect However, the situation changes at higher energies. (often described by the triple Pomeron vertex) will In Table 2, we show experimentally measured mean probably become more important at higher energies. multiplicities at Tevatron energies (first row). They All these features are somehow implemented in the are compared with the results of the IPPI model in well-known Monte Carlo programs PYTHIA [41], the second row, where the IPPI predictions at 14 TeV HERWIG [42], and DPM–QGSM [9, 11]. However, (LHC) and 100 TeV are also shown. According to for the latter one, for example, the multiplicity dis- the above hypothesis, these values should coincide tribution for a single ladder is given by the Poisson with QCD predictions (17) at a factor of 2 smaller distribution of emission centers (resonances) con- energy. The latter ones are presented in the third voluted with their decay properties, and probabilities row. The difference between them and experimentally wj contain several adjustable parameters. It differs measured values of the first row is demonstrated in from the IPPI model. The latter approach proposes the fourth row for Tevatron energies, while at higher a more economical way with a smaller number of energies the difference of QCD and IPPI predictions such parameters. What concerns the further devel- is shown. opment of the event generator codes, it is tempting to It is seen that both experimental and theoretical incorporate there the above approach with an NBD of values of average multiplicity coincide pretty well in − particles created by a single parton pair and confront pp¯ and e+e up to 1 TeV. However, already at Teva- the results with a wider set of experimental data. We tron energy 1.8 TeV, the rescaled pp¯ multiplicity is intend to do this later to learn how it influences other lower by 3.4 charged particles. This difference be- characteristics. tween rescaled predictions of the IPPI model for pp¯ PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 5 2005 QCD AND MODELS ON MULTIPLICITIES 767 40 – pp(pp) Data (a) pp(pp– ) ()s/2 30 e+e– Data + – 〉 Fit to e e /2 PHOBOS part N 〈 20 PHOBOS interp. / 〉 ch NA49 N 〈 E895 10 0 (b) 1.0 Fit – e + e / 〉 ch N 〈 0.5 100 101 102 103 s, GeV Fig. 9. The energy dependence of average multiplicities for various processes (a) and their ratio to QCD prediction for e+e− collisions (b) [44]. and QCD for e+e− becomes extremely large at LHC e+e− [2, 31] and models of pp/pp¯ [26] can fit these (26.6), and even more so at higher energies. observations. With these observations, one tends to claim (if There is, however, one definite QCD prediction, at all!) the approximate quantitative similarity of all which allows one to ask the question whether QCD processes up to 1 TeV and “a universal mechanism and, e.g., the IPPI model are compatible, in principle. of particle production in strongly interacting systems This is the asymptotic behavior of Hq moments in as 2 controlled mainly by the amount of energy available QCD. They should behave [18] as Hq =1/q .One for particle production” [44] only at energies below can also determine the asymptotics of Hq moments the highest Tevatron values. The situation becomes in the IPPI model and compare both approaches [46]. even worse if we compare other features of multiplicity These values are noticeably larger than QCD predic- distributions. tions of 1/q2. Thus, QCD and the IPPI model have Hq moments. First, we have calculated [45] Hq different asymptotics. It is an open question whether moments for experimental multiplicity distributions other asymptotic relations for probabilities of multi- in various high-energy processes. They are shown in parton interactions different from those adopted in the Fig. 10. These moments weakly depend on energy in IPPI model can be found which would lead to the + − the energy regions available to present experiments. same behavior of Hq moments in pp¯ and e e col- Their oscillating shape is typical for all processes. lisions. Only then can one hope to declare an analogy However, one notices that the amplitudes of Hq os- between these processes. cillations in Fig. 10 differ in various reactions. They Fractal properties. The particle density within are larger for processes with participants possessing the phase space in individual events is much more more complicated internal structure. For example, the structured and irregular than in the sample averaged − amplitudes in e+e are about two orders of magni- distribution. However, even for the latter ones, fluc- tude smaller than those in pp. Both QCD applied to tuations depend on the phase space of a sample. The PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 5 2005 768 DREMIN Hq 0.04 0.02 610 0 24 8 12 q –0.02 –0.04 e+e– –0.06 ep pp –0.08 pAl 32S238U –0.10 Fig. 10. Oscillations of Hq moments for various processes [45]. dq 0.20 (a)(b) (c) 0.15 e+e–, HRS hh, NA22 16O + AgBr, KLM e+e–, DELPHI hh, UA1 32S + AgBr, KLM µp, EMC 0.10 0.05 0 13 5135135 Order q Fig. 11. Anomalous fractal dimensions for various processes [28]. smaller the phase space, the larger the fluctuations. (δy)−φ(q) for δy → 0 and φ(q) > 0 would correspond This can be described by the behavior of normal- to an intermittent phenomenon well known from tur- ized factorial moments F = F /nq as functions of bulence. This also shows that the created particles are q q distributed in the phase space in a fractal manner. The the amount of phase space available. In the one- anomalous fractal dimension dq is connected with dimensional case of the rapidity distributions with- the intermittency index φ(q) by the relation φ(q)= in the interval δy, the powerlike behavior Fq(δy) ∝ (q − 1)(d − dq),whered is the ordinary dimension PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 5 2005 QCD AND MODELS ON MULTIPLICITIES 769 Table 2. The mean multiplicities at Tevatron and higher energies √ s,GeV 300 546 103 1.8 × 103 1.4 × 104 105 npp,¯ exp 25.4 30.5 39.5 45.8 – – npp,¯ theor 24.1 30.0 39.4 45.7 71.6 97.0 ne+e−,theor 25.3 31.8 39.9 49.2 98.2 180 ne+e− − npp¯ –0.1 1.3 0.4 3.4 26.6 83 of a sample studied (d =1for the one-dimensional Moreover, QCD and the considered models of hadron rapidity plot). It has been calculated in QCD [47–49] interactions predict different asymptotics for some only in the lowest order perturbative approximations. characteristics of e+e− and pp(pp¯).Wehavecon- The qualitative features of the behavior of factorial sidered here just multiplicity distributions. Other moments as functions of the bin size observed in inclusive and exclusive characteristics also seem to experiment are well reproduced by QCD. be different in these processes. The anomalous dimensions as functions of the In conclusion, multiplicity distributions in various order q derived from experimental data are shown high-energy processes possess some common qual- in Fig. 11 for various collisions [28]. They are quite itative features but differ quantitatively. Theoretical large for e+e− and become smaller for hh and even approaches to their description have very little in more so for AA collisions. This shows that the more common. Thus, it is premature to claim their common structured the colliding partners, the more strongly dynamical origin independently of our belief in QCD smoothed the density fluctuations in the phase space. as a theory of strong interactions. In some way, this observation correlates with the enlarged amplitudes of Hq oscillations mentioned above. ACKNOWLEDGMENTS In any case, this is another indication that it Iam grateful to V.A. Nechitailo for collabora- is premature to claim the similarity of e+e−-and tion on many aspects of the problems discussed. hadron-initiated processes. We have compared just Special thanks are to the PHOBOS Collaboration some features of multiplicity distributions. There are for providing Fig. 9. This work has been supported many more characteristics of the processes that can in part by the Russian Foundation for Basic Re- be compared, but this is beyond the scope of the search, project nos. 02-02-16779, 04-02-16445-a, present paper. and NSH-1936.2003.2. APPENDIX 7. CONCLUSIONS We have briefly described the theoretical ap- The higher the rank of the moment, the higher proaches to collisions of high-energy particles. They the multiplicities that contribute to it. Therefore, the seem to be quite different for various processes. high-rank moments are extremely sensitive to the No direct similarity in multiplicity distributions has high-multiplicity tail of the distribution. At the same been observed. According to experimental data, the time, the energy-momentum conservation and exper- energy dependence of average multiplicities in dif- imental statistics limitations are important at the tail ferent collisions can be similar in the energy region of the distribution. Therefore, the question about the above 10 GeV if some rescaling procedures are used. limits of applicability of the whole approach is quite However, this is just the first moment of multiplicity reasonable. distributions. To speak about their similarity, one Let us estimate the range of validity of considering should compare other moments. Again, the quali- large-q values of Hq moments imposed by some cut- tative features of moment oscillations are somewhat offs (see also [35]). It is well known that experimental similar, but quantitatively they differ. The same can be cutoffs of multiplicity distributions due to the limited said about the fractal properties of particle densities statistics of an experiment can influence the behavior within the phase space. No deep reasoning for the of Hq moments. Consequently, they impose some corresponding rescaling has been promoted. Thus, limits on q values allowed to be considered when a besides our general belief that the QCD Lagrangian comparison is done. Higher rank moments can be is at the origin of all these processes, we cannot evaluated if larger multiplicities have been measured. present any serious arguments in favor of similar To estimate the admissible range of q,weusethe schemes applicable to the dynamics of the processes. results obtained in QCD. Characteristic multiplicities PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 5 2005 770 DREMIN that determine the moment of the rank q can be 23.V.N.Akimov,D.S.Chernavsky,I.M.Dremin,and found. By inverting this relation, one can write the I.I.Royzen,Nucl.Phys.B14, 285 (1969). asymptotic expression for the characteristic range of 24.I.M.DreminandA.M.Dunaevsky,Phys.Rep.18, q [20]. This provides the bound qmax ≈ Cnmax/n, 159 (1975). where C ≈ 2.5527. However, it underestimates the 25. A. Giovannini and L. Van Hove, Z. Phys. C 30, 391 factorial moments. Moreover, the first moment is not (1986). properly normalized (it becomes equal to 2/C instead 26. I. M. Dremin and V. A. Nechitailo, hep-ph/0402286. of 1). Strongly overestimated values (however, with 27. B. Andersson, The Lund Model, Cambridge Monogr. a correct normalization of the first moment) are ob- Part. Phys. Nucl. Phys. Cosmol. (1997), Vol. 7. tained if C is replaced by 2. Hence, one can say that 28. E. A. De Wolf, I. M. Dremin, and W.Kittel, Phys. Rep. the limiting values of q are given by the inequalities 270, 1 (1996). 29. I. M. Dremin and V. A. Nechitailo, Mod. Phys. Lett. 2nmax/n PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 5 2005 Physics of Atomic Nuclei, Vol. 68, No. 5, 2005, pp. 771–778. From Yadernaya Fizika, Vol. 68, No. 5, 2005, pp. 804–811. Original English Text Copyright c 2005 by Voloshin. Relative Yield of Charged and Neutral Heavy Meson Pairs in e+e− Annihilation near Threshold* M. B. Voloshin William I. Fine Theoretical Physics Institute, University of Minnesota, Minneapolis, MN, and Institute of Theoretical and Experimental Physics, Bol’shaya Cheremushkinskaya ul. 25, Moscow, 117259 Russia Received May 13, 2004 Abstract—The subject of the charged-to-neutral yield ratio for BB¯ and DD¯ pairs near their respective thresholds in e+e− annihilation is revisited. As previously argued for the B mesons, this ratio should exhibit a substantial variation across the Υ(4S) resonance due to interference of the resonance scattering phase with the Coulomb interaction between the charged mesons. Asimple alternative derivation of the expression describing this effect is presented here, and the analysis is extended to include the D-meson production in the region of the ψ(3770) resonance. The available data on kaon production at the φ(1020) resonance are also discussed in connection with the expected variation of the charged-to-neutral yield ratio. c 2005 Pleiades Publishing, Inc. 1. INTRODUCTION bears considerable similarity, although specificfac- The near-threshold resonances φ(1020), ψ(3770), tors contributing to the ratio in each case are slightly and Υ(4S), decaying, respectively, to pairs of pseu- different. In a way, the simplest case is presented ¯ ¯ ¯ by the threshold B-pair production, where the de- doscalar mesons KK, DD,andBB, are well-known c/n “factories” for production of these mesons in e+e− viation of R from one is essentially entirely de- annihilation. In a number of experimental studies, it termined by the Coulomb interaction between the charged B mesons, since the mass difference between is important to know the relative yield of the pairs − ± of charged and neutral mesons in the corresponding the B mesons is very small: mB0 mB+ =0.33 ¯ resonance region, i.e., the ratio 0.28 MeV[8].FortheDD pair production at the + − → + − ψ(3770) peak, the difference in the P -wave kinemat- c/n σ(e e P P ) 3 R = − , (1) ical factor p due to the substantial mass difference σ(e+e → P 0P¯0) between the charged and neutral D mesons is the where P stands for the pseudoscalar meson, i.e., K, main source of deviation of Rc/n from one, and the D,orB. The knowledge of this ratio is of partic- Coulomb interaction effect is somewhat smaller, but ular importance for the studies of the B mesons at is still of a measurable O(10%) magnitude. Finally, in the B factories, and dedicated measurements have the φ(1020)-resonance region in addition to the mass been done at the Υ(4S) resonance by CLEO [1, 2], difference and the Coulomb interaction effects, the + − BABAR [3, 4], and Belle [5]. The central values of amplitudes of production of K K and KLKS also the ratio Rc/n found in these measurements typically differ due to a nonresonant isovector, I =1,contribu- range from 1.01 to 1.10 with the latest result [4] tion coming from the electromagnetic current of the u being 1.006 ± 0.036(stat.) ± 0.031(syst.). The same and d quarks. Another technical difference between ratio for the production of D-meson pairs is likely the heavy mesons and the kaons is that the former to be studied in detail at the ψ(3770) resonance by can safely be considered nonrelativistic at energies in the imminent CLEO-c experiment [6]. For the kaon the region of the corresponding resonance (the veloc- pair production, the most detailed available scan of ities of the mesons at the corresponding resonances ≈ + ≈ production of charged and neutral mesons in the are v(B)/c 0.06 and v(D )/c 0.13), while the φ-resonance region has been done by the SND Col- velocity of the charged kaons at the φ resonance is + 2 2 laboration [7]. v(K )/c ≈ 0.25,andtheO(v /c ) relativistic effects Rc/n can show up at some level of intended accuracy in The theoretical treatment of the ratio at these c/n three thresholds near the corresponding resonances R . The kinematical effect of the mass difference, and ∗This article was submitted by the author in English. of the nonresonant isovector amplitude for the case 1063-7788/05/6805-0771$26.00 c 2005 Pleiades Publishing, Inc. 772 VOLOSHIN of kaon production, leads to factors in the Rc/n, leads to the expression for the O(α/v) term found which are rather smoothly varying within the width there. In Section 3 are presented specific estimates of the corresponding resonance. The behavior of the of the behavior of the ratio Rc/n across the ψ(3770) Coulomb interaction effect is, however, quite different. resonance, which take into account both the mass Namely, as recently pointed out [9], with a proper difference between the D mesons and the effect of the treatment of the strong resonant interaction between Coulomb interaction. the mesons, there is a phase interference between It can be noticed that the very fact of the exis- the Coulomb interaction and the resonance (Breit– tence of the variation of Rc/n within the resonance Wigner) scattering phase. Since the latter phase width is essentially model independent and is a direct changes by π across the resonance, the effect of consequence of a somewhat standard application of the Coulomb interaction in Rc/n should exhibit a the quantum-mechanical scattering theory. Never- substantial variation with energy within the width theless, there has been some skepticism expressed of the resonance. Thus, in particular, a comparison regarding the existence of the predicted [9] rapid vari- between different measurements of the charged-to- ation of Rc/n at the resonance, most notably in the neutral ratio at the Υ(4S) peak can only be mean- introductory part of the recent experimental paper [4]. ingful with proper account of the differences in the Although the measurement reported in [4] has been specific experimental setups, such as the beam energy done at just one point in the energy at the peak of spread, stability, and calibration. Needless to say, the Υ(4S) resonance, the paper implicitly puts the the best approach would be a dedicated scan of the prediction of the variation in doubt by arguing that Υ(4S) peak. The details of this variation depend [9] “such rapid variation in the charged-to-neutral ratio on the specifics of the transition from the strong has not been observed in scans across the φ(1020) interaction region at short distances to the long- resonance” (with a reference to the scans by the SND range Coulomb interaction and also on such details Collaboration [7]). Thus, it appears to be useful to as the nonresonant scattering phase for the mesons. point out the difference between the predictions for Adedicated scan of the charged-to-neutral ratio the B production at Υ(4S) and the kaon production thus could possibly provide information on rather fine at φ(1020) following from the discussed approach. properties of strong interaction between the heavy The main difference is that the amplitude of the vari- mesons, which are not likely to be accessible by other ation at the φ is expected to be suppressed by a means. It may well be that a more experimentally factor of approximately 0.25 due to about four times feasible object for such study is provided by the larger velocity of the kaons at the relevant energy, ψ(3770) resonance, and possibly can be explored in and additional differences in the details arise through detail in the forthcoming CLEO-c experiment. The different parameters (e.g., widths) of the φ and Υ(4S) main motivation of the present paper is to extend resonances. Acomparison of a “representative” the- the analysis of [9] to the case of the DD¯ production oretical curve with the behavior of Rc/n across the in the region of this resonance, and to estimate the φ resonance extracted from the scan data [7] is pre- magnitude of the expected effects, to the extent that it sented in Section 4. Given the existing theoretical and can be done within the present (rather approximate experimental uncertainties, a meaningful comparison at best) understanding of the details of the strong can only be done at a mostly qualitative level, and interaction between heavy mesons. it is left entirely up to the reader to assess from the The typical amplitude of the varying part of the presented comparison whether the scan data indeed ratio Rc/n due to the Coulomb interaction is set by exclude a variation with expected amplitude of the charged-to-neutral ratio across the φ resonance. the Coulomb parameter α/v for the charged mesons, which tells us that, for the D mesons at ψ(3770), the amplitude of the variation should be approxi- 2. COULOMB INTERACTION CORRECTION TO PRODUCTION OF CHARGED MESONS mately two times smaller than for the B mesons at Υ(4S) (and by another factor of two smaller for the If the mesons were point particles devoid of kaons at φ). In either case, the Coulomb parameter strong interaction and produced by a point source, is sufficiently small to justify limiting the consider- the Coulomb interaction between charged mesons ation to only the linear term in the Coulomb inter- would enhance their production cross section by the action treated as a perturbation. Asimple derivation textbook factor πα 2 2 of this Coulomb interaction correction to production Fc =1+ + O(α /v ), (2) of a charged meson pair with the exact treatment 2v of the strong interaction scattering is presented in where v is the velocity of each of the mesons in the Section 2. This derivation is slightly different from center-of-mass frame, which was the early predic- that given in [9] and, hopefully, more transparently tion [10] for the ratio Rc/n at the Υ(4S) resonance. It PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 5 2005 RELATIVE YIELD OF CHARGED AND NEUTRAL HEAVY MESON PAIRS 773 was, however, then realized that the effects of the me- r>a, the mesons interact via the potential V (r).The son electromagnetic form factor [11] and of the form meson pair is produced by a source localized at dis- factor in the decay vertex Υ(4S) → BB¯ [12, 13] tend tances shorter than a (this obviously corresponds to to reduce the enhancement. In all these approaches, the production in the e+e− annihilation). The problem the Coulomb rescattering of the produced mesons is to find the correction of the first order in V to the is considered, assuming that the propagation of the production rate. meson pair between the production vertex and the In order to solve the formulated problem, we con- rescattering as well as after the Coulomb rescattering sider the radial part of the wave function of a sta- is described by a free propagator, and thus ignoring tionary P -wave scattering state, written in the form the fact that the strong interaction introduces the R(r)=χ(r)/r. According to the general scattering scattering phase factors exp(iδ) in these propaga- theory [14], the asymptotic form of χ(r) at r →∞in tors. This would be a reasonable approximation if the absence of the long-range potential V (r) is δ the strong phase were small. In the region of a iδ ipr −iδ −ipr strong resonance, however, this assumption is def- χ(r)=2cos(pr + δ)=e e + e e . (4) initely invalid, since the phase changes by π within Furthermore, in the absence of the potential V ,the the width of the resonance. The calculation described function χ(r) satisfies the Schrodinger¨ equation for in [9] allows one to completely take into account free motion in the P wave, the strong scattering phase as well as the Coulomb 2 interaction between the mesons. In the first order in χ (r)+ p2 − χ(r)=0, (5) the Coulomb rescattering, the expression [9] for the r2 Coulomb correction factor has a simple form at all distances r outside the region of strong interac- ∞ 1 i 2 tion, i.e., at r>a. Thus, the solution valid at all such F =1+ Ime2iδ e2ipr 1+ V (r)dr, distances and having the asymptotic form (4) can be c v pr a written explicitly, − ∗ (3) χ(r)=eiδf(pr)+e iδf (pr), (6) where p is the meson momentum and V (r) is the po- in terms of the function tential for the rescattering interaction. Clearly, up to i a possible form factor, V (r)=−α/r for the Coulomb f(pr)= 1+ eipr (7) interaction. The short-distance cutoff parameter a in pr Eq. (3) accounts for the fact that, in the region of and of its complex conjugate f ∗.Atdistancesr PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 5 2005 774 VOLOSHIN The perturbation δχ of the scattering state wave is explicitly real.1) In other words, at r →∞,the function can be found in the standard way by writ- correction changes only the scattering phase by ing it as a sum of outgoing and incoming waves, adding the “Coulomb” phase to δ. − δχ(r)=δχ+(r)+δχ (r), similarly to Eq. (6), and At r = a, the perturbation δχ+(a) is found from where δχ+(r) contains at r →∞ only an outgo- ∗ the same expression (12). In this case, one has r PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 5 2005 RELATIVE YIELD OF CHARGED AND NEUTRAL HEAVY MESON PAIRS 775 + − p0. However, for the DD¯ pairs, the D D threshold 3. ESTIMATES OF THE EXPECTED is 9.6 MeV higher than that for D0D¯ 0,thedifference VARIATION OF Rc/n ACROSS THE ψ(3770) of which is substantial at the energy of the ψ(3770) RESONANCE resonance. Here, we apply the formulas of the previous sec- It should be noticed that neglecting the higher tion, specifically Eqs. (17), (18), and (16), to an es- terms of expansion in the momenta for the parameters timate of the scale of the variation in the ratio Rc/n γ and δ is justified only inasmuch as the momenta 1 for the DD¯ pair production at the ψ(3770) resonance. are small in the scale of the size of the strong inter- In doing so, it should be clearly understood that, action region a . In the calculation of the Coulomb s at present, one can only guess (in some reasonable interaction effect, it is essential that the distance pa- range) the appropriate values of the scattering phase rameter a for the onset of the Coulomb interaction not δ1 and the cutoff parameter a for the Coulomb in- be smaller than as: a ≥ as. Otherwise, no restriction on the value of the product pa is implied. In [9], teraction (or, more generally, make a guess about an these parameters were reasonably assumed to be ap- appropriate model for the onset at short distances proximately equal. However, it should be emphasized of the Coulomb interaction between the charged D that, generally, the applicability of formula (3) for the mesons). Coulomb correction is separate from the applicabil- Asdifferent from the case of the B-meson pro- ity of the first terms of the threshold expansion in duction at the Υ(4S), in the charged-to-neutral ratio Eq. (16), and these two issues can be studied sepa- for the D-meson production at ψ(3770), the Coulomb 3 rately in experiments.2) effect multiplies the ratio of the p factors: In conclusion of this section, we write the explicit 3 c/n p+ formula for the correction described by Eq. (3) in the R = Fc . (19) resonance region in terms of real quantities [9]: p0 α F =1+ (17) Also, according to the Particle Data Tables [8], the c v width of ψ(3770) is somewhat larger than that of ∆2 − γ2 the Υ(4S): Γ(ψ )=23.6 ± 2.7 MeV as opposed to × (A cos 2δ1 + B sin 2δ1) ± ∆2 + γ2 14 5 MeV, which tends to smoothen the variation of Rc/n near the ψ(3770) peak. 2γ∆ − (B cos 2δ1 − A sin 2δ1) . Rc/n ∆2 + γ2 Asample expected behavior of the ratio for the D pair production near the ψ(3770) resonance is The coefficients A and B are found as the imaginary shown in Fig. 1 for some representative values of the and the real parts of the integral in Eq. (3) with the parameters a and δ1(E0). For comparison, the same Coulomb potential: curves for the B pairs in the vicinity of the Υ(4S) ∞ 1 2cos2u du peak are also shown in a separate plot in Fig. 1. It − − should be noted in connection with this comparison A = 1 2 sin 2u + (18) u u u that, generally, it is not expected that the phase δ is pa 1 the same in these two cases, although the parameter π cos 2pa sin 2pa = − + − Si(2pa), a viewed as characterizing the electric charge struc- 2 pa 2(pa)2 ture in a heavy meson is likely to be quite similar ∞ for D and B mesons, if one considers both c and b 2sin2u 1 du B = − 1 − cos 2u quarks as asymptotically heavy. With all the present u u2 u uncertainty in the knowledge of the strong interaction pa parameters, one can see from this comparison that cos 2pa sin 2pa Rc/n = + − Ci(2pa), the expected variation of the ratio is quite less 2(pa)2 pa prominent for D mesons at the ψ(3770) peak than for where B mesons at the Υ(4S). Hopefully, the amplitude of z ∞ the variation of Rc/n at the ψ(3770) is still sufficient dt dt Si(z)= sin t and Ci(z)=− cos t . for a study in the upcoming CLEO-c experiment. t t 0 z 4. DISCUSSION OF THE φ(1020) DATA 2)In fact, an attempt at detecting terms higher than p3 in the AND SUMMARY width parameter for the Υ(4S) resonance has been done by ARGUS Сollaboration [15]. However, the deviation from So far, the most detailed scan data for the pro- the cubic behavior turned out to be too small within the duction cross section of charged and neutral mesons experimental accuracy. are available only for the kaons in the vicinity of PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 5 2005 776 VOLOSHIN Rc/n Rc/n D mesons B mesons 0.8 1.2 0.7 1.1 0.6 1.0 0.5 0.9 3750 3760 3770 3780 3790 10.572 10.576 10.580 10.584 10.588 E, MeV E, GeV Fig. 1. The energy dependence of the ratio Rc/n for D-andB-meson pair production in the region of the respective resonance: ψ(3770) and Υ(4S) (the center positions are assumed at E0 =3.770 and 10.580 GeV, respectively) for some values of a −1 ◦ −1 ◦ −1 and δ1(E0): a = 200 MeV, δ1(E0)=30 (solid curve); a = 200 MeV, δ1(E0)=−30 (dashed curve); a = 300 MeV, ◦ −1 ◦ δ1(E0)=30 (dash-dotted curve); and a = 300 MeV, δ1(E0)=−30 (dotted curve). the φ(1020) resonance [7], and it is quite natural to as well as of its general variation with energy on a discuss whether any hint at the discussed variation scale somewhat larger than the width of φ. Areal fitof of Rc/n is indicated by those data. Aconsideration the involved parameters, including the rapid variation of this ratio at the φ peak, however, encounters cer- of the Coulomb factor Fc discussed here, requires tain peculiarities. As mentioned above, the Coulomb knowledge of the raw data and of the correlation in interaction effect is relatively smaller for production the errors. For this reason, here in Fig. 2 is shown of K mesons at the φ resonance, and also the rela- a comparison of the data sets listed in [7] with a tivistic effects can play a certain role. The analysis of representative theoretical curve. the ratio Rc/n is further complicated by the fact that More specifically, in [7] are listed (in Table IX) the production amplitude also receives an isovector the scan data for the cross section for production − + − contribution, which has opposite sign for K+K and of separately the K K pairs and the KLKS pairs KLKS and thus changes the charged-to-neutral ra- acquired in two different scans (called PHI9801 and tio. (In the vector dominance model this contribution PHI9802). Furthermore, the detection of the KS is considered as the “tail” of the ρ resonance.) Ne- mesons was done by two separate methods, i.e., by glecting a smooth variation of the nonresonant I =1 detecting their decay into charged pions and into neu- amplitude and also of the nonresonant part of the tral pions. Accordingly, paper [7] lists two separate I =0amplitude (e.g., the tail of the ω resonance), one sets of data for the KLKS production measured by c/n each of these two methods. The systematic error for can approximate the formula for R in this case as − the K+K cross section is listed at 7.1% and for the 3 − 2 c/n p+ 1+(A0 A1)(∆ + iγ) neutral kaons at 4 and 4.2% for the two identification R = Fc , techniques used. The statistical errors are listed for p0 1+(A0 + A1)(∆ + iγ) (20) each individual entry for the cross section. The data points in the plot of Fig. 2 are the charged-to-neutral where A1 and A0 stand for the appropriately normal- ratios calculated from the data listed in [7] with the ized relative contribution of the nonresonant I =1 errors corresponding to the listed statistical errors and I =0 amplitudes. The charged K mesons are only. The representative theoretical curve corresponds lighter than the neutral ones; thus, the cube of the to 1.38Fc. In other words, the absolute normalization ratio of the momenta is larger than one and decreases is set rather arbitrarily, given the described theoretical towards one at energies far above the threshold. The uncertainty in calculating the absolute normalization data [7], however, display a slight general increase Rc/n at the φ resonance and given the systematic in the ratio with energy, which shows that the effect uncertainty in the data. Also, any overall variation of A1 is not negligible. This makes any “absolute” of the kinematical and amplitude factors in the prediction of Rc/n in this case rather troublesome, shown energy interval is neglected. Changing the PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 5 2005 RELATIVE YIELD OF CHARGED AND NEUTRAL HEAVY MESON PAIRS 777 Rc/n 16 15 14 13 12 1016 1020 1024 1028 E, MeV Fig. 2. The SND data [7] at the φ(1020) resonance represented as the charged-to-neutral yield ratio for the two reported scans (PHI9801 and PHI9802) in [7] and for two methods of identifying the KS mesons by their decay into neutral or into charged pions: PHI9801 neutral (stars), PHI9801 charged (diamonds), PHI9802 neutral (triangles), and PHI9802 charged (squares). The solid curve is explained in the text. normalization of the curve and also including the measuring the discussed effect for B mesons at the overall energy dependence would result in a vertical Υ(4S) resonance and for D mesons at the ψ(3770) shift and a slight tilt of the curve. The behavior of resonance might provide information on these details, the Coulomb correction factor in the curve shown which are not accessible by other means. The esti- in Fig. 2 is calculated assuming E0 = 1019.5 MeV, mated effect for D mesons in the region of the ψ(3770) −1 ◦ Γ=4.26 MeV, a = 200 MeV, and δ(E0)=40 . is less prominent than for B mesons at the Υ(4S) TheonlypurposeofthetheoreticalcurveinFig.2 but possibly is still measurable. The expected effect is to illustrate the approximate magnitude of the of the rapid variation of the charged-to-neutral ratio expected effect of the variation of Rc/n at the φ(1020) for the kaon production at the φ(1020) resonance is resonance, and by no means should it be considered the smallest, and, conservatively, the available data as a detailed prediction. appear to be not conclusive enough to confirm or Due to large experimental and theoretical uncer- exclude such variation with a statistical significance. tainties as explained in the text and can be seen from Fig. 2, it is not entirely clear whether the available ACKNOWLEDGMENTS data [7] exclude or rather suggest a variation with the expected magnitude of the ratio Rc/n within the width I thank J. Rosner for stimulating questions re- of the φ resonance. Perhaps, a detailed global fitof garding the estimates of the behavior of Rc/n at the the raw data could provide a statistically significant ψ(3770) resonance and A. Vainshtein for a useful dis- evaluation of the amplitude of such variation. cussion of the kaon production in the φ(1020) region. In summary, the strong interaction phase signif- This work is supported in part by the DOE, grant icantly modifies the behavior of the cross section for no. DE-FG02-94ER40823. production of pairs of charged pseudoscalar mesons near threshold in e+e− annihilation. Asimple calcu- lation of this effect in the first order in the Coulomb REFERENCES interaction is presented in Section 2. The existence 1. J. P. Alexander et al. (CLEO Collab.), Phys. Rev. of a strong near-threshold resonance gives rise to a Lett. 86, 2737 (2001). rapid variation of the charged-to-neutral yield ratio 2. S. B. Athar et al. (CLEO Collab.), Phys. Rev. D 66, due to the interference of the resonance scattering 052003 (2002). phase with the Coulomb phase. The specific behavior 3. B. Aubert et al. (BABAR Collab.), Phys. Rev. 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(ARGUS Collab.), Z. Phys. C 65, (1990). 619 (1995). PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 5 2005 Physics of Atomic Nuclei, Vol. 68, No. 5, 2005, pp. 779–783. From Yadernaya Fizika, Vol. 68, No. 5, 2005, pp. 812–816. Original English Text Copyright c 2005 by Kaidalov. The Problem of “Saturation” in DIS and Heavy-Ion Collisions* A. B. Kaidalov** Institute of Theoretical and Experimental Physics, Bol’shaya Cheremushkinskaya ul. 25, Moscow, 117259 Russia Received August 3, 2004 Abstract—The problem of saturation of parton densities in deep inelastic scattering (DIS)in the limit x → 0 and in heavy-ion collisions at very high energies is investigated in the framework of Reggeon theory. It is emphasized that the shadowing effects are important in a definite kinematical region. The model which includes unitarity effects and gives a good description of structure functions and diffractive processes in DIS is discussed. In the limit x → 0,the“saturation” of parton densities is achieved. The Gribov approach to interaction with nuclei allows one to describe structure functions of nuclei in the small-x region. It is shown how the prediction of the Glauber model for the density of hadrons produced in the central rapidity region in nucleus–nucleus interactions is modified due to shadowing of small-x partons. Calculations show that shadowing effects are important at RHIC energies, but the situation is far from saturation. Production of particles with large transverse momenta and jets is discussed. c 2005 Pleiades Publishing, Inc. 1. INTRODUCTION condensate” [5]. However, the pattern of saturation is much more general. Investigation of the deep inelastic scattering (DIS) in the region of small x-Bjorken gives important in- In this paper, I first shall review theoretical ap- formation on the behavior of partonic densities in the proaches to the origin of the Pomeron in QCD. In regime when these densities are large and nonlinear Section 3, I shall consider the problem of small-x be- effects in their evolution are important. Very large havior in DIS and shall discuss the model [6] based on densities can be obtained in heavy-ion collisions at Reggeon theory and the dipole picture of interaction very high energies and the dynamics of these pro- of virtual photons with nucleons. This model gives a cesses has many features common to DIS in the limit self-consistent description of structure functions of a → x 0. Experimental data on small-x DIS obtained at proton and diffraction dissociation of a photon in a HERA [1] show that distributions of quarks and glu- broad region of virtualities Q2. I discuss the pattern of ons have a fast increase as x decreases up to values −4 saturation in this model. It is shown in Section 4 that ∼10 , which can be obtained at HERA. In Reggeon the Gribov approach to interactions with nuclei al- theory, this increase is related to an intercept of the lows one to calculate shadowing effects in the small-x ∼ leading Pomeron singularity αP(0): xq(x) or xg(x) region for structure functions of nuclei. Production of ∆ ≡ − 1/x ,where∆ αP(0) 1 (see, for example, [2]). hadrons and jets in heavy-ion collisions is described There are good reasons to believe (for reviews on in Section 5. It is emphasized that the coherence this subject, see [2, 3])that this fast increase will condition strongly limits the kinematic region where be modified for even smaller values of x due to the the shadowing effects for inclusive cross sections are → influence of unitarity effects. In the limit x 0,par- important. In the same approach which leads to de- tonic distributions xq(x)(xg(x)) reach the limit of scription of nuclear structure functions, it is shown saturation and increase only as powers of ln(1/x). that shadowing effects in heavy-ion collisions lead The problem of saturation is even more important to a sizable reduction of inclusive particle spectra in in heavy-ion collisions [4], where densities of par- comparison with the Glauber model even at RHIC tons in colliding nuclei are very large. This problem energies and agree with existing experimental data. has been studied in QCD perturbation theory [4, 5], It is emphasized, however, that at RHIC (and even and partonic configurations, which include nonlinear LHC)energies, the situation is far from the saturation effects in QCD evolution, were called “color glass limit. The effects which can lead to suppression of ∗This article was submitted by the author in English. production for particles with large p⊥ at RHIC are **e-mail: [email protected] discussed. 1063-7788/05/6805-0779$26.00 c 2005 Pleiades Publishing, Inc. 780 KAIDALOV 2. POMERON AND SMALL-x PHYSICS Wustho¨ ff (G-BW)[14]. In the Pomeron theory, the multi-Pomeron contributions are related to diffractive Scattering amplitude of a virtual photon on a pro- 2 production processes, and for self-consistency, it is ton target at fixed virtuality Q and very high c.m. en- important to describe simultaneously total cross sec- ergy W is described in Reggeon theory by exchange tions (structure functions)and di ffractive processes. of the Pomeron pole and multi-Pomeron cuts. An At high energies, virtual photons dissociate first to 2 ≈ increase in the structure function F2(x, Q ) as x qq¯ pairs, which interact with a target. For small rel- 2 2 → ∼ ∆ Q /W 0 in the pole approximation 1/x ,where ative transverse distance r⊥ between q and q¯,such ≡ − 2 2 ∆ αP(0) 1.AtlargeQ , adipolehasasmall(∼ r ) total interaction cross ⊥ F (x, Q2) ∼ e2x(q (x, Q2)+q (x, Q2)), section and small diffraction dissociation cross sec- 2 i i i tion. On the other hand, for a large-size pair, these i cross sections are not small. As a result, the role 2 2 where qi(x, Q ) is the distribution of quarks of fla- of multi-Pomeron rescatterings decreases as Q in- vor i. Thus, for ∆ > 0, the density of partons in- creases. Explicit models which take these effects into creases as x → 0. The value of ∆ was calculated in account have been constructed in the framework of QCD perturbation theory. Summation of the leading the Reggeon theory. The contribution of small-size ln(1/x) leads to the BFKL [7] Pomeron with the value configurations σ(tot) has been described in QCD per- ≈ s of ∆=(12ln2/π)αs 0.5, which corresponds to a turbation theory in a form similar to the one used very fast increase in parton densities. In this leading in [14]: approximation, the Pomeron corresponds to a cut in (tot)T (L) 2 the j plane. In the next-to-leading approximation, σs (s, Q ) (1) 1) the leading singularity is the Regge pole and the r0 1 value of ∆ substantially decreases to the values ∆= 2b d2r dz| T (L) r, z, Q2 |2 0.15−0.25 [8]. This leading pole strongly depends =4 d Ψ ( ) on the nonperturbative region of small momentum 0 0 2 transfer [9]. A connection of the Pomeron with the × σs(b, r, s, Q ), spectrum of glueballs and the role of nonperturba- tive effects were studied in [10], using the method where ΨT (L)(r, z, Q2) are wave functions of qq¯ pair of vacuum correlators [11]. It was shown [10] that for transverse (longitudinal)photons. Integration on qq¯ confinement effects and mixing of gluonic and - transverse size r was limited by the value r0 ≈ 0.2 fm. Regge trajectories are important for the value of the intercept of the Pomeron trajectory. The Pomeron Contributions of large-distance dipoles were de- pole with ∆ > 0 leads for s ≡ W 2 →∞to a violation scribed in the same framework as hadronic inter- of s-channel unitarity. It is well known (see, for ex- actions. An important difference from the G-BW ample, [2])that the unitarity is restored if the multi- model is that the model [6] corresponds to sum- Pomeron exchanges in the t channel are taken into mation of a certain class of Reggeon diagrams and account. In the following, I will use for this intercept formulas for cross sections σ(b, r, s, Q2) are given in the value ∆ ≈ 0.2, found from analysis of experimen- the impact parameter representation. Also, the fan- tal data on high-energy hadronic interactions with type diagrams with triple Pomeron interaction are account of multi-Pomeron contributions [12]. included [6]. This leads to a qualitative difference from the G-BW model in the pattern of saturation for very small x (x 10−5)2). In particular, in the saturation 3. MODELS FOR SMALL-x PHYSICS region, cross sections of qq¯ pairs with a target do not Experimental data at HERA [1] show that distri- tend to a constant (as in the G-BW model), but have butionsofquarksandgluonshaveafastriseasx the Froissart-type increase ∼ ln2(1/x). This is due to 2 decreases. For parametrization of F2(x, Q ) at small a logarithmic increase in the interaction radius and x ∼ 1/x∆, data demonstrate that ∆ increases with is beyond the scope of QCD perturbation theory. It is Q2 and is close to 0.2 at Q2 ∼ 10 GeV2. This was worth emphasizing that the model of [6] predicts that interpreted in [13] as a consequence of the decrease an approach to the unitarity limit or saturation regime in multi-Pomeron contributions as Q2 increases. A is very slow and, for large Q2 (Q ∼ 102 GeV2),itis similar interpretation in terms of saturation of parton not achieved even for x ∼ 10−12. densities has been proposed by Golec-Biernat and 2)Models of [6], as well as the G-BW model [14], give a good 1) There is a sequence of poles in the j plane concentrated at description of HERA data on structure functions F2 and point j =1. diffractive production. PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 5 2005 THE PROBLEM OF “SATURATION” IN DIS 781 2 4. STRUCTURE FUNCTIONS OF NUCLEI × fA(tmin)/σγ∗N (x, Q ) . IN THE SHADOWING REGION Using information on diffraction dissociation in γ∗N Study of nuclear structure functions in the collisions, it is possible to calculate nuclear structure small-x region can provide important information on functions in good agreement with experimental data. mechanisms of shadowing. The total cross section for These calculations show that the shadowing effects a virtual photon–nucleus interaction can be written increase as x decreases in the region x<0.1.The as a series in number of rescatterings on nucleons of (2) (1) a nucleus shadowing correction σ /σ increases with atomic ∼ 1/3 ∼ −2 (tot) (1) (2) number A , but in the region of x 10 ,rel- σ ∗ = σ ∗ + σ ∗ + .... γ A γ A γ A (2) evant for RHIC energies (see below), and for Q2 ∼ 2 The first term corresponds to a sum of incoherent 1 GeV , a decrease in F2A due to shadowing is less interactions and is equal to3) than 25% even for heavy nuclei (Au). (1) Thus, the shadowing effects for distributions of σ ∗ = Aσ ∗ ; γ A γ N (3) quarks (which are directly measured in DIS)can be the second term describes the shadowing and accord- well described theoretically and prediction for smaller ing to Gribov theory [15] can be expressed in terms x, relevant for LHC energies, looks reliable. Informa- of the cross section of diffraction dissociation of a tion on shadowing effects for gluons is more limited. photon on a nucleon target This information can be obtained from the data on “gluonic content” of the Pomeron [19]. Existing in- (2) − 2 2 −3 σ = 4π d bTA(b) (4) formation indicates that, at least for x 10 ,shad- owing for gluons is of similar size as the one for DD dσ ∗ (t =0) quarks. × dM 2 γ N f (t ), dM 2dt A min where TA(b)= dzρA(r) is the nuclear profile func- 5. HEAVY-ION COLLISIONS tion, ρ(r) is nuclear density ( ρ (r)d3r = A),and A In heavy-ion collisions, the problem of shadowing 1 · of soft partons of colliding nuclei is very important. f (t)= eiq rρ (r)d3r A A A This phenomenon takes place only in the kinematical region where longitudinal momentum fractions of soft is the nuclear form factor. The minimal value of mo- partons satisfy the condition x 1/(m R )(i = mentum transfer to nucleons i N A 2 2 1, 2) discussed in the previous section. Values of xi − 2 2 Q + M x can be determined from the kinematical conditions tmin = xPmN ,xP = = , (5) s β 2 x1x2 = M⊥/s, x1 − x2 = xF, where β = Q2/(Q2 + M 2). It is important to em- where M⊥ is the transverse mass of the system pro- phasize that the shadowing effects are different from duced in the collision and xF is its Feynman x. zero only in the region of very small xP or x,where √ 2 tmin 1/R (RA is the nuclear radius). The same For example, at RHIC energy s = 200 GeV in A ≈ condition of coherence corresponds to lifetimes of the central rapidity region (xF 0), the shadowing the initial hadronic fluctuation τh ∼ 1/(mN x) much for Au–Au collisions is possible only for transverse larger than the radius of a nucleus. These well-known momenta less than several GeV.In the forward region conditions for existence of shadowing are also im- (x2 x1 ≈ xF), the shadowing is absent for partons portant for heavy-ion collisions (see below). Higher of the nucleus 1 and extends to p⊥ ∼ 10 GeV for order rescatterings in Eq. (4)are model dependent. partons of nucleus 2. In [16, 17], they were calculated in the Schwimmer It is possible to calculate the shadowing effects model [18], where for inclusive densities of produced particles using T (b) the Gribov approach [15] to nuclear interactions. F /F = d2b A , (6) 2A 2N 1+F (x, Q2)T (b) For models of the Glauber type, which do not take A into account interactions between Pomerons, there is with the Abramovsky–Gribov–Kancheli (AGK)rule [20], 2 2 DD 2 which expresses inclusive spectra in A1A2 collisions F (x, Q )=4π dM dσγ∗N (t =0)/dM dt in terms of spectra in NN collisions 3) dσA1A2 dσNN In the small-x region, it is possible to neglect a difference = TA1A2 (b) , (7) between interaction of γ∗ with p and n. dy dy PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 5 2005 782 KAIDALOV where for xA2 1/(mN RA2 ), the lifetime for the initial- state configuration of a nucleus A1 is larger than 2 b − s s TA1A2 (b)= d sTA1 ( )TA2 ( ). the size of a nucleus A2. In this case, conditions for coherence are satisfied and usual AGK cutting For density of particles, we obtain from Eq. (7) rules [20] are valid. For xA2 1/(mN RA), the space- time picture is close to final-state reinteractions of dn (b) dn A1A2 NN produced particles with formation time [26]. The last = nA1A2 (b) , (8) dy dy situation corresponds to a picture used in most of where the existing models of heavy-ion collisions. Note (tot) that it is valid only in the nonshadowing region. TA1A2 (b)σNN This situation is very general. For J/ψ production nA1A2 (b)= σ(tot) in hadron–nucleus interactions, it was discussed A1A2 in detail in [27]. It has important implications for is the number of collisions in the Glauber model. production of particles and jets with large p⊥ at ≈ − Note that Eqs. (7), (8) are valid in the Glauber RHIC. For xF =0√ and p⊥ 1 2 GeV, the shad- model for xF ≈ 0 in the limit s →∞.Forfinite ener- owing effects at s = 200 GeV are small, as was gies, there are kinematical corrections due to energy– emphasized below and is demonstrated by the data momentum conservation effects [21]. on D–Au collisions [28]. However, in this region of In Reggeon theory, the shadowing effects dis- xAi 1/(mRA),thefinal-state interaction effects cussed above are related to diagrams with interac- are important. In particular, account of the Cronin tions between Pomerons. They lead to a substantial effect and interaction of a particle (jet)with comovers decrease in particle densities at high energies. On the allowed one to describe [29] a suppression of particle other hand, different cuttings of the same diagrams production at large p⊥ observed at RHIC [24] for Au– correspond to reinteractions between particles pro- Au collisions. The data in D–Au collisions [28] can duced in NN subcollisions and can lead to equilibra- also be understood in the same model. tion in the system produced in heavy-ion collisions. In string models, interactions between Pomerons can be considered as interactions of strings, which can lead 6. CONCLUSIONS to percolation in the string system [22]. The Pomeron theory of high-energy interactions The same model, which has been used for de- is able to describe a large class of physical processes scription of shadowing effects for nuclear structure including small-x DIS and nuclear interactions. functions, was used for prediction of inclusive par- An important problem for high-energy physics is ticle densities in heavy-ion collisions [21]. In this the nature of the Pomeron. The analysis of this prob- approach, expression (8)for densities of particles in lem in QCD with inclusion of both nonperturbative A A collisions is modified as follows [21]: 1 2 and perturbative effects shows that the Pomeron has dnA1A2 dnNN a very rich dynamical structure. = nA1A2 (b) γA1 γA2 , (9) dy dy The problem of saturation of parton densities is where related in the Pomeron theory to interactions be- tween Pomerons. These effects are especially impor- 1 T (b) 2 A tant for heavy-ion collisions. The shadowing effects γA = d b 2 . (10) A 1+F (x, Q )TA(b) are clearly seen in the RHIC data, but the saturation is not achieved at RHIC energies. For LHC, Eqs. (9), (10) predict a decrease in particle densities by a factor of ∼4 compared to the Glauber model [21], while for RHIC the suppression ACKNOWLEDGMENTS is ≈2. Detailed predictions for shadowing suppression are given in [17]. This approach agrees [2] with the I would like to thank N. Armesto, K. Boreskov, results of experiments at RHIC [23, 24]. The depen- A. Capella, V. Fadin, E.G. Ferreiro, O.V. Kancheli, dence of number of produced particles on number of V.A. Khoze, E. Levin, L.N. Lipatov, A. Martin, participant nucleons is also in agreement with da- C. Merino, C.A. Salgado, K.A. Ter-Martirosyan, and ta [25]. J. Tran Thanh Van for useful discussions. 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Original English Text Copyright c 2005 by Ebert, Faustov, Galkin, Martynenko. Properties of Doubly Heavy Baryons in the Relativistic Quark Model* D. Ebert1),R. N. Faustov 2),V. O. Galkin 2),and A. P. Martynenko 3) Received May 5, 2004; in final form, November 2, 2004 Abstract—Mass spectra and semileptonic decay rates of baryons consisting of two heavy (b or c)and one light quark are calculated in the framework of the relativistic quark model. The doubly heavy baryons are treated in the quark–diquark approximation. The ground and excited states of both the diquark and quark–diquark bound systems are considered. The quark–diquark potential is constructed. The light quark is treated completely relativistically, while the expansion in the inverse heavy-quark mass is used. The weak transition amplitudes of heavy diquarks bb and bc going, respectively, to bc and cc are explicitly expressed through the overlap integrals of the diquark wave functions in the whole accessible kinematic range. The relativistic baryon wave functions of the quark–diquark bound system are used for the calculation of the decay matrix elements, the Isgur–Wise function, and decay rates in the heavy-quark limit. c 2005 Pleiades Publishing, Inc. 1. INTRODUCTION Then the HQET expansion in the inverse heavy- quark mass can be used. The main distinction of qQQ qQ¯ QQ The description of baryons within the constituent the baryon from the meson is that the color source, though being almost localized, still quark model and quantum chromodynamics (QCD) is a composite system bearing integer spin values is a very important problem. Since the baryon is a (0, 1,...). Hence, it follows that the interaction of the three-body system, its theory [1] is much more com- heavy diquark with the light quark is not pointlike, plicated compared to the two-body meson system. but is smeared by the form factor expressed through Even now, it is not clear which of the two main the overlap integrals of the diquark wave functions. QCD models, Y law or ∆ law, correctly describes Besides this, the diquark excitations contribute to the the nonperturbative (long-range) part of the quark baryon excited states. interaction in the baryon [2, 3]. The popular quark– Recently, the first experimental indications of the diquark picture of a baryon is not universal and does existence of doubly charmed baryons were published not work in all cases [4]. The success of the heavy- by SELEX [14]. Although these data need further ex- quark effective theory (HQET) [5] in predicting some perimental confirmation and clarification, this man- ¯ properties of the heavy-light qQ mesons (B and ifests that, in the near future, the mass spectra and D) suggests applying these methods to heavy-light decay rates of doubly heavy baryons will be measured. baryons too. The simplest baryonic systems of this This gives additional grounds for the theoretical in- kind are the so-called doubly heavy baryons (qQQ)[1, vestigation of the doubly heavy baryon properties. The 6–12].Thetwoheavyquarks(b or c)composein energies necessary to produce these particles have this case a bound diquark system in the antitriplet already been reached. The main difficulty remains color state, which serves as a localized color source. in their reconstruction, since these particles have in The light quark orbits around this heavy source at general a large number of decay modes and thus high amuchlarger(∼1/mq) distance than the source statistics is required [15]. size (∼2/mQ). Thus, the doubly heavy baryons look In previous approaches to the calculation of dou- effectively like a two-body bound system and strongly bly heavy baryon properties, the expansion in inverse resemble the heavy-light B and D mesons [2, 13]. powers not only of the heavy-quark (diquark) mass mQ (Md) but also of the light-quark mass mq was ∗This article was submitted by the authors in English. 1) carried out. The estimates of the light-quark velocity Institut fur¨ Physik, Humboldt-Universitat¨ zu Berlin, Ger- in these baryons show that the light quark is highly many. ∼ − 2)Scientific Council for Cybernetics, Russian Academy of Sci- relativistic (v/c 0.7 0.8). Thus, the nonrelativistic ences, Russia. approximation is not adequate for the light quark. 3)Samara State University, Samara, Russia. Here, we present a consistent treatment of doubly 1063-7788/05/6805-0784$26.00 c 2005 Pleiades Publishing, Inc. PROPERTIES OF DOUBLY HEAVY BARYONS 785 W d d' q BQQ BQ'Q Fig. 1. Weak-transition matrix element of the doubly heavy baryon in the quark–diquark approximation. heavy baryon properties in the framework of the rela- of Ξcc, Ξbb, Ωcc,andΩbb baryons. We consider the tivistic quark model, based on the quasipotential wave excitations of both the quark–diquark system and the equation without employing the expansion in 1/mq; diquark. The mixing between excited baryon states namely, the light quark is treated fully relativistically. with the same total angular momentum and parity Concerning the heavy diquark (quark), we apply the is discussed. For Ξcb and Ωcb baryons, composed of expansion in 1/Md (1/mQ). Then we use the calcu- heavyquarksofdifferent flavors, we give predictions lated wave functions for calculating the semileptonic only for ground states, since the excited states of the decay rates of doubly heavy baryons in the quark– cb diquark are unstable under the emission of soft diquark approximation. The covariant expressions for gluons [11]. A detailed comparison of our predictions the semileptonic decay amplitudes of the baryons with with other approaches is given. We reveal the close the spin 1/2, 3/2 are obtained in the limit mc,mb → similarity of the excitations of the light quark in a dou- ∞ and compared with the predictions of HQET. The bly heavy baryon and a heavy-light meson. We also calculation of semileptonic decays of doubly heavy test the fulfillment of different relations between mass baryons (bbq) or (bcq) to doubly heavy baryons (bcq) splittings of baryons with two c or b quarks, as well as or (ccq) can be divided into two steps (see Fig. 1). the relations between splittings in the doubly heavy The first step is the study of form factors of the weak baryons and heavy-light mesons, following from the transition between initial and final doubly heavy di- heavy-quark symmetry. Then we apply our model to quarks. The second one consists in the inclusion of the investigation of the heavy-diquark transition ma- the light quark in order to compose a baryon with spin trix elements in Section 6. The transition amplitudes 1/2 or 3/2. of heavy diquarks are explicitly expressed in a covari- ant form through the overlap integrals of the diquark The paper is organized as follows. In Section 2, wave functions. The obtained general expressions re- we describe our relativistic quark model, giving spe- produce in the appropriate limit the predictions of cial emphasis to the construction of the quark–quark heavy-quark symmetry. Section 7 is devoted to the interaction potential in the diquark and the quark– construction of transition matrix elements between diquark interaction potential in the baryon. In Sec- doubly heavy baryons in the quark–diquark approx- tion 3, we apply our model to the investigation of imation. The corresponding Isgur–Wise function is the heavy-diquark properties. The cc-andbb-diquark determined. In Section 8, semileptonic decay rates of mass spectra are calculated. We also determine the doubly heavy baryons are calculated in the nonrela- diquark interaction vertex with the gluon, using the tivistic limit for heavy quarks. Section 9 contains our quasipotential approach, and calculate diquark wave conclusions. functions. Thus, we take into account the internal structure of the diquark, which considerably modi- fies the quark–diquark potential at small distances 2. RELATIVISTIC QUARK MODEL and removes fictitious singularities. In Section 4, we construct the quasipotential of the interaction of a In the quasipotential approach and quark–diquark light quark with a heavy diquark. The light quark is picture of doubly heavy baryons, the interaction of two treated fully relativistically. We use the expansion in heavy quarks in a diquark and the light-quark inter- inverse powers of the heavy-diquark mass to sim- action with a heavy diquark in a baryon are described plify the construction. First, we consider the infinitely by the diquark wave function (Ψd) of the bound d heavy diquark limit and, then, include the 1/MQQ quark–quark state and by the baryon wave function corrections. In Section 5, we present our predictions (ΨB) of the bound quark–diquark state, respectively, for the mass spectra of the ground and excited states which satisfy the quasipotential equation [16] of the PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 5 2005 786 EBERT et al. Schrodinger¨ type [17] with b2(M) p2 V 2 µ ν (p, q; M)= αsDµν (k)γ1 γ2 − Ψd,B(p) (1) 3 2µR 2µR 1 µ 1 3 V S d q + Vconf(k)Γ1 Γ2;µ + Vconf(k); = V (p, q; M)Ψ (q), 2 2 (2π)3 d,B (b) for quark–diquark (qd) interaction, where the relativistic reduced mass is V (p, q; M) (6) E E M 4 − (m2 − m2)2 1 2 1 2 | | µR = = 3 , (2) d(P ) Jµ d(Q) 4 ν E1 + E2 4M = u¯q(p) αsDµν (k)γ uq(q) 2 Ed(p)Ed(q) 3 and E1 and E2 are given by ∗ µ V 2 − 2 2 2 − 2 2 + ψd(P )¯uq(p)Jd;µΓq Vconf(k)uq(q)ψd(Q) M m2 + m1 M m1 + m2 E1 = ,E2 = . ∗ S 2M 2M + ψd(P )¯uq(p)Vconf(k)uq(q)ψd(Q), (3) where αs is the QCD coupling constant; the color Here, M = E1 + E2 is the bound-state (diquark or factor is equal to 2/3 for quark–quark and 4/3 baryon) mass; m1,2 are the masses of heavy quarks for quark–diquark interactions; d(P )|Jµ|d(Q) is (Q1 and Q2), which form the diquark, or of the heavy the vertex of the diquark–gluon interaction, which diquark (d) and light quark (q), which form the doubly is discussed in detail below [P =(Ed, −p) and heavy baryon (B); and p is their relative momentum. Q =(Ed, −q)]; Dµν is the gluon propagator in the In the center-of-mass system, the relative momen- Coulomb gauge: tum squared on the mass shell reads 00 4π 2 2 2 2 k − [M − (m + m ) ][M − (m − m ) ] D ( )= 2 , (7) b2(M)= 1 2 1 2 . k 4M 2 4π kikj (4) Dij(k)=− δij − ,D0i = Di0 =0, k2 k2 V (p, q; M) The kernel in Eq. (1) is the quasipo- and k = p − q;andγ and u(p) are the Dirac matri- tential operator of the quark–quark or quark–diquark µ ces and spinors: interaction. It is constructed with the help of the off-mass-shell scattering amplitude, projected onto (p)+m 1 the positive energy states. In the following analysis, uλ(p)= · χλ, (8) we closely follow the similar construction of the 2 (p) (σ p) quark–antiquark interaction in mesons, which were (p)+m extensively studied in our relativistic quark model [18, 1 λ = ± , 19]. For the quark–quark interaction in a diquark, we 2 use the relation VQQ = V ¯ /2, arising under the as- QQ 2 2 sumption about the octet structure of the interaction with (p)= p + m . from the difference of the QQ and QQ¯ color states. The diquark state in the confining part of the An important role in this construction is played by the quark–diquark quasipotential (6) is described by the Lorentz structure of the confining interaction. In our wave functions analysis of mesons, while constructing the quasipo- 1, for scalar diquark, tential of quark–antiquark interaction, we adopted ψd(P )= (9) that the effective interaction is the sum of the usual εd(p), for axial-vector diquark, one-gluon-exchange term with the mixture of long- where the four-vector range vector and scalar linear confining potentials, · · where the vector confining potential contains the (εd p) (εd p)p εd(p)= , εd + , (10) Pauli terms. We use the same conventions for the Md Md(Ed(p)+Md) construction of the quark–quark and quark–diquark εd(p) · p =0, interactions in the baryon. The quasipotential is then defined as follows [10, 18]: is the polarization vector of the axial-vector di- quark with momentum p, E (p)= p2 + M 2,and (a) for the quark–quark (QQ) interaction, d d εd(0) = (0, εd) is the polarization vector in the di- V (p, q; M) (5) quark rest frame. The effective long-range vector =¯u1(p)¯u2(−p)V(p, q; M)u1(q)u2(−q), vertex of the diquark can be represented in the form PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 5 2005 PROPERTIES OF DOUBLY HEAVY BARYONS 787 (P + Q)µ , for scalar diquark, 2 E (p)E (q) J = d d (11) d;µ (P + Q) µ − iµd ν ˜ Σµkν, for axial-vector diquark, 2 Ed(p)Ed(q) 2Md where k˜ =(0, k) and we neglected the contribution 3. DIQUARKS IN THE RELATIVISTIC QUARK of the chromoquadrupole moment of the axial-vector MODEL diquark, which is suppressed by an additional power of k/M . Here, the antisymmetric tensor The quark–quark interaction in the diquark con- d sists of the sum of the spin-independent and spin- ν − ν − ν (Σρσ)µ = i(gµρδσ gµσδρ ), (12) dependent parts: and the axial-vector diquark spin S is given by SI SD d VQQ(r)=VQQ(r)+VQQ (r). (16) (Sd;k)il = −iεkil. We choose the total chromomag- 2 2 netic moment of the axial-vector diquark µd =2[10, The spin-independent part with account of v /c cor- 20]. rections including retardation effects [23] is given by The effective long-range vector vertex of the quark 2 α (µ2) 1 is defined by [18, 19] V SI (r)=− s + (Ar + B) (17) QQ 3 r 2 iκ Γ (k)=γ + σ k˜ν, (13) 1 1 1 2 α (µ2) µ µ 2m µν + + ∆ − s 8 m2 m2 3 r where κ is the Pauli interaction constant charac- 1 2 terizing the anomalous chromomagnetic moment of 1 + (1 − ε)(1 + 2κ)Ar quarks. In the configuration space, the vector and 2 scalar confining potentials in the nonrelativistic limit 1 2 α (p · r)2 reduce to − s 2 + p + 2 V − 2m1m2 3 r r W Vconf(r)=(1 ε)Vconf(r), (14) 1 1 − ε ε 1 1 V S (r)=εV (r), − conf conf + 2 + 2 2 2m1m2 4 m m with 1 2 (p · r)2 V (r)=V S (r)+V V (r)=Ar + B, × 2 − conf conf conf (15) Ar p 2 r W where ε is the mixing coefficient. 1 1 − ε ε 1 1 All the parameters of our model, like quark masses, − 2 + 2 + 2 Bp , parameters of linear confining potential A and B, 2 2m1m2 4 m1 m2 ε mixing coefficient , and anomalous chromomagnetic where {...} denotes the Weyl ordering of operators quark moment κ,werefixed from the analysis of W and heavy-quarkonium masses [18] and radiative de- 4π cays [21]. The quark masses mb =4.88 GeV, mc = 2 αs(µ )= 2 2 (18) 1.55 GeV, ms =0.50 GeV, mu,d =0.33 GeV and the β0 ln(µ /Λ ) 2 parameters of the linear potential A =0.18 GeV with µ fixed to be equal to twice the reduced mass. and B = −0.30 GeV have standard values of quark models. The value of the mixing coefficient of vector The spin-dependent part of the quark–quark po- and scalar confining potentials ε = −1 has been tential can be represented in our model [18] as follows: determined from the consideration of the heavy- V SD(r)=aL · S˜ (19) quark expansion [22] and meson radiative decays [21]. QQ 3 Finally, the universal Pauli interaction constant κ = + b (S · r)(S · r) − (S · S ) −1 has been fixed from the analysis of the fine splitting r2 1 2 1 2 3 of heavy-quarkonium PJ -states [18] and also from + c S1 · S2 + d L · (S1 − S2), the heavy-quark expansion [22]. Note that the long- range magnetic contribution to the potential in our 2 2 2 model is proportional to (1 + κ) and thus vanishes for 1 m1 + m2 2 αs(µ ) a = 1+ 3 (20) the chosen value of κ = −1. m1m2 4m1m2 3 r PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 5 2005 788 EBERT et al. 1 m2 + m2 A 1 − 1 2 + (1 + κ) As a result, we obtain the mass formula 2 4m1m2 r 2 2 b (M) ˜ 2 = W + a L · S (22) (m1 + m2) A 2µ × (1 − ε) , R 2m1m2 r 3 · · − · + b 2 (S1 r)(S2 r) (S1 S2) 2 r 1 2αs(µ ) 1 2 − A b = 3 + (1 + κ) (1 ε) , + c S1 · S2 + d L · (S1 − S2) , 3m1m2 r 2 r where 2 8πα (µ2) c = s δ3(r) 2 SI p 3m1m2 3 W = VQQ + , 2µR 1 A 1 + (1 + κ)2(1 − ε) , L · S˜ = (J˜(J˜+1)− L(L +1)− S˜(S˜ +1)), 2 r 2 3 2 − 2 2 (S · r)(S · r) − (S · S ) 1 m2 m1 2 αs(µ ) − 1 A r2 1 2 1 2 d = 3 m1m2 4m1m2 3 r 2 r 6(L · S˜ )2 +3L · S˜ −2S˜(S˜ +1)L(L +1) = − , 1 m2 − m2 A 2(2L − 1)(2L +3) + (1 + κ) 2 1 (1 − ε) , 2 2m1m2 r 1 3 S · S = S˜(S˜ +1)− , S˜ = S + S , 1 2 2 2 1 2 where L is the orbital momentum and S1,2, S˜ = S1 + S2 are the spin momenta. For κ = −1, the form of and a , b , c , d are the appropriate averages over the spin-dependent potential (19) agrees with the radial wave functions of Eq. (20). We use the notation expression based on QCD [24, 25]. 2S˜+1 for the heavy-diquark classification: n LJ˜,where Now we can calculate the mass spectra of heavy n =1, 2,... is a radial quantum number, L is the diquarks with account of all relativistic corrections angular momentum, S˜ =0, 1 is the total spin of two 2 2 (including retardation effects) of order v /c .For heavy quarks, and J˜ = L − S,L,L˜ + S˜ is the total this purpose, we substitute the quasipotential, which ˜ ˜ is a sum of the spin-independent (17) and spin- angular momentum (J = L + S), which is considered dependent (19) parts, into the quasipotential Eq. (1). as the spin of a diquark (Sd) in the following section. Then we multiply the resulting expression from the The first term on the right-hand side of the mass left by the quasipotential wave function of a bound formula (22) contains all spin-independent contri- state and integrate with respect to the relative mo- butions, the second and the last terms describe the mentum. Taking into account the accuracy of the spin–orbit interaction, the third term is responsible calculations, we can use for the resulting matrix for the tensor interaction, and the fourth term gives elements the wave functions of Eq. (1) with the static the spin–spin interaction. The results of our numer- potential ical calculations of the mass spectra of cc and bb 2 α (µ2) 1 V NR(r)=− s + (Ar + B). (21) QQ 3 r 2 Table 2. Mass spectrum and mean squared radii of the bb diquark Table 1. Mass spectrum and mean squared radii of the cc State Mass, GeV r2 1/2,fm diquark 3 1 S1 9.778 0.37 2 1/2 3 State Mass, GeV r ,fm 2 S1 10.015 0.71 3 3 1 S1 3.226 0.56 3 S1 10.196 0.98 3 3 2 S1 3.535 1.02 4 S1 10.369 1.22 3 1 3 S1 3.782 1.37 1 P1 9.944 0.57 1 1 1 P1 3.460 0.82 2 P1 10.132 0.87 1 1 2 P1 3.712 1.22 3 P1 10.305 1.12 1 1 3 P1 3.928 1.54 4 P1 10.453 1.34 PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 5 2005 PROPERTIES OF DOUBLY HEAVY BARYONS 789 diquarks are presented in Tables 1 and 2. The mass of F(r) the ground state of the bc diquark in the axial-vector 1.0 3 (1 S1) state is A 0.8 Mbc =6.526 GeV 1 0.6 and in the scalar (1 S1) state is S Mbc =6.519 GeV. 0.4 In order to determine the diquark interaction with 0.2 the gluon field, which takes into account the diquark structure, it is necessary to calculate the correspond- ing matrix element of the quark current between di- 0 0.2 0.4 0.6 0.8 quark states. This diagonal matrix element can be r, fm parametrized by the following set of elastic form fac- tors: Fig. 2. The form factors F(r) for the cc diquark. The solid S curve is for the 1S state, the dashed curve for the 1P state, (a) scalar diquark ( ): the dash-dotted curve for the 2S state, and the dotted 2 curve for the 2P state. S(P )|Jµ|S(Q) = H+(k )(P + Q)µ; (23) (b) axial-vector diquark (A): for different cc and bb diquark states are given in A(P )|J |A(Q) (24) µ Tables 3 and 4. As we see, the functions F(r) vanish = −[ε∗(P ) · ε (Q)]H (k2)(P + Q) + H (k2) → d d 1 µ 2 in the limit r 0 and become unity for large values × ∗ · · ∗ of r. Such a behavior can be easily understood [εd(P ) Q]εd;µ(Q)+[εd(Q) P ]εd;µ(P ) intuitively. At large distances, a diquark can be well 1 2 ∗ · · approximated by a pointlike object and its internal + H3(k ) 2 [εd(P ) Q][εd(Q) P ](P + Q)µ, MV structure cannot be resolved. When the distance to the diquark decreases, the internal structure plays a where k = P − Q and εd(P ) is the polarization vector of the axial-vector diquark (10). more important role. As the distance approaches zero, the interaction weakens and goes to zero for r =0, In our quark model, we find the following relation since this point coincides with the center of gravity between these diquark transition form factors in the nonrelativistic limit [26]: of the two heavy quarks forming the diquark. Thus, the function F(r) gives an important contribution 2 2 2 2 H+(k )=H1(k )=H2(k )=2F(k ), (25) to the short-range part of the interaction of the light 2 H3(k )=0, quark with the heavy diquark in the baryon and can be √ neglected for the long-range (confining) interaction. E M d3p F(k2)= d d It is important to note that the inclusion of such a E + M (2π)3 3 d d function removes a fictitious singularity 1/r at the 2m origin arising from the one-gluon-exchange part of × ¯ Q2 ↔ Ψd p + k Ψd(p)+(1 2) , the quark–diquark potential when the expansion in Ed + Md inverse powers of the heavy-quark mass is used. where Ψd are the diquark wave functions in the rest reference frame. We calculated corresponding form factors F(r)/r, which are the Fourier transforms of Table 3. Parameters ξ and ζ for ground and excited states F(k2)/k2, using the diquark wave functions found of the cc diquark by numerically solving the quasipotential equation. In Fig. 2, the functions F(r) for the cc diquark in the 1S, State ξ,GeV ζ,GeV2 1P , 2S, 2P states are shown as an example. We see that the slope of F(r) decreases with the increase in 1S 1.30 0.42 the diquark excitation. Our estimates show that this 2S 0.67 0.19 form factor can be approximated with a high accuracy 3S 0.57 0.12 by the expression − − 2 1P 0.74 0.315 F(r)=1− e ξr ζr , (26) 2P 0.60 0.155 which agrees with previously used approxima- tions [27]. The values of the parameters ξ and ζ 3P 0.55 0.075 PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 5 2005 790 EBERT et al. → 2 − 2 2 Table 4. Parameters ξ and ζ for ground and excited states q(p) Eq =(M Md + mq)/(2M) in the Dirac of the bb diquark spinors (8) [19]. Such a simplifying substitution is widely used in quantum electrodynamics [28–30] and State ξ,GeV ζ,GeV2 introduces only minor corrections of order of the ratio of the binding energy V to E . This substitution 1S 1.30 1.60 q makes the Fourier transformation of the potential (27) 2S 0.85 0.31 local. In contrast with the heavy-light meson case, 3S 0.66 0.155 no special consideration of the one-gluon-exchange term is necessary, since the presence of the diquark 4S 0.56 0.09 structure described by an extra function F(k2) in 1P 0.90 0.59 Eq. (27) removes the fictitious 1/r3 singularity at the 2P 0.65 0.215 origin in configuration space. The resulting local quark–diquark potential for 3P 0.58 0.120 Md →∞can be represented in configuration space 4P 0.51 0.085 in the following form: Eq + mq V →∞(r)= (28) Md 2E 4. QUASIPOTENTIAL OF THE INTERACTION q OF A LIGHT QUARK WITH A HEAVY 1 DIQUARK × VCoul(r)+Vconf(r)+ 2 (Eq + mq) The expression for the quasipotential (6) can, in principle, be used for arbitrary quark and diquark × V − S masses. The substitution of the Dirac spinors (8) and p[VCoul(r)+Vconf(r) Vconf(r)]p diquark form factors (23) and (24) into (6) results in an extremely nonlocal potential in the configuration − Eq + mq V − space. Clearly, it is very hard to deal with such poten- ∆Vconf(r)[1 (1 + κ)] 2mq tials without any simplifying expansion. Fortunately, 2 in the case of the heavy-diquark–light-quark picture + V (r) − V S (r) − V V (r) r Coul conf conf of the baryon, one can carry out (following HQET) the expansion in inverse powers of the heavy-diquark Eq Eq + mq mass Md. The leading terms then follow in the limit × − 2(1 + κ) l · Sq , Md →∞. mq 2mq where VCoul(r)=−(4/3)αsF(r)/r is the smeared 4.1. Infinitely Heavy Diquark Limit Coulomb potential. The prime denotes differentiation In the limit Md →∞, the heavy-diquark ver- with respect to r; l is the orbital momentum and tices (23) and (24) have only the zeroth component, Sq is the spin operator of the light quark. Note that and the diquark mass and spin decouple from the the last term in (28) is of the same order as the first consideration. As a result, we get in this limit the two terms and thus cannot be treated perturbatively. quasipotential for the light quark similar to the one It is important to note that the quark–diquark po- in the heavy-light meson in the limit of an infinitely tential VMd→∞(r) almost coincides with the quark– heavy antiquark [19]. The only difference consists in antiquark potential in heavy-light (B and D) mesons 2 the extra factor F(k ),defined in (25), in the one- for mQ →∞[19]. The only difference is the presence gluon-exchange part, which accounts for the heavy- of the extra factor F(r) in VCoul(r),whichaccounts diquark structure. The quasipotential in this limit is for the internal structure of the diquark. This is the given by consequence of the heavy-quark (diquark) limit, in which its spin and mass decouple from the consider- − 4 F 2 4π 0 ation. V (p, q; M)=¯uq(p) αs (k ) γq (27) 3 k2 In the infinitely heavy diquark limit, the quasipo- tential Eq. (1) in configuration space becomes κ + V V (k) γ0 + γ0(γ · k) + V S (k) u (q). 2 − 2 2 conf q q conf q Eq mq p 2mq − ΨB(r)=VMd→∞(r)ΨB(r), 2Eq 2Eq The resulting interaction is still nonlocal in configu- (29) ration space. However, taking into account that dou- bly heavy baryons are weakly bound, we can replace and the mass of the baryon is given by M = Md + Eq. PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 5 2005 PROPERTIES OF DOUBLY HEAVY BARYONS 791 Solving (29) numerically, we get the eigenvalues Table 5. The values of Eq in the limit Md →∞(in GeV) Eq and the baryon wave functions ΨB. The obtained results are presented in Table 5. We use the notation Baryon state Eq (ccq) Es (ccs) Eq (bbq) Es (bbs) ndLnql(j) for the classification of baryon states in the infinitely heavy diquark limit. Here, we first give the 1S1s(1/2) 0.491 0.638 0.492 0.641 radial quantum number nd and the angular momen- 1S1p(3/2) 0.788 0.906 0.785 0.904 tum L of the heavy diquark. Then the radial quan- 1S1p(1/2) 0.877 0.968 0.880 0.969 tum number nq, the angular momentum l,andthe value j of the total angular momentum (j = l + Sq) 1S2s(1/2) 0.987 1.080 0.993 1.084 of the light quark are shown. We see that the heavy- 1P 1s(1/2) 0.484 0.633 0.489 0.636 diquark spin and mass decouple in the limit Md →∞, and thus we get the number of degenerated states 1P 1p(3/2) 0.793 0.909 0.789 0.906 in accord with the heavy-quark symmetry prediction. 1P 1p(1/2) 0.873 0.965 0.876 0.967 This symmetry also predicts an almost equality of 1P 2s(1/2) 0.980 1.075 0.984 1.078 corresponding light-quark energies Eq for bbq and ccq baryons and their nearness in the same limit to the 2S1s(1/2) 0.481 0.631 0.486 0.634 light-quark energies E of B and D mesons [19]. The q 2S1p(3/2) 0.794 0.909 0.791 0.908 small deviations of the baryon energies from values of the meson energies are connected with the differ- 2S1p(1/2) 0.871 0.963 0.874 0.965 ent forms of the singularity smearing at r =0in the 2S2s(1/2) 0.979 1.074 0.982 1.076 baryon and meson cases. 2P 1s(1/2) 0.479 0.630 0.481 0.631 3S1s(1/2) 4.2. 1/Md Corrections 0.478 0.630 0.480 0.630 The heavy-quark symmetry degeneracy of states is broken by 1/Md corrections. The corrections of 1 1 order 1/Md to the potential (28) arise from the spatial + V (r) − V (r)+(1+κ) 3 r Coul Coul components of the heavy-diquark vertex. Other con- tributions at first order in 1/M come from the one- 1 d × V V (r) − V V (r) gluon-exchange potential and the vector confining r conf conf potential, while the scalar potential gives no contri- 3 bution at first order. The resulting 1/Md corrections × −S · S + (S · r)(S · r) q d r2 q d to the quark–diquark potential (28) are given by the following expressions: 2 (a) scalar diquark: + ∆V (r)+(1+κ)∆V V (r) S · S , 3 Coul conf d q 1 δV1/Md (r)= p [VCoul(r) (30) EqMd where S = Sq + Sd being the total spin, Sd being 2 the diquark spin (which is equal to the total angular l 1 +V V (r) p + V (r) − ∆V V (r) momentum J˜ of two heavy quarks forming the di- conf Coul 2r 4 conf quark). The first three terms in (31) represent spin- 1 1+κ independent corrections, the fourth and the fifth terms + V (r)+ V V (r) l · S ; r Coul r conf q are responsible for the spin–orbit interaction, the sixth one is the tensor interaction, and the last one is (b) axial-vector diquark: the spin–spin interaction. It is necessary to note that the confining vector interaction gives a contribution 1 to the spin-dependent part, which is proportional to δV1/Md (r)= p [VCoul(r) (31) EqMd (1 + κ). Thus, it vanishes for the chosen value of 2 κ = −1, while the confining vector contribution to the V l − 1 V +Vconf(r) p + VCoul(r) ∆Vconf(r) spin-independent part is nonzero. 2r 4 1 1+κ + V (r)+ V V (r) l · S In order to estimate the matrix elements of spin- r Coul r conf q dependent terms in the 1/Md corrections to the 1 1 1+κ quark–diquark potential (30) and (31), as well as + V (r)+ V V (r) l · S 2 r Coul r conf d different mixings of baryon states, it is convenient to PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 5 2005 792 EBERT et al. Table 6. Mass spectrum of Ξcc baryons State Mass, GeV State Mass, GeV (n Ln l)J P (n Ln l)J P d q this work [11] d q this work [11] (1S1s)1/2+ 3.620 3.478 (1P 1s)1/2− 3.838 3.702 (1S1s)3/2+ 3.727 3.61 (1P 1s)3/2− 3.959 3.834 (1S1p)1/2− 4.053 3.927 (2S1s)1/2+ 3.910 3.812 (1S1p)3/2− 4.101 4.039 (2S1s)3/2+ 4.027 3.944 (1S1p)1/2− 4.136 4.052 (2P 1s)1/2− 4.085 3.972 (1S1p)5/2− 4.155 4.047 (2P 1s)3/2− 4.197 4.104 (1S1p)3/2− 4.196 4.034 (3S1s)1/2+ 4.154 4.072 Table 7. Mass spectrum of Ξbb baryons State Mass, GeV State Mass, GeV (n Ln l)J P (n Ln l)J P d q this work [11] d q this work [11] (1S1s)1/2+ 10.202 10.093 (2S1s)1/2+ 10.441 10.373 (1S1s)3/2+ 10.237 10.133 (2S1s)3/2+ 10.482 10.413 (1S1p)1/2− 10.632 10.541 (2S1p)1/2− 10.873 (1S1p)3/2− 10.647 10.567 (2S1p)3/2− 10.888 (1S1p)5/2− 10.661 10.580 (2S1p)1/2− 10.902 (1S1p)1/2− 10.675 10.578 (2S1p)5/2− 10.905 (1S1p)3/2− 10.694 10.581 (2S1p)3/2− 10.920 (1S2s)1/2+ 10.832 (2P 1s)1/2− 10.563 10.493 (1S2s)3/2+ 10.860 (2P 1s)3/2− 10.607 10.533 (1P 1s)1/2− 10.368 10.310 (3S1s)1/2+ 10.630 10.563 (1P 1s)3/2− 10.408 10.343 (3S1s)3/2+ 10.673 (1P 1p)1/2+ 10.763 (3P 1s)1/2− 10.744 (1P 1p)3/2+ 10.779 (3P 1s)3/2− 10.788 (1P 1p)5/2+ 10.786 (4S1s)1/2+ 10.812 (1P 1p)1/2+ 10.838 (4S1s)3/2+ 10.856 (1P 1p)3/2+ 10.856 (4P 1s)1/2− 10.900 use the following relations: and | − J+l+Sd+Sq J, j = ( 1) (33) |J, j = (−1)J+l+Sd+Sq (32) Jd S Sd l Jd × (2Jd + 1)(2j +1) |J, Jd , S S S Sq J j × (2S + 1)(2j +1) d q |J, S l J j where J = j + Sd is the baryon total angular mo- PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 5 2005 PROPERTIES OF DOUBLY HEAVY BARYONS 793 Table 8. Mass spectrum of Ωcc baryons State Mass, GeV State Mass, GeV (n Ln l)J P (n Ln l)J P d q this work [32] d q this work [32] (1S1s)1/2+ 3.778 3.594 (1P 1s)1/2− 4.002 3.812 (1S1s)3/2+ 3.872 3.730 (1P 1s)3/2− 4.102 3.949 (1S1p)1/2− 4.208 4.050 (2S1s)1/2+ 4.075 3.925 (1S1p)3/2− 4.252 4.102 (2S1s)3/2+ 4.174 4.064 (1S1p)1/2− 4.271 4.145 (2P 1s)1/2− 4.251 4.073 (1S1p)5/2− 4.303 4.134 (2P 1s)3/2− 4.345 4.213 (1S1p)3/2− 4.325 4.176 (3S1s)1/2+ 4.321 4.172 Table 9. Mass spectrum of Ωbb baryons State Mass, GeV State Mass, GeV (n Ln l)J P (n Ln l)J P d q this work [32] d q this work [32] (1S1s)1/2+ 10.359 10.210 (2S1s)1/2+ 10.610 10.493 (1S1s)3/2+ 10.389 10.257 (2S1s)3/2+ 10.645 10.540 (1S1p)1/2− 10.771 10.651 (2S1p)1/2− 11.011 (1S1p)3/2− 10.785 10.661 (2S1p)3/2− 11.025 (1S1p)5/2− 10.798 10.670 (2S1p)1/2− 11.035 (1S1p)1/2− 10.804 10.700 (2S1p)5/2− 11.040 (1S1p)3/2− 10.821 10.720 (2S1p)3/2− 11.051 (1S2s)1/2+ 10.970 (2P 1s)1/2− 10.738 10.617 (1S2s)3/2+ 10.992 (2P 1s)3/2− 10.775 10.663 (1P 1s)1/2− 10.532 10.416 (3S1s)1/2+ 10.806 10.682 (1P 1s)3/2− 10.566 10.462 (3S1s)3/2+ 10.843 (1P 1p)1/2+ 10.914 (3P 1s)1/2− 10.924 (1P 1p)3/2+ 10.928 (3P 1s)3/2− 10.961 (1P 1p)5/2+ 10.937 (4S1s)1/2+ 10.994 (1P 1p)1/2+ 10.971 (4S1s)3/2+ 11.031 (1P 1p)3/2+ 10.986 (4P 1s)1/2− 11.083 mentum, j = l + Sq is the light-quark total angular quantum number of the light quark (nq =1, 2, 3,...) momentum, S = Sq + Sd is the baryon total spin, and and its orbital momentum by a lowercase letter Jd = l + Sd. (l = s,p,d,...), and at the end the total angular momentum J and parity P of the baryon. The presence of the spin–orbit interaction propor- 5. MASS SPECTRA OF DOUBLY HEAVY · BARYONS tional to l Sd and of the tensor interaction in the quark–diquark potential at 1/Md order (31) results in For the description of the quantum numbers of a mixing of states, which have the same total angular P baryons, we use the notation (ndLnql)J ,where momentum J and parity, but different light-quark to- we first show the radial quantum number of the tal momentum j. For example, the baryon states with diquark (nd =1, 2, 3,...) and its orbital momentum a diquark in the ground state and light quark in the p by a capital letter (L = S,P,D,...), then the radial wave (1S1p)forJ =1/2 or 3/2 have different values PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 5 2005 794 EBERT et al. M, GeV 4.8 3/2+ 4.6 3/2– 1/2+ 2S2s 3/2– 1/2– 1/2– 1P2s 5/2– 3/2+ 2S1p(1/2) 3/2– + 4.4 + 5/2 1/2– 3/2 1P1p(1/2) 1/2+ 2S1p(3/2) 3/2+ 1/2+ 1P1p(3/2) 1/2+ 1S2s 4.2 3/2– 5/2– 1S1p(1/2) 1/2– 3/2– – 2 2 1S1p(3/2) 1/2 S s 3/2+ 4.0 1P1s 3/2– 1/2+ 1/2– 3.8 1S2s 3/2+ + 3.6 1/2 Fig. 3. Mass spectrum of Ξcc baryons. The horizontal dashed line shows the ΛcD threshold. of the light-quark angular momentum j =1/2 and For the bbq baryon, we get the mixing matrix for J = 3/2, which mix between themselves. In the case of the 1/2 ccq baryon, we have the mixing matrix for J =1/2 −18.0 −2.4 [MeV], (38) −55.6 −7.3 − − [MeV] (34) 2.8 12.6 − − 8.5 37.9 which has eigenvalues − with the following eigenvectors: |(1S1p)1/2 =0.349|j =3/2 +0.937|j =1/2 , − |(1S1p)1/2 = −0.334|j =3/2 +0.943|j =1/2 , (39) − (35) |(1S1p)1/2 =0.915|j =3/2 +0.402|j =1/2 , | − | | and for J =3/2, the mixing matrix is (1S1p)1/2 =0.925 j =3/2 +0.380 j =1/2 . For the ccq baryon with J =3/2, the mixing matrix is −7.45.9 given by [MeV], (40) 7.16.3 −23.018.1 [MeV], (36) so that − 21.318.9 |(1S1p)3/2 =0.341|j =3/2 +0.940|j =1/2 , and the eigenvectors are equal to (41) | − | | − (1S1p)3/2 =0.917 j =3/2 +0.400 j =1/2 . |(1S1p)3/2 =0.343|j =3/2 +0.939|j =1/2 , The quasipotential with 1/m corrections is given (37) Q by the sum of Vm →∞(r) from (28) and δV (r) | − | − | Q 1/mQ (1S1p)3/2 =0.919 j =3/2 0.394 j =1/2 . from (30) and (31). By substituting it into the PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 5 2005 PROPERTIES OF DOUBLY HEAVY BARYONS 795 M, GeV 3/2+ 1/2+ 3/2– 2S2s 11.0 1/2– 1P2s 3/2– 5/2– 2S1p(1/2) 1/2– + 3/2– 3/2+ 3/2 – 1 1 (1/2) + 1/2 1/2+ P p 1/2 2S1p(3/2) 5/2+ 10.8 1S2s + 1P1p(3/2) 3/2 1/2+ 3/2– 1S1p(1/2) 1/2– 5/2– 3/2– 1/2– 10.6 1S1p(3/2) 2S1s 3/2+ 1P1s 1/2+ – 10.4 3/2 1/2– 1S1s 3/2+ 10.2 1/2+ Fig. 4. Mass spectrum of Ξbb baryons. The horizontal dashed line shows the ΛbD threshold. quasipotential Eq. (1) and treating the 1/mQ cor- above. Since the diquark has spin 1, the states with j =1/2 split into two different states with J =1/2 rection term δV1/mQ (r) using perturbation theory, we are now able to calculate the mass spectra of or 3/2, while the states with j =3/2 split into three Ξcc, Ξbb, Ξcb, Ωcc, Ωbb, Ωcb baryons with account of different states with J =1/2, 3/2,or5/2.Thefine 1/mQ corrections. In Tables 6–9, we present mass splitting between p levels turns out to be of the same spectra of ground and excited states of doubly heavy order of magnitude as the gap between j =1/2 and baryons containing both heavy quarks of the same j =3/2 degenerate multiplets in the infinitely heavy flavor (c and b). The corresponding level orderings are diquark limit. The inclusion of 1/Md corrections leads schematically shown in Figs. 3–6. In these figures, also to the relative shifts of the baryon levels, further we first show our predictions for doubly heavy baryon decreasing this gap. As a result, some of the p levels spectra in the limit when all 1/Md corrections are from different (initially degenerate) multiplets overlap; neglected (denoting baryon states by ndLnql(j)). We however, the heavy-diquark spin-averaged centers see that, in this limit, the p-wave excitations of the remain inverted. The resulting picture for the diquark light quark are inverted. This means that the mass of in the ground state is very similar to the one for heavy- the state with higher angular momentum j =3/2 is light mesons [19]. The purely inverted pattern of p smaller than the mass of the state with lower angular levels is observed only for the B meson and Ξbb, Ωbb momentum j =1/2 [13, 31]. The similar p-level baryons, while in other heavy-light mesons (D, Ds, inversion was found previously in the mass spectra Bs) and doubly heavy baryons (Ξcc, Ωcc) p levels from of heavy-light mesons in the infinitely heavy-quark different j multiplets overlap. The absence of the p- limit [19]. Note that the pattern of levels of the light level overlap for the Ωbb baryon in contrast to the quark and level separation in doubly heavy baryons Bs meson (where we predict a very small overlap of and heavy-light mesons almost coincide in these these levels [19]) is explained by the fact that the d limits. Next, we switch on 1/Md corrections. This ratio ms/Mbb is approximately two times smaller than results in splitting of the degenerate states and mixing ms/mb andthusitisofordermq/mb. As was argued of states with different j, which have the same total in [19], these ratios determine the applicability of the angular momentum J and parity, as was discussed heavy-quark limit. PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 5 2005 796 EBERT et al. M, GeV 4.8 3/2+ 1/2+ 3/2– – 1/2– 3/2 2S2s 5/2– 1/2– 4.6 – 1P2s 3/2+ 3/2 5/2+ 2S1p(1/2) 1/2– + 1/2+ 3/2 3/2+ 2S1p(3/2) 1P1p(1/2) 1/2+ 1/2+ 4.4 1P1p(3/2) 1S2s 3/2– 5/2– 1/2– – 1 1 (1/2) 3/2 S p 1/2– 4.2 2S1s + 1S1p(3/2) 3/2 1P1s 3/2– 1/2+ 4.0 1/2– 1S1s 3/2+ 3.8 1/2+ Fig. 5. Mass spectrum of Ωcc baryons. The horizontal dashed line shows the ΛcDs threshold. In Tables 6–9, we compare our predictions for the smaller than in our model. The other main source of ground- and excited-state masses of Ξcc, Ξbb and Ωcc, the difference is the expansion in inverse powers of the Ωbb baryons with the results of [11, 32]. As we see light-quark mass, which was used in [11, 32] but is from these tables, our predictions are approximately not applied in our approach, where the light quark is 50−150 MeV higher than the estimates of [11, 32]. treated fully relativistically. One of the reasons for the difference between these In Table 10, we compare our model predictions two predictions for the masses of Ξcc and Ωcc baryons for the ground-state masses of doubly heavy baryons (which is the largest) is the difference in the c-quark with some other predictions [6, 9, 11, 33] as well as masses. The mass of the c quark in [11] is deter- our previous prediction [10], where the expansion in mined from fitting the charmonium spectrum in the inverse powers of the heavy- and light-quark masses quark model, where all spin-independent relativis- was used. In general, we find reasonable agreement tic corrections were ignored. However, our estimates within 100 MeV between different predictions [6, 9– show that, due to the rather large average value of 11, 33] for the ground-state masses of the doubly v2/c2 in charmonium,4) such corrections play an im- heavy baryons. The main advantage of our present portant role and give contributions to the charmo- approach is the completely relativistic treatment of nium masses of order of 100 MeV. As a result, the the light quark and account of the nonlocal composite c-quark mass found in [11] is approximately 70 MeV structure of the diquark. The authors of [33] do not ex- less than in our model. For the calculation of the ploit the assumption about the quark–diquark struc- diquark masses, we also take into account the spin- ture of the baryon. The consideration in [9] is based on independent corrections (17) to the QQ potential. We some semiempirical regularities of the baryon mass find that their contribution is less than that in the spectra and the Feynman–Hellmann theorem. The predictions of [6] are obtained within the nonrelativis- case of charmonium, since VQQ = VQQ¯ /2.Thus,the cc-diquark masses in [11] are approximately 50 MeV tic quark model. For the Ξcb and Ωcb baryons, containing heavy 4)The spin-dependent relativistic corrections, which are of the quarks of different flavors (c and b), we calculate only same order in v2/c2, produce level splittings of the order of the ground-state masses. As was argued in [11], the 120 MeV. excited states of heavy diquarks, composed of the PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 5 2005 PROPERTIES OF DOUBLY HEAVY BARYONS 797 M, GeV 3/2+ 11.2 1/2+ 3/2– 1/2– 2S2s 3/2– 5/2– 1P2s 1/2– – + 3/2 11.0 3/2+ 3/2 2S1p(1/2) 1/2– 1/2+ 1/2+ 5/2+ 2S1p(3/2) 1P1p(1/2) 3/2+ 1/2+ 1S2s 1P1p(3/2) 3/2– 1/2– 10.8 5/2– 1S1p(1/2) 3/2– 1/2– 1S1p(3/2) 2S1s 3/2+ + 10.6 1P1s 1/2 3/2– 1/2– 1S1s 10.4 3/2+ 1/2+ Fig. 6. Mass spectrum of Ωbb baryons. The horizontal dashed line shows the ΛbDs threshold. quarks with different flavors, are unstable under the symmetry predicts simple relations between the spin- emission of soft gluons, and thus the calculation of averaged masses of doubly heavy baryons with the the excited baryon (cbq and cbs) masses is not justified accuracy of order 1/Md: in the quark–diquark scheme. We get the following ∆M¯ ≡ M¯ (Ξ ) − M¯ (Ξ ) predictions for the masses of the ground-state cbq 1,2 1 bb 1 cc (42) baryons: = M¯2(Ξbb) − M¯2(Ξcc)=M¯1(Ωbb) − M¯1(Ωcc) + ¯ − ¯ d − d ≡ d (1S1s)1/2 states with the axial-vector and = M2(Ωbb) M2(Ωcc)=Mbb Mcc ∆M , scalar cb diquarks, respectively: where the spin-averaged masses are M(Ξcb)=6.933 GeV,M(Ξcb)=6.963 GeV; ¯ M1 =(M1/2 +2M3/2)/3, ¯ (1S1s)3/2+ state: M2 =(M1/2 +2M3/2 +3M5/2)/6, ∗ M(Ξcb)=6.980 GeV; MJ are the masses of baryons with total angular momentum J,andM d are the masses of diquarks and for cbs baryons: QQ in definite states. The numerical results are presented (1S1s)1/2+ states with the axial-vector and in Table 11 (only the states below threshold are con- scalar cb diquarks, respectively: sidered). We see that the equalities in Eq. (42) are satisfied with good accuracy for the baryons in the M(Ωcb)=7.088 GeV,M(Ωcb)=7.116 GeV, ground and excited states. It follows from the heavy-quark symmetry that the (1S1s)3/2+ state: hyperfine mass splittings of initially degenerate light- ∗ M(Ωcb)=7.130 GeV. quark states ∆M(ΞQQ) ≡ M3/2(ΞQQ) − M1/2(ΞQQ), (43) Now we compare our results with the model- ≡ − independent predictions of HQET. The heavy-quark ∆M(ΩQQ) M3/2(ΩQQ) M1/2(ΩQQ) PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 5 2005 798 EBERT et al. Table 10. Comparison of different predictions for mass spectra of ground states of doubly heavy baryons (in GeV) ({QQ} denotes the diquark in the axial-vector state and [QQ] denotes the diquark in the scalar state) Baryon Quark content J P This work [11] [10] [9] [6] [33] + Ξcc {cc}q 1/2 3.620 3.478 3.66 3.66 3.61 3.69 ∗ { } + Ξcc cc q 3/2 3.727 3.61 3.81 3.74 3.68 + Ωcc {cc}s 1/2 3.778 3.59 3.76 3.74 3.71 3.86 ∗ { } + Ωcc cc s 3/2 3.872 3.69 3.89 3.82 3.76 + Ξbb {bb}q 1/2 10.202 10.093 10.23 10.34 10.16 ∗ { } + Ξbb bb q 3/2 10.237 10.133 10.28 10.37 + Ωbb {bb}s 1/2 10.359 10.18 10.32 10.37 10.34 ∗ { } + Ωbb bb s 3/2 10.389 10.20 10.36 10.40 + Ξcb {cb}q 1/2 6.933 6.82 6.95 7.04 6.96 + Ξcb [cb]q 1/2 6.963 6.85 7.00 6.99 ∗ { } + Ξcb cb q 3/2 6.980 6.90 7.02 7.06 + Ωcb {cb}s 1/2 7.088 6.91 7.05 7.09 7.13 + Ωcb [cb]s 1/2 7.116 6.93 7.09 7.06 ∗ { } + Ωcb cb s 3/2 7.130 6.99 7.11 7.12 should scale with the diquark masses: for the ground-state hyperfine splittings of mesons ∼ and baryons, we obtain in the approximation M d = ∆M(Ξcc)=R∆M(Ξbb), (44) QQ 2mQ ∆M(Ωcc)=R∆M(Ωbb), 3 mQ ∼ 3 where R = M d /M d is the ratio of diquark masses. ∆M(ΞQQ)= ∆MB,D = ∆MB,D, (45) bb cc 2 M d 4 Our model predictions for these splittings are dis- QQ played in Table 12. Again, we see that heavy-quark ∼ 3 ∆M(Ω ) = ∆M ,Q= b, c, symmetry relations are satisfied with high accuracy. QQ 4 Bs,Ds The close similarity of the interaction of the light where the factor 3/2 is just the ratio of the baryon quark with the heavy quark in the heavy-light mesons and meson spin matrix elements. The numerical ful- and with the heavy diquark in the doubly heavy fillment of relations (45) is shown in Table 13. baryons produces very simple relations between the meson and baryon mass splittings [7, 34, 35]. In fact, 6. HEAVY-DIQUARK TRANSITION Table 11. Differences between spin-averaged masses of FORM FACTORS doubly heavy baryons definedinEq.(42)(inGeV) Now we can apply the calculated masses and wave functionsofheavydiquarksanddoublyheavybaryons Baryon ¯ ¯ ¯ ¯ d ∆M1(Ξ) ∆M2(Ξ) ∆M1(Ω) ∆M2(Ω) ∆M for studying their semileptonic decays. We use the state two-step picture schematically shown in Fig. 1. First, 1S1s 6.534 6.538 6.552 we consider the heavy-diquark weak transition and, 1S1p 6.512 6.532 6.519 6.552 then, include the light quark in consideration. The form factors of the subprocess d(QQs) → 1P 1s 6.476 6.486 6.484 d (Q Qs)eν¯, where one heavy quark Qs is a spectator, 2S1s 6.480 6.492 6.480 are determined by the weak decay of the active heavy PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 5 2005 PROPERTIES OF DOUBLY HEAVY BARYONS 799 Table 12. Hyperfine splittings of the doubly heavy baryons for the states with the light-quark angular momentum j =1/2 (in MeV) Baryon state R ∆M(Ξbb) R∆M(Ξbb) ∆M(Ξcc) ∆M(Ωbb) R∆M(Ωbb) ∆M(Ωcc) 1S1s [3/2–1/2] 3.03 35 106 107 30 91 94 1S1p [3/2–1/2] 3.03 19 58 60 17 52 54 1P 1s [3/2–1/2] 2.87 40 115 121 34 98 100 2S1s [3/2–1/2] 2.83 41 116 117 35 99 99 Table 13. Comparison of hyperfine splittings (in MeV) in doubly heavy baryons and heavy-light mesons (experimental values for the hyperfine splittings in mesons are taken from [36]) 3 exp 3 exp 3 exp 3 exp ∆M(Ξcc) ∆M ∆M(Ξbb) ∆M ∆M(Ωcc) ∆M ∆M(Ωbb) ∆M 4 D 4 B 4 Ds 4 Bs 107 106 35 34 94 108 30 35 quark Q → Q eν¯. The local effective Hamiltonian is Relativistic four-momenta of the particles in the ini- given by tial and final states are defined as follows: 3 GF √ ¯ ¯ ± (i) i Heff = VQQ Q γµ(1 − γ5)Q lγµ(1 − γ5)νe , p1,2 = 1,2(p)v n (v)p , (49) 2 i=1 (46) P v = , Mi = 1(p)+ 2(p), where GF is the Fermi constant and VQQ is the CKM Mi matrix element. In the relativistic quark model, the 3 (i) i transition matrix element of the weak current between q1,2 = 1,2(q)v ± n (v )q , two diquark states is determined by the contraction of i=1 the wave functions Ψ of the initial and final diquarks d Q with the two-particle vertex function Γ, v = , Mf = 1(q)+ 2(q), Mf d (Q)|JW |d(P ) (47) µ and n(i) are three four-vectors defined by 3 3 d pd q λσ λσ,ρω ρω = Ψ¯ (q)Γ (p, q)Ψ (p), (i) i ij 1 i j 6 d ,Q µ d,P n (v)= v ,δ + v v . (2π) 1+v0 1 λ, σ, ρ, ω = ± . After making necessary transformations, the expres- 2 sion for Γ should be continued in Mi and Mf to the Here, we denote mass, energy, and four-velocity of values of initial Mi and final Mf bound-state masses. the initial diquark (QbQs,indexb stands for the initial active quark and index s for spectator) by Mi, Ei = 0 W W Miv ,andv = P/Mi, and mass, energy, and four- velocity of the final diquark (QaQs,indexa denotes 0 the final active quark) by Mf , Ef = Mf v ,andv = b a b a Q/Mf . = + .... The leading contribution to the vertex function Γµ comes from the diagram in Fig. 7 [22, 37, 38] (we s ss s ΓΓ(1) explicitly show spin indices): λσ,ρω (1) λ − Γµ (p, q)=Γµ =¯ua(q1)γµ(1 γ5) (48) Fig. 7. The leading-order contribution Γ(1) to the diquark × ρ σ ω 3 − σω ub (p1)¯us (q2)us (p2)(2π) δ(p2 q2)δ . vertex function Γ. PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 5 2005 800 EBERT et al. The transformation of the bound-state wave func- ˆ − ˆ × εˆp˜ − vˆ √ 1 p˜εˆ 5 tions from the rest frame to the moving one with four- γ0γ ΦA(p). s(p)+ms 2 2 b(p)+mb momenta P and Q is given by [22, 37] √ ρω 1/2,ρα αβ Here, Φ (p) ≡ Ψ (p)/ 2M is the diquark wave Ψ (p)=D (RW )D1/2,ωβ(RW )Ψ (p), d d,0 d d,P b LP s LP d,0 function in the rest frame, normalized to unity. The (50) four-vector λσ ετ +1/2,ελ W +1/2,τσ W Ψ¯ (q)=Ψ¯ (q)D (R )D (R ), · · d ,Q d ,0 a LQ s LQ (p P) (p P)P p˜ = LP (0, p)= , p + (55) W M M(E + M) where R is the Wigner rotation, LP is the Lorentz boost from the diquark rest frame to a moving one, has the following properties: and the rotation matrix D1/2(R) is defined by p˜2 = −p2, (56) · − · 10 1/2 W −1 (ε p˜)= (ε p), D (R )=S (p1,2)S(P)S(p), (51) 01 b,s LP (˜p · v)=0. where 3 − The presence of δ (p2 q2),withmomentap2 and q (p)+m (α · p) 2 given by Eq. (49), in the decay matrix element (52) S(p)= 1+ leads to the following additional relations valid up to 2m (p)+m the terms of order 1/mQ: is the usual Lorentz transformation matrix of the four- w (p) − (q) p˜ = s s (wv − v ) (57) spinor. µ 2 − µ µ w 1 Using relations (49)–(51), we can express the 2 p matrix element (47) in the form of the trace over = √ (wv − v ), 2 − µ µ spinor indices of both particles. As a result, we get w 1 the covariant (see [37]) expression for the transition w s(q) − s(p) q˜ = (wv − v ) (58) matrix element: µ w2 − 1 µ µ 3 3 2 W d pd q q d (Q)|J |d(P ) =2 MiMf (52) = √ (wv − v ), µ (2π)3 2 − µ µ w 1 × { ¯ − } 3 − 2 2 Tr Ψd (Q, q)γµ(1 γ5)Ψd(P, p) δ (p2 q2), s(q)=w s(p) − w − 1 p , (59) 2 where the amplitudes Ψd for the scalar (S) and axial- 2 2 2 q = w − 1 s(p) − w p , vector (A)diquarks(d)aregivenby (p)=w (q) − w2 − 1 q2, (60) (p)+m s s b b 2 ΨS(P, p)= (53) 2 2 2 2 b(p) p = w − 1 s(q) − w q , − q p p × s(p)+ms vˆ √+1 vˆ √ 1 which allow one to express either through or + through q. The argument of the δ function can then 2 s(p) 2 2 2 2 be rewritten as p˜2 vˆ +1 × − √ s(p)+ s(q) p2 − q2 = q − p − (v − v), (61) ( b(p)+mb)( s(p)+ms) 2 2 w +1 1 vˆ − 1 1 where w =(v · v ). × + √ p˜ˆ γ0ΦS(p), s(p)+ms 2 2 b(p)+mb Calculating traces in Eq. (52) and using rela- tions (57)–(60), one can see that the spectator-quark contribution factors out in all decay matrix elements. b(p)+mb s(p)+ms Indeed, all transition matrix elements have a common ΨA(P, p)= (54) 2 (p) 2 s(p) factor b vˆ +1 vˆ − 1 p˜2 s(p)+ms s(q)+ms × √ εˆ+ √ εˆ (62) ( (p)+m )( (p)+m ) 2 s(p) 2 s(q) 2 2 2 2 b b s s − · ˆ w − 1 p2 q2 − vˆ √ 1 2(ε p˜)p˜ vˆ √+1 × 1 − + + w +1 (p)+m (q)+m 2 2 ( b(p)+mb)( s(p)+ms) 2 2 s s s s PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 5 2005 PROPERTIES OF DOUBLY HEAVY BARYONS 801 2 2 − · ∗ · p q 2 (v ε )(v ε)[h5(w)vµ + h6(w)vµ], + = Is(p, q). [ s(q)+ms][ s(p)+ms] w +1 A (v ,ε ))|JA|A (v,ε) f µ i If the δ function is used to express q through p or p = i µαβγ (70) MAi MAf through q,thenIs(p, q)=Is(p) or Is(p, q)=Is(q), respectively, with × β ∗γ α − α ε ε [h7(w)(v + v ) + h8(w)(v v ) ] √ 2 2 w s(p) − w − 1 p β γ · ∗ α · ∗α I (p)= (63) + v v [h9(w)(v ε )ε + h10(w)(v ε)ε ] . s (p) s The transition form factors are expressed through the w − 1 × θ (p) − m − (p)+m overlap integrals of the diquark wave functions: s s w +1 s s h+(w)=h1(w)=h7(w) (71) m + s 3 3 √ 2 d pd q ¯ − 2 − 2 = Φd (q) s(p)[w s(p) w 1 p ] w +1 (2π)3 − w 1 a(q)+ma b(p)+mb × θ s(p)+ms − s(p) − ms . × w +1 2 a(q) 2 b(p) 2 2 The weak-current matrix elements have the fol- × − p q lowing covariant decomposition: 1 Φd(p) [ a(q)+ma][ b(p)+mb] (a) scalar-to-scalar diquark transition (bc → bc) (p)+ (q) | V | × 3 − s s − Sf (v ) Jµ Si(v) Is(p, q)δ p q + (v v) , = h+(w)(v + v )µ (64) w +1 MSi MSf − hV (w)=h3(w)=h4(w) (72) + h−(w)(v v )µ; 2 d3pd3q (b) scalar-to-axial-vector diquark transition = Φ¯ (q) w +1 (2π)3 d (bc → cc) | V | (q)+m (p)+m A(v ,ε) Jµ S(v) ∗α β γ × a a b b √ = ihV (w) µαβγ ε v v , (65) M M 2 a(q) 2 b(p) A S 2 2 A(v ,ε )|JA|S(v) × w +1 q p √ µ ∗ 1+ − + = hA1 (w)(w +1)εµ (66) w 1 a(q)+ma b(p)+mb MAMS − · ∗ − · ∗ 2 2 hA2 (w)(v ε )vµ hA3 (w)(v ε )vµ; p q + Φd(p) [ a(q)+ma][ b(p)+mb] (c) axial-vector-to-scalar diquark transition → (bb bc) 3 s(p)+ s(q) × Is(p, q)δ p − q + (v − v) . | V | w +1 S(v ) Jµ A(v,ε) α β γ √ = ihV (w) µαβγ ε v v , (67) MAMS Explicit formulas for other form factors are given in [39]. S(v )|JA|A(v,ε) √ µ If we consider the spectator quark to be light and = hA1 (w)(w +1)εµ (68) MAMS then take the limit of an infinitely heavy active-quark mass, m →∞, we can explicitly obtain heavy- − h˜ (w)(v · ε)v − h˜ (w)(v · ε )v ; a,b A2 µ A3 µ quark symmetry relations for the decay matrix ele- (d) axial-vector-to-axial-vector diquark transition ments of heavy-light diquarks, which are analogous (bb → bc, bc → cc) to those of heavy-light mesons [22, 40]: V ˜ Af (v ,ε))|J |Ai(v,ε) h+(w)=hV (w)=hA1 (w)=hA3 (w)=hA3 (w) µ (69) = h1(w)=h3(w)=h4(w)=h7(w)=ξ(w), MAi MAf ∗ − ˜ = −(ε · ε)[h1(w)(v + v )µ + h2(w)(v − v )µ] h (w)=hA2 (w)=hA2 (w)=h2(w)=h5(w) ∗ ∗ · · = h6(w)=h8(w)=h9(w)=h10(w)=0, + h3(w)(v ε )εµ + h4(w)(v ε)εµ PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 5 2005 802 EBERT et al. − F(w) hA2 (w)= 2f(w)F (w); 1.0 (c) axial-vector-to-scalar diquark transition: 0.8 ˜ hV (w)=hA3 (w)=[1+(w +1)f(w)]F (w), (76) − 0.6 hA1 (w)=[1+(w 1)f(w)]F (w), h˜ (w)=0; 0.4 A2 (d) axial-vector-to-axial-vector diquark transi- 0.2 tion: h1(w)=h7(w)=F (w), (77) 1.00 1.02 1.04 1.06 w h2(w)=h8(w)=−(w +1)f(w)F (w), Fig. 8. The function F (w) for the bb → bc quark transi- h3(w)=h4(w)=(1+(w +1)f(w))F (w), tion. h5(w)=h9(w)=2f(w)F (w), h6(w)=h10(w)=0, F(w) 1.0 where 1 0.8 F (w)= (78) w(w +1) 0.6 1/2 m × 1+ a 0.4 m2 +(w2 − 1)m2 a s 0.2 d3p 2m 1.00 1.05 1.10 1.15 1.20 × ¯ s − 3 Φf p + (v v) Φi(p) w (2π) w +1 → and Fig. 9. The function F (w) for the bc cc quark transi- m tion. f(w)= s . (79) 2 2 − 2 ma +(w 1)ms + ma with the Isgur–Wise function The appearance of the terms proportional to the func- tion f(w) is the result of the account of the spectator- 2 d3pd3q ξ(w)= Φ¯ (q)Φ (p) (73) quark recoil. Their contribution is important and dis- w +1 (2π)3 f i tinguishes our approach from previous considera- tions [41, 42]. We plot the function F (w) for bb → bc 3 s(p)+ s(q) × Is(p, q)δ p − q + (v − v) . and bc → cc diquark transitions in Figs. 8 and 9. w +1 The diquark transition matrix element should be mul- tiplied by a factor of 2 if either the initial or final 7. DOUBLY-HEAVY-BARYON TRANSITIONS diquark is composed of two identical heavy quarks. The second step in studying weak transitions of For the heavy-diquark system, we can now apply doubly heavy baryons is the inclusion of the spectator the v/c expansion. First, we perform the integration light quark in the consideration. We carry out all over q in the form factors (71), (72) and use rela- further calculations in the limit of the infinitely heavy- →∞ tions (59). Then, applying the nonrelativistic limit, we diquark mass, Md , treating the light quark rel- get the following expressions for the form factors: ativistically. The transition matrix element between doubly heavy baryon states in the quark–diquark (a) scalar-to-scalar diquark transition: approximation (see Figs. 1 and 7) is given by [cf. h+(w)=F (w), (74) Eqs. (47) and (48)] h−(w)=−(w +1)f(w)F (w); B (Q)|JW |B(P ) (80) µ d3pd3q (b) scalar-to-axial-vector diquark transition: =2 M M Ψ¯ (q) i f (2π)3 B ,Q hV (w)=[1+(w +1)f(w)]F (w), (75) × | W | 3 − − d (Q) Jµ d(P ) ΨB,P (p)δ (pq qq), hA1 (w)=hA3 (w)=[1+(w 1)f(w)]F (w), PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 5 2005 PROPERTIES OF DOUBLY HEAVY BARYONS 803 × β γ − where ΨB,P (p) is the doubly heavy baryon wave func- [ihV (w) µαβγ v v gµαhA1 (w +1) tion; p and q are the relative quark–diquark momen- ˜ ˜ ¯ + vµvαhA2 (w)+vµvαhA3 (w)]UΞ (v ) ta; and pq and qq are light-quark momenta, expressed Q Qs in the form similar to (49). × α α (γ + v )γ5UΞQQs (v)η(w); The baryon ground-state wave function ΨB,P (p) ∗ ∗ (c) Ξ → Ξ and Ξ → Ξ transi- is a product of the spin-independent part ΨB(p), sat- QQs Q Qs QQs Q Qs isfying the related quasipotential Eq. (1), and the spin tions: ∗ W part UB(v): Ξ (v )|J |Ξ (v) Q Qs µ QQs Ψ (p)=Ψ (p)U (v). (81) (86) B,P B B 2 MiMf The baryon spin wave function is constructed from β γ − =[ihV (w) µαβγ v v gµαhA1 (w +1) the Dirac spinor uq(v) of the light spectator quark and ¯ α∗ + vµvαhA2 (w)+vµvαhA3 (w)]UΞ (v ) the diquark wave function. The ground-state spin- Q Qs 1/2 baryons can contain either the scalar or axial- × UΞ (v)η(w), QQs vector diquark. The former baryon is denoted by Ξ QQ W ∗ | | Ξ (v ) Jµ Ξ (v) and the latter one by ΞQQ . The ground-state spin- Q Qs QQs (87) 3/2 baryon can be formed only from the axial-vector 2 MiMf ∗ diquark and is denoted by ΞQQ . To obtain the cor- β γ − =[ihV (w) µαβγ v v gµαhA1 (w +1) responding baryon spin states, we use in the baryon ˜ ˜ ¯ matrix elements the following replacements: + vµvαhA2 (w)+vµvαhA3 (w)]UΞ (v ) Q Qs → × α∗ uq(v) UΞ (v), (82) UΞ (v)η(w); QQ QQs i √ [εµ(v)uq(v)] → (γµ + vµ)γ5UΞ (v), (d) ΞQQ → ΞQ Q transition: spin-1/2 3 QQ s s µ | W | µ → ΞQ Qs (v ) Jµ ΞQQs (v) 1 [ε (v)uq(v)]spin-3/2 UΞ∗ (v), − QQ = (88) 2 MiMf 3 where baryon spinor wave functions are normalized ∗ × − by UBUB =1(B =Ξ, Ξ)andtheRarita–Schwinger gρλ[h1(w)(v + v )µ + h2(w)(v v )µ] ∗µ ∗ − wave functions are normalized by UΞ∗ UΞ µ = 1. − h (w)g v − h (w)g v + v v [h (w)v Then the decay amplitudes of doubly heavy baryons 3 µρ λ 4 µλ ρ ρ λ 5 µ in the infinitely heavy diquark limit are given by the + h (w)v ]+i gβgγ [h (w)(v + v )α following expressions. 6 µ µαβγ ρ λ 7 − α β γ α (a) Ξ → Ξ transition: + h8(w)(v v ) ]+v v [h9(w)gρ vλ QQs Q Qs W Ξ (v )|J |Ξ (v) α ¯ λ λ Q Qs µ QQs + h10(w)g v ] UΞ (v )γ5(γ + v ) (83) λ ρ Q Qs 2 M M i f × ρ ρ (γ + v )γ5UΞQQs (v)η(w); − ¯ =[h+(w)(v + v )µ + h−(w)(v v )µ]UΞ (v ) Q Qs ∗ ∗ → → (e) ΞQQs ΞQ Q and ΞQQ ΞQ Qs transi- × UΞ (v)η(w); s s QQs tions: ∗ → → | W | (b) Ξ ΞQ Q and ΞQQ Ξ transi- Ξ (v ) Jµ ΞQQs (v) i QQs s s Q Qs Q Qs = −√ (89) tions: 2 MiMf 3 W ΞQ Q (v )|J |Ξ (v) i s µ QQs √ = (84) × gρλ[h1(w)(v + v )µ + h2(w)(v − v )µ] 2 MiMf 3 × [ih (w) v βvγ − g h (w +1) − h (w)g v − h (w)g v + v v [h (w)v V µαβγ µα A1 3 µρ λ 4 µλ ρ ρ λ 5 µ ¯ β γ α + vµvαhA2 (w)+vµvαhA3 (w)]UΞ (v ) Q Qs + h6(w)vµ]+i µαβγ gρ gλ[h7(w)(v + v ) α α × γ5(γ + v )U (v)η(w), ΞQQ − α β γ α s + h8(w)(v v ) ]+v v [h9(w)gρ vλ | W | Ξ (v ) J ΞQQs (v) i Q Qs µ α λ = √ (85) ¯ ∗ + h10(w)gλ vρ] UΞ (v ) 2 MiMf 3 Q Qs PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 5 2005 804 EBERT et al. × ρ ρ (γ + v )γ5UΞQQs (v)η(w), form factors hi(w) obey relations (74)–(77). In this ∗ limit, the baryon transition matrix elements contain | W | ΞQ Qs (v ) Jµ Ξ (v) i QQs = −√ (90) the common factor F (w)η(w) (cf. [41]). 2 M M 3 i f 8. SEMILEPTONIC DECAY RATES × g h w v v h w v − v ρλ[ 1( )( + )µ + 2( )( )µ] OF DOUBLY HEAVY BARYONS − − The exclusive differential rate of the doubly heavy h3(w)gµρvλ h4(w)gµλvρ + vρvλ[h5(w)vµ baryon semileptonic decay B → B eν¯ can be written β γ α + h6(w)vµ]+i µαβγ gρ gλ[h7(w)(v + v ) in the form √ 2 | |2 3 2 − − α β γ α dΓ GF Vbc Mf w 1 + h8(w)(v v ) ]+v v [h9(w)gρ vλ = Ω(w), (93) dw 48π3 α ¯ + h10(w)g v ] UΞ (v ) · 2 2 − 2 λ ρ Q Qs where w =(v v )=(Mi + Mf k )/(2MiMf ), k = P − Q Ω(w) × λ λ ρ ,andthefunction is the contraction γ5(γ + v )UΞ∗ (v)η(w); QQs of the hadronic transition matrix elements and the ∗ ∗ leptonic tensor. For the massless leptons, the differ- (f) Ξ → Ξ transition: QQs Q Qs Ξ → Ξ ential decay rates of the transitions QQs Q Qs ∗ W ∗ | | → Ξ (v ) Jµ Ξ (v) and ΞQQs Ξ are as follows: Q Qs QQs (91) Q Qs 2 2 2 MiMf dΓ G |VQQ | → F (ΞQQ ΞQ Qs )= (94) dw s 72π3 = − gρλ[h1(w)(v + v )µ + h2(w)(v − v )µ] × 2 − 1/2 3 3 2 2 − − (w 1) (w +1) Mf 2(Mi + Mf h3(w)gµρvλ h4(w)gµλvρ + vρvλ[h5(w)vµ + h (w)v ]+i gβgγ [h (w)(v + v )α w − 1 6 µ µαβγ ρ λ 7 − 2M M w) h2 (w)+ h2 (w) i f A1 w +1 V + h (w)(v − v )α]+v βvγ [h (w)gαv 8 9 ρ λ + (M − M w)h (w)+(w − 1) α λ ρ f i A1 ¯ ∗ ∗ + h10(w)gλ vρ] UΞ (v )UΞ (v)η(w). Q Qs QQs 2 × 2 Here, η(w) is the heavy-diquark–light-quark Isgur– (Mf hA2 (w)+MihA3 (w)) η (w), Wise function, which is determined by the dynamics of the light spectator quark q and is calculated simi- 3 larly to Eqs. (73) and (62): dΓ Mf (Ξ → Ξ )= QQs Q Qs 3 (95) 2 d3pd3q dw Mi η(w)= Ψ¯ B(q)ΨB(p) (92) w +1 (2π)3 dΓ × (Ξ → Ξ ,M ↔ M ,h → h˜ , dw QQs Q Qs i f A2 A2 3 q(p)+ q(q) × Iq(p, q)δ p − q + (v − v) . → ˜ w +1 hA3 hA3 ). We plot the Isgur–Wise function η(w) in Fig. 10. In The differential decay rates for other transitions are the nonrelativistic limit for heavy quarks, the diquark given in [39]. The semileptonic decay rates of doubly heavy η baryons are calculated in the nonrelativistic limit for (w) heavy quarks and presented in Tables 14 and 15. 1.0 Our results for the semileptonic decay rates of doubly heavy baryons Ξbb and Ξbc are compared with previ- 0.8 ous predictions. It is seen from Table 14 that results of different approaches vary substantially. Most of the 0.6 previous papers [42–44] give their predictions only for 1.00 1.05 1.10 1.15 1.20 selected decay modes. Their values agree with ours w in order of magnitude. Our predictions are smaller than the QCD sum rule results [44] by a factor of ∼2. Fig. 10. The Isgur–Wise function η(w) of the light- This can be a result of our treatment of the heavy- quark–heavy-diquark bound system. spectator-quark recoil in the diquark. On the other PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 5 2005 PROPERTIES OF DOUBLY HEAVY BARYONS 805 Table 14. Semileptonic decay rates of doubly heavy Table 15. Semileptonic decay rates of doubly heavy −14 −14 baryons Ξbb and Ξbc (in 10 GeV) baryons Ωbb and Ωbc (in 10 GeV) Decay This work [45] [42] [44] [43] Decay Γ Decay Γ → → → Ξbb Ξbc 1.64 4.28 Ωbb Ωbc 1.66 Ωbc Ωcc 1.90 → → → ∗ Ξbb Ξbc 3.26 28.5 8.99 Ωbb Ωbc 3.40 Ωbc Ωcc 3.66 → ∗ → ∗ → Ξbb Ξbc 1.05 27.2 2.70 Ωbb Ωbc 1.10 Ωbc Ωcc 4.95 ∗ → ∗ → → ∗ Ξbb Ξbc 1.63 8.57 Ωbb Ωbc 1.70 Ωbc Ωcc 1.48 ∗ → ∗ → ∗ → Ξbb Ξbc 0.55 52.0 Ωbb Ωbc 0.57 Ωbc Ωcc 0.80 ∗ → ∗ ∗ → ∗ ∗ → ∗ Ξbb Ξbc 3.83 12.9 Ωbb Ωbc 3.99 Ωbc Ωcc 5.76 → Ξbc Ξcc 1.76 7.76 → ∗ Ξbc Ξcc 3.40 28.8 originates from the internal structure of the diquark, which is important at small distances. The first- Ξ → Ξ 4.59 8.93 4.0 8.87 0.8 bc cc order contributions to the heavy-(di)quark expansion → ∗ Ξbc Ξcc 1.43 14.1 1.2 2.66 explicitly depend on the values of the heavy-diquark Ξ∗ → Ξ 0.75 27.5 and heavy-quark spins and masses. This results in bc cc the different number of levels to which the initially ∗ → ∗ Ξbc Ξcc 5.37 17.2 degenerate states split, as well as their ordering. Our model respects the constraints imposed by heavy- quark symmetry on the number of levels and their hand, the authors of [45], where the Bethe–Salpeter splittings. equation is used, give more decay channels. Their results are substantially higher than ours; for some We find that the p-wave levels of the light quark decays, the difference reaches almost two orders of with j =1/2 and j =3/2, are inverted in the infinitely magnitude, which seems quite strange. For example, heavy (di)quark limit. The origin of this inversion ( ,∗) is the following. The confining potential contribu- for the sum of the semileptonic decays Ξ → Ξ , bb bc tion to the spin–orbit term in (28) exceeds the one- ∼ × −13 the authors of [45] predict 6 10 GeV, which gluon-exchange contribution. Thus, the sign before almost saturates the estimate of the total decay rate the spin–orbit term is negative [46, 47], and the level Γtotal ∼ (8.3 ± 0.3) × 10−13 GeV [12] and thus is Ξbb inversion emerges. However, the 1/Md corrections, unlikely. which produce the hyperfine splittings of these multi- plets, are substantial. As a result, the purely inverted pattern of p levels for the heavy diquark in the ground 9. CONCLUSIONS state occurs only for the doubly heavy baryons Ξbb and In this paper, we calculated the masses and Ωbb.ForΞcc and Ωcc baryons, the levels from these semileptonic decay rates of the doubly heavy baryons multiplets overlap. A similar pattern was previously on the basis of the quark–diquark approximation in found in our model for the heavy-light mesons [19]. the framework of the relativistic quark model. The Then we used the baryon wave functions for the mass spectra of orbital and radial excitations of both calculation of semileptonic decay rates of doubly the heavy diquark and the light quark were consid- heavy baryons. The weak transition matrix elements ered. The main advantage of the proposed approach between heavy-diquark states were calculated with consists in the fully relativistic treatment of the light- the self-consistent account of the spectator-quark re- quark (u, d, s) dynamics and in account of the internal coil. It was shown that recoil effects lead to additional structure of the diquark in the short-range quark– contributions to the transition matrix elements. Such diquark interaction. We apply only the expansion in terms were missed in the previous quark-model cal- inverse powers of the heavy-(di)quark mass, which culations. If we neglect these recoil contributions, the considerably simplifies calculations. The infinitely previously obtained expressions [41, 42] for heavy- heavy-(di)quark limit, as well as the first-order 1/Md diquark transition matrix elements are reproduced. spin-independent and spin-dependent contributions, It was found that these recoil terms, which are was considered. A close similarity between excita- proportional to the ratio of the heavy-spectator to tions of the light quark in the doubly heavy baryons the final active-quark mass [see Eqs. (74)–(79)], give and heavy-light mesons was demonstrated. In the important contributions to transition matrix elements infinitely heavy-(di)quark limit, the only difference of doubly heavy diquarks even in the nonrelativistic PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 5 2005 806 EBERT et al. limit. In this limit, these weak-transition matrix 10. D. Ebert, R. N. Faustov, V.O. Galkin, et al.,Z.Phys. elements are proportional to the function F (w) (78), C 76, 111 (1997). which is expressed through the overlap integral of the 11. S. S. 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M. J. White and M. J. Savage, Phys. Lett. B 271, 410 47. Yu. S. Kalashnikova, A. V. Nefediev, and Yu. A. Si- (1991). monov,Phys.Rev.D64, 014037 (2001). PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 5 2005 Physics of Atomic Nuclei, Vol. 68, No. 5, 2005, pp. 808–813. From Yadernaya Fizika, Vol. 68, No. 5, 2005, pp. 841–846. Original English Text Copyright c 2005 by Wang, Ping, Qing, Goldman. Multiquark States* F. W a n g 1),J.L.Ping1), 2),D.Qing3), and T. Goldman4) Received May 21, 2004; in final form, November 24, 2004 Abstract—The pentaquark state recently discovered has been discussed based on various quark model calculations.Odd parity for the state cannot be ruled out theoretically because the contributions related to nontrivial color structures have not been studied completely.Other multiquark states, especially dibaryons, have been discussed also.A strangeness −3 NΩ dibaryon has been shown to have a width as small as 12–22 keV and should be detectable in Ω-high-productivity reactions such as at RHIC, COMPAS, and the planned JHF and FAIR projects. c 2005 Pleiades Publishing, Inc. 1.MULTIQUARK STATE SEARCH the NA49 Collaboration claimed that they found the antidecuplet partner Ξ−− of Θ+ [3].The HERA-H1 Multiquark states were studied even before the ad- Collaboration claimed that they found the charm vent of QCD.The development of QCD accelerated pentaquark Θc [4]. multiquark studies because it is natural in QCD that there should be multiquark states, including glueballs If we had not had bitter previous experience in and quark–gluon hybrids.Prof.Yu.A.Simonov is one the search for multiquark states, one would have ac- of the pioneers of multiquark studies; he led an ITEP cepted that the pentaquark state has been discovered. group that developed the quark compound bag model Taking into account the historical lessons that many in the early 1980s to study hadron interactions and low-statistics multiquark signals eventually disap- multiquark states [1]. peared, the consensus is that high-statistics data are necessary to confirm this state. For a long time, multiquark states were only a the- oretical speculation; experimental searches had not Moreover, there are weaknesses among these obtained definite evidence, even though there were measurements.Some “pentaquark signals” of the various claims of the discovery of multiquark states. real or virtual photon reactions might be due to the kinematical reflection of the following normal meson- production processes [5]: 2.DISCOVERY OF THE PENTAQUARK − + γn → f2n → K K n, (1) Eleven groups [2] claimed recently that they γp → a+n → KKn,¯ + 2 found a pentaquark state, now called Θ , with mass → → ¯ ∼1540 MeV and width Γ < 25 MeV.Five measure- γp f2p KKp. p d ments used a real or virtual photon–nucleus ( , ,or Some measurements only measure the charged π+, other nuclei) reaction and the resonance is inferred π− decay products of K0 to obtain the pK0 invariant from the final state nK+ or pK0 invariant mass. S S S mass and there might be kinematical reflections in A reanalysis of 1986 bubble chamber K–nucleus + + these events too [6].Reanalysis of the vast K n and reaction data also found the Θ .Three most recent 0 + experiments used pp or p–nucleus reactions.One KLp scattering data could not find the Θ signal with used old neutrino reaction data and the eleventh a width greater than or equal to a few MeV [7], except used deep-inelastic ep scattering data.In addition, the recent time delay and speed plot analysis, where a signal appears around 1.57 GeV in the P01 (with the ∗ This article was submitted by the authors in English. notation LI(2J)) channel [8].In addition, a very tiny 1) Department of Physics and Center for Theoretical Physics, bump had appeared in 1973 CERN K+p → pK0 π+ Nanjing University, Nanjing, China. S 2)Department of Physics, Nanjing Normal University, Nan- inelastic-scattering data [9]. jing; School of Physics and Microelectronics, Shandong The NA49 claim has been challenged by another University, Jinan, China. 3)CERN, Geneva, Switzerland. CERN group based on Ξ-spectroscopy data with 4)Theoretical Division, Los Alamos National Laboratory, Los higher statistics [10].From HERA-B p–nucleus re- Alamos, USA. action data, one has not found the Θ+ [11].BES 1063-7788/05/6805-0808$26.00 c 2005 Pleiades Publishing, Inc. MULTIQUARK STATES 809 J/ψ-decay data analysis has not found the Θ+ ei- be symmetric with respect to the overall diquark ther [12].There are other groups that have not found exchange and this dictates the relative motion of the the Θ+ signal but did not claim so publicly. two diquarks in a P wave.In this way, Ja ffeand Wilczek get a positive-parity Θ+ state, the same as the chiral soliton quark model one.This model gives 3.WHAT IS THE PENTAQUARK an antidecuplet spectrum different from the chiral Theoretical studies based on the chiral soliton soliton quark model ones and predicts a pentaquark model played an important role in triggering the Θ+ octet based on phenomenological assumptions.Si- searches [13].In the chiral soliton quark model, the monov did a quantitative calculation based on the Θ+ is a member of the antidecuplet rotational excita- Jaffe–Wilczek configuration by means of the effec- tion following the well-established octet and decuplet tive Hamiltonian approach.This method obtained 1 + a quantitative fit of single-baryon masses, but the baryons.The model predicted an I =0, JP = calculated Θ+ mass is about 400 MeV higher than 2 state with mass about 1540 MeV and width less the observed 1540 MeV [18].Dudek and Close [19] than 15 MeV, quite close to the later experimental pointed out that there should be a spin–orbit partner 3 + results.The QCD background of this model has been Θ∗ with JP = not too far from the Θ+ mass. criticized by Jaffe and Wilczek [14], and the difficulties 2 of the flavor SU(3) extension in the description of An even more serious problem is the decay width. the meson–baryon scattering with strangeness was Buccella and Sorba [20] suggested an approximate discussed by Karliner and Mattis [15]. flavor symmetry selection rule to explain the narrow Various quark models have been proposed to un- width of Θ+ based on the Jaffe–Wilczek model. derstand the Θ+, mainly aimed at explaining the low However, if the Jaffe–Wilczek correlated diquark– mass and narrow width, which is not a serious prob- diquark wave function is totally antisymmetrized, lem in the chiral soliton quark model but hard to un- then there will still be an SU(6) totally symmetric derstand in the usual quark model.First, the ground component and no flavor symmetry selection rule to state should be an S-wave one in the naive quark forbid the pentaquark to decay to a KN final state. model [16, 17] and this means Θ+ should be a neg- Diakonov and Petrov [21] even raised the criticism ative parity state.The S-wave uudds¯ configuration that the nonrelativistic quark model might not make will have S-wave KN components.However, both any sense for light flavor pentaquark states. the I =0, 1 KN S-wave phase shifts are negative in Stancu, Riska, and Glozman [22] gave another the Θ+ energy region and this means that Θ+ cannot explanation of the special properties of the pentaquark be an S-wave KN resonance.On the other hand, state using their Goldstone boson exchange (GBE) since the I =0 KN P-wave P01 phase shifts are quark model.The flavor–spin dependence of the GBE positive, there might be a resonance in this channel qq interaction favors a totally symmetric flavor–spin 1 + wave function of the four light quarks.The overall and this is consistent with the JP = predicted by 2 color singlet dictates the four light quarks to be in a the chiral soliton quark model of the spin–parity of color triplet state, i.e., in a color SU(3) [211] repre- Θ+.If the Θ+ is confirmed to be a positive-parity state sentation which is the same as discussed in the Jaffe– and there is no negative-parity pentaquark state with Wilczek model.The Pauli principle further dictates lower energy, then it will provide another example of the four light quarks to be in a [31] representation of an inverted energy-level structure which has been a SO(3); therefore, they must be in a s3p configuration. weak point of the naive quark model of the baryon The kinetic-energy increase might be compensated spectrum: the first excited states of N and ∆ have by a potential-energy decrease.However, the GBE positive parity instead of negative parity as predicted interaction itself might not be enough to form the by the naive quark model. Θ+ and Stancu, Riska, and Glozman [22] have to Jaffe and Wilczek [14] proposed a {ud}{ud}s¯ introduce an additional qs¯interaction due to η-meson {ud} exchange, which was argued not to exist in the GBE structure for the pentaquark state, where means + a diquark in the color antitriplet, spin-0, isospin-0 approach [23].The narrow width of Θ has not been S-wave state.They assume a strong color –spin force discussed. to argue for such a diquark structure, a color Cooper Jennings and Maltman [24] did a comparative pair.The overall color singlet condition dictates that study of the three models (chiral soliton model, di- thefourquarksmustbeinacolortripletstate(in quark model, and GBE model) of the Θ+ mentioned a color SU(3) [211] representation), which is anti- above and related exotics.They concluded that the symmetric with respect to the diquark exchange in three models appear to be different but might describe color space.The two diquarks (quasibosons) should the same physics.They also pointed out that the PHYSICS OF ATOMIC NUCLEI Vol.68 No.5 2005 810 WANG et al. + a narrow width of Θ may be explained by the small Here, λi (a =1,...,8) is the color SU(3)-group gen- overlap of the five-quark-model wave function and the erator.For a single hadron, qq¯mesons, or q3 baryons, KN one.But Jennings –Maltman avoid discussion of such a modeling can be achieved by adjusting the the mass of the pentaquark because it is dependent on strength constant a of the confinement potential.The estimates of the confinement and kinetic energy. color factor λi · λj gives rise to a strength ratio 1/2 for Our group has done three-quark-model calcula- baryon and meson, which is almost the ratio for the tions.The first one is an application of the Fock space minimum length of the flux tube to the circumference expansion model which we developed to explain the of the triangle formed by three valence quarks of a nucleon spin structure [25].The naive quark model baryon. assumes that the baryon has a pure valence q3 con- Howtoextendtheconfinement potential to mul- figuration.This is certainly an approximation.One tiquark systems is an open question.There are a expects there should be higher Fock components, few lattice QCD calculations of pentaquarks which B = aq3 + bq3qq¯+ .... (2) obtained an S-wave ground state [27].However, the lattice calculations have not given the color flux The nucleon spin structure discovered in polarized tube or string structure as the Simonov group did lepton–nucleon deep-inelastic scattering can be ex- for mesons, baryons, and glueballs.From general plained by allowing the nucleon ground state to have SU(3) color group considerations, there might be about a 15% q3qq¯ component, where the qq¯ parts the following color structures: q3(1)qs¯(1), q3(8)qs¯(8), have pseudoscalar-meson quantum numbers.In the qq(3)¯ qq(3)¯¯ s(3)¯ , etc.Here, the number in parenthe- Θ+-mass calculation, we assume Θ+ isapureuudds¯ five-quark state but with channel coupling.Our pre- ses denotes the color SU(3) representation labeled liminary results are listed in the table.In the table, by its dimensions.The first one is a color singlet meson–baryon channel; the second is the hidden K8N8 means that the K and N are both in color octets but coupled to an overall color singlet.These color meson–baryon channel; the third is the color numerical results are under further check.However, structure used in the Jaffe–Wilczek model.New color the following results will not change.The S-wave structures will give rise to additional interactions state will have a lower mass than that of the P wave; which have not been taken into account in the quark the channel coupling plays a vital role in reducing model calculations of the pentaquark so far.And our the calculated S-wave Θ+ mass; and it is possible to first model calculation mentioned above shows that obtain a mass close to the observed mass 1540 MeV hidden color channel coupling reduces the calculated of the Θ+ if the overestimation of the K mass in this pentaquark mass. model is taken into account. In the 1990s, based on the quark cluster model, Quite possibly, confinement is due to gluon flux we developed a model, called the quark delocal- tube (or gluon string) formation in the QCD vacuum. ization, color screening model (QDCSM), to take Simonov proposed a field correlator method to study into account the additional interaction in multiquark the non-Abelian nonperturbative properties of QCD systems induced by various color structures, which and found a Y -shaped gluon flux tube (or string), but are not possible for a qq¯ meson and q3 baryon, by not a ∆-shaped one, in a baryon.The ground-state a reparametrization of color confinement [28].This energy can be approximated by a potential [26] model explains the existing baryon–baryon interac- NN NΛ NΣ − 1 tion data (bound-state deuteron and , , V3q = A3q + σ3qLmin + C3q, (3) scatterings) well with all model parameters fixed in |ri − rj| i PHYSICS OF ATOMIC NUCLEI Vol.68 No.5 2005 MULTIQUARK STATES 811 The calculated mass of Θ+ (in MeV) ∗ ∗ Pure KN KN + K N KN + K8N8 KN + K N + K8N8 S01 parity = − 2282 2157 1943 1766 P01 parity =+ 2357.1 2356.3 2357.0 2336.8 of 1615 MeV is obtained in an adiabatic approxi- recur.The low-lying, even parity rotational excitation mation.More precise dynamical calculation might of nuclei is hard to explain by the naive Mayer– reduce the calculated mass further.For the P -wave Jenson nuclear shell model; A.Bohr and B.Mot- channels, we only obtain spin-averaged effective KN telson had to introduce the rotational excitation of interactions because the spin–orbit coupling has not a deformed liquid-drop model.Later, nucleon Cooper been included yet.In the I =0 channel,thereisa pairs were introduced because of the strong short- strong attraction, as wanted in other quark models. range pairing correlation.In the 1970s –1980s, an S– However, in our model, the P -wave attraction is not D Cooper pair interaction boson model was developed strong enough to overcome the kinetic-energy in- and the collective rotation was rederived from this crease to make the P -wave state be a ground state. model, which is based on Mayer–Jenson’s nuclear This is consistent with the lattice and QCD sum-rule shell model but with nucleon pair correlation.In the results [27, 30].In the I =1channel, only a very weak description of the pentaquark, one has introduced attraction is obtained.This rules out the I =1Θ+ the chiral soliton rotational excitation, quark color again. Cooper pairs, and much more.The historical lessons In a third model, we use the {ud}{ud}s¯ configu- of nuclear structure study might be a good mirror to ration.The color structure is the same as the Ja ffe– light the way for the study of hadron structure. Wilczek one, but the four nonstrange quarks are to- tally antisymmetrized.The space part is fixed to be an equilateral triangle with the two diquarks sitting 4.TETRAQUARKS, HYBRIDS, at the bottom corners and the s¯at top.The height and AND GLUEBALLS the length of the bottom side of the triangle are taken as variational parameters in addition to the quark By the end of the 1970s, Jaffe [31] had already delocalization.A three-body variational calculation suggested that the scalar mesons with masses less with the QDCSM has been done.The minimum of than 1 GeV (σ(600), f0(980), a0(980), κ(900)) might this variational calculation is around 1.3–1.4 GeV, be qqq¯q¯ states [31].The “discovery” of a pentaquark corresponding to a color screening parameter µ = enhances this possibility.In addition, there are a few 1.0−0.8 fm−2.This gives rise to an e ffective attraction new candidates for tetraquarks: the BABAR experi- with a minimum at around 50–150 MeV,qualitatively ment observed a resonance of M = 2317 MeV, Γ < + 0 similar to our second model result but with the pos- 10 MeV in the DS π invariant mass analysis [32]. sibility of further reducing the calculated Θ+ mass to CLEO confirmed this resonance and observed a new + be closer to the observed value. DSJ(2463 MeV) state [33].These might be qqc¯ s¯ As mentioned before, these numerical results have tetraquark states.BELLE observed a resonance of to be checked further and the QDCSM is only a 3872 MeV in the J/ψπ+π− channel [34] in B-meson model of QCD.Based on these results, we cannot decay.CDF-II con firmed this resonance in pp¯ colli- definitively claim that the ground state of the Θ+ is sions [35].This might be a D0D¯ ∗0 molecular state. 1 − IJP =0 .However, this possibility has not been There have been glueball and hybrid candidates. 2 In general, these states will be mixed and also mixed ruled out because color confinement contributions of with tetraquark states.The “discovery” (not con- various nontrivial color structures for a pentaquark firmed) of a pentaquark state will make the identifi- system have not been studied thoroughly.The re- cation of glueballs and hybrids even harder [36]. sults of QDCSM can be checked by devising pen- taquark interpolating field operators with these non- trivial color structures in lattice QCD calculations. 5.DIBARYONS Suppose the Θ+ is finally verified to be a 1540-MeV 1 + Immediately after the development of MIT bag narrow-width (∼1MeV)IJP =0 state.Then 2 model, Jaffe predicted the deeply bound dibaryon H, an interesting scenario similar to that of nuclear an I = S =0uuddss flavor singlet state [37].Long- structure study from the 1940s to the 1950s will term extensive searches for the H cannot confirm PHYSICS OF ATOMIC NUCLEI Vol.68 No.5 2005 812 WANG et al. its existence.Simonov [38] proposed the quark com- range 2.1–2.2 GeV, which might make the identifi- pound bag model and predicted various dibaryon res- cation of the d∗ even harder, and all of them might onances.Lomon [39] extended the R-matrix formal- be the origin of the observed broad resonance of ism to dibaryon studies.The R-matrix formalism the pp and np total cross section in that energy re- model explains the NN-scattering data well because gion.The strangeness −3 NΩ state is a very narrow it has the aid of the boundary condition model of the dibaryon resonance and might be detected in Ω-high- NN interaction [40].It predicted a 2.7-GeV NN productivity reactions such as at RHIC, COMPAS, resonance and an experimental signal was claimed. and the planned JHF in Japan and FAIR in German. A direct extension of the naive quark model [16, 17] to baryon–baryon interactions only gives rise to 6.SUMMARY the short-range repulsion of the NN interaction but not the intermediate-range attraction nor the well- Multiquark states have been studied for about + established long-range tail (due to one-π exchange) 30 years.The Θ ,iffurtherconfirmed, will be the in the meson-exchange model.This means that im- first example.Once the multiquark Pandora’s box portant physics has been missed in the application is opened, the other multiquark states, tetraquark, of the naive quark model to the baryon–baryon in- hexaquark (or dibaryon), etc., cannot be kept inside. teraction.One possibility arises from spontaneously One expects that they will be discovered sooner or broken chiral symmetry and the related GBE [41]. later.A new landscape of hadron physics will appear Obukhovsky and Kusainov [42] applied such a hy- and it will not only show new forms of hadronic matter brid model, with both gluon and GBE, to the NN but will also exhibit new features of low-energy QCD. interaction.Li et al. [43] extended this hybrid model Nonperturbative and lattice QCD have revealed from SU(2) to SU(3) and used it to predict dibaryon the color flux tube (or string) structure of the qq¯ and resonances.Such a direct extension with a univer- q3 states.The multiquark system will have more color sal σ−u, d, s quark coupling will overestimate the structures.How do these color structures interplay σ-meson contribution in high-strangeness channels within a multiquark state? Nuclear structure seems and makes the predicted mass of the di-Ω too low [44]. to be understood in terms of colorless nucleons within Kopeliovich et al. [45] studied the dibaryon with a nucleus.We emphasized that other color structures the Skyrmion model and predicted dibaryons up to should be studied and the multiquark systems provide high strangeness −6.But after taking into account a good field to do that.The low mass and narrow width + the Casimir energy, these dibaryons become un- of the Θ might be related to such new structures bound [45]. instead of to residual interactions. We have studied the dibaryon candidates in the u, d, s three-flavor world systematically with both non- ACKNOWLEDGMENTS relativistic and relativistic versions of the QDCSM in We apologize to those authors whose contribu- an adiabatic approximation [46].This model has been tions in this field have not been mentioned due to the mentioned before.The main new ingredients are as limitation of space. follows: Baryon distortion or internal excitation in the course of interaction is included by allow- This work is supported by the NSFC under grants ing the quark mutual delocalization between two nos.90103018, 10375030 and in part by the US interacting baryons and a variational method is Department of Energy under contract W-7405- developed to let the interacting baryon choose its ENG-36. own favorable configuration; a new parametriza- tion of color confinement is assumed to take into REFERENCES account the contribution of nontrivial color struc- 1. 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Jr.,Phys.Rev.Lett. 59, 627 (1987); H.R.Pang, 26.Yu.A.Simonov, Nucl.Phys.B 307, 512 (1988); J.L.Ping, F.Wang, et al., nucl-th/0306043; D.S.Kuzmenko, V.I.Shevchenko, and Yu.A.Si- H.R.Pang, J.L.Ping, L.Z.Chen, et al., Phys.Rev. monov, hep-ph/0310190. C 70, 035201 (2004). PHYSICS OF ATOMIC NUCLEI Vol.68 No.5 2005 Physics of Atomic Nuclei, Vol. 68, No. 5, 2005, pp. 814–827. Translated from Yadernaya Fizika, Vol. 68, No. 5, 2005, pp. 847–859. Original Russian Text Copyright c 2005 by Blinnikov. Supernovae and Properties of Matter in the Densest and Most Rarefied States S. I. Blinnikov* Institute of Theoretical and Experimental Physics, Bol’shaya Cheremushkinskaya ul. 25, Moscow, 117218 Russia and MPA, Garching, D-85741, Germany Received June 2, 2004; in final form, November 3, 2004 Abstract—An overview of the relationship between the astrophysics of supernovae and fundamental physics is given.It is shown how astronomical observations of supernovae are used to determine the parameters of matter in the most rarefied states (“dark energy”); it is also revealed that the mechanism of supernovae explosion is related to the properties of matter in the densest states.The distinction between thermonuclear and collapsing supernovae is explained.Some problems that arise in the theory of powerful cosmic explositions— supernovae and gamma-ray bursts—and which require new physics for solving them are indicated. c 2005 Pleiades Publishing, Inc. 1.INTRODUCTION physics—specifically, supernovae, which are explod- ing objects, will be the focus of attention here. Still being an upperclassman in secondary school, I eagerly awaited the appearance of each successive At the beginning of the article, it is explained issue of the “Feynman Lectures on Physics,” trans- how parameters of cosmological models can be de- lated into Russian by a group of physicists, including termined with the aid of a constant-power source of Yu.A.Simonov.In one of his lectures, R.Feynman light, so-called cosmological standard candle.Fur- said approximately the following.If asked to choose ther, it is shown that some subtle points in employ- one clause that would explain the maximum number ing supernovae as cosmological standard candles are of phenomena in nature, he would say, “Matter con- often treated incorrectly by physicists.For example, sists of atoms.” I recall that I was deeply impressed by Dolgov writes, in his very useful review article [1], these words.Since the main subjects of my scienti fic that type Ia supernovae are cosmological standard activities are associated with modern astrophysics, I candles.But in fact, these supernovae are not stan- have been dealing primarily with processes that are dard candles! Nevertheless, the acceleration of the explained by the properties of atoms and ions in var- expansion of the Universe was discovered with the aid ious states.Modern theoretical physics focuses pri- of precisely those objects. marily on more fundamental things—the properties There is no doubt that a thermonuclear explosion of fields, strings, or some other geometric objects, on is the mechanism underlying the explosion of type the basis of which theorists try to explain the entire Ia supernovae.This mechanism leads to a complete micro- and macrocosm.(Yet, it cannot be stated that disruption of a star.Supernovae of other types stem allofthetheoreticalproblems—especially those that from the collapse of the core of a star.In contrast are needed for applications, in atomic physics, for to thermonuclear supernovae, there are many more example—have already been solved.) unclear points here in the explosion mechanism.The Astrophysics is flourishing at the present time. concluding part of this brief review article is devoted Since the end of the twentieth century, astrophysics mostly to collapsing supernovae. and classical observational astronomy have provided ever more data that can crucially affect the develop- ment of the most fundamental branches of physics.In 2.REDSHIFT IN COSMOLOGY particular, it has become clear to date that about 95% of the mean density of the observed Universe does In 1929, E.Hubble, who worked at a new 2.4-m not consist of ordinary matter—thatis,ofatoms.In telescope of the Mount Wilson Observatory near Los this review article, I will consider but a small num- Angeles, was able to estimate the distance to galaxies bers of facts concerning the impact of astronomy on for the first time.By plotting the redshifts z of the spectral lines of those galaxies against the estimated *E-mail: [email protected] distances d, he constructed a graph that, later on, was 1063-7788/05/6805-0814$26.00 c 2005 Pleiades Publishing, Inc. SUPERNOVAE AND PROPERTIES OF MATTER 815 called the Hubble diagram.Fitting a straight line to where a is the radius of curvature of the 3D space. these data, he obtained the dependence Writing ds2 = c2dt2 − d 2 for four-dimensional space- time, we will arrive at the metric of the world in λ − λ H0d z = lab = , the form discovered by Friedmann and used in the λlab c cosmological section of the textbook by Landau and which is presently referred to as the Hubble law.Here, Lifshitz [5].A somewhat di fferent form is obtained λ is the observed value of the wavelength for some upon the substitution sin χ = r, in which case dχ2 = structure in the spectrum of a specific object, while dr2/(1 − r2); as a result, the interval can be written λlab is the laboratory value for the same structure.The in the form value of the Hubble constant H0 was overestimated ds2 = c2dt2 − a2(t) by Hubble himself by almost an order of magnitude in dr2 relation to present-day data [about 70 km/(s Mpc)], × + r2(dθ2 +sin2 θdϕ2) . but the very fact of the growth of the redshift with dis- 1 − kr2 tance was established correctly.This fact is referred r to as the recession of masses of luminous matter (We note that, here, is a dimensionless quantity!) (galaxies and their clusters), in which case z is in- This is the Friedmann metric in the Robertson– k =1 terpreted as a manifestation of the Doppler effect (z = Walker form.At , we have a closed Friedmann universe (where the curvature of 3D space is positive), v/c at v c and v = H d).I will try to show that, 0 while, at k =0, there arises a 4D world with a flat 3D although it is legitimate to interpret unambiguously space.In the latter case, the quantity a(t) loses the the Hubble law as a manifestation of the expansion of meaning of the radius of curvature; therefore, it would the observed Universe, the treatment of the redshift be better to refer to it as a scale factor for all of the as the Doppler effect at large cosmological distances versions.It can easily be veri fied that the case of k = leads to logical difficulties and is, strictly speaking, −1 corresponds to yet another world, that where 3D unacceptable (sometimes, the cosmological redshift space is of negative curvature, featuring Lobachevski is associated in the literature with the Doppler effect geometry [3].It is obtained from the case of k =+1 by definition [1]). by means of the substitution sin χ → sinh χ.In all of Theoretically, the expansion of the Universe was these cases, the point r =0can be chosen arbitrarily discovered by Friedmann [2, 3] long before Hubble’s in a uniform space, in just the same way as a pole studies.(De Sitter [4] derived his solutions still earlier, on a uniform 2D sphere.We assume that galaxies but his treatment of these solutions was purely for- are points associated with fixed values of r, θ,andϕ mal; only the studies of Friedmann [2, 3] established (comoving coordinates) and that the scale factor a(t) the physics pattern of a nonstationary universe—that determines the expansion of the Universe.According is, they created the framework within which modern to present-day data, it is better, strictly speaking, to physical cosmology develops.) Friedmann’s line of take the centers of mass of galaxy clusters (rather reasoning can be traced in the following way. than galaxies) for “fixed” points, since galaxies can A two-dimensional sphere specified by the metric move within clusters at peculiar velocities of about d 2 = a2(dθ2 +sin2 θdϕ2) is curved, but, since it is 1000 km/s. the surface of the ball x2 + y2 + z2 = a2, the sphere is Friedmann did much more than just a number of obviously isotropic and uniform, in just the same way coordinate transformations.He showed (taking into as a two-dimensional (2D) plane, whose metric can account the possibility that the cosmological Λ term be represented in the form d 2 = a2(dθ2 + θ2dϕ2), in nonzero) that all versions of the metrics that he where the polar radius is denoted by θ instead of considered can provide exact solutions to the Ein- conventional r. stein equations for a reasonable choice of equation of If we now take a flat three-dimensional (3D) space state for matter.It is only necessary to determine the and replace its metric behavior of the scale factor a(t) for a nonstationary world with allowance for this equation of state. 2 2 2 2 2 2 d = da + a (dθ +sin θdϕ ) From the Einstein equations (see, for example, [5]) by or directly from the Hilbert variational principle (see [6]), one can easily obtain d 2 = a2[dχ2 +sin2 χ2(dθ2 +sin2 θdϕ2)], (1) a˙ 2 8πG kc2 = N E− , (2) then, by analogy with a 2D sphere, we obtain an a 3c2 a2 isotropic and uniform 3D space in the form of the ≡ surface of the 4D ball, where a˙ da/dt, GN is the Newtonian gravitational E 2 2 2 2 2 constant and is the density of all forms of energy. w + x + y + z = a , This relation is referred to as the Friedmann equation. PHYSICS OF ATOMIC NUCLEI Vol.68 No.5 2005 816 BLINNIKOV The physical meaning of this equation can be un- whose density is given by derstood by using prerelativistic physics.If we intro- c4Λ duce the mass M within the radius R = a, E = . vac 8πG 4πEa3 N −E 2 = M, We arrive at P = vac, provided that the thermo- 3c 2 dynamic identity d (Evac/ρ) − Pdρ/ρ =0,whereρ the result obtained from the Friedmann equation is is the baryon-number density, holds.Thus, we have identical to that in the Newtonian case under the w = −1.Below, we will not distinguish between E condition that the whole energy density is due to vac and EDE, but, in general, w = −1 for the case of dark nonrelativistic baryons—that is, to matter of pressure energy.Since the matter energy density and the ra- P =0. diation energy density both decrease as the Universe Assuming that the mean matter density ρ is uni- expands, the vacuum energy, if any, may appear to be form at large scales (R>10 Mpc in the present-day dominant in the dynamics of expansion. Universe), we find that the mass within the radius 3 The velocity of expansion is measured by the Hub- R is M =4πρR /3, and Newton’s laws lead to the ble parameter energy-conservation law a˙ u2 G M H = . − N = −const, a 2 R The value of the Hubble parameter in the present era which holds as long as u ≡ R˙ c,sothat is referred to as the Hubble constant H0 (it is constant in space rather than in time in uniform models featur- (R˙ )2 4πG ρR2 − N = −const. ing a synchronized time t [7]).Since H0 =(˙a/a)now, 2 3 we can see that the constant k in the Friedmann This relation is equivalent to the Friedman equa- equation is positive or negative, depending on the tion (2), and this is good, since, on small scales, the ratio world (spacetime) is flat, so that the general theory E 3H2 Ω ≡ , ρ ≡ 0 . of relativity must reduce to nonrelativistic mechanics 2 where c ρcc 8πGN according to the correspondence principle. If Ω > 1,thenk =1.At Ω=1,3Dspaceisflat: k = But in fact, matter in the Universe is likely to 0;forΩ < 1,wehavek = −1. involve, in addition to baryons, dark matter, which is cold in all probability—that is, nonrelativistic.More- For matter (that is, for ordinary and dark matter ≡E 2 over, the results of observations require introducing taken together), we will write Ωm m/ρcc .For the a new substance, which was referred to, not quite vacuum or dark energy, we introduce the notation 2 appropriately, as dark energy (DE).This may be either ΩΛ ≡EDE/ρcc . a constant nonzero vacuum energy density or a new Astronomers also introduce the dimensionless de- field [1].In order to explain observations, it is only celeration parameter necessary that an equation of state of the form P = E −aa¨ w DE be valid, with the coefficient w being nega- q = 2 , tive (in the first approximation, it is close to −1)— a˙ that is, in contrast to ordinary gases, which have a which measures the rate at which the velocity of positive pressure, this substance must be in a state expansion changes.For historical reasons, this pa- of tension, as stretched elastic rubber (but stretched rameter was defined with a minus sign, since it was isotropically—that is, uniformly in all directions of 3D believed that deceleration was natural, but, according space!). to present-day data, q<0;thatis,a>¨ 0,andthe In the Friedman equation (2), one can immediately expansion of the Universe is accelerated. take into account a nonzero vacuum energy density It can be shown that the photon frequency satisfies (dark energy) by assuming that the energy density E the relation E E E involves two components, = m + DE,wherethe ωa(t)=const. (3) first component is the energy density of matter, while the second component is the vacuum (dark) energy The simplest way to derive this relation is to go over density. to the conformal time η specified by the equation dη = In the simplest form, the inclusion of a constant cdt/a(t) (see, for example, [5]). vacuum energy density is equivalent to supplement- The metric ds2 = c2dt2 − a(t)2d 2 is nonstatic, ing the Einstein equations with a cosmological con- but, written in terms of the time coordinate η, it as- stant Λ.The vacuum energy behaves as an ideal liquid sumes the form ds2 = a(η)2(dη2 − d 2).The spatial PHYSICS OF ATOMIC NUCLEI Vol.68 No.5 2005 SUPERNOVAE AND PROPERTIES OF MATTER 817 part d 2 is independent of η, so that zero geodesic δx =¨x(δt)2/2.The correct answer arises only upon lines are ds =0—that is, light rays in the coordinates supplementing the Doppler effect with the Einstein 2 2 η and are identical for different initial instants ηe of effect of violet shift, zEin = δφ/c = −r /2 (here, δφ the emission of light signals, and all intervals ∆η be- is the change of the gravitational potential between tween the signals are constant.This means that, if the the objects).We then have z = z + z = r2/2,in interval of the physical time between the instants of Dop Ein emission of two light signals at one point is ∆t ,then which case the derivation of the cosmological redshift e according to [8] is inapplicable, this being indicated the interval between the instants of reception, ∆tr, varies in proportion to a,sincedt ∝ a(t)dη,whence there.In the general case of a variable density, the we obtain Eq.(3). Einstein shift effect cannot be singled out in a pure form, so that the equations of photon propagation ω A photon emitted with a frequency 1 will be ob- must be solved directly [10]. served with a lower frequency ω0 if the scale factor grows: Despite a strong temptation to attribute the cos- ω0 a1 mological redshift to the relative motion of an ob- = . ω1 a0 server and an emitter, this is possible at large dis- tances only in a flat world.In a curved space, this can In cosmology, the subscript “0” is always reserved for be done only at the instant when an observer and an the modern era, so that a is an earlier (and smaller) 1 emitter occur at the same point, moving relative to value of the scale factor than a0.The redshift mea- sured by astronomers is then given by each other.The problem is that, in a curved spacetime onecannottransportavector(thevelocityofafar λ − λ a z = 0 1 = 0 − 1. object here) to the observation point: the result of λ1 a1 a parallel transportation depends on the path along which this is done.By way of illustration, we take an It should be emphasized that this redshift does not arbitrary vector at the emission time t at the point r reduce to the conventional Doppler effect, although 1 1 t relation (3) can sometimes be deduced, under some and transport it parallelly toward an observer at 0 at assumptions, from the Doppler effect for close ob- the point r0.In doing this, we fix the angular coor- servers.The relative velocity of close observers (oc- dinates θ and ϕ and compare two possible “natural” curring at a small distance of cδt)isv = Hcδt = paths of the transportation of this vector: acδt/a˙ = cδa/a.According to the Doppler e ffect, we (i) that which first goes from r1 to r0 at constant then have t = t1 and then from t = t1 to t0 at constant r = r0; δω v δa = − = − , t t ω c a (ii) that which first goes from 1 to 0 at constant r = r1 and then from r1 to r0 at constant t = t0. whence we obtain Eq.(3) upon integration.This derivation is given by Zeldovich and Novikov in [8], It is clear that the results will be different for the but the following comments are in order here: first, different paths in a curved spacetime, the difference this derivation does not work in all of the cases where being proportional to the area within the contour the above derivation in terms of the conformal time η formed by these two paths (that is, it increases with from [5] is valid; second, it is hazardous to apply the increasing difference of t1 and t0 and with increasing Doppler interpretation of the cosmological redshift z difference of r1 and r0).It should be emphasized that at large distances.This issue was discussed in detail a nonempty 4D world is curved even if k =0 and by Harrison [7, 9], predominantly from the philosoph- 3D space is flat! Therefore, the interpretation of the ical point of view.I will give more physical arguments. redshift in terms of the Doppler effect loses physical Let us consider the closed Friedmann universe meaning at cosmological distances, since the concept (k =+1), and let t0 correspond to the instant of of a relative velocity becomes meaningless.(Davis largest expansion: a˙(t0)=0.Suppose that an ob- and Lineweaver [11] discussed the inapplicability of server who measures redshifts has the coordinate the formulas of the special theory of relativity in what r =0.In the first order in r, neighboring objects is concerned with the Doppler effect in cosmology, but are immobile; however, the velocity of recession of they did not even touch upon a greater danger—the objects at a distance cδt = a(t0)r is nonzero in the fact that the concept of a relative velocity loses phys- second order, and calculations reveal that the Doppler ical meaning.) It is safe to say that, owing to the re- 2 redshift is zDop = r .At the same time, the correct lation ωa(t)=const, the redshift is a geometric effect result according to Eq.(3) is z = r2/2.This distinc- of the expansion of the Universe, but it is illegitimate tion is explained trivially by the difference in the ex- to associate the change in the photon frequency with pressions for the velocity δv =¨xδt and the coordinate the relative velocity of the objects being considered. PHYSICS OF ATOMIC NUCLEI Vol.68 No.5 2005 818 BLINNIKOV 3.PHOTOMETRIC DISTANCE the factor (1 + z), and the frequency of photon arrival The redshift can be measured: if we know the also decreases by the factor (1 + z).Therefore, we laboratory wavelengths of various spectral lines in have the spectra of distant objects, we can say how their L S = 2 2 2 wavelengths changed from the emission instant t1 to 4πa0r (1 + z) t the observation instant 0.From here and from (3), or we know the ratio of the scale factors at these two instants: dL = a0r(1 + z)=d(t0)(1 + z). (5) a(t0) =1+z. The photometric distance dL isaquantityacces- a(t1) sible to measurement if we have at our disposal an How can we measure distances? astrophysical source whose absolute luminosity L is known (cosmological standard candle).But r is not We can introduce a formal definition of a proper observable, so that it is necessary to get rid of it.On distance: the zero geodesic line where dθ = dϕ =0,wehave d(t)=a(t)r; (4) a2 0=ds2 = c2dt2 − dr2 it would be measured (at k =0) at the coordinate- 1 − kr2 1 time instant t by a set of observers with rigid rulers 1 between the points having the radial coordinates 0 or and r [10, 12].(Recall that r is dimensionless!) t0 r cdt dr The definition of a comoving distance, d = a(t)χ, = 1 . c − 2 1/2 a(t) (1 kr1) where χ is the Friedmann radial coordinate intro- t 0 duced in Eq.(1) (see, for example, [13]), is of impor- 1 tance in theoretical cosmology. We now make the transformations We cannot measure directly the “distances” d and t0 a0 1 dt dt da a dt a0 dc to far objects in the expanding Universe with the = = − d aid of rules or radio detection and ranging.Instead, a(t) da a a0 da a a various definitions are introduced in cosmography t1 1 a0/a1 that depend on those measurements that can actu- a0/a1 a dt a ally be performed (for example, the angular distance = d 0 , determined by measuring the transverse angular size a0 da a of a standard ruler).Details are nicely explained by 1 Weinberg [12].I also follow the text of Carroll [14], whence we obtain but I would like to emphasize that it is not easy to r z+1 z find, in standard textbooks, a complete derivation of dr d(z +1) dz a 1 = c 1 = c 1 , a formula that would relate distances to cosmological 0 − 2 1/2 (1 kr1) H H parameters.Therefore, I give such a derivation in the 0 1 0 present article. so that everything has reduced to observables like The so-called photometric distance 1/H (equal a/a˙ ), z (with the aid of the relation 1/2 L a0/a1 = z +1), and so on.In this way, we can get dL = , rid of r in the expression for d by expressing r in 4πS L r − 2 −1/2 terms of the elementary integral 0 (1 kr1) dr1 where L is the absolute luminosity (light power) of and z dz /H. asourceandS is the flux measured by an observer 0 1 (energy arriving within a unit time per unit area of the In order to do this, we make use of the Fried- receiver), is the most valuable for us.This de finition man equation, taking simultaneously into account corresponds to the statement that, in flat space, the the possibility of a nonzero vacuum energy. flux at distance d from the source is S = L/(4πd2).In We write the Friedmann equation (2) in the form the Friedmann universe, however, it would be incor- E 2 2 8πGN kc rect to substitute here naively d = a r from (4), where H = − , 0 3c2 a2 a0 is the scale factor at the instant when photons were observed at the comoving coordinate r measured from which is equivalent to the source. 2 2 3 H = H0 [Ωm(1 + z) +ΩΛ The point is that the flux decreases because of two − − 2 effects: individual photons undergo a phase shift by +(1 Ωm ΩΛ)(1 + z) ], PHYSICS OF ATOMIC NUCLEI Vol.68 No.5 2005 SUPERNOVAE AND PROPERTIES OF MATTER 819 2 2 where Ωm = Em/ρcc and ΩΛ = EDE/ρcc are the pa- obtained from observations of distant supernovae.We rameters that were introduced above to characterize, will now proceed to consider these extremely interest- respectively, the density of nonrelativistic matter [its ing objects. 3 density Em varies in proportion to (1 + z) in accor- dance with the variation of the comoving volume] and the vacuum-energy density (this density is constant 4.SUPERNOVAE OF VARIOUS TYPES in the simplest case of the Λ term).Substituting z Supernovae (Supernova = SN, Supernovae = H into the expression 0 dz1/H and expressing r SNe) are explosive bursts of stars whose luminosity r − 2 −1/2 ∼ 10 in terms of 0 (1 kr1) dr1 only for the case of (that is, radiation power) is L 10 L or higher 33 k = −1 (the cases of other k values are obtained for a few weeks.Here, L ≈ 4 × 10 erg/sisthe automatically by means of an analytic continuation), luminosity of the Sun—that is, one supernova devel- we arrive at the required formula for the photometric ops, within some time interval, the same light power distance: as a mean galaxy consisting of billions of stars.It is z this power that makes it possible to use supernovae c 1 3 dL = (1 + z)√ sinh Ωk Ωm(1 + z) in cosmography.A larger part of the star mass is H0 Ωk disintegrated and ejected.Supernovae are among 0 the strongest explosions in the Universe: the ejected 51 2 −1/2 kinetic energy is E ∼ 10 erg. +ΩΛ +Ωk(1 + z) dz . The energy of an explosion is estimated as follows. From the widths of spectral lines, it can be found Ω ≡ 1 − Ω − Ω Ω < 0 Here, k m Λ;for k , the hyperbolic that the speed in the atmosphere is v ∼ 104 km/s sinh sin sine ( √ ) goes over to the trigonometric sine ( ), or higher.On the basis of a simulation of the light while Ω goes over to |Ω |.For Ω → 0,the k k k curves—that is, the dependences L(t)—the mass Mej limit can easily be evaluated, sinh disappearing from is estimated at about 1M to a few tens of M . the expression for dL, where there only remains the z −1/2 From here, one obtains estimates for the kinetic integral 0 [...] dz. energy of supernova ejection for all types of supernova The dependence dL(z) can now be expressed in outbursts [the Ia,Ib/c, and II types are introduced terms of cosmological parameters.We can see from in accordance with the special features of respective Eq.(5) that dL is very simply related both to d from spectra (for details, see below)]: on average, E ∼ Eq.(4) and to the Friedmann radial coordinate χ from 1051 erg ≡ 1FOE [10 raised to the power of 51 or Eq.(1). Fifty-One Ergs (FOE)], although there occur devi- For some important particular cases, we can write ations of an order of magnitude above and below this simple analytic expressions for the proper distance value.Within a few ten thousand years, this energy d = d(z), which, according to Eq.(5), di ffers from dissipates in the circumstellar medium, heating it dL only by the factor (1 + z).By way of example, and generating x rays and cosmic rays—that is, it we indicate that, for the vacuum-dominated de Sitter generates a gaseous remnant of a supernova. Ω =1 Ω =0 universe ( Λ , m ), the result is The aforementioned ejecta enrich the medium in H0d = cz. heavy elements.Shock waves gather the circumstel- lar medium into dense clouds, and this leads to the For an empty universe (Ω =0, Ω =0), we have Λ m formation of young stars.The history of investigations c 1 devoted to supernovae is outlined in the monograph of H d = (1 + z) − . 0 2 1+z Shklovsky [15] and in the review articles [16, 17]. Owing to the use of state-of-the-art astrophysical In the case of a flat (parabolic) Friedmann universe methods, a vast body of observational data on super- (Ω =0), Ω =1: Λ m novae has been obtained over all ranges of electro- 1 magnetic radiation—from the radio-wave to the x-ray H0d =2c 1 − √ . 1+z range.Also, the first nonelectromagnetic signals— neutrinos from the SN 1987А—have been recorded. In all of the above cases, these formulas reduce to Despite all this, the mechanisms of supernova ex- the Hubble law H0d = cz at small z,buttheydiffer plosions have yet to be clarified conclusively.In re- significantly at large value of the redshift z. cent years, there have appeared indications that some Intermediate values of ΩΛ ≈ 0.7 and Ωm ≈ 0.3 cosmic gamma-ray bursts are related to supernovae. correspond to the modern standard model of the Uni- Possibly, the origin of gamma-ray bursts is also re- verse (“concordance model”).These values were first lated to the origin of supernova explosions. PHYSICS OF ATOMIC NUCLEI Vol.68 No.5 2005 820 BLINNIKOV Cosmic gamma-ray bursts are irregular pulses of powerful emission from oxygen.Nonthermal radia- photons whose energies range between about 0.1 and tion from SN Ib/c was discovered, and these objects 1 MeV or take even higher values, the pulse duration are likely to be correlated with the regions of active being a few tenths of a second to a few minutes. star formation.In all probability, these supernovae The exposure corresponding to the weakest of such explode through the collapse of cores of massive stars pulses is S ∼ 10−7 erg/cm2 ∼1 photon/cm2,but (in just the same way as SN II), whereas SN Ia are many of them are much stronger.Gamma-ray bursts thermonuclear explosions of white dwarfs in binary are surveyed in [18, 19].However, any surveys are systems that have lost hydrogen by the instant of the always behind the latest advances in the vigorously explosion (no neutron stars or black holes arise in this developing science that explores powerful explosions case).Thus, the classical astronomical classi fication in the Universe, this especially concerning the most of supernovae does not take fully into account the recent discoveries in the realms of gamma-ray bursts. special features of the mechanism governing the Theory proposes various explanations for the ori- explosion, which occurs in the interior of the star— gin of the supernova energy and seeks mechanisms it is more adequate to the structure of the outer layers underlying supernova explosions.In thermonuclear of a supernova. scenarios, an explosion begins because of the devel- The SN II phenomenon can arise at the end of opment of a thermal instability in the degenerate stel- the lifetime of a single massive star that preserved lar core upon the ignition a carbon–oxygen mixture hydrogen in its envelope.The outbursts of SN I b/c (helium in some scenarios). can occur in the collapse of the core of a single mas- The gravitational collapse of a star into a neutron sive star that lost hydrogen.If the SN I c subtype can star or a black hole—this seems a more efficient actually be distinguished from SN Ib, this means that mechanism of a star explosion—was described long SN Ic massive presupernovae lost not only hydrogen ago.Theoretically, the energy released in this case but also helium.In SN I a, approximately half the may be an order of magnitude higher than the ther- ejected mass is due to elements of the iron peak, monuclear energy—specifically, it must be about 10% while, in SN Ib/c, the bulk of these elements went into of the star-core mass.For a core of mass 1M , this the collapse.Therefore, SN I b/c ejection is dominated 53 energy is about 10 erg; for a more massive core, it by elements like oxygen, whereby the distinction be- is naturally still higher.But in the case of supernova tween the spectra of supernovae belonging to different explosions, the bulk of energy is carried away by types is explained. the neutrino flux.In gamma-ray bursts, gamma-ray 54 The affiliation of a star with a binary system plays a photons alone carry up to about 10 erg! This value crucial role in the evolution of type Ia presupernovae. is obtained from the observed flux S by the formula 2 Binarity effects seem responsible for the properties Egamma-ray burst =4πSdL,wheredL is the photomet- of some peculiar type II supernovae as well.The ric distance to the gamma-ray burst under study.It is possibility of a gamma-ray burst in a binary system of course clear that, in this formula, the flux S must leads to interesting effects—this is one of the possible be multiplied by some small solid angle rather than by models for the afterglow of a gamma-ray burst [20]. 4π.The energy will then be lower, but it is necessary Moreover, the idea that gamma-ray bursts can be to reveal concurrently the reasons behind the release generated at cosmological distances in the merger of of energy in the form of a narrow beam or jet within a neutron-star binary was put forth long ago [21]. this small angle.Possibly, the asymmetry of explo- sions is related to some phenomena in supernovae. The traditional astronomical classification par- 5.TYPE I a SUPERNOVAE: titions supernovae into two classes: type-I (SN I, STANDARDIZATION OF A CANDLE whose spectrum does not contain hydrogen lines in Owing to a number of factors, type Ia supernovae the vicinity of the maximum light) and type-II (SN II, (SN Ia) are convenient for measuring distances and whose spectrum features hydrogen lines there) super- for determining the geometry of the Universe.First, novae.Later, this classi fication was refined.The first these are very bright objects, so that we can obtain class was divided into subclasses: SN Ia and SN Ib/c. rich information about them even if they explode in The spectra of SN Ib/c in the vicinity of the maximum very distant galaxies characterized by large redshifts light do not exhibit a silicon line, which is pronounced z.Second, the spectra of SN I a and the shapes of their in SN Ia.The spectra of SN I a and SN Ib/c show the light curves seem to suggest, at first glance, that they most glaring distinctions within a late era, t 250 form quite a uniform class.Formerly, it was assumed days after the explosion, where the spectra of SN Ia that they are cosmological standard candles in the are formed largely by the lines of ionized iron, while sense that the maxima of the absolute luminosity are the spectra of SN Ib/c are dominated by an extremely identical for different SN Ia. PHYSICS OF ATOMIC NUCLEI Vol.68 No.5 2005 SUPERNOVAE AND PROPERTIES OF MATTER 821 However, this is not so! A closer inspection of Values of χ2 in comparing data on 157 SN Ia with various SN Ia revealed distinctions within this class of ob- models jects. Long ago, Pskovskiı˘ showed [22] that the flux Model χ2 maxima in the SN I spectra are not identical, and his colleagues revealed clear-cut distinctions even within Ωm =0.27, ΩΛ =0.73 171 the subclass of thermonuclear SN Ia [23]. Pskovskiı˘ [22] also found the interplay between the Ωm =1.00, ΩΛ =0.00 497 maximum luminosity of SN Ia and the rate of the sub- sequent weakening of flux.The flux of more powerful Ωm =0.00, ΩΛ =0.00 196 bursts decreases more slowly than the flux of their less Gray dust (at Ω =1.00, Ω =0.00) 293 powerful counterparts.Later on, this dependence was m Λ vigorously studied by many astronomers interested in Recent dust (at Ωm =1.00, ΩΛ =0.00) 168 SN Ia—especially meticulous studies on the subject Weakening in proportion to z were performed by Phillips and his colleagues [24, 241 25] on the basis of observations of close supernovae (at Ωm =1.00, ΩΛ =0.00) characterized by moderately small z. When astronomers discover a supernova char- of its radioactive decay.On the other hand, a large acterized by a large redshift z, they determine the amount of nickel is expected to increase the nontrans- rate of the decrease in its flux after the maximum; parency of matter greatly.The di ffusion of radiation only after the application of the Pskovskiı–˘ Phillips through a stellar medium takes a longer time, and the dependence, which provides the only way to perform light curve becomes more gently sloping.However, the “standardization of a candle”, can one estimate the decrease on the light curve depends not only on the luminosity of the supernova and, hence, the pho- the amount of nickel but also on its distribution (and tometric distance dL to it.However, the Pskovski ı–˘ Phillips dependence is of a correlation rather than a on the distribution of other heavy elements as well) functional character; therefore, each individual mea- within the expanding star and on the velocity of the surement may involve large errors. expansion of matter.In turn, this distribution and this velocity depend on how burning propagated through An unexpected result was obtained in the studies the star. of two groups [26, 27]: from observational data on The theory of burning in supernovae has actively distant supernovae, it follows quite reliably that Ω > Λ developed since the studies of Arnett [29], Ivanova 0; that is, the expansion of the Universe accelerates. et al. [30], and Nomoto et al. [31], but many problems A vast body of observational data has been accu- in it have yet to be solved (see, for example, [32, 33]). mulated since then.The table presents the cosmo- A great number of models have been proposed logical parameters obtained in one of the recent stud- to describe the explosion of SN Ia characterized by ies [28] on SN Ia.The authors of [28] show that the various masses, various regimes of burning (det- d (z) observed distribution of L can be explained only onation, deflagration, and various combinations of by assuming a nonzero vacuum energy or a rather these mechanisms), various energies of the explosion, artificial systematic effect: the emergence of dust in and various velocities of matter expansion.In these a recent era. theoretical models, chemical elements originate from It should be noted that, in all of the studies burning in markedly different proportions, their dis- devoted to high-z supernovae, use was made of tributions over the star also being different.This leads relations of the Pskovskiı–˘ Phillips type (maximum to different theoretical light curves.By comparing the luminosity–rate of the decrease in flux relation), resulting light curves with their observed counter- which were obtained from an analysis of close objects. parts, one can find out which explosion models are But even for close SN Ia, deviations from such depen- more realistic. dences for individual objects cannot be explained by By considering the possibility of employing SN Ia the errors of the observations exclusively. in cosmology, it was concluded in [34] that, at the From the theoretical point of view, the flux de- present time, the statistics of distant supernovae are creases more slowly with increasing maximum lumi- insufficient for drawing definitive conclusions on the nosity because both these quantities are controlled geometry of the Universe.Terrestrial experiments re- primarily by the amount of 56Ni formed in the ex- vealed that the regime of burning in an explosion can- plosion.The maximum of the luminosity of SN I a not always be predicted in advance.For supernovae, is determined by the amount of 56Ni since the light the situation is similar: it is quite feasible that the curve is formed predominantly by the contribution distinction between initial conditions only changes PHYSICS OF ATOMIC NUCLEI Vol.68 No.5 2005 822 BLINNIKOV the probabilities of various scenarios of burning, but is responsible for a supernova outburst.Even detailed that this does not pinpoint a specific scenario.Since computer calculations yield contradictory results be- the regime of burning affects strongly the shape of cause of uncertainties in the equation of state for the light curve, the rate of the decrease cannot be superdense matter, in the rates of weak-interaction reliably predicted if only initial conditions are known. reactions, and in the fundamental properties of the The probability of one value of the rate of the de- neutrinos (for example, their oscillations), as well as crease in flux or another—recall that this quantity because of difficulties in describing neutrino transport plays a significant role in determining cosmological and because of the emergence of convection. parameters—can be established only upon collecting If, in the main sequence, a star had a mass in the sufficiently vast observational statistics of SN Ia at range 8M M 20M , then, at the end of its evo- various values of z. lution, there arises a partly degenerate core of mass Upon an increase in statistics, it would be possible close to the Chandrasekhar limit.At the same time, 3 to reveal subtler effects—in particular, to answer the the density becomes so high (109−1010 g/cm )that, question of whether the dark-energy density is con- owing to a large chemical potential (Fermi energy) of stant and to establish the relevant equation of state. electrons, the neutronization reactions For example, there are even presently attempts [35] − at extracting, from observations of SN Ia,thez de- e +(A, Z) → (A, Z − 1) + νe (6) pendence of the coefficient w in the equation P = begin to proceed actively even at zero temperature. wEDE.These attempts are as yet premature, since they take no account of the fact that the properties of As a matter of fact, the temperature at these stages supernovae themselves or the regimes of burning in reaches of few tens of kiloelectronvolts, this enhanc- ing electron-capture reactions.Since electrons are them and their light curves can evolve with the age relativistic at such densities, the adiabatic exponent of the Universe.Moreover, there are also problems in the very procedure for extracting the dependence is close to the critical value of 4/3.With increas- Y w = w(z) from observations (see, for example, the ing density, the number of electrons per baryon, e, decreases, and the pressure begins, at some instant, article of Jonsson et al. [36], who criticize the re- 4/3 sults reported in [35]).In addition, it should be noted growing more slowly than in proportion to ρ ; this that no significant evolution of dark energy can be means that the gravitational force grows faster than revealed [37] by combining data on supernovae with the pressure force, with the result that a catastrophic data on cosmic microwave background radiation and compression (collapse) develops [16].At the initial x-ray radiation from galaxy clusters. mass of a star in the region M>20M , the mass and temperature are substantially higher, and a col- lapse begins owing to the photon-induced splitting 6.CORE-COLLAPSE SUPERNOVAE of nuclei.At a still higher mass, M 60M ,the − Let us now briefly touch upon the as-yet-unresolved production of e+e pairs begins contributing to the problem of explaining the explosion in the core col- reduction of the elasticity of matter and to the loss lapse and indicate what may be here in common with of stability.It should be borne in mind that the above the problem of gamma-ray bursts. mass values are very rough because the modern the- At the end of the lifetime of massive stars, a core ory takes very roughly into account a number of im- collapse must develop upon the depletion of the nu- portant phenomena, including a continuous loss of clear fuel. star mass, the rotation of stars, and the fact that they As far back as the 1930s, Baade and Zwicky [38] form binary systems. put forth the idea that there is a relationship between As the collapse reaches the dynamical stage, the supernovae outbursts and the formation of neutron central regions of the star are compressed, within a −1/2 stars; however, the quantitative theory of the ex- hydrodynamic time of thyd ∼ (GNρ) , which is a plosion mechanism in the collapse is still far from few tenths of a second, to nuclear-matter densities. completion.From a simple estimate of the gravita- Within so short a time, photon diffusion and the elec- E − ∼− 2 tional energy g = GN mdm/r GNM /R,it tron thermal conductivity are unable to remove heat 53 follows that an energy of about |Eg|∼10 erg is re- efficiently; therefore, temperature grows almost adia- leased in the formation of a neutron star of mass M ≈ batically at first.The majority of the nucleons remain 6 1M and radius R ≈ 10 cm.However, this energy bound in nuclei up to densities at which nuclei begin is released predominantly in the form of neutrinos touching one another.Only at such densities does rather than in the form of cosmic rays or photons, the elasticity of matter increase sharply, and there can as was hypothesized by Baade and Zwicky.It is not occur the termination of the collapse if the mass does easy to estimate the energy that is transferred to the not exceed some limit.The reverse motion (bounce) envelope around the emerging neutron star and which of matter generates a shock wave at a distance of PHYSICS OF ATOMIC NUCLEI Vol.68 No.5 2005 SUPERNOVAE AND PROPERTIES OF MATTER 823 about 50 km from the center, and this shock wave process νν¯ → e− + e+ was proposed by Berezinskii heats matter strongly.There then appear many free and Prilutskii [40] for explaining gamma-ray bursts nucleons (because of the disintegration of nuclei).As before the commencement of its applications in the a result, processes like physics of supernovae. − e + p → n + νe, (7) + 7.ASYMMETRY OF THE EXPLOSION e + n → p +¯νe, (8) Since spherically symmetric model calculations and the annihilation of electron–positron pairs into of collapsing presupernovae have not yet provided neutrionos, a successful pattern of explosions, it is necessary − e + e+ → ν +¯ν (9) to seek mechanisms that do not feature symmetry. These mechanisms may be operative in gamma-ray (this is one the most important processes at later bursts as well.If they produce a radiation flux into stages of the evolution of massive stars), come into a solid angle Ω, the requirements on the energy of play. gamma-ray bursts are relaxed by the factor Ω/(4π). An order of magnitude estimate of weak-interaction Observations have given many indications that su- ∼ 2 2 pernova explosions are asymmetric: cross sections is σ GFE ,whereE is the charac- teristic energy of a given process and G = G /(c)3, (i) Radiation from collapsing supernovae is polar- F F ized to a considerable extent.The degree of this polar- with GF being the Fermi constant.If one measures the particle energy in megaelectronvolts, it is con- ization grows with decreasing mass of the hydrogen venient to write G2 =5.3 × 10−44 cm2/MeV2.At envelope, reaching a maximum for SN Ib/c, which are F deprived of hydrogen.A spectacular example is pro- temperatures of a few tens of MeV,which are attained vided by a record polarization of the type Ic SN 1997X in the case of collapse, an estimate of the cross section (in all probability, such supernovae are deprived of not shows that neutrinos are vigorously produced, and only the hydrogen but also the helium envelope, and it seems that they can readily transfer energy to the this means that the ejected mass must be especially envelope.In the formation of a neutron star, neutrinos 53 small and that the asymmetry of the explosion must carry away more than 10 erg—that is, about 10% be the most pronounced in such objects). of the mass of the Sun! If one percent of this energy were captured by the envelope of a star, the problem of (ii) In many cases (maybe even always), the ex- the core-collapse mechanism of supernova explosion plosion of a collapsing supernova is followed by the would be solved. formation of a neutron star (known examples are pro- vided by pulsars in the Crab nebula and in the Vela σ From the estimate of , it can be seen that, at den- remnant).Many radio pulsars are observed to have sities above a value of about 1012 g/cm3, the neutrino velocities of up to 1000 km/s.A high momentum cor- mean free path is indeed short—it may become five to responding to such velocities is possibly associated six orders of magnitude smaller than the dimensions with an asymmetry of the respective explosion. of a hot neutron star.In deep layers, the mean free path is determined primarily by the reactions that are (iii) Observations of SN 1987A revealed that inverse to the processes in (7) and (8).In the vicinity (a) in the course of the explosion, radioactive mat- of the neutrinosphere and above it, coherent neu- ter was very fast transferred to outer layers (also, a trino scattering on surviving nuclei is more important. considerable mixing of 56Ni is required for explaining Because of short mean free paths, neutrinos diffuse the SN 1987A light curves); slowly, lose energy, and cannot eject its envelope. (b) the infrared lines of oxygen, iron, nickel, and For heating and ejecting outer layers of a collaps- hydrogen exhibit a considerable asymmetry of the ing stellar core, the process profiles; − ν +¯ν → e + e+ → γ, (c) light was polarized; which is inverse to that in (9), may also be of im- (d) photographs from the Hubble cosmic telescope portance.Neutrino –antineutrino pairs of all neutrino show a manifest asymmetry of ejecta, and the Chan- flavors must be copiously produced in the collapse. dra x-ray observatory recorded jets. The detailed neutrino spectra were first calculated (iv) In the vicinity of the young remnant of the by Nadyozhin [39].Unfortunately, the neutrinos are supernova ∼1680 Cassiopeia А (Cas A), there are overly soft for this process to be of importance for quickly moving lumps of matter rich in oxygen beyond supernovae.But if hard neutrinos escape into a vac- the main envelope of the remnant (maybe, there are uum, it can produce many photons there! In fact, the also two oppositely directed jets). PHYSICS OF ATOMIC NUCLEI Vol.68 No.5 2005 824 BLINNIKOV Three-dimensional images of the Cas A remnant The evolution of a neutron-star binary has not yet show that the flocculent distribution of calcium, sul- been calculated in detail.In just the same way as in fur, and oxygen is not symmetric in the direction to- some models of gamma-ray bursts, there can occur a ward an observer.No simple spherical envelopes can direct merger involving the formation of a black hole be seen.This and other remnants have a systematic and jets induced by the accretion disk.An alternative velocity of up to 900 km/s with respect to a local possibility was considered in [21].When the major circumstellar medium.All of these asymmetries are half-axis a of the orbit of a binary star becomes signif- expected to be associated with asymmetric flows of icantly smaller than its initial value, the less massive type Ib/c presupernovae leading to explosions that component(whoseradiusislarger)willfill its Roche produce remnants like Cas A—that is, stars of the lobe.This may lead to a signi ficant flow to a massive Wolf–Rayet type. satellite [21].A neutron star of mass satisfying the The latest x-ray observations of Cas A from the condition M 7.2. Exploding Neutron Stars in Binary Systems 7.3. Prospects of Verifying the Mechanisms For a neutron-star binary (NS + NS), an evolu- of the Explosion of Collapsing Supernovae tion scenario that leads to the explosion of one of the and Possible Relation to Gamma-Ray Bursts components and to a possible gamma-ray burst was proposed in [21].The evolution of such a binary sys- Per average galaxy, the outbursts of core-collapse tem is determined by gravitational radiation, which supernovae are severalfold more frequent than the leads to the merger of the components.A similar outbursts of thermonuclear supernovae; however, the process of the merger of white dwarfs may be one of understanding of the former has not yet reached the the possible ways toward the explosion of SN Ia.It is level of understanding of the latter.The reasons for then natural to address the question of how frequently this are the following: the physics of matter at densi- such events may occur in the Milky Way Galaxy. ties in excess of the nuclear-matter density is more This question was explored in [44], and it was shown uncertain, it is necessary to take into account all there that the frequency of the (NS + NS) mergers interactions (including relativistic gravity) at such of neutron-star binaries, RNS, is approximately equal densities, hydrodynamic flows are multidimensional to unity per about 3000 years if there is no recoil in in a collapse, and so on. the formation of neutron stars.This frequency falls We will briefly indicate the main difficulties for em- to unity per about 10 000 years at a recoil velocity of ploying the aforementioned promising mechanisms of 400 km/s. the explosion of core-collapse supernovae. PHYSICS OF ATOMIC NUCLEI Vol.68 No.5 2005 SUPERNOVAE AND PROPERTIES OF MATTER 825 (i) An explosion under the effect of neutrino radi- substantially different from the properties of the lat- ation in a collapse requires developing an elaborate ter; however, the constraints that were obtained from formalism that would describe neutrino transport in astrophysical data [55] should be respected.There is a three-dimensional convective flow for all neutrino yet another exotic possibility, that which is associated flavors.It is necessary to take into account rotation with the process involving the formation of quark and magnetic fields; possibly, it is also necessary to cores of neutron stars in a collapse.This process was allow for the emission of gravitational waves.For an proposed long ago in [56] (for relevant references and overview of the current state of this sector of super- new ideas, see [57]). novae theory, the interested reader is referred to [47], for example. In the near future, there will arise the possibility (ii) At the present time, the magnetorotational of recording the spectra of neutrinos and gravita- mechanism of a supernova outburst [41, 43] is the tional waves from star collapses.Together with ter- most successful.Here, the theory takes fully into restrial astronomical observations of supernovae and account rotation and magnetic fields, which form the gamma-ray bursts and their observations performed basis of the mechanism; however, other physics— beyond the atmosphere over all ranges of the electro- for example, neutrino transport—has so far been de- magnetic spectrum, this will contribute to solving the scribed quite roughly.An insu fficiently fast rotation of most challenging problems of modern astrophysics thecoresofthemajorityofstarsmaybeoneofthe and the physics of fundamental interactions. difficulties in this model [48]—there is still no clarity in this issue. (iii) Imshennik’s mechanism [45, 49] also depends 8.CONCLUSION on the value of the moment of momentum of central star regions before the collapse, and the emerging I have tried to show how supernovae aid funda- neutron star cannot disintegrate if this value is overly mental physics.On one hand, they demonstrate that small.Here, there has remained one more as-yet- the properties of a cosmological vacuum are non- unexplored issue.The merger of a neutron-star bi- trivial.On the other hand, problems associated with nary [21] must inevitably proceed under the effect of explaining the mechanism of the explosion of core- gravitational radiation (we know this from observa- collapse supernovae require performing a thorough tions of binary pulsars in the Milky Way Galaxy), but it is unclear at the present time whether the same analysis of the properties of matter at supernuclear radiation of gravitational waves (which efficiently car- densities and taking into account effects of all known ries away the moment of momentum) may in princi- interactions.Possibly, resort to new exotic particles is ple cause the disintegration of a hot neutron proto- also necessary here. star [50]. Per average galaxy, the frequency of gamma-ray bursts is two or three orders of magnitude less than ACKNOWLEDGMENTS the frequency of supernova outbursts; however, we cannot rule out the possibility that gamma-ray bursts The idea of writing this review article resulted accompany the collapse of massive stars if some spe- from my participation in the remarkable seminar cial, quite exotic, conditions hold simultaneously (see, organized by Yu.A. Simonov at the Institute of for example, [51, 52]).Since the mechanism of su- Theoretical and Experimental Physics for discussing pernova explosion in the collapse has yet to be clar- articles of topical interest from the literature.I am ified, the theory of gamma-ray burst has to overcome grateful to him and to M.I. Vysotsky, V.S.Imshennik, more significant difficulties.We cannot rule out the D.K. Nadyozhin, L.B. Okun, I.V. Panov, K.A. Post- possibility that the exoticism in a collapse does not nov, P.V. Sasorov, Е.I. Sorokina, V.P. Utrobin, concern the prevalent conditions exclusively: possi- N.N. Chugai, and other colleagues for numerous bly, unknown exotic particles emerge under special discussions and cooperation.The article was written conditions of a collapse, and it is these exotic particles predominantly in Garching (Germany) under support that generate bursts [19].If, for example, axion-like of the MPA (Max-Planck-Institut fur¨ Astrophysik) particles whose decay involves photon emission off guest program.I am indebted to W.Hillebrandt and the stellar core were formed in a collapse, then we R.A. Sunyaev for the kind hospitality extended to me would observe a supernova outburst in the presence there. of a massive envelope [53] or a powerful gamma- ray burst in the rare case of the absence of an en- This work was supported in part by the Russian velope [54].Such particles cannot be “conventional” Foundation for Basic Research (project nos.05-02- hypothetical axions, since their properties must be 17480, 04-02-16793). PHYSICS OF ATOMIC NUCLEI Vol.68 No.5 2005 826 BLINNIKOV REFERENCES 28. A.G.Riess, L.-G.Strolger, J.Tonry, et al.,Astro- 1.A.D.Dolgov, in 18th Les Rencontre de Physique phys.J. 607, 665 (2004). de la Valle d’Aosta. La Truile, Aosta Valley, Italy, 29. W.D.Arnett, Astrophys.Space Sci. 5, 180 (1969). 2004; hep-ph/0405089. 30. L.N.Ivanova, V.S.Imshennik, and V.M.Chechetkin, 2. A.Friedmann, Z.Phys. 10, 377 (1922). Astrophys.Space Sci. 31, 497 (1974). 3. A.Friedmann, Z.Phys. 12, 326 (1924). 31. K.Nomoto, D.Sugimoto, and S.Neo, Astrophys. Space Sci. 39, L37 (1976). 4. 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[Astron.Lett. 26, 730 (2000)]. Phys.B (Proc.Suppl.) 113, 268 (2002); Astrophys. 53. D.N.Schramm and J.R.Wilson, Astrophys.J. 260, J. 586, 1250 (2003). 868 (1982). 54. Z.Berezhiani and A.Drago, Phys.Lett.B 473, 281 (2000). Translated by A. Isaakyan PHYSICS OF ATOMIC NUCLEI Vol.68 No.5 2005 Physics of Atomic Nuclei, Vol. 68, No. 5, 2005, pp. 828–832. From Yadernaya Fizika, Vol. 68, No. 5, 2005, pp. 860–864. Original English Text Copyright c 2005 by Dolgov, Kawasaki. Cosmological Model with Dynamical Cancellation of Vacuum Energy and Dark Energy* A. D. Dolgov1), 2) and M. Kawasaki2) Received May 21, 2004 Abstract—We propose a model with a compensating scalar field whose back reaction to the cosmological curvature cancels possible vacuum energy density down to the terms of the order of the time-dependent critical energy density.Thus, the model simultaneously solves the mystery of the compensation of vacuum energy with an accuracy of 120 orders of magnitude and explains the existence of the observed dark energy. At an early stage, the suggested cosmological model might experience exponential expansion without an additional inflaton field.However, the solution found is unstable with respect to small perturbations. The stability can be ensured by introducing nonanalytical terms depending upon the absolute value of the curvature scalar R.Unfortunately, stable solutions do not describe realistic cosmology at the matter- dominated stage. c 2005 Pleiades Publishing, Inc. The problem of vacuum energy (or what is the indeed exists some dark energy in the universe which same, cosmological constant) seems to be the most induces accelerated expansion [3] and contributes serious one in contemporary fundamental physics. about 70% to the total energy density [4].However, Any reasonable estimate of the magnitude of this though several models with different concrete mech- energy gives a nonzero result which is 50–100 or- anisms of the compensation have been explored [5– ders of magnitude larger than the value allowed by 14], none of them could satisfactorily describe real- astronomy and just by our existence.The potential istic cosmology.In particular, the universe expansion importance of this problem was indicated in the 1930s regime was not related to the matter content of the by several people after formulation of quantum field universe.The list of papers quoted above is probably theory, but a more serious attitude was stimulated incomplete and more references and discussions can much later by papers [1].Several mechanisms have be found in reviews [15]. since been discussed to explain this mysterious and Inthiswork,wetakeascalarfield φ asacom- extremely precise cancellation, but none has yet been pensating agent, though higher spin fields are also satisfactory.A quite promising one is based on the possible [5, 10, 11].For simplicity, we con fine our idea [2] that there may exist a compensating field model to the action which depends only on the cur- whose back reaction to the curvature of spacetime vature scalar R, but in higher orders in curvature, the induced by vacuum energy would cancel the latter action may depend upon the Ricci, Rµν , and Riemann and change the expansion regime from the de Sitter (Riemann–Christoffel), Rµανβ, tensors.The general to the Friedmann one.A generic feature of scenarios action containing only first derivatives of φ and the curvature scalar R can be written as based on this idea is that vacuum energy is never √ compensated down to zero, but a noncompensated 4 1 2 2 A = d x g − (R +2Λ)+F1(R) (1) remnant is of the order of mPl/t ,wheremPl is the 2 Planck mass and t is the cosmological time.Such 1 µ µ models predicted that there should exist in the uni- + F2(R)DµφD φ + F3(R)DµφD R − U(φ, R) , verse a new form of matter with an energy density 2 ∼ 2 2 close to the critical one ρc mPl/t with possibly a where the metric has the signature (+, −, −, −); Dµ is quite unusual equation of state [2].In a sense, it is the covariant derivative in this metric; g = − det[gµν ]; good old “news” because it was found later that there Fj are some functions of the curvature scalar—in par- ∗ ticular, the Hilbert–Einstein action term (the first one This article was submitted by the authors in English. in this expression) is separated out and the function 1)INFN, Sezione di Ferrara, Italy; ICTP, Trieste, Italy; ITEP, Moscow, Russia. F1(R) contains possible terms nonlinear in curvature; 2)Research Center for the Early Universe, Graduate School of and the function U is a generalization of the φ po- Science, University of Tokyo, Tokyo, Japan. tential, which may also depend upon the curvature. 1063-7788/05/6805-0828$26.00 c 2005 Pleiades Publishing, Inc. COSMOLOGICAL MODEL WITH DYNAMICAL CANCELLATION 829 2 − We took the units such that mPl/8π =1.The cosmo- Taking the trace over (µ ν) in this equation, we obtain logical constant Λ is expressed through the vacuum ρ =Λm2 /8π =Λ 2 energy as vac Pl .Rede fining the field 1 2 → −R +3 (Dαφ) − 4[U(φ)+ρ ] (5) by φ K(R)φ, one can always bring its kinetic term R vac to the canonical form DµφDµφ/2 or annihilate the 2 2 term proportional to DµR. − 2 − 1 (Dαφ) µ 6D 2C1R = Tµ . In what follows, we will not use the general form of R R the action but will take a simplified version with F = 3 If the usual matter is absent, i.e., T =0,then 0, U = U(φ), F =(1/R)2,andF = C R2.With µν 2 1 1 Eqs.(3) and (5) together with the relation this choice of F1, the terms proportional to C1 do not enter the trace equation for constant R [see below R = −6 2H2 + H˙ (6) Eq.(5)].In this form, the action is similar to that con- sidered in [14] with the difference that, in the quoted form a complete system which allows the solution 2n 3) 2 paper, F2 =(1/R) with n>3/2. According to H ∼ h/t and R ∼ 1/t .That is, instead of or after the results of [14], the vacuum energy could indeed be some de Sitter exponential regime of expansion, we compensated by the field φ, but the compensation is arrive at the Friedmann one.To see this, let us look at very slow, so that the expansion regime changes from the equation of motion of φ (3).This is the equation the exponential to an almost exponential one, a(t) ∼ of Newtonian dynamics with the force proportional to exp(βtκ),with2/3 ≤ κ<1.In this regime, the en- R2U and “liquid” friction term (3H − 2R/R˙ ).The ergy density of the usual matter is quickly inflated friction force would explode at R =0 if R˙ =0 or, away and we arrive at an empty universe not only R˙ R without vacuum energy but also devoid of the usual more generally, if vanished more slowly than . R matter.A more detailed discussion of the properties of This is the case if reaches zero in finite time.It implies that the field φ would stick at the stable point such a model is presented in [16].As we see in what 2 follows, a smaller n drastically changes the behavior of this equation at R U =0 (or, better to say, φ of possible solutions [17]. would asymptotically approach the stable point).So, in equilibrium, either U =0 or R =0.If one may The equation of motion for the field φ, as follows 2 2 neglect the time derivative terms in Eq.(5), then from the action (1), with F1 = C1R , F2 =(1/R) , R = −4[ρ + U(φ)] . (7) and F3 =0, can be written as vac 2 So at least when the motion is slow and with a proper µ 1 Dµ D φ + U (φ)=0. (2) initial value of φ, the system could evolve toward R = R 0 and hence to a vanishing effective vacuum energy (eff) In the cosmological Friedmann–Robertson–Walker ρvac = ρvac + U(φ0)=0,whereφ0 is the value of φ (eff) (FRW) background, this equation takes the form for which the condition of vanishing of ρvac is realized. ˙ One can check that the approach to the equilib- R 2 φ¨ +3Hφ˙ − 2 φ˙ + R U (φ)=0. (3) rium point asymptotically for t 1 and |φ − φ0| R |U (φ0)/U (φ0)| 1 can be described by the law The Einstein equations are modified as 2 2 φ − φ0 ≈ a/t ,R≈ r/t ,H≈ h/t, (8) 1 1 2 R − g R − C 4R R − g R2 − and the constant coefficients satisfy the equations µν 2 µν 1 µν µν R − − 2 (4) ≡ h(2h 1)(6h 1) 1 1 k(h) 2 = . (9) 1 1 2 4R (1 + 3h) 6 U (φ0) × D φD φ + (D φ)2 g + µν µ ν 2 R α µν R The other coefficients are a =18U h2(2h − 1)2/(1 + 2 3h) and r = −6h(2h − 1).We excluded the evident − gµν [U(φ)+ρvac] − 2 gµν D − DµDν solution a = r = h =0,whichcorrespondstoφ = φ0 1 2 (D φ)2 when the system stays at the equilibrium point. × 2C R − α = T , 1 R R µν The function k(h) on the right-hand side of Eq.(9) is positive if 0 PHYSICS OF ATOMIC NUCLEI Vol.68 No.5 2005 830 DOLGOV, KAWASAKI approximately 0.0243. So the solution in this interval could strongly change the behavior of the vacuum 2 exists if |U (φ0)| > 2.6. solution if ρm drops slower than 1/t ,i.e.,forh<2/3. There could also be another regime of approach If, at some stage, the universe were dominated to equilibrium which, at least asymptotically, agrees by relativistic matter and the vacuum energy were with equations of motion (3), (5), and (6).In this vanishingly small, then, according to Eqs.(3) and regime, the last term on the right-hand side of Eq.(5) (1), φ would be zero or a constant and the cosmology proportional to D2 becomes nonnegligible and the would be the usual Friedmann one.It is well known, 2 however, that, in the course of expansion and cooling solution takes the form φ˙ ≈−U (φ0)R t.In this case, − 2 ∼ down, several phase transitions could take place in both R and φ φ0 tend to zero faster than 1/t , R cosmological plasma.Typically, in the course of phase 2+σ − ∼ 2+2σ 1/t and φ φ0 1/t , and the Hubble pa- transition, vacuum energy changes, and if it were rameter decreases as 1/t1+σ with σ satisfying σ(σ + zero initially, it would become nonvanishing and, if 2 1) = 1/(6(U ) ).If such a regime were realized, the φ remained constant, the newly created ρvac would universe would reach a stationary state with a con- ultimately dominate the total energy density, because stant scale factor.We assume that such a pathologi- ρvac does not decrease in the course of expansion. cal situation does not exist. However, when ρvac becomes cosmologically essen- One can check that there are several branches tial, the compensation mechanism described above of the solution.For example, the determination of R should be switched on and the solution (8) would be approached.However, in this case, the initial condi- from Eq.(5) through φ˙ and (U + ρ ), in the case vac tions would probably demand h>1/2.Amorede- that the last term on the right-hand side of (5) may be tailed investigation is in progress. neglected, can be done by solution of a cubic algebraic equation which has three possible roots.The choice The simple solution (8) does not describe the ob- of a certain root is dictated by initial conditions.We served universe acceleration, but it may be because it have found numerically that there exists a solution is not precise, and a more accurate treatment could for which Ht → const ≈ 0.5 and R tends to zero reveal the accelerated behavior.The “optimistic” pic- somewhat faster than 1/t2.So the system approaches ture presented above is based on the properties of the expansion regime typical for relativistic matter.A aspecific power law solution (8) to the equation of detailed numerical study of different solutions will be motion.For such a solution, the highest derivative presented elsewhere. term in Eq.(5) is subdominant and can indeed be neglected.However, the neglect of the highest deriva- The cosmological expansion regime is determined tive could be justified if the corresponding solution by the value of the constant h according to a(t) ∼ is stable.Unfortunately, as we have checked both th.The evolution of the energy density of the usual numerically and analytically, this is not so and the matter is governed by the covariant conservation of account of D2 in Eq.(5) leads to an essential insta- its energy–momentum tensor: bility of the solution.We have found that the scalar curvature R changes sign, becoming positive, and the ρ˙m = −3H(ρm + pm), (10) Hubble parameter H tends to infinity in finite time where ρm and pm are, respectively, energy and pres- (see Figs.1 and 2). sure densities of matter.So, as is well known, the en- To avoid such an unpleasant cosmological sin- ∼ ergy density of relativistic matter decreases as ρrel gularity, we have suggested a toy model with the 4 ∼ 1/a , while that of nonrelativistic evolves as ρnr substitution [19] 1/a3.Thus, if initially the energy density of matter (Dφ)2 (Dφ)2 were subdominant with respect to the vacuum energy →− . (11) 2 | | and h<1/6, the solution presented above should R R R change its character because at a certain moment In this case, the solution drastically changes its be- the usual matter would start to dominate.Probably, havior.In particular, H does not explode to infinity the energy density of matter and not compensated but, instead, tends to radiation dominance value H ∼ remnant of vacuum energy would manifest tracking 1/(2t), while R quickly oscillates around zero with behavior as suggested in [18].One more comment gradually decreasing amplitude (see Fig.3).So the µ is worth making here.For relativistic matter, Tµ =0 cosmology looks similar to the radiation dominated and the solution found above should be valid in the one, though not exactly the same.In particular, an presence of such matter.However, even a small ad- addition of nonrelativistic matter does not essentially mixture of matter violating the condition of vanishing change the expansion regime.A more detailed dis- µ of Tµ , either by the existence of massive particles, cussion of the stability problem and the suggested even with m PHYSICS OF ATOMIC NUCLEI Vol.68 No.5 2005 COSMOLOGICAL MODEL WITH DYNAMICAL CANCELLATION 831 –1 U' = 2.5 10 U' = 2.5 (a) 100 |H| H 10–3 φ 10–2 –5 10 |R| R < 0 φ |R| H > 0 –4 H < 0 10–7 10 R < 0 0 1 2 0 10 10 10 t 10 (b) U' = 2.5 Fig. 1. Evolution of the scalar φ, Hubble H,andthe –1 curvature R for the potential U =2.5 and ρm =0.For 10 Ht initial conditions, we take φ(0) = 0.01, H(0) = 0.01, R(0) = −0.01, φ(0) = 10−5,andR(0) = 0. 10–2 U' = 2.5 | | U – ρ H 10–3 vac 100 10–4 R < 0 R < 0 |φ| 100 101 102 103 10–2 t | | H > 0 H < 0 R Fig. 3. (a) Evolution of the scalar φ, Hubble H,andthe curvature R for the modified kinetic term Eq.(11).We φ φ > 0 < 0 take U =2.5 and ρm =0.The initial values are φ(0) = 10–4 0.001, H(0) = 0.01, R(0) = −0.01, φ(0) = 10−4,and R (0) = 0.(b) Evolution of h = Ht and U − ρvac. 100 101 102 t sense, inflation is unnatural, demanding quite strong fine-tuning.On the other hand, since the regions with Fig. 2. Same as Fig.1 except for ρm(0) = 1. an unsuppressed unstable mode would remain cos- mologically (and maybe even microscopically) small, the anthropic principle might help.The exit from in- The model presented here may be compatible with flation could be triggered by the unstable mode and the inflationary scenario without an additional inflaton the matter might be created by the gravity itself.As field.If initially ρvac =0and the feedback (or, better was shown many years ago [20], particle produc- to say, “kill-back”)effect of φ were sufficiently slow tion by the Planck scale gravitational field could be (this could be achieved by initial conditions and the quite significant.Moreover, in the model suggested shape of U(φ)), there would be a period of superlumi- here, the particle production would be much more nous (inflationary) expansion.However, again the in- efficient because the curvature scalar oscillates with flationary solution suffers from instability.Moreover, the Planck frequency. the modification of the model according to Eq.(11) We do not intend here to present a detailed discus- does not enforce stability to the inflationary regime. sion of cosmology, including big bang nucleosynthe- It can be shown that the exponential expansion a ∼ sis, large scale structure formation, etc., in this short exp(HI t),withHI ≈ const, is a solution to the equa- paper.It should be a subject of a much longer work. tions of motion of the model if U is sufficiently small We only want to demonstrate that, in the considered but there is an unstable rising mode, δ ∼ exp(cHI t) scenario, there exists a solution (maybe unstable) for with c>1 (c ≈ 2).Thus, to realize su fficiently long which the vacuum energy is automatically canceled ∼ 2 2 inflation, this rising mode should be eliminated by out with an accuracy of the order of ρc mPl/t initial conditions with an exponential accuracy.In this and the usual matter could survive in the course of PHYSICS OF ATOMIC NUCLEI Vol.68 No.5 2005 832 DOLGOV, KAWASAKI cosmological evolution (it was a strong drawback of 8. S.M.Barr and D.Hochberg, Phys.Lett.B 211,49 previously considered models).Moreover, the non- (1988). compensated remnant of the vacuum energy could be 9. Y.Fujii and T.Nishioka, Phys.Rev.D 42, 361 (1990); an excellent candidate for the observed dark energy Phys.Lett.B 254, 347 (1991), and references therein. today.The model may encounter serious problems 10.A.D.Dolgov, Yukawa Int.Preprint YITP/K-924 with quantization, but we think that, at this stage, it is (1991) (unpublished). too early to go to quantum physics; even demonstra- 11. A.D.Dolgov, Phys.Rev.D 55, 5881 (1997). tion of the existence of a classical model with auto- 12. V.A.Rubakov, Phys.Rev.D 61, 061501 (2000). matic cancellation of vacuum energy and emergence 13. A.Hebecker and C.Wetterich, Phys.Rev.Lett. 85, of dark energy in the same frameworks would be an 3339 (2000). interesting result. 14. S.Mukohyama and L.Randall, hep-th/0306108. 15.S.Weinberg, Rev.Mod.Phys. 61, 1 (1989); A.D.Dolgov, in Proceedings of the XXIV Rencontre ACKNOWLEDGMENTS de Moriond; Series: Moriond Astrophysics Meet- ings, Ed.by J.Adouse and J.Tran Thanh Van (Les A.D. Dolgov is grateful to the Research Center for the Early Universe of the University of Tokyo for the Arcs, France, 1989), p.227; astro-ph/9708045; P.Bi- netruy, Int.J.Theor.Phys. 39, 1859 (2000); V.Sahni hospitality during the time when this work was done. and A.Starobinsky, Int.J.Mod.Phys.D 9, 373 (2000); A.D.Dolgov, hep-ph/0203245; N.Strau- REFERENCES mann, astro-ph/0203330; T.Padmanabhan, Phys. Rep. 380, 235 (2003); J.Yokoyama, gr-qc/0305068. 1.Y.B.Zeldovich, Pis’ma Zh. Eksp.Teor.Fiz.´ 6, 833 16.S.Mukohyama, hep-th/0306208. (1967) [JETP Lett. 6, 316 (1967)]; Usp.Fiz.Nauk 95, 209 (1968) [Sov.Phys.Usp. 11, 381 (1969)]. 17. A.D.Dolgov and M.Kawasaki, astro-ph/0307442. 2. A.D.Dolgov, in The Very Early Universe,Ed.by 18. 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Shevchenko** Institute of Theoretical and Experimental Physics, Bol’shaya Cheremushkinskaya ul. 25, Moscow, 117259 Russia Received July 15, 2004; in final form, November 1, 2004 Abstract—The structure of leading nonperturbative corrections to the static Coulomb potential at small distances in Abelian and non-Abelian theories is analyzed. Related problems of validity of the Dirac quantization condition for running charges in Abelian theory and significance of the quantity A2 are briefly discussed. c 2005 Pleiades Publishing, Inc. a − a 1. INTRODUCTION For non-Abelian charges (gTD; gTD) in the repre- sentation D of SU(N) interacting with soft nonper- Much attention has been given recently to the turbative fields, the result is given by [10, 11] question about next-to-leading terms for the static 2 potential in confining theory such as QCD at short c−1 = −α˜s = −CD(g /4π); (4) distances (see [1–6] and references therein). This a a c1 =0; c2 = γα˜s Gµν Gµν Tg; ..., problem is closely related to the structure of OPE in confining theories, despite the fact that, to define the where CD is an eigenvalue of quadratic Casimir op- a a potential, one has to go to the large-time limit and erator, g is strong interaction constant, Gµν Gµν therefore leave the region of applicability of the stan- is nonperturbative gluon condensate [8], and non- dard OPE [7]. In particular, the appearance of new perturbative correlation length Tg characterizes the terms in the so-called SVZ condensate expansion [8] falloff of the gauge-invariant two-point correlator of is discussed in [2]. Phenomenologically, the static the field-strength tensors normalized to the gluon potential between charge and anticharge is given at condensate at the origin (see review [12] and refer- small distances as the following expansion: ences therein). The eigenvalue of quadratic Casimir operator reads c−1 2 3 V (r)= + c0 + c1r + c2r + O(r ), (1) a a r CD = TrDTDTD; TrD1D =1. (5) where the coefficients ci may also depend on r (but In the expressions above, γ remain for some dimen- only logarithmically in four-dimensional renormal- sionless numerical factors, whose values are of no izable theories). We discuss some typical patterns relevance for our discussion. What attracts attention below.1) in all mentioned cases is the absence of a linear cor- rection proportional to c1. This is also true for nother The first example is given by the Abelian charges models [13] which we have not included in the list. (e; −e)ininfinite space beyond tree level: However, one cannot propose strong theoretical 2 e (r) arguments why c1 should vanish in the context of c− = − ; c = c = ...=0. (2) 1 4π 1 2 QCD or another theory with complicated vacuum structure. Moreover, there are at least two indepen- If the Abelian charges (e; −e) are put into a cavity of dent sets of lattice data indicating that c1 =0 [14, size L, one gets at the tree level [9] 15]. Different models explaining small-distance linear e2 correction have been proposed (see [1–6] and refer- c−1 = − ; c1 =0; (3) ences therein). They can be divided into three groups. 4π In the first group, there are models [2, 5, 6] which are 2 −(i+1) ≥ ci = γie L for i 2. based on different corrections to the perturbative two- point correlation function either in the form of gluon ∗This article was submitted by the author in English. a b propagator Aµ(x)Aν (y) or in the form of gauge- **e-mail: [email protected] 1) invariant path-dependent field strength correlator In principle, one could consider a situation with noninteger powers of distance r entering (1), but this case is beyond our Tr Gµν (x)Φ(x, y)Gρσ (y)Φ(y,x) . The second group analysis. is represented by the “dynamical bag” model [1], 1063-7788/05/6805-0833$26.00 c 2005 Pleiades Publishing, Inc. 834 SHEVCHENKO which essentially explores (3) but with L = L(r) After integration over the matter fields, the parti- instead of a constant. Finally, studies of the static tion function of effective low-energy theory is given by potential at small distances in Abelian confining e D theories [3] form the third group. We briefly comment Z[jµ]= Aµ (6) on the latter one at the end of the paper, while our at- tention is focused on the first and second groups. One × − 1 d −1 µ ν − exp d pAµ(p)[D ]ν (p)A ( p) of the main points of this paper is that, despite many 2 common points, it is possible to make a distinction − d e µ − on the lattice between models of the first and of the Seff[A]+i d pjµ(p)A ( p) , second group by studying the small-distance static potential for charges in higher representations of the where Seff[A] contains interaction terms for the field −1 µ gauge group. Aµ. The function [D ]ν (p) can be decomposed as − The paper consists of two relatively independent [D 1]µ(p)=δµd (p2)+pµp d (p2), (7) parts. In the first part (Sections 2 and 3), we discuss ν ν 0 ν 1 some elements of short-distance physics in Abelian where the function d1(p) is gauge-dependent if the theories. In particular, we briefly discuss the following theory in question is a gauge theory. In the latter question: does the Dirac quantization condition [16] case, the theory can be rewritten in the field-strength for bare charges eg =2π stay intact when loop effects formulation [20] as are included? There has long been discussion of this e D ˜ question in the literature starting from [17] (see re- Z[jµν ]= Fµν δ(∂ν Fµν ) (8) views [18, 19] and references therein). An affirmative answer is given and the physics behind it is explained. 1 × exp − ddpF µν (p)∆ρσ (p)F (−p) We will also point the reader’s attention in Section 3 8 µν ρσ to the fact that the local average of the square of the × − d e µν − photon field Aµ(x)Aµ(x) can be defined in a covari- exp Seff[F ]+i d pjµν (p)F ( p) , ant gauge in the context of theory with boundaries. ˜ 1 The second part of the paper (Sections 4 and 5) is where Fµν = 2 µναβ Fαβ and concentrated on the non-Abelian theory. We demon- ∆ρσ (p)=(δρδσ − δσδρ)∆ (p2) (9) strate the difference between dynamical bag models µν µ ν µ ν 0 ρ σ − σ ρ σ ρ − ρ σ 2 incorporating the Meissner effect and modified prop- +(pµp δν pµp δν + pνp δµ pνp δµ)∆1(p ). agator models in Section 4, which makes it possible to distinguish these scenarios. Section 5 presents our The relation 2 2 2 2 2 conclusions. d0(p )=p (∆0(p )+p ∆1(p )) (10) In the rest of the paper, unless explicitly stated provides the correlator’s matching between the two otherwise, we work in Euclidean space with the formulations. notation for the four-vectors k =(k1,k2,k3,k4) and ν µ The field-strength formulation (8) allows one to scalar product kp = kµp δν . The three-vectors are introduce monopoles into the theory, performing the denoted as k =(k1,k2,k3) and the Wick rotation shift corresponds to the replacement k → ik . As usual, 4 0 ν ˜ → ν ˜ − m = c =1. δ(∂ Fµν ) δ(∂ Fµν jµ ). (11) If one may neglect Seff[F ] (see discussion below), 2. ELEMENTS OF SHORT-DISTANCE straightforward integration gives the resulting mono- PHYSICS IN ABELIAN THEORIES pole partition function WITH MONOPOLES m e D Z[jµ ,jµν =0]= Bµ (12) Since we are interested in the physics of interac- tions at small distances, as a warm-up example, we 1 δµp2 − pµp × exp − ddpB (p) ν ν Bν(−p) consider here the theory of Abelian vector field Aµ µ 2 2 ∆0(p ) interacting with the massive electrically charged mat- ter field and external monopole currents. An integral d m µ − + i d pjµ (p)B ( p) . part of this section is of a review type, since most of the discussed results can be found in the literature. 2 2 2 We believe, however, that a consistent presentation of For the free theory, d0(p )=p , ∆0(p )=1,and 2 them can be of some use. ∆1(p )=0, which means that (6) coincides with (12) PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 5 2005 ON THE SHORT-DISTANCE POTENTIAL 835 up to the interchange of electric and magnetic cur- vertex in Seff, contributing to the monopole poten- rents (charges); in other words, pure photodynamics tial (14) at small distances, contains the factor gkek = is exactly self-dual. If we switch on the interactions, (2πn)k, while the analogous contribution to (13) is the inverse propagator gets renormalized and runs k k 2k 2 2 2 2 2 suppressed as e e = e 1. On the other hand, with p as d0(p )=p (1 + Π(p )),whereΠ(p ) is the k>4 correlators, describing the processes of the corresponding polarization operator. Let us as- multiphoton scattering, do not have 3) contributions sume for the time being that the dynamics keeps the logarithmically rising with p2. It means that, for large gauge invariance of the theory intact, which means, in 2 particular, that Π(0) = 0. When computing the static p and not too large a value of the product eg,onecan potential in d =3+1dimensions, one gets formally indeed take the one-loop result (15) as a correct first in the electric case approximation. With the product eg getting large, however, higher order terms in the non-Gaussian part dp 1 − 2 · Seff become more and more important and their ef- Vee¯(r)= e 3 2 exp(ip r), (13) (2π) d0(p ) fect might overcome the leading kinematical one-loop logarithm and spoil (16). However, this is not what while the corresponding expression in the magnetic happens. Since the effective Lagrangian is formed in case reads √ this case at distances l ∼ r/ eg, which are for eg dp d (p2) V (r)=−g2 0 exp(ip · r), (14) 1 smaller than the typical scale of the field change r mm¯ (2π)3 p4 (see, e.g., [21, 22]), the constant field approximation can be used. The Lagrangian in a constant magnetic where the charges e, g are physical renormalized (but | | field is given (we use Minkowski metric conventions not running) charges and r = r . here) at the first order by the following expression [23]: It is seen from (13), (14) that the leading correc- H2 α eH tions to the Coulomb potential have different signs in − − L = 1 log 2 , (17) the electric and magnetic cases: 2 3π m 2 | | 1 − 1 where 4πα = e and H = H . Since the pointlike 2 δVee¯(r)= 2 δVmm¯ (r) (15) e g magnetic charge is defined microscopically via diver- dp Π(p2) gence of H,divH(r)=gδ(r), one gets from (17) for = exp(ip · r). (2π)3 p2 the running magnetic charge with logarithmic accu- racy Sometimes, this is considered as a proof 2) of the α 1 g(r)=g 1 − log . (18) fact that Dirac quantization condition eg =2πn holds 3π (mr)2 also for running charges (see discussion and refer- ences in [19]) if it does for bare charges: On the other hand, pointlike electric charge is defined D(r)=eδ(r) D = e g = eg = e(r)g(r)=2πn, (16) via Maxwell’s equation div ,where 0 0 ∂L/∂E, which leads to the standard Uehling and where e ,e,ande(r) denote the bare, the renormal- Serber result [24] (again with logarithmic accuracy) 0 ized, and the running charge, respectively. The sit- α 1 uation is much more subtle, however. The problem e(r)=e 1+ log 2 . (19) is that, introducing monopoles by (11), one sets the 3π (mr) scale of magnetic fields by the large coupling g ∼ 1/e, This is precisely the same answer one can get m since jµ contains the factor g 1. As an imme- from (15) with the standard one-loop expression for diate result, all higher order irreducible correlators Π(p2). of magnetic fields interacting with the electrically At first glance, the coincidence between (15) and charged matter field described by Seff canplayarole. (18), (19) seems rather surprising, since the latter two One should distinguish three different kinematical expressions were obtained from the exact strong-field regions. For large distances rm 1,wherem is Lagrangian (17), which sums up an infinite number electron mass, the corrections to the static potential of graphs, while the former one is a trivial outcome are exponentially damped with r for both Vee¯(r) and of Gaussian integrations with a one-loop two-point Vmm¯ (r). Of main interest is the small-distance region function. The reason for this result is the remarkable rm 1. The leading term in the k-point irreducible correspondence between QED at small distances and 2) Actually, only at the next-to-leading order in electric cou- 3)As is well known, at k =4, the amplitude formally diverges pling e2, which is assumed to be a small parameter. Notice logarithmically, but this divergence is exactly cancelled in the that Π(p2) is O(e2). sum of all diagrams. PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 5 2005 836 SHEVCHENKO QED in strong fields, studied in the series of pa- large distances rm 1, the potential (21) has linear pers [22]. It follows directly from the minimal coupling asymptotics and describes confinement. principle for the gauge and charged matter fields: We are to address the question of applicability → − pµ Pµ = pµ eAµ, which gives a hint that large pµ of (21) at small distances. Obviously, expression (21) and large eAµ can describe one and the same physics. −1 breaks down at r mH since at such small dis- The above example shows a rather nontrivial manifes- tances the scalar field condensate gets excited and tation of this correspondence. It is worth noticing that virtual loops of scalar particles contribute to the po- no topological objects like Dirac strings have been larization of the vacuum. If one takes (20) as some involved in our analysis. low-energy effective theory, the ultraviolet cutoff im- If Π(˜ p2) ≡ p2Π(p2) does not vanish when p2 → 0, plicitly enters through higher dimensional operators. the theory leaves the Coulomb phase and one can- Moreover, if the Higgs boson in this effective theory is not introduce external monopoles simply by (11). A composite in terms of underlying microscopic theory well-known example of such a theory is given by the degrees of freedom, the vector bosons with energy Abelian Higgs model, where the electrically charged higher than the corresponding binding energy start field is condensed: to resolve the constituents. In either case, the poten- 1 µν 1 2 2 2 2 tial (21) gets modified. Let us consider the simplest L = F F + |D φ| + λ(|φ| − η ) . (20) 2 4 µν 2 µ possible modification, which is to replace m in (21), ˜ 2 Attempts to write down an exact expression for the (22) by vector-boson self-energy Π(p ) such that 2 confining potential in this theory encounter a serious Π(0)˜ = m . Such replacements are in line with (13), problem. The reason for that is a physical one: since (14) and can be justified4) in terms of perturbation the confining string is created between the particles theory in the electric coupling e. Needless to say, (external monopole–antimonopole pair), its quantum the high-momentum asymptotics of Π(˜ p2) will be dynamics should be properly taken into account. The different for different microscopic scenarios outlined confining interaction cannot therefore be described in above. terms of particle exchanges. Moreover, even in the The leading short-distance contribution to (21) is effective Abelian theory framework, there exists no an attractive Coulomb potential, while there are two consistent procedure to perform corresponding cal- kinds of subleading corrections: culations analytically in terms of strings at the mo- ment. The best one can do is to compute the Wilson × g2 V [r T ](r)=− + δV (1) (r) (23) loop for a particular choice of the confining string mm¯ 4πr mm¯ world-sheet geometry. The conventional choice is the (2) O 2 4 minimal surface S for the given contour C,theflat + δVmm¯ (r)+ (g e ). one if the contour is rectangular (as it is for the static The second correction term comes from the second potential). In the London limit, the Higgs boson is Yukawa-like term in (21) and formally reads much heavier than the vector boson, mH m = eη, ˜ 2 and such a defined potential reads in this limit (see, 1 (2) dp Π(p ) · δVmm¯ (r)= exp(ip r). (24) e.g., [25]) g2 (2π)3 p4 2 [S] 2 1 The low-p divergence is artificial and can be easily Vmm¯ (r)=g lim (21) T →∞ T regularized. The term with the double integral in (21) 4 4 1 produces the following correction: × d x d y Σµν (x)∆m(x − y)Σµν (y) 4 3 ˜ 2 2 1 (1) d k Π k /r dp exp(ip · r) δVmm¯ (r)=r (25) − g2 , g2 (2π)3 k2 3 2 2 (2π) p + m 2 × 4sin (k3/2) where the surface current Σ (x) and the kernel 2 , µν (k3) ∆m(x − y) are given by the expressions whereforconvenienceweputtheWilsoncontourin (4) Σµν (x)= dσµν (ξ)δ (x(ξ) − x); (22) the (3, 4) plane. It is seen that the dependence of the r S correction (25) on is completely determined by the ˜ 2 2 d4p m2 behavior of Π(p ) asafunctionofp . In particular, if ∆ (x)= exp(ipx). ˜ 2 m (2π)4 p2 + m2 Π(p ) is such that the r dependence of the integral is weak enough, the correction to the potential will We have indicated by superscript [S] that the potential depends on the profile of the string chosen in (22). At 4)Of course, Π(0)˜ = 0 in the perturbative QED context. PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 5 2005 ON THE SHORT-DISTANCE POTENTIAL 837 be approximately linear. If Π(˜ p2) ≡ m2,theYukawa- effective action depending on this parameter does not like correction is linear in r and negative: δV (2) (r)= depend on ξ. By way of example, let us study pho- mm¯ todynamics with the Casimir-type static boundary −(g2m2r)/(8π) (we have omitted the irrelevant con- conditions (see review [29] and references therein): stant term). This is the only correction (up to the change g → e) one would get for the static electric e D Z[jµ]= Aµδ[BC] (27) charges without taking into account the string dy- namics. The “confining” correction (25) is positive 1 × exp − ddx F (x)F µν (x) in this case, but logarithmically divergent in the ul- 4 µν traviolet region. This divergence will be cured if the ˜ 2 1 2 e µ correct form of Π(p ) in the high-momentum regime + (∂µAν (x)) + ijµ(x)A (x) , is taken. Presumably, the relative minus sign between 2ξ (1) (2) δVmm¯ (r) and δVmm¯ (r) persists in this case as well. where the boundary conditions read It is worth putting an emphasis on the fact that the δ[BC]=δ(nµF˜µν (x1,x2,x3 = a1,x4)) physics behind δV (1) (r) and δV (2) (r) is very differ- mm¯ mm¯ × δ(nµF˜µν (x1,x2,x3 = a2,x4)). ent. In a sense, the former one describes the process of string tension formation, which starts essentially at Here, nµ =(0, 0, 1, 0) isaunitvectorinthex3 di- some ultraviolet scale in this model and goes all the rection and a1, a2 mark the x3 positions of parallel way up to the distance scale given by m−1.Sincethe infinitely thin ideally conducting plates. The distance corresponding formation speed is only logarithmic, between plates is given by a = |a2 − a1|. one can indeed have an approximately linear potential The gauge field propagator can be readily obtained at small distances—some kind of “ultraviolet confine- from (27). It consists of two parts: ment,” a phenomenon which apparently goes beyond (0) − the conventional OPE. Aµ(x)Aν (y) = Dµν (x, y)=Dµν (x y; ξ) + D¯µν (x, y; a1,a2), 3. A COMMENT ON THE QUANTITY A2 (0) µ where the term Dµν (x − y; ξ) is the standard gauge- The linear correction to the potential implies the dependent tree-level photon propagator, while the ¯ existence of a local d =2 parameter in the theory, term Dµν (x, y; a1,a2) encodes the information about proportional to c1. It cannot be a local polynomial the boundaries. The exact form of the latter was (0) function of original fields of the theory since the only first obtained in [30]. The function Dµν (x − y; ξ) is candidate, the average of vector potential squared translation-invariant but gauge-dependent, while the 2 TrAµ(x) , is not gauge-invariant. Moreover, there function D¯µν (x, y; a1,a2) depends on x3 and y3 sep- is a gauge defined by the infinitesimal transforma- arately, but is ξ-independent (not to be confused with µ tion U(x)=1+igAµ(x0)(x − x0) + ... such that the gauge-invariance!). Moreover, it has a finite limit U a A (x0)=0; i.e., one always has locally when x approaches y. Therefore, one can address the µ µ issue of the Aµ(x)A (x) condensate in the Casimir U a 2 vacuum exactly in the same way as one computes the min Aµ(x) =0, (26) U∈G energy density in this problem, namely, subtracting the boundary-independent part: where G is the gauge group, for Abelian or non- Abelian theory. In the Abelian case, however, the 2 def µ A (x) = lim Aµ(x)A (y) (28) 2 y→x minimum of the integral of Aµ(x) over the whole space can be rewritten in terms of gauge-invariant − A (x)Aµ(y) (0) = D¯ µ(x, x; a ,a ). quantities [26]. The physical relevance of such a non- µ µ 1 2 local object for the non-Abelian case is under de- It is convenient to introduce the notation z = a1 + bate [27, 28]. a2 − 2x3. Then, using the exact expression for We would like to point the reader’s attention to the D¯µν (x, y; a1,a2),onegets fact that, though it seems to be impossible to assign πz any gauge-invariant meaning to the local quantity 2 1 3 2 A (z) = 2 1+ tan . (29) A2 , one can have in some cases a ξ-independent 12a 2 2a definition of A2 ,whereξ is the covariant gauge- Such a defined local two-dimensional “condensate” fixing parameter. It happens when one studies the is ξ-independent, strictly positive, and diverges on the theories which contain some external parameter of boundaries. In a completely analogous way, one can the dimension of mass/length and if the part of the define the A2 condensate at finite temperature. PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 5 2005 838 SHEVCHENKO 11 2 α2 In the context of the Casimir problem, one is usu- = − N − N s + O(α3). ally interested in studying the changes in physical 3 3 f 2π s quantities like components of the energy–momentum tensor with respect to their vacuum values. The above In principle, the average (30) depends on the ac- example demonstrates that a “nonphysical” quantity tual state of the system; for example, one can study 2 its density or temperature dependence. In terms of like A is also nontrivially modified. It is worth not- 2 2 ing that (29) is not related to any linear correction to invariant functions D(z ) and D1(z ), the vacuum the potential, since the potential for a dipole between gluon condensate is given as [10] 2 the mirrors is of the form (3) with L = a. g TrGµν (0)Gµν (0) = 24(D(0) + D1(0)), (32) whereonlythefunctionD(z2) is responsible for con- 4. DYNAMICAL BAG VERSUS MODIFIED finement. In particular, the deconfinement transition PROPAGATOR is characterized by the condition D(z2) ≡ 0,with the smooth behavior of D1(z) over the phase transi- One of the main physical assumptions of the SVZ tion [33]. framework can be formulated as follows: the nonper- One can raise a question about the status of (30) turbative vacuum of the theory is not perturbed by the in a color field of an external source. Owing to SU(3) external sources. In the case of large virtualities, it is and O(4) invariance, such an average can be defined sometimes justified using the language of condensed as matter physics: since the time scale of the hard pro- α G (r)= s Ga (r)Ga (r) (33) cess we wish to study is much smaller than typical 2 π µν µν relaxation times characterizing such a medium as 2 (G (r)G (r)) · exp i A dz a nonperturbative QCD vacuum, it is reasonable to g Tr µν µν Tr P C ρ ρ = 2 , suggest that the vacuum state remains unchanged. 2π Tr P exp i C Aρdzρ It is of interest to relax this assumption and to where C stays for the (closed) world line of the source. study the corresponding physics. An example of the If the point r is far enough from C, the average situation when one has to do it is provided by the reaches the vacuum value: G2(r) → G2. Correspond- monopole–antimonopole pair in the superconductor. ingly, one defines (r)= (E(r)) via (30). To the best It is well known that the condensate must be broken of the author’s knowledge, no systematic lattice anal- along some line connecting the particles, no matter ysis of the behavior of (r) in a strong external field how small their charge is (see review [19] and ref- has been performed. This problem is of direct rele- erences therein). The reason for that, eventually, is vance for our discussion. Namely, let us put a single the nonperturbative nature of the interaction between static charge in the nonperturbative QCD vacuum. magnetic and electric particles. Due to the confinement property, there should be First, we rederive the results of the dynamical bag, an anticharge somewhere in the space to end the or dynamical cavity, model [1] in a more quantitative confining string, and therefore the field distribution way. After that, we will come to the modified propa- will not be spherically symmetric. Suppose, however, gator model [2]. Since we consider gluodynamics in that we look at the gluon fields in a thin spherical thissection,weswitchtodualterminology,sothat layer between r and r +∆r,wherer is smaller than particles which are confined (quarks) carry electric the typical nonperturbative nonlocality scale given by and not magnetic charge. The microscopic descrip- Tg (it is worth mentioning that the same quantity tion of confinement as the dual Meissner effect via Tg defines the width of the confining string). Then monopole condensation refers to the Abelian projec- it is natural to assume that the Faraday flux lines of tion procedure [31]. We adopt the physical picture of thechargeareaffected only weakly by the opposite confinement as the dual Meissner effect but will make charge sitting on the other end of the string. On the no use of the Abelian Higgs model language in this other hand, they are affected by the nonperturbative section. As is well known, the energy density of non- vacuum and act back on it. Correspondingly, we take perturbative fields is given by the energy–momentum the energy of the gluon fields in the volume V as a sum tensor trace anomaly [32]: of perturbative and nonperturbative parts 1 1 β(αs) a a E = E + E = drEa(r) · Ea(r)+ dr (r), = θµµ = Gµν (0)Gµν (0) , (30) p np 4 16αs 2 V V where (34) 2 a dαs(µ ) where E (r) is the perturbative electric field of the β(α )=µ2 (31) s dµ2 charge. In principle, the coordinate dependence of PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 5 2005 ON THE SHORT-DISTANCE POTENTIAL 839 a E (r) and (r)= (E(r)) on r can be very complex, that 2 coincides with the vacuum energy density in resembling the formation of an intermediate state in our approach) the vicinity of the charge, as happens in ordinary su- − − − − × −3 4 perconductors. It can be energetically advantageous 1 2 = κ =(1 κ)(4 8) 10 GeV . to squeeze slightly perturbative flux lines in one part (38) 5) of the volume and to rarefy them in the rest of it Thevalueofthepositivefactorκ = 1/ 2 < 1 is un- [this increases the first term in (34)], but compensate known and enters as a free parameter. Taking α˜s = the excess of energy by the nonperturbative energy 0.47 (corresponding to the charges in the fundamen- gain from the rest of the volume, which can be possi- tal representation and αs(Mτ )), we finally get ble if (r) decreases towards its vacuum value when − −1/4 − Ea(r) decreases in this part of the volume. This is rc =(1 κ) (0.2 0.3) fm. (39) the essence of the Meissner effect. Let us simplify the Unless κ is unnaturally close to unity, this rather matter further by approximating (34) as small value of r is compatible with the discussed c 2 2 2 picture—were it much larger than T , the analysis E 1 g 4πr g ∆ = CD 2 (35) would be meaningless. 2 4πr S1(r) Let us, following [1], consider now a small color × S (r)∆r + S (r)∆r + S (r)∆r + O(∆r2). 1 1 1 2 2 dipole of the size R. The expression for the energy (34) We have taken (r) as constants 1 and 2 in the parts stays intact, but the physics of critical distance rc of the volume S1∆r and S2∆r, respectively. Notice is different. Since there is no flux of a perturbative 2 that S1(r)+S2(r)=4πr . Expression (35) assumes field through the surface of the volume with the dipole that the parts of the space occupied by the nonper- inside, it is advantageous to confine flux lines inside turbative fields with the (vacuum) energy density 2 some finite volume. Suppose that we have chosen the completely expel the perturbative field flux lines into volume of some fixed size L R around the dipole the regions where the energy density of nonperturba- such that vacuum energy density 2 = outside is tive fields is given by 1. The rescaling of the electric reduced to 1 > 2 inside it. The perturbative electric perturbative field of the point charge reflects the flux field felt on the boundary of this volume is proportional conservation. to R/L3. It is important that it go to zero when R 2 Since S1(r) can be never greater than 4πr ,there decreases. It means that, if we make R small enough, exists a critical radius [corresponding to the station- the total energy can also be lowered by appropriate ary point of (35)] decreasing of L, i.e., L = L(R). Indeed, when we go from L to L − ∆L, the perturbative energy increase 4 α˜s rc = − . (36) due to the rearrangement of the perturbative field is 8π( 1 2) given by When r reaches rc from above, the region of unper- 1 g 2 turbed vacuum S shrinks to zero. Physically, the ∆Ep = CD (40) 2 2 4π model describes the formation of a bag of the ra- L dius rc, where perturbative fields are strong enough R2 (R · r)2 to change the nonperturbative vacuum state signifi- × dr +3 r6 r8 cantly. We are not addressing an interesting question L−∆L at the moment of whether there is a local deconfine- 2 ment transition (in any sense of the word) in this R 2 =˜αs ∆L + O(∆L ), volume (which is natural to think of as being based on L4 an Abelian Meissner effect analogy). Let us estimate while the nonperturbative energy gain is rc numerically. We use different sets of data [34] and take for the gluon condensate ∆E =4πL2( − )∆L + O(∆L2). (41) np 1 2 αs a a 4 G2 = G G =0.014−0.026 GeV . (37) This defines the stationary point π µν µν 2 1/6 Hence, for the nonperturbative vacuum energy in one α˜sR Lc = − . (42) loop choosing N =3, Nf =2, we obtain (remember 4π( 1 2) 5)Since we are working at small distances, the total pertur- We can now estimate the nonperturbative correction bative flux (i.e., the number of the flux lines) is conserved. to the energy of the dipole as More accurately, because of asymptotic freedom it is not, but 2 its violation due to asymptotic freedom is a weak logarithmic E ≈ R − 4π 3 δ (R) α˜s 3 +( 1 2) Lc (43) effect. 3Lc 3 PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 5 2005 840 SHEVCHENKO 4 onegetsintermsof(45)λ2 = −(1 − κ)1/2(0.4− = πα˜s( 1 − 2)R. 3 0.6) GeV2, which is surprisingly close to the value This correction is linear in R, as one can alternatively found in [2]. The important difference, however, is a see, inserting (42) into (3). Numerically, we get different CD dependence. The remarkable property of (43) is its square-root dependence on the eigen- 4 1/2 2 πα˜s( 1 − 2)=(1− κ) (0.10−0.14) GeV . value of Casimir operator C . This result is known in 3 D (44) the context of the MIT bag model [36], where it takes place for the slope of the confining linear potential. The discussed picture reminds one of the old MIT As such, the square-root law was definitely ruled bag model [35]. This similarity is formal, in some out by recent precise studies of the static potential sense. While in the MIT bag model the bag has the at large distances in different representations of the typical hadron size, the dynamical cavity of the size gauge group [37, 38]. Instead, the Casimir scaling Lc(R) given by (42) is an object which has physical phenomenon (proportionality of VD(r) to CD)was meaning only at very small distances. Since Lc ∼ confirmed (see [39] for a review). However, there 1/3 R , at small enough R, we are always in the Tg exist no calculations of static potential for charges Lc R regime, where all the picture is meaningful. in higher representations at small distances. Such a With R and Lc rising and reaching Tg,theconfining lattice simulation will confront in a nontrivial way (43) string formation process starts and we leave the do- and (46). main of qualitative applicability of the model. From√ the general point of view, the Casimir scaling We can now come to the modified propagator and CD law for the leading small-distance correc- models [2, 5, 6]. Actually, their analysis is simpler, and tion would correspond to rather different physical pic- we take as an example the model of [2]. It is suggested tures. Both cases physically describe perturbative– that one modify the one-gluon exchange propagator nonperturbative interference and require modification by adding a “tachyon”-mass term to the gluon, i.e. of the standard condensate ideology in QCD. How- (in Minkowski metric and Feynman gauge), ever, in the latter case, this modification must be by far 1 1 λ2 more radical. Roughly speaking, the former case cor- Aa (k)Ab (−k) = η δab → η δab + . µ ν µν k2 µν k2 k4 responds to a new kind of nonperturbative corrections (45) to the perturbative propagator, which are “harder” than those studied before, while the latter case indi- Notice that (45) is a large-momentum expansion, cates that the nonperturbative vacuum structure itself so one actually never gets closer to the artificial is strongly affected by the perturbative field, when it tachyon pole. The analysis of the existing phe- reaches some critical value. λ2 ≈− . 2 nomenology leads to an estimate 0 5 GeV for To be self-contained, let us briefly mention the the “tachyon” mass [2]. However, the physical origin analysis of the classical Abelian Higgs model on of the λ2 term in the context of QCD has not been 2 2 the shortest distances performed in [3]. The small- clarified. Alternative scenarios suggest a µ /z con- distance correction to the potential was found nu- 6) 2 tribution to the function D1(z ) [5], strong coupling merically and fitted by a linear function with a good constant freezing [6], and infrared renormalons [1]. accuracy. This behavior was linked to the Dirac veto, It is easy to see that all these proposals lead to the i.e., the condition for the charged condensate φ to Casimir scaling law for the corresponding correction; vanish along some line (the Dirac string) between the for example, with expression (45), one gets for the dual charges. The problem mentioned in Section 2 static potential between charges in the representation still persists in this case, because the position of the D of the gauge group Dirac string can be chosen in an arbitrary way and 1 λ2r the answer for the short-distance potential depends V (r)+const = −α˜ + + O(r2), (46) D s r 2 on this choice. Moreover, it is not quite clear what the Dirac string of effective Abelian theory corresponds 2 where α˜s = CDαs = CD(g /4π); i.e., VD(r) demon- to in terms of the original non-Abelian theory. In strates Casimir scaling. the dynamical cavity model, the linear behavior (43) The magnitudes of corrections are close to each also results from the Meissner effect but without other in both models; for example, taking value (44), any special role of singularities like Dirac strings. Roughly speaking, the effect in [3] comes from the 6)It can be shown that the 1/z2 term in the function D(z2) infinitely thin line between the charges, while in would produce ultraviolet divergence on the world sheet of the discussed case all the volume of the size Lc 2 the confining string, while such a term in D1(z ) is safe. contributes to it. PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 5 2005 ON THE SHORT-DISTANCE POTENTIAL 841 5. CONCLUSION 14. G. Burgio, F.Di Renzo, G. Marchesini, and E. Onofri, There are a few models in the literature that aim Phys. Lett. B 422, 219 (1998). to reproduce the linear dependence on distance of the 15. G. S. Bali, Phys. Lett. B 460, 170 (1999). leading nonperturbative correction to the static po- 16. P.A. M. Dirac, Phys. Rev. 74, 817 (1948). tential at small distances. Comparing different mod- 17. J. Schwinger, Phys. Rev. 144, 1087 (1966); 173, 1536 els, we call the reader’s attention to the fact that a (1968). possible way to test them is to check the dependence of this correction on the eigenvalue of the quadratic 18. M. Blagojevic and P.Senjanovic, Phys. Rep. 157, 233 Casimir operator for higher representations of the (1988). charges. Such a computation can shed new light on 19. M. N. Chernodub, F. V. Gubarev, M. I. Polikarpov, strong interaction physics and the problem of con- and V.I. Zakharov, Yad. Fiz. 64, 615 (2001) [Phys. At. finement. Nucl. 64, 561 (2001)]. 20. M. Halpern, Phys. Rev. D 19, 517 (1979). ACKNOWLEDGMENTS 21. G.J.GoebelandM.T.Thomas,Phys.Rev.D30, 823 It is a pleasure for me to express my deep respect (1984). to Prof. Yurii Antonovich Simonov, the physicist and ´ the man whose quest for the truth in science and life 22. V. I. Ritus, Zh. Eksp. Teor. Fiz. 69, 1517 (1975) [Sov. constantly inspires people around him. Phys. 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D 62, 034509 (2000); Nucl. H. Leutwyler, Phys. Lett. B 98B, 447 (1981); Phys. B (Proc. Suppl.) 73, 587 (1999). J.S.BellandR.A.Bertlmann,Nucl.Phys.B 39. V. I. Shevchenko and Yu. A. Simonov, Int. J. Mod. 187, 285 (1981); I. I. Balitsky, Nucl. Phys. B 254, Phys. A 18, 127 (2003). 166 (1985). PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 5 2005 Physics of Atomic Nuclei, Vol. 68, No. 5, 2005, pp. 842–860. From Yadernaya Fizika, Vol. 68, No. 5, 2005, pp. 874–893. Original English Text Copyright c 2005 by Savkli, Gross, Tjon. Nonperturbative Dynamics of Scalar Field Theories through the Feynman–Schwinger Representation* C. Savkli1),F.Gross2),andJ.Tjon2), 3) Received May 13, 2004; in final form, October 29, 2004 Abstract—We present a summary of results obtained for scalar field theories usingthe Feynman – Schwinger (FSR) approach. Specifically, scalar QED and χ2φ theories are considered. The motivation behind the applications discussed in this paper is to use the FSR method as a rigorous tool for testing the quality of commonly used approximations in field theory. Exact calculations in a quenched theory are presented for one-, two-, and three-body bound states. Results obtained indicate that some of the commonly used approximations, such as Bethe–Salpeter ladder summation for bound states and the rainbow summation for one-body problems, produce significantly different results from those obtained from the FSR approach. We find that more accurate results can be obtained usingother, simpler, approximation schemes. c 2005 Pleiades Publishing, Inc. 1. INTRODUCTION Therefore, any reaction that involves quarks will nec- essarily involve bound states in the initial and/or final In the study of hadronic physics, one has to face states. This implies that, even at high-momentum the problem of determiningthe quantum-dynamical transfers, where QCD is perturbative, formation of properties of physical systems in which the inter- quarks into a bound state necessitates a nonpertur- action between the constituents is of a nonpertur- bative treatment. Therefore, it is essential to develop bative nature. In particular, such systems support new methods for doingnonperturbative calculations bound states and clearly nonperturbative methods in field theory. are needed to describe their properties. Assuming that such systems can be described by a field the- The plan of this article is as follows. In the fol- ory, one has to rely on some approximation scheme. lowingsection, a brief review of the particle trajectory One common approximation is known as perturba- method in field theory will be given. The Feynman– tion theory. Perturbation theory involves makingan Schwinger representation will be introduced through expansion in the couplingstrengthof the interaction. applications to scalar fields. In particular, the em- The Green’s function in field theory can be expanded phasis will be on comparingvarious nonperturbative in powers of the couplingstrength. In order to be results obtained by different methods. It will be shown able to obtain a bound-state result, one must sum the with examples that nonperturbative calculations are interactions to all orders. Most practical calculations interestingand exact nonperturbative results could to date have been done within the Bethe–Salpeter significantly differ from those obtained by approxi- framework, where the resultingkernel is perturba- mate nonperturbative methods. tively truncated. In this paper, we will discuss another Results for scalar quantum electrodynamics are method, one which is based on the path integral for- discussed in Section 3. In particular, a cancellation mulation of Feynman and Schwinger [1, 2]. It was is found between the vertex corrections and self- initiated by Simonov and collaborators [3–9] in their energy contributions, so that the exact result can be study of quantum chromodynamics (QCD). described by essentially the sum of only the gen- With the discovery of QCD, nonperturbative cal- eralized exchange ladders. In Section 4, the bound culations in field theory have become even more es- states are obtained for a χ2φ theory in a quenched sential. It is known that the buildingblocks of mat- approximation. Exact results are presented for the ter, quarks and gluons, only exist in bound states. bindingenergiesof two- and three-body bound states ∗ and compared with relativistic quasi-potential predic- This article was submitted by the authors in English. tions. Section 5 deals with the stability of the χ2φ 1)Lockheed Martin Space Operations, Greenbelt, USA. 2)Jefferson Lab, Newport News, USA. theory. It is argued that the quenched theory does not 3 3)Department of Physics, University of Maryland, College suffer from the well-known instability of a φ theory. Park, USA. The paper closes with some concludingremarks. 1063-7788/05/6805-0842$26.00 c 2005 Pleiades Publishing, Inc. NONPERTURBATIVE DYNAMICS OF SCALAR FIELD 843 2. FEYNMAN–SCHWINGER of exact nonperturbative calculations will be shown REPRESENTATION with explicit examples. Nonperturbative calculations can be divided into The basic idea behind the FSR approach is to two general categories: (i) integral equations and transform the field-theoretical path integral (1) into (ii) path integrals. Integral equations have been used a quantum-mechanical path integral over particle for a longtime to sum interactions to all orders with trajectories. When written in terms of trajectories, various approximations [10–13]. In general, a com- the exact results decompose into separate parts and plete solution of field theory to all orders can be pro- permit us to study the individual role and numerical vided by an infinite set of integral equations relating size of exchange, self-energy, and vertex corrections. vertices and propagators of the theory to each other. This,inturn,alowsustostudydifferent approxima- However, solvingan in finite set of equations is beyond tions to field theory and, in some cases, prove new our reach and usually integral equations are truncated results. by various assumptions about the interaction kernels To illustrate these ideas, we consider the appli- and vertices. The most commonly used integral equa- cation of the FSR technique to scalar QED. The tions are those that deal with few-body problems. Minkowski metric expression for the scalar QED La- The Bethe–Salpeter equations [10] are the starting grangian in Stueckelberg form is given by point of those investigations. Approximation schemes 2 2 1 2 1 2 2 have extensively been studied, where in addition to LSQED = −m χ − F + µ A (2) solutions of the Bethe–Salpeter equation [14–18] 4 2 ∗ µ µ 1 2 also three-dimensional reductions [19–24] have been +(∂µ − ieAµ)χ (∂ + ieA )χ − λ (∂ · A) , explored for the N-particle free particle Green’s func- 2 tion. In most calculations, the ladder approximation where Aµ is the gauge field of mass µ, χ is the has been used for the kernel of the resultingequa- charged field of mass m, F µν = ∂µAν − ∂νAµ is the tions. Issues of convergence of these schemes remain. 2 µ gauge field tensor, and, for example, A = AµA .The Therefore it is important to have exact solutions of presence of a mass term for the exchange field breaks field-theory models available to test these approx- gauge invariance and was introduced in order to avoid imations. One promisingway to reconstruct exact infrared singularities that arise when the theory is solutions of field theory is the path-integral method. applied in 0+1 dimensions. For dimensions larger Path integrals provide a systematic method for than n =2, the infrared singularity does not exist and summinginteractions to all orders. The Green’s func- therefore the limit µ → 0 can be safely taken to insure tion in field theory is given by the path-integral ex- gauge invariance. pression The path integral is to be performed in Euclidean 0|T [φ(x1)φ(x2) ···φ(xn)]|0 (1) metric. Therefore, we perform a Wick rotation: D ··· 4 4 4 [ φ]φ(x1)φ(x2) φ(xn)exp i d xL(x) exp i d xLM −→ exp − d xLE . (3) = . [Dφ]exp i d4xL(x) The Wick rotation for coordinates is obtained by ∂ x →−ix ,∂= → i∂ . (4) While path integrals provide a compact expression 0 0 0 ∂x0 0 for the exact nonperturbative result for propagators, evaluation of the path integral is a nontrivial task. The transformation of field A under Wick rotation is found by noting that, under a gauge transformation, In general, field-theoretical path integrals must be evaluated by numerical integration methods, such as Aµ → Aµ + ∂µΛ. (5) Monte Carlo integration. The best known numerical Then, under a Wick rotation, integration method is lattice gauge theory. Lattice → gauge theory involves a discretization of space–time A0 iA0, (6) and numerical integrations over field configurations and the Wick-rotated Lagrangian for SQED be- are carried out usingMonte Carlo techniques. A more comes efficient method of performingpath integrals in field L ∗ 2 − 2 − · theory has been proposed and it consists of explic- SQED = χ [m ∂ 2ieA ∂ (7) 2 2 itly integrating out the fields. It has been demon- − ie∂ · A + e A ]χ + LA. strated to be highly successful for the case of simple scalar interactions [6, 25–33]. This method is known The exchange field part of the Lagrangian is given by as the Feynman–Schwinger representation (FSR). 1 1 L ≡ A (µ2g − λ∂ ∂ )A + F 2 (8) Through applications of the FSR, the importance A 2 µ µν µ ν ν 4 PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 5 2005 844 SAVKLI et al. 1 2 The Green’s function (16) includes contributions = Aµ (µ − )gµν +(1− λ)∂µ∂ν Aν . 2 comingfrom all possible interactions. The determi- We employ the Feynman gauge λ =1, which yields nant in Eq. (16) accounts for all matter (χχ¯) loops. Settingthis determinant equal to unity (det S → 1, 1 2 LA = Aν(µ − )Aν . (9) referred to as the quenched approximation) eliminates 2 all contributions from these loops (illustrated in The two-body Green’s function for the transition from Fig. 1) and greatly simplifies the calculation. an initial state Φi(x, x¯) to final state Φf (y,y¯) is given Analytical calculation of the path integral over the by gauge field A in Eq. (16) seems difficult due to the A S(x, y) G(y,y¯|x, x¯) (10) nontrivial dependence in . In more compli- cated theories, such as QCD, integration of the gauge ∗ ∗ − = N Dχ Dχ DAΦ Φ e SE , field, as far as we know, cannot be done analytically. f i Therefore, in QCD, the only option is to do the gauge- field path integral by using a brute force method. where Here, one usually introduces a discrete space–time 4 lattice and integrates over the values of the field com- SE = d xLSQED, (11) ponents at each lattice site. However, for the simple and a gauge-invariant two-body state Φ is defined by scalar QED interaction under consideration, it is in ∗ fact possible to go further and eliminate the path Φ(x, x¯)=χ (x)U(x, x¯)χ(¯x). (12) integral over the field A. In order to be able to carry out the remaining path integral over the exchange field A, The gauge link U(x, y) which insures gauge invari- it is desirable to represent the interactingpropagator ance of bilinear product of fields is defined by in the form of an exponential. This can be achieved y by usinga Feynman representation for the interacting U(x, y) ≡ exp −ie dzA(z) . (13) propagator. The first step involves the exponentiation of the denominator in Eq. (17): x ∞ One can easily see that, under a local gauge transfor- −sm2 | − | mation S(x, y)= dse y exp[ sH] x . (19) χ(x) → eieΛ(x)χ(x), (14) 0 This expression is similar to a quantum-mechanical Aµ(x) → Aµ(x)+∂µΛ(x), propagator with s = it and a Hamiltonian H which Φi(x, x¯) remains gauge invariant: is a covariant function of 4-vector momenta and co- Φ(x, x¯) (15) ordinates. It is known how to represent a quantum- mechanical propagator as a path integral. The rep- x¯ resentation is in terms of the Lagrangian, and a co- ∗ → exp −ieΛ(x)+ieΛ(¯x) − ie dzµ∂µΛ χ (x) variant Lagrangian can easily be obtained from the Hamiltonian (18) x 2 0 H(ˆz,pˆ)=(ˆp + ieA(ˆz)) =⇒ L(z,z˙) (20) × U(x, x¯)χ(¯x)=Φ(x, x¯). z˙2 = − iez˙ · A(z). Performingpath integrals over χ and χ∗ fields in 4 Eq. (10), one finds Using this Lagrangian, the path-integral representa- tion for the interactingpropagatorbecomes G(y,y¯|x, x¯)=N DA(detS)U(x, x¯) (16) ∞ D × U ∗(y,y¯)S(x, y)S(¯x, y¯)e−S[A], S(x, y)= ds ( z)xy (21) 0 where the interactingone-body propagator S(x, y) is s defined by 1 × − 2 − 2 exp sm dτsz˙ (τs) 1 4 ≡ 0 S(x, y) y 2 x (17) m + H(ˆz,pˆ) s with − ie dτszA˙ (z(τs)) , 2 H(ˆz,pˆ) ≡ (ˆp + ieA(ˆz)) . (18) 0 PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 5 2005 NONPERTURBATIVE DYNAMICS OF SCALAR FIELD 845 where zi(τs) is a particle trajectory which is a para- xy metric function of the parameter τs,withs ≥ τs ≥ 0 and endpoints zi(0) = xi, zi(s)=yi,andi =1 to 4. This representation allows one to perform the remaining path integral over the exchange field A. The final result for the two-body propagator involves a quantum-mechanical path integral that sums up contributions comingfrom all possible trajectories of the two charged particles: x– y– ∞ ∞ − D Fig. 1. The dashed curves represent exchanges of the G = ds ds¯ ( z)xy (22) gauge field with mass µ and the solid curves the propa- 0 0 gation of the matter fields with mass m. Note the matter loop in one of the middle exchanges. All loops of this −K[z,s]−K[¯z,s¯] kind are neglected in the quenched approximation (when × (Dz¯)x¯y¯e W (C), det S → 1). where the parameter τ is rescaled, so that τ = sτ; s s the kinetic term K is defined by z(τ) 1 xy 1 K[z,s]=m2s + dτz˙2(τ); (23) 4s 0 and the Wilson loop average W (C) is given by W (C)≡ DAexp − ie dzA(z) (24) Ð τ– C – z ( ) – x y 1 4 2 2 − d zA(z)(µ − ∂ )A(z) , Fig. 2. The contour C, known as a Wilson loop, that 2 arises in Eq. (24). where the contour of integration C (shown in Fig. 2) follows a clockwise trajectory x → y → y¯ → x¯ → x as parameters τ and τ¯ are varied from 0 to 1.TheA Equation (22) has a very nice physical interpre- − integration in the definition of the Wilson loop aver- tation. The term ∆µν (za zb,µ) describes the prop- age is of standard Gaussian form and can be easily agation of gauge field interactions between any two performed to obtain points on the particle trajectories, and the appearance of these interaction terms in the exponent means that W (C) (25) the interactions are summed to all orders with arbi- e2 trary orderingof the points on the trajectories. Self- = exp − dz dz¯ ∆ (z − z,µ¯ ) , interactions come from terms with the two points 2 µ ν µν C C za and zb on the same trajectory, generalized ladder exchanges arise if the two points are on different d4p eipx ∆µν (x, µ)=gµν . (26) trajectories, and vertex corrections arise from a com- (2π)4 p2 + µ2 bination of the two. Because the particles formingthe When it is necessary to regulate the ultraviolet sin- two-body bound state carry opposite charges, it fol- gularities in (26), a double Pauli–Villars subtraction lows that the self-energy and exchange contributions will be used, so that (26) will be replaced by have different signs. ∆ (x, µ) (27) µν The bound-state spectrum can be determined d4p eipx(Λ2 − µ2)(Λ2 − µ2) from the spectral decomposition of the two-body = g 1 2 . µν (2π)4 (p2 + µ2)(p2 +Λ2)(p2 +Λ2) Green’s function 1 2 ∞ Through the results given in Eqs. (22), (25) and −mnT G(T )= cne , (28) either (26) or (27), the path integration expression n=0 involving fields has been transformed into a path inte- gral representation involving trajectories of particles. where T is defined as the average time between the PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 5 2005 846 SAVKLI et al. Bubble sum Dyson–Schwinger Feynman–Schwinger Fig. 3. Various interactions included in each approach are shown. initial and final states, Theself-energyforthesimplebubblesum(in0+1 1 dimension) is given by T ≡ (y +¯y − x − x¯ ). (29) 2 4 4 4 4 ∞ dk 1 In the limit of large T , the ground-state mass is given Σ(p)=−e2 (33) 2π (k2 + µ2) by −∞ D −S[Z] − 2 − d − ZS [Z]e (2p k) m0 = lim ln[G(T )] = − . × − 1 . T →∞ dT DZe S[Z] [(p − k)2 + m2] (30) The self-energy integral in this case is trivial and can be performed analytically, and the dressed mass is 3. SCALAR QUANTUM ELECTRODYNAMICS determined from Eq. (32). In this section, we take a closer look at one-body The rainbow Dyson–Schwinger equation sums mass pole calculations for the case of SQED. Two more diagrams than the simple bubble summation popular methods frequently used to find the dressed (Fig. 3). The self-energy of the rainbow Dyson– massofaparticlearetodoasimplebubblesum- Schwinger equation involves a momentum-dependent mation or to solve the one-body Dyson–Schwinger mass: ∞ equation in the rainbow approximation. It is interest- dk 1 ingto compare results givenby the bubble summa- Σ(p)=−e2 (34) 2π (k2 + µ2) tion and the Dyson–Schwinger with the exact FSR −∞ result. Below, we first give a quick overview of how (2p − k)2 dressed masses can be obtained in bubble summation × − 1 . and the Dyson–Schwinger equation approaches. For − 2 2 [(p k) + m +Σ( k )] technical simplicity, the one-body discussion will be limited to 0+1dimension. In this case, the self-energy is nontrivial and it must The simple bubble summation involves a summa- be determined by a numerical solution of Eq. (34). tion of all bubble diagrams to all orders. The dressed The dressed mass is determined by the logarithmic propagator is given by derivative of the dressed propagator in coordinate 1 space ∆d(p)= . (31) p2 + m2 +Σ(p) d M = − lim log[∆d(t)]. (35) The dressed mass M is determined from the self- T →∞ dT energy using The type of diagrams summed by each method is M = m2 +Σ(iM). (32) shown in Fig. 3. Note that the matter loops do not PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 5 2005 NONPERTURBATIVE DYNAMICS OF SCALAR FIELD 847 M, GeV 2.0 FSR Bubble sum 1.8 Dyson–Schwinger Bubble (unphys.) 1.6 1.4 1.2 1.0 0 0.2 0.4 0.6 e2, GeV3 Fig. 4. The function M(e2),calculatedin0+1 dimension with values of m = µ =1GeV, usingthe FSR approach, the Dyson–Schwinger equation in the rainbow approximation, and the bubble summation. While the exact result is always real, the rainbow Dyson–Schwinger and the bubble summation results become complex beyond a critical coupling. Γ 3(p, q) p–q p qq Γ 4(p, q, k) p–q–k p qk qk Fig. 5. Because of the interactions, the one-particle irreducible vertex functions Γn (n =3, 4,...) depend on the external momenta. give any contribution, as explained earlier. Results display similar behavior. While the exact result pro- obtained by these three methods are shown in Fig. 4. vided by the FSR linearly increases for all coupling It is interestingto note that the simple bubble sum- strengths, both the simple bubble summation and the mation and the rainbow Dyson–Schwinger results rainbow Dyson–Schwinger results come to a critical PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 5 2005 848 SAVKLI et al. Γ Γ 4 3 dimensions. We start with an initial bare mass m and calculate the full two-body bound-state result with the inclusion of all interactions: generalized ladders, self-energies, and vertex corrections. Let us denote the result for the exact two-body bound- tot 2 state mass by M2 (e ,m), since it will be a function of the couplingstrength e and the bare input mass m, and the superscript “tot” implies that all inter- actions are summed. Next, we calculate the dressed 2 one-body mass M1(e ,m). Then, usingthe dressed Γ 2 5 mass value M1(e ,m), we calculate the bound-state exch 2 mass M2 (e ,M1) by summingonly the gener- alized exchange-interaction contributions. In this Fig. 6. Exact computation of the two-body bound-state last calculation, we sum only exchange interactions propagator requires the summation of all particle self- (generalized ladders), but the self-energy is approxi- energies, vertex corrections, and ladder and crossed lad- der exchanges. mately taken into account since we use the (constant) dressed one-body mass as input. However, the vertex corrections and wave-function renormalization are point beyond which solutions for the dressed masses completely left out since we use the original vertex become complex. This example very clearly shows provided by the Lagrangian. In order to compare the that conclusions about the mass poles of propagators full result, where all interactions have been summed, based on approximate methods such as the rainbow with the result obtained by two dressed particles Dyson–Schwinger equation can be misleading. interacting only by generalized ladder exchanges, In general, a consistent treatment of any non- we plot the bound-state masses obtained by these perturbative calculation must involve summation of methods. Numerical results are presented in Fig. 7. all possible vertex corrections. Vertex corrections are This result is qualitatively similar to that obtained those irreducible diagrams that surround an interac- analytically for SQED in 0+1dimension [33]. tion vertex. The elementary vertex is the three-point The numerical results presented here yield the fol- vertex, Γ3, but the particle interactions will lead to lowingprescription for bound-state calculations: In the appearance of nth-order irreducible vertices, Γn, order to get the full result for bound states, it is a as illustrated in Fig. 5. The propagation of a bound good approximation to first solve for dressed one- state therefore involves a summation of all diagrams body masses exactly (summingall generalized rain- with the inclusion of higher order vertices (Fig. 6). A bow diagrams) and then use these dressed masses rigorous determination of all of these vertices is not and the bare interaction vertex provided by the La- feasible. In the literature on bound states, Γn>3 in- grangian to calculate the bound-state mass by sum- teraction vertices are usually completely ignored. The mingonly generalized ladder interactions (leavingout three-point vertex Γ3 can be approximately calculated vertex corrections). In terms of Feynman graphs, this in the ladder approximation [34]. However, a rigorous prescription can be expressed as in Fig. 8. determination of the exact form of the three-point The significance of the results presented above vertex is not possible, for this requires the knowledge rests in the fact that the problem of calculatingexact of even higher order vertices. results for bound-state masses in SQED has been re- In order to be able to make a connection between duced to that of calculatingonly generalized ladders. the exact theory and predictions based on approxi- Summation of generalized ladders can be addressed mate bound-state equations, it is essential that the within the context of bound-state equations [22, 23, role of interaction vertices be understood. The FSR is 35]. In this paper, we have shown the connection a useful technique for this purpose. between the full prediction of a Lagrangian and the We now present the interestingoutcome that the summation of generalized ladder diagrams. Our re- full bound-state result dictated by a Lagrangian can sults are rigorous for SQED, but are only suggestive be obtained by summingonly generalized ladder di- for more general theories with spin or internal sym- agrams (“generalized” ladders include both crossed metries. Since we have neglected charged particle ladders and, in theories with an elementary four- loops (our results are in the quenched approximation) point interaction, both overlappingand nonoverlap- and the current is conserved in SQED, it is perhaps ping “triangle” and “bubble” diagrams). not surprisingthat the bare couplingis not renor- We adopt the followingprocedure for determin- malized, but the fact that the momentum dependence ingthe contribution of vertex corrections in 3+1 of the dressed mass and vertex corrections seem to PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 5 2005 NONPERTURBATIVE DYNAMICS OF SCALAR FIELD 849 M2/m 3.2 2.8 2.4 2.0 2 4 6 8 10 e2/(4π) exch Fig. 7. Two-body bound-state mass for SQED in 3+1dimensions. Closed triangles are 2M1, open squares are M2 ,and tot closed inverted triangles are M2 .Here,µ/m =0.15, Λ1/µ =3,andΛ2/µ =5. The smooth curves are fits to the “data.” Note exch tot that M2 = M2 to within errors. Γ 3(p, q) p p' p 1 p–q ⇒ S(p, m(p)) S(p', m(p')) S0(p, M) S0(p–q, M) qq Γ (p, q, k) p 4 p–q–k p 1 p–q–k ⇒ q k qk Γ Γ ⇒ 5, 6, ... 0 Fig. 8. The correct two-body result can be obtained by simply usinga dressed constituent mass and a bare vertex and ignoring the contributions of higher order vertices. cancel is unexpected. If we were to unquench our lation that occurs in the one-body calculations. The calculation or to use a theory without a conserved exact self-energy shown in Fig. 7 (and also in Fig. 4 current, it is reasonable to expect that both the bare for different parameters) is nearly linear in e2 [29]. This interaction and the mass would be renormalized. remarkable fact implies that the exact self-energy is Finally, we call attention to a remarkable cancel- well approximated by the lowest order result from PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 5 2005 850 SAVKLI et al. z(τ) 4. SCALAR χ2φ INTERACTION V0 x y WITH THE FSR APPROACH V12 We consider the theory of charged scalar particles χ of mass m interacting through the exchange of a V0 neutral scalar particle φ of mass µ. The Euclidean z–(τ– ) x– y– Lagrangian for this theory is given by ∗ 1 Fig. 9. Sample trajectories with self-energy and ex- L = χ m2 − ∂2 + gφ χ + φ(µ2 − ∂2)φ. (40) change interactions. 2 The two-body propagator for the transition from ∗ the initial state Φi = χ (x)χ(¯x) to final state Φf = perturbation theory. It is instructive to see how this ∗ comes about. If we expand the self-energy to fourth χ (y)χ(¯y) is given by order, expandingeach term about the bare mass m, G(y,y¯|x, x¯) (41) we have ∗ ∗ −1 D D D − 4 L 2 2 − 2 2 = N χ χ φΦf Φiexp d x . Sd (p )=m p +Σ(p ) (36) = m2 − p2 +Σ +(p2 − m2)Σ +Σ , 2 2 4 After the usual integration of matter fields is done, the 2 Green’s function reduces to where Σ =Σ(m ) is the contribution of order e 2 2 2 2 p = m Σ = dΣ(p )/dp −L [φ] evaluated at , evaluated at G(y,y¯|x, x¯)=N Dφ(detS)S(x, y)S(¯x, y¯)e 0 , p2 = m2, and the formula is valid for p2 − m2 e2. Expandingthe dressed mass in a power series in e2 (42) 2 2 2 2 M1 = m + m2 + m4 + ..., (37) with the free Lagrangian L0 and the interactingprop- agator S(x, y) defined by where m2 is the contribution of order e,andsubsti- 1 tutingit into Eq. (36) gives L [φ] ≡ d4zφ(z)(µ2 − ∂2)φ(z), (43) 0 2 z M 2 = m2 +Σ +Σ Σ +Σ + .... (38) 1 2 2 2 4 1 S(x, y) ≡ y x , The mass is then m2 + H(ˆz,pˆ) 2 − 2 Σ2 4m [Σ2Σ2 +Σ4] Σ2 with the Hamiltonian M1 = m + + 3 + .... 2m 8m ≡ 2 − (39) H(ˆz,pˆ) pˆ gφ(ˆz). (44) The linearity of the exact result implies that the As in the case of scalar QED, we employ the → fourth-order term in Eq. (39) must be zero (or very quenched approximation: detS 1. small), and this can be easily confirmed by direct We exponentiate the denominator by introducing calculations! an s integration along the imaginary axis with an ' The cancellation of the fourth-order mass cor- prescription rection (and all higher orders) is reminiscent of the i∞ cancellations between generalized ladders, which ex- 2 −s(m +i) | − | plains why quasipotential equations are more effec- S(x, y)= ds e y exp[ sH] x . (45) tive than the ladder Bethe–Salpeter equation in ex- 0 plainingtwo-body bindingenergies.It shows that a This representation should be compared with the rep- simple evaluation of the second-order self-energy at the bare-mass point is more accurate than solution resentation used earlier in SQED Eq. (19). Here, the of the Dyson–Schwinger equation in the rainbow integration is done along the imaginary axis because H approximation. is not positive definite. The general lesson seems to be that attempts to Again, a quantum-mechanical path-integral rep- sum a small subclass of diagrams exactly is often less resentation can be constructed by recognizing that accurate than the approximate summation of a larger Lagrangian corresponding to the H of Eq. (44) is class of diagrams. given by In the next section, we consider the application of z˙2 L(z,z˙)= + gφ(z). (46) the FSR approach to scalar χ2φ interaction. 4 PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 5 2005 NONPERTURBATIVE DYNAMICS OF SCALAR FIELD 851 The path-integral representation for the interacting Ims propagator is therefore ∞ S(x, y)=−i ds Dz exp is(m2 + i') (47) 0 s s Γ i is0 + − dτz˙2(τ)+ig dτφ(z(τ)) . 4 0 0 This representation allows the elimination of the inte- gral over the exchange field φ. The two-body propa- (0, 0) Res gator reduces to Fig. 10. Wick rotation in the s integration. ∞ ∞ G = − ds ds¯ (Dz)xy (48) In order to be able to compute the path integral over 0 0 trajectories, a discretization of the path integral is iK[z,s]+iK[¯z,s¯] × (Dz¯)x¯y¯e Iφ, needed: N −1 2N 4 where mass and kinetic term is given by (Dz)xy → (N/4πs) d zi. (54) 1 i=1 1 K[z,s]=(m2 + i')s − dτz˙2(τ). (49) The s dependence is crucial for correct normalization. 4s After discretization, the one-body propagator takes 0 the followingform: The field integration Iφ is a standard Gaussian inte- ∞ N−1 gration: N 2N ds G = i dz (55) s 4π i s2N i=1 0 Iφ ≡ Dφexp +ig dτφ(z(τ)) (50) k2 0 × exp im2s − i − s2v[z] , s¯ 4s + dτφ¯ (¯z(¯τ)) − L0[φ] where v[z] was defined in Eq. (51). This is an oscil- 0 latory and regular integral and it is not convenient for ≡ exp(−V0[z,s] − 2V12[z,z,s,¯ s¯] − V0[¯z,s¯]), Monte Carlo integration. The origin of the oscillation is the fact that the s integral was defined alongthe where V0 and V12 (self- and exchange-energy contri- imaginary axis, butions in Fig. 9) are defined by Rep. 1: S(x, y) (56) V0[z,s] (51) −i∞ 1 1 2 2 2 g 2 2 = y ds exp[−s(m − ∂ + gφ + i')] x . = s dτ dτ ∆(z(τ) − z(τ ),µ) ≡ s v[z], 2 0 0 0 V12[z,z,s,¯ s¯] (52) In earlier works [4, 27], a nonoscillatory FSR was 1 1 used, 2 g − S(x, y) = ss¯ dτ dτ¯∆(z(τ) z¯(¯τ),µ). Rep. 2: (57) 2 ∞ 0 0 2 2 = y ds exp[−s(m − ∂ + gφ)] x . It should be noted that the interaction terms explicitly depend on the s variable, which was not the case for 0 SQED. The interaction kernel ∆ is given by Rep. 2 leads to a nonoscillatory and divergent result d4p eip·x ∆(x, µ)= (53) ∞ (2π)4 p2 + µ2 2 ∝ ds − 2 − k 2 µ G 2N exp m s + s v[z] , (58) = K (µ|x|). s 4s 4π2|x| 1 0 PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 5 2005 852 SAVKLI et al. –1 + s2 – g2 s3 C 1.4 2 1 1.3 –1 1 234 1.2 –1 s –2 1.1 –3 1.0 Fig. 11. As the effective couplingstrength g2 is increased, 20 40 60 80 100 120 the stationary point disappears. g2, GeV2 Fig. 13. The dependence of the peak of the s distribution on the couplingstrengthfor the two-body bound state z1 z2 is shown. The peak location is given by s0 = CT/2m. zN = y There is no real solution for C beyond g2 = 100 GeV2. z0 = x With this factor, the integral in Eq. (59) is modified: ∞ Fig. 12. Number of steps a particle takes between initial ds k2 s2v[z] and final coordinates is discretized. The space–time is ∝ 2 − − G 2N exp im s i 2 . (62) continuous and there are no space–time boundaries. s 4s R (s, s0) 0 This modification allows us to make a Wick rotation and the large-s divergence was regulated by a cutoff since the contribution of the contour at infinity now Λ. This is not a satisfactory prescription since it relies vanishes. However, this procedure relies on the fact on an arbitrary cutoff. Later, it was shown [28, 30] that there exists a stationary point. It can be seen from that the correct procedure is to start with Rep. 1 the original expression Eq. (59) that this is not always and make a Wick rotation such that the final result true. According to the original integral, the stationary is nonoscillatory and regular. The implementation of point is given by the following equation: Wick rotation, however, is nontrivial. Consider the k2 s-dependent part of the integral for the one-body im2 + i − 2sv[z]=0. (63) propagator 4s2 ∞ Introducing s = isssk/(2m) and g2 = k v[z]/m3, this ds k2 equation becomes G ∝ exp im2s − i − s2v[z] . (59) s2N 4s −1+s2 − g2s3 =0. (64) 0 The stationary point is determined by the first in- It is clear that a replacement of s → is leads to a tersection of a cubic plot with the positive s axis as divergent result. The problem with the Wick rotation shown in Fig. 11. The plot in Fig. 11 shows that, (Fig. 10) comes from the fact that the s integral is 2 infinite both alongthe imaginary axis and alongthe as the effective couplingstrength g is increased, the s contour at infinity. These two infinities cancel to yield curve no longer crosses the positive s axis. Therefore, a finite integral along the real axis. As g → 0,the beyond a critical couplingstrength,the stationary dominant contribution to the s integral in Eq. (59) point vanishes and mass results should be unstable. comes from the stationary point Limitingdiscussion to cases where the originalex- pression Eq. (59) has a critical point, we now turn to k perform a Wick rotation on the modified expression s = is i . (60) 0 2m Eq. (62). The Wick rotation in Eq. (62) amounts to a simple replacement s → is, and a regular, nonoscil- Therefore, one might suppress the integrand away latory integral is found: from the stationary point by introducinga damping ∞ factor R: ds k2 s2v[z] G ∝ exp −m2s − + . (65) 2 2 2N 2 R(s, s0) ≡ 1 − (s − is0) /Γ . (61) s 4s R (is, s0) 0 PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 5 2005 NONPERTURBATIVE DYNAMICS OF SCALAR FIELD 853 At first glance, it seems that the new integral always Samplingof trajectories is done usingthe standard has a stationary point determined by the following Metropolis algorithm, which insures that configura- equation: tions sampled are distributed accordingto the weight −S[Z] k2 2sv[z] (R2(is, s )) e . In samplingtrajectories, the final-state (spa- − 2 − 2 0 cial) coordinates of particles can be integrated over, m + 2 + 2 s v[z] 2 =0. 4s R (is, s0) R (is, s0) which puts the system at rest and projects out the (66) S-wave ground state. As trajectories of particles are The key point to remember is that the stationary point sampled, the wave function of the system can be we find after the Wick rotation should be the same determined simply by storingthe final-state config- stationary point we had before the Wick rotation. This urations of particles in a histogram. is required to make sure that the physics remains In samplingtrajectories, the first step is thermal- the same after the Wick rotation. Self-consistency ization. In order to insure that the initial configuration therefore requires that the stationary point after the of trajectories has no effect on results, the first 1000 Wick rotation be at s = is0. In that case, R(is0,s0)= or so updates are not taken into account. Statistical 2 independence of subsequent samplings is measured 1 and (R (is0,s0)) =0and Eq. (66) determiningthe critical point reduces to the earlier original Eq. (63). by the correlation function X(n),defined as m(i)m(i + n)− m2 The regularization of the ultraviolet singularities X(n) ≡ , (70) is done usingPauli –Villars regularization, which m2 is particularly convenient for numerical integration where m(i) is the mass measurement at the ith up- since it only involves a change in the interaction date. kernel In order to insure that the location of the station- ∆(x, µ) −→ ∆(x, µ) − ∆(x, αµ). (67) ary point is self-consistent, as discussed earlier, its location must be determined carefully. The stationary 2 Calculations of the χ φ interaction in 3+1 di- point can be parametrized by s0 = CT/(2m),where mension require numerical Monte Carlo integration. T/(2m) is the location of the stationary point when The first step is to represent the particle trajectories the couplingstrength g goes to zero. As the coupling byadiscretenumberofN +1points with boundary strength is increased, the stationary point moves to conditions given by larger values of s0 (recall Fig. 11) and C increases. z = x =(x ,x ,x , 0), (68) Eventually, a critical value of the couplingconstant 0 1 2 3 is reached beyond which there is no self-consistent zN = y =(y1,y2,y3,T). stationarypoint.InordertobeabletodoMonteCarlo The discretization employed in the FSR is for the integrations, an initial guess must be made for the number of time steps a particle takes in going from location of the stationary point. Self-consistency is realized by insuringthat the peak location of the s the initial time to the final time alonga trajectory in distribution in the Monte Carlo integration agrees a four-dimension coordinate space, as illustrated in with the initial guess for the stationary point [30]. In Fig. 12. This is very different from the discretization Fig. 13, the dependence of the location of the sta- employed in lattice gauge theory. Contrary to lattice tionary point on the couplingstrengthfor two-body gauge theory, in the FSR approach, space–time is bound states is shown. The figure shows that, beyond continuous and rotational symmetry is respected. An g2 100 2 C additional important benefit is the lack of space–time the critical point GeV , goes to infinity, lattice boundaries, which allows calculations of arbi- implyingthat there is no stationary point. A similar trarily large systems using the FSR approach. This critical behavior was also observed in [36, 37] within feature provides an opportunity for doingcomplex the context of a variational approach. In Fig. 14, FSR applications such as calculation of the form factors of two-body bound-state mass results are shown for m =1 GeV, µ =0.15 GeV. These results are all large systems. χ φ for a Pauli–Villars mass of 3µ. Also shown are the In doingMonte Carlo sampling,we sample trajec- predictions of various integral equation calculations. tories (lines) rather than gauge-field configurations The FSR calculation sums all ladder and crossed (in a volume). This leads to a significant reduction ladder diagrams and excludes the self-energy con- in the numerical cost. The ground-state mass of the tributions. Accordingto Fig.14, all bound-state Green’s function is obtained using equations underbind. Amongthe manifestly covariant DZS[Z]e−S[Z] equations, the Gross equation (labeled GR in Fig. 14) m = . (69) gives the closest result to the exact calculation 0 D −S[Z] Ze obtained by the FSR method. This is due to the fact PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 5 2005 854 SAVKLI et al. m, GeV 2.0 1.8 FSR BSLT BSE ET GR NR MW 1.6 40 80 120 160 2 g2, GeV Fig. 14. The coupling-constant dependence of the two-body bound-state mass is shown. Beyond the critical coupling strength of g2 = 100 GeV2, the two-body mass becomes unstable. The Bethe–Salpeter equation in the ladder approximation gives the smallest binding. The other models are described in the text. C m, GeV 1.4 3.0 2.9 1.2 2.8 FSR 2.7 Gross 1.0 Schrodinger¬ 2.6 20 40 60 80 100 g2, GeV2 30 40 50 60 70 80 2 2 Fig. 15. The dependence of the peak of the s distribu- g , GeV tion on the couplingstrengthfor the three-body bound- state mass is shown. The peak location is given by s0 = Fig. 16. Three-body bound-state results for three equal- CT/2m. There is no real solution for C beyond g2 = mass particles of mass 1 GeV. 81 GeV2. results away from the exact results (Mandelzweig– that, in the limit of infinitely heavy–light systems, the Wallace equation [24], labeled MW in Fig. 14). Gross equation effectively sums all ladder and crossed In particular, the Bethe–Salpeter equation in the ladder diagrams. The equal-time equation (labeled ladder approximation (labeled BSE in Fig. 14) gives ET in Fig. 14) also produces a strong binding, but the the lowest binding. Similarly, the Blankenbecler– inclusion of retardation effects pushes the equal-time Sugar–Logunov–Tavkhelidze equation [20, 21] (la- PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 5 2005 NONPERTURBATIVE DYNAMICS OF SCALAR FIELD 855 1.6 0 –1.6 –1.4 1.6 –1.4 0 0 0 1.4 –1.6 1.4 1.6 1.6 0 0 –1.6 –1.6 –1.4 –1.4 0 0 1.4 1.4 Fig. 17. The four panels show the evolution of the probability distribution for the third particle as the distance between the two fixed particles is increased. When the fixed particles are very close to each other, the third particle sees them as a single particle (upper left panel). As the fixed particles are separated, the third particle starts penetratingbetween them (upper rightand lower left panels), and when two fixed particles are far apart (lower right panel), the third particle is most likely to be found between the two fixed particles. beled BSLT in Fig. 14) gives a very low binding. A sees them as a single particle. However, as the fixed comparison of the ladder Bethe–Salpeter, Gross, and particles are separated from each other, the third the FSR results shows that the exchange of crossed particle starts havinga nonzero probability of beingin ladder diagrams plays a crucial role. between the two fixed particles (separation increases In Fig. 15, the dependence of the location of the as we go from the upper right to lower left panels in stationary point on the couplingstrengthfor three- Fig. 17). Eventually, when the two fixed particles are body bound states is shown. In Fig. 16, the three- far away from each other, the third particle is most body bound-state results for three equal-mass par- likely to be at the origin (the lower right panel in ticles of mass 1 GeV are shown. For the three-body Fig. 17). case, the only available results are for the Schrodinger¨ Up to this point, the FSR method has been derived and Gross equations. Accordingto the results pre- and various applications to nonperturbative problems sented in Fig. 16, the bound-state equations under- have been presented. In the next section, we discuss bind for the three-body case too. The Gross equation the stability of the χ2φ theory. gives the closest result to the exact FSR result. Determination of the wave function of bound 5. STABILITY OF THE SCALAR χ2φ states is done by keepingthe final-state configura- INTERACTION tions of particles in a histogram. For example, for a † three-body bound-state system, the probability dis- Scalar field theories with a χ χφ interaction tribution of the third particle for a given configuration (which we will subsequently denote simply by χ2φ) of first and second particles is shown in Fig. 17. In have been used frequently without any sign of insta- the upper left panel of Fig. 17, the two fixed particles bility, despite an argument in 1952 by Dyson [38] sug- are very close to each other, so that the third particle gesting instability and a proof in 1960 by Baym [39] PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 5 2005 856 SAVKLI et al. √ 2 2 showingthat the theory is unstable. For example, with Em(k)= m + k . The equal-time commuta- it is easy to show that, for a limited range of cou- tion relations are plingvalues 0 ≤ g2 ≤ g2 , the simple sum of bubble † 3 3 crit [a(k),a (k )] = (2π) · 2Em(k)δ (k − k ). (76) diagrams for the propagation of a single χ particle leads to a stable ground state, and it is shown in [13] The Lagrangian for the χ2φ theory is that a similar result also holds for the exact result † 1 in the “quenched” approximation. However, if the L = χ [∂2 − m2 + gφ]χ + φ(∂2 − µ2)φ, (77) scalar χ2φ interaction is unstable, then this instability 2 should be observed even when the couplingstrength g and the Hamiltonian H is a normal ordered product of 2 → + is vanishingly small g 0 , as pointed out recently interacting(or dressed) fields φd and χd: by Rosenfelder and Schreiber [36] (see also [40]). 2 Both the simple bubble summation and the quenched ∂χd H[φ ,χ ,t]= d3r : (78) calculations do not exhibit this behavior. Why do the d d ∂t simple bubble summation and the exact quenched ∇ 2 2 2 calculations produce stable results for a finite range +( χd) + m χd of couplingvalues? 2 1 ∂φd ∇4 2 2 2 − 2 A clue to the answer is already provided by the + ( φd) + µ φd gχdφd : . simplest semiclassical estimate of the ground-state 2 ∂t energy. In this approximation, the ground-state en- This Hamitonian conserves the difference between ergy is obtained by minimizing the number of matter and the number of antimat- 2 2 1 2 2 − 2 ter particles, which we denote by n0. Eigenstates of E0 = m χ + µ φ gφχ , (71) | 2 the Hamiltonian will therefore be denoted by n0,λ , where λ represents the other quantum numbers that where m is the bare mass of the matter particles, and define the state. Hence, allowingfor the fact that the µ the mass of the “exchanged” quanta, which we will eigenvalue may depend on the time, refer to as the mesons. The minimum occurs at 4 H[φd,χd,t]|n0,λ = Mn ,λ(t)|n0,λ. (79) 2 2 − 2 χ 0 E0 = m χ g 2 . (72) 2µ In the absence of an exact solution of (79), we may This is identical to a χ4 theory with a couplingof the estimate it from the equation wrongsign,as discussed by Zinn-Justin [41]. The Mn ,λ(t)= n0,λ|H[φd,χd,t]|n0,λ (80) ground state is therefore stable under fluctuations of 0 | −1 | the χ field about zero, provided = n0,λU (t, 0)H[φ, χ, 0]U(t, 0) n0 ,λ ≡ | | 2m2µ2 n0,λ,tH[φ, χ, 0] n0,λ,t , g2 PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 5 2005 NONPERTURBATIVE DYNAMICS OF SCALAR FIELD 857 where stabilty are qualitatively unrelated. For example, the N † † cubic interaction is also unstable in dimensions lower 0(k, k )= a (k)a(k )+b (k)b(k ) , (82) than four, where there is no need for regularization. The ultraviolet regularization would have an effect N1(k, k )=N0(k, k ) on the behavior of functions f(p) and g(p), which + a†(k)b†(−k)+a(−k)b(k) , are left unspecified in this discussion except for their normalization. and ω(k)= µ2 + k2. To evaluate the matrix ele- The matrix element (80) can now be evaluated. ment (80), we express the the eigenstates as a sum of Assumingthat f(k)=f(−k) and g(k)=g(−k),it free particle states with n0 matter particles, npair pairs becomes of χχ¯ particles, and 6 mesons: M (t)={n +2L(t)} m˜ + G(t)˜µ (89) n0,λ 0 |n ,λ,t≡|n ,α(t),β(t) 0 0 (83) − gV {n +2L(t)+2L (t)} G (t), ∞ ∞ 0 1 1 1 = α (t)β (t)|n ,n ,6, where the constants m˜ , µ˜,andV are γ(t) npair 0 pair n =0 pair =0 2 m˜ ≡ dkE˜ m(k)f (k), (90) where γ(t) is a normalization constant (defined be- low), the time dependence of the states is contained in µ˜ ≡ dpE˜ (p)g2(p), thetimedependenceofthecoefficients α(t) and β(t), µ and ˜ ˜ − dkmdkmf(k)f(k )g(k k ) |n ,n ,6 (84) V ≡ , 0 pair m2 +(k − k)2 | k1,...,kn1 ; q1,...,qn2 ; p1,...,p ≡ . and the time-dependent quantities are (n0 + npair)!npair!6! ∞ n α2 (t) pair npair In this equation, n1 = n0 + npair, n2 = npair,and L(t)= , (91) α2(t) n1 n2 npair=0 ˜ ∞ = dkif(ki) dq˜jf(qj) dp˜lg(pl), (85) 2 6β (t) i=1 j=1 G(t)= , l=1 β2(t) =0 where f(k) and g(p) are momentum wave functions ∞ √ √ ˜ n0 + npair npairαnpair (t)αnpair−1(t) and the particle masses in dk and dp˜ have been sup- L (t)= , 1 α2(t) pressed; their values should be clear from the context. n =1 The normalization of the functions f(p) and g(p) is pair √ ∞ chosen to be 6β(t)β−1(t) G1(t)= 2 . 2 2 β (t) dkf˜ (k)= dpg˜ (p) ≡ 1, (86) =1 Note that L and G are the average number of matter which leads to the normalization pairs and mesons, respectively, in the intermediate n ,n ,6|n0,n ,6 = δ δ δ , (87) state. 0 pair pair n0,n0 npair,npair , n ,λ,t|n ,λ,t =1, The variational principle tells us that the correct 0 0 mass must be equal to or larger than (89). This if γ(t)=α(t)β(t) with inequality may be simplified by usingthe Schwarz ∞ inequality to place an upper limit on the quantities L1 α2(t)= α2 (t)=α(t) · α(t), (88) and G1. Introducingthe vectors npair √ √ n =0 pair f1 = {α1, 2α2,...} = { nαn}, (92) ∞ √ √ { } 2 2 · f2 = n0 +1α0, n0 +2α1,... β (t)= β (t)=β(t) β(t). √ = { n + nα − }, =0 √ 0 n 1 √ { } { } { } { } The expansion coefficients αnpair (t) and β(t) are h = β1, 2β2,... = 6β , vectors in infinite-dimensional spaces. we may write In principle, the scalar cubic interaction in four 2 2 dimensions requires ultraviolet regularization. How- f1(t) · f2(t) f (t)f (t) L (t)= ≤ 1 2 (93) ever, the issue of regularization and the question of 1 α2(t) α2(t) PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 5 2005 858 SAVKLI et al. = L(t){n0 +1+L(t)}, given qualitatively in Eq. (65). The detailed expres- sion, needed in the followingdiscussion, is h(t) · β(t) h2(t)β2(t) ≤ ∞ G1(t)= 2 2 = G(t). 2N N −1 β (t) β (t) N 4 G(x, y)= ds d zi (98) Hence, suppressingexplicit reference to the time de- 4πs 0 i=1 pendence of L and G and usingthe inequalities (93), Eq. (89) can be converted into an inequality × exp −K[z,s] − V [z,sr] . M (t) ≥ (n +2L)˜m + Gµ˜ (94) n0,λ 0 √ √ Here, the integrations are over all possible particle 2 − − gV n0 +1+L + L − 1 G. trajectories (discretized into N segments with N 1 variables zi and boundary conditions z0 = x and Minimization of the ground-state energy with respect zN = y), and the kinetic and self-energy terms are to the average number of mesons G occurs at N 2 N 2 √ K[z,s]=m s + (zi − zi−1) , (99) gV 2 4s G = n +1+L + L − 1 . (95) i=1 0 2˜µ 0 g2s2 N V [z,s]=− ∆(δz ,µ) , (100) At this minimum point, the ground-state energy is 2N 2 ij bounded by i,j=1 ≥{ } − Mn0,λ(t) n0 +2L m˜ µG0. (96) where ∆(z,µ) is the Euclidean propagator of the me- 1 son (suitably regularized), δz = (z + z − − z − If we continue with the minimization process, we ij 2 i i 1 j z − ),and would obtain M (t) →−∞as L →∞,providing j 1 n0,λ s s no lower bound and hence suggesting that the state is s ≡ = . (101) r − 2 2 unstable. However, if L is finite, this result shows that R(s, s0) 1+(s s0) /Γ the ground state is stable for couplings in the interval (The need for the substitution s → s was discussed 2 2 r 0 PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 5 2005 NONPERTURBATIVE DYNAMICS OF SCALAR FIELD 859 Straight trajectory: Continuons deformation x y x 0 1 z(τ) n y 0 1 2 n z (τ) Folded trajectory: f x y τ x 0 1 zf ( ) n 0 y 1 2 n Fig. 18. It is possible to create particle–antiparticle pairs τ usingfolded trajectories. However, folded trajectories are z'f ( ) suppressed by the kinematics, as discussed in the text. Fig. 19. A folded trajectory at the end point of the path and a similar one with z1 closer to z0. Eq. (102), we choose integration points zi = iT/N uniformly spaced alonga line. The classical trajectory connects these points without doublingback, so that smaller and the potential energy is larger, and there- they increase monotonically with proper time τ.How- fore ever, since the integration over each z is independent, − − i K[zf ,s] V [zf ,s] >K[zf ,s] V [zf ,s]. (106) there also exist trajectories where zi does not increase monotonically with τ. In fact, for every choice of It is clear that the larger the folding in the trajectory, integration points zi, there exist trajectories with zi the less energetically favorable the path, and the most monotonic in τ and trajectories with zi nonmonotonic favorable path is again an unfolded trajectory with no in τ. The latter double back in time and describe Z points outside of the limits z0 PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 5 2005 860 SAVKLI et al. 6. CONCLUSIONS 10. E. E. Salpeter and H. A. Bethe, Phys. Rev. 84, 1232 (1951). In this paper, we have given a summary of results 11. F.Gross and J. Milana, Phys. Rev. D 43, 2401 (1991). for scalar interactions obtained with the use of the 12. P. C. Tiemeijer and J. A. Tjon, Phys. Rev. C 49, 494 FSR representation. The FSR approach uses a co- (1994). variant path-integral representation for the trajecto- 13.C.SavkliandF.Gross,Phys.Rev.C63, 035208 ries of particles. Reduction of field-theoretical path (2001); hep-ph/9911319. integrals to path integrals involving particle trajecto- 14. M. Levine, J. Wright, and J. A. Tjon, Phys. Rev. 154, ries reduces the dimensionality of the problem and the 1433 (1967). associated computational cost. 15. N. Nakanishi, Prog. Theor. Phys. Suppl. 43, 1 (1969). Applications of the FSR approach to one- and 16. N. Nakanishi, Prog. Theor. Phys. Suppl. 95, 1 (1988). two-body problems, in particular, shows that un- 17. T. Nieuwenhuis and J. A. Tjon, Few-Body Syst. 21, controlled approximations in field theory may lead to 167 (1996). significant deviations from the correct result. Our re- 18. C. Savkli and F. Tabakin, Nucl. Phys. A 628, 645 sults indicate that use of the Bethe–Salpeter equation (1998); hep-ph/9702251. in the ladder approximation to solve the two-body 19. E. E. Salpeter, Phys. Rev. 87, 328 (1952). bound-state problem is a poor approximation. For the 20. A. A. Logunov and A. N. Tavkhelidze, Nuovo Cimento scalar theories examined here, a better approximation 29, 380 (1963). to the two-body problem is obtained usingthe Gross 21. R. Blankenbecler and R. Sugar, Phys. Rev. 142, 1051 equation in the ladder approximation. Similarly, use of (1966). the rainbow approximation for the one-body problem 22. F. Gross, Phys. Rev. 186, 1448 (1969). gives a poorer result than simply calculating the self- 23. F.Gross,Phys.Rev.C26, 2203 (1982). energy to second order! In all of these cases, the 24. S. J. Wallace and V. B. Mandelzweig, Nucl. Phys. A explanation for these results seems to be that the 503, 673 (1989). crossed diagrams (such as crossed ladders) play an 25. Taco Nieuwenhuis, Yu. A. Simonov, and J. A. Tjon, essential role, cancelingcontributions from higher Few-Body Syst. Suppl. 7, 286 (1994). order ladder or rainbow diagrams. 26. Taco Nieuwenhuis and J. A. Tjon, Phys. Lett. B 355, 283 (1995). ACKNOWLEDGMENTS 27. Taco Nieuwenhuis and J. A. Tjon, Phys. Rev. Lett. 77, 814 (1996); hep-ph/9606403. We thank Prof. Yu. Simonov for his leadership 28.C.Savkli,F.Gross,andJ.Tjon,Phys.Rev.C60, in this field and for many useful discussions. It is a pleasure to contribute to this volume celebratinghis 055210 (1999); hep-ph/9906211. 70th birthday. 29.C.Savkli,F.Gross,andJ.Tjon,Phys.Rev.D62, This work was supported in part by the DOE grant 116006 (2000); hep-ph/9907445. DE-FG02-93ER-40762 and DOE contract DE- 30. C. Savkli, Comput. Phys. Commun. 135, 312 (2001); AC05-84ER-40150, under which the Southeastern hep-ph/9910502. Universities Research Association (SURA) operates 31.F.Gross,C.Savkli,andJ.Tjon,Phys.Rev.D64, the Thomas Jefferson National Accelerator Facility. 076008 (2001); nucl-th/0102041. 32. C. Savkli, Czech. J. Phys. 51B, 71 (2001); hep- REFERENCES ph/0011249. 1. R. P.Feynman, Phys. Rev. 80, 440 (1950). 33. C. Savkli, F. Gross, and J. Tjon, Phys. Lett. B 531, 2. Julian S. Schwinger, Phys. Rev. 82, 664 (1951). 161 (2002); nucl-th/0202022. 3. Yu.A.Simonov,Nucl.Phys.B307, 512 (1988). 34. B.-F. Ding, Nucl. Phys. B (Proc. Suppl.) 90, 127 4. Yu.A.Simonov,Nucl.Phys.B324, 67 (1989). (2000); nucl-th/0008048. 5. Yu. A. 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PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 5 2005 Physics of Atomic Nuclei, Vol. 68, No. 5, 2005, pp. 861–869. From Yadernaya Fizika, Vol. 68, No. 5, 2005, pp. 894–903. Original English Text Copyright c 2005 by Karanikas, Ktorides. Theoretical Evidence for a Tachyonic Ghost-State Contribution to the Gluon Propagator in High-Energy, Forward Quark–Quark “Scattering”* A. I. Karanikas** and C. N. Ktorides*** Physics Department Nuclear & Particle Physics Section, University of Athens, Panepistimiopolis, Athens, Greece Received May 13, 2004; in final form, September 20, 2004 Abstract—Implications stemming from the inclusion of nonperturbative confining effects, as contained in the stochastic vacuum model of H. Dosch and Yu.A. Simonov, are considered in the context of a (hypothetical) quark–quark “scattering process” in the Regge kinematical region. In a computation wherein the nonperturbative input enters as a correction to established perturbative results, a careful treatment of infrared divergences is shown to imply the presence of an effective propagator associated with the existence of a linear term in the static potential. An equivalent statement is to say that the modified gluonic propagator receives a contribution from a tachyonic ghost state, an occurrence which is fully consistent with earlier suggestions made in the context of low-energy QCD phenomenology. c 2005 Pleiades Publishing, Inc. 1. INTRODUCTION The construction of the SVM is motivated by the intention to incorporate established observa- From the theoretical point of view, forward scatter- tions/results regarding the structure of the QCD ing at very high energies (Regge kinematics) presents vacuum [8] into a well-defined theoretical framework. a situation where one can readily apply eikonal ap- In particular, it summarizes all that is known and/or proximation techniques [1]. In this context, such pro- surmised about its properties through a set of three cesses provide grounds for exploring long-distance axioms, which are expressed in terms of field strength, properties of the underlying fundamental theory for as opposed to field potential, correlators. The underly- the implicated interaction. For the particular case of ing stochasticity assumption for the vacuum state fa- QCD, long-distance behavior constitutes a funda- cilitates the application of the cumulant expansion [9], mental issue whose exploration is not only relevant to which describes the factorization rules for higher high-energy processes but also to low-energy phe- order gluon field strength correlators in terms of two- nomenology. point ones. One of the first results arrived at through In the present paper, we revisit the (idealized) the SVM is the deduction of the area law for the static problem of quark–quark “scattering” in the Regge Wilson loop, i.e., confinement. Specific applications limit, which has been extensively studied within the of the SVM scheme, including comparisons with framework of pQCD [1–5], with the aim to extend the lattice results, can be found, e.g., in [10]. aforementioned analyses in a direction which takes A concrete, as well as practical, way to apply into account confining aspects of the theory. Specif- the SVM scheme to specific situations has been ically, we shall rely on the premises of the stochas- suggested by Simonov [11]. The idea is to use the tic vacuum model (SVM), pioneered by Dosch and background gauge fixing method [12] and assign the Simonov [6] for the explicit purpose of accomodating background gauge fields with the task of becoming the confinement property of QCD. A similar approach the agents of the nonperturbative dynamics. Specifi- a has been pursued by Nachtmann [7] by employing cally, one employs the gauge potential splitting Aµ = adifferent methodology from the one we shall adopt a a a αµ + Bµ with the αµ being associated with the usual in this work. Different will also be the scope of the perturbative-field modes. The Ba, on the other hand, present analysis. µ enter as dynamical fields, assigned with the task of ∗This article was submitted by the authors in English. carrying the nonperturbative physics through field **e-mail: [email protected] strength correlators which adhere to the cumulant ***e-mail: [email protected] expansion rules. 1063-7788/05/6805-0861$26.00 c 2005 Pleiades Publishing, Inc. 862 KARANIKAS, KTORIDES Following our previous work of [5], we find it 2. PRELIMINARY CONSIDERATIONS convenient to employ the FFS-worldline casting of QCD, originally pioneered by Fock [13], Feyn- Consider an idealized quark–quark scattering man [14], and Schwinger [15] and most recently process in the Regge limit, defined by s/m2 →∞, 2 2 revived in path-integral versions (see [16–18]). The s t(= −q = q⊥). The amplitude is given by reason for such a choice is that the eikonal approxi- 2 2 2 T (s/m ,q /λ ) (1) mation acquires a straightforward realization in this ii jj ⊥ scheme, since, in the perturbative context at least, it − · 2 iq b 2 2 2 can be readily implemented by restricting one’s con- = d be Eii jj (s/m , 1/b λ ), siderations to straight worldline paths. As it will turn out, the inclusion of input from the SVM will produce where λ is an “infrared” scale; b is the impact distance a kind of deformation of the eikonal paths, with low- in the transverse plane to the direction of the colliding energy consequences, which represent subleading, quarks (assumed to be traveling along the x3 axis); perturbative–nonperturbative interference effects. and Eiijj , which incorporates the dynamics of the process, is specified, in Euclidean spacetime, by [5] The conditions which justify the relevant computation will be made explicit in the text. Suffice it to say, ∞ at this point, that it was explicitly demonstrated E = P exp ig dτv · A(v τ) (2) in [19] that the FFS-worldline formulation of QCD, ii jj 1 1 −∞ in combination with the background gauge field ii ∞ splitting, is ideally suited for providing a framework conn within which one can directly and efficiently apply the ×P exp ig dτv2 · A(v2τ + b) . premises of the SVM. −∞ A jj The main result of this paper is the following: The v1 and v2 are constant four-velocities character- Once (subleading) contributions incorporating non- izing the respective eikonal lines of the quarks partic- perturbative, as induced by the SVM, corrections ipating in the scattering process and “conn” stands to the perturbative expressions are taken into ac- for “connected.” In Minkowski space (and in light count, then a consistent analysis of the infrared is- + cone coordinates), one has v1 (v1 , 0, 0⊥)√, v2 sues entering the quark–quark high-energy forward- − + − 1 (0,v , 0⊥), b (0, 0, b⊥) with v v √ s/m. “scattering” process reveals that the gluonic prop- 2 1 2 2 agator exhibits a behavior which can be interpreted a a Employing the gauge field splitting Aµ = αµ + in terms of the presence of a “tachyonic mass pole.” a Bµ, the expectation values with respect to field con- Arguments in favor of such an occurrence, with im- ··· ··· portant phenomenological as well as theoretical im- figurations acquire the form A = α,B.Ex- plications, have been promoted, from different per- panding in terms of powers of the perturbative field components, one obtains spectives, in several papers [20–22]. ∞ The organization of the paper is as follows. In E = P exp ig dτv · B(v τ) (3) the next section, we present the basic formulas re- ii jj 1 1 −∞ lated to the “amplitude” for the high-energy “quark– ii ∞ quark scattering process” in the forward direction conn 2 and display their FFS-worldline form. Section 3 fo- ×P exp ig dτv2 · B(v2τ + b) − g cuses its attention on infrared issues associated with −∞ B jj the pertubative–nonperturbative interference effects under consideration in this study. The resulting ex- ∞ ∞ ∞ pression for the amplitude is shown to be equivalent × ds2 ds1 P exp ig dτv1 · B(v1τ) to the introduction of an effective propagator, which −∞ −∞ s is associated with the presence of a linear term in 1 ik s the static potential. Section 4 extends the implica- 1 tions of the aforementioned result to issues related to × P exp ig dτv1 · B(v1τ) renormalization: modified running coupling constant −∞ li and summation of leading logarithms via the Callan– ∞ Symanzyk equation. Finally, the technical manipu- × · lations leading to the main result of Section 3 are P exp ig dτv2 B(v2τ + b) displayed in the Appendix. s2 jm PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 5 2005 THEORETICAL EVIDENCE FOR A TACHYONIC GHOST-STATE CONTRIBUTION 863 s 2 integrals, given that the endpoints of the two trajec- × · a b ∞ −∞ P exp ig dτv2 B(v2τ + b) tkltmn tories can now be joined at + and at . These, −∞ of course, do not correspond to boundary conditions nj for a scattering process per se; however, our only × ba O 4 4 objective in this work is to extract long-distance im- v2µv1ν iGµν (v2s2 + b, v1s1) + (g α ). plications based on the exchanges taking place in the B immediate vicinity of the points of closest approach Our objective in this paper is to study the behavior of between the two worldlines. The study of realistic theamplitudeas|b|→0. Accordingly, contributions situations involving the scattering of physically ob- attributed exclusively to the nonperturbative, back- servable particle entities in the Regge limit, using ground terms will be ignored given that, by definition, the presently adopted methodology, is under current they are finite in this limit. This narrows the expres- consideration and the relevant analysis will be pre- sion of computational interest to the following one: sented in the near future. Finally, concerning the issue of gauge fixing for the B-field sector, our choice is ∞ prompted by the intention to rely on field strength cor- 2 Nc a a E = −g t t ds (4) relators for describing nonperturbative dynamics [11]. ii jj 2 − ii jj 2 Nc 1 −∞ It, accordingly, becomes convenient to work in the ∞ Fock–Schwinger (FS) gauge [23, 24]. The latter is specified by × 1 ds1v2µv1ν TrAiGµν (v2s2 + b, v1s1) x Nc B −∞ a − a Bµ(x)= duν(∂µuρ)Fρν (u) (6) + O(g4 α4 )+O(b2). x0 1 The propagator = −(x − x ) dααF (x + α(x − x )). ba ≡ b a 0 ν µν 0 0 iGµν (v2s2 + b, v1s1) αµ(v2s2 + b)αν(v1s1) 0 acquires the following worldline expression: Of course, the arbitrary point x0 should not enter any ba gauge-invariant expression. iG (v2s2 + b, v1s1) (5) µν ∞ T 1 3. INFRARED ISSUES ASSOCIATED = dT Dx(t)exp− dtx˙ 2(t) 4 WITHTHEPROPAGATION 0 x(0) = v1s1 0 OF GLUONIC MODES IN A CONFINING x(T )=v2s2 + b ENVIRONMENT ba T T Consider the quantity defined by × P exp g dtJ · F + ig dtx˙ · B . 1 I(l) ≡ v v iTr G (l) (7) 2 − 2µ 1ν A µν B 0 0 µν Nc 1 ∞ In the above formula, (J · F ) = JαβF ,withJαβ N µν µν αβ µν = c v v dT Dx(t) 2 − 2µ 1ν being the generators for the spin-1 representation Nc 1 1) 0 x(0) = 0 of the Lorentz group. It should also be noted that x(T )=l ba c c ab we have employed the notation Bµ = Bµ(tG) = T −Bc f abc 1 1 µ . × exp − dtx˙ 2(t) Tr P N A To close this section, let us briefly comment on 4 c 0 gauge-symmetry-related issues. Given the “ideal- T T ized” process under consideration, our handling of × · · gauge invariance will be to let the two worldlines of exp g dtJ F + ig dtx˙ B , the “colliding” quarks extend to infinity in both direc- 0 0 µν B tions and impose the boudary conditions Aµ[x(t)] → where we have introduced lµ ≡ v2µs2 − v1µs1 − bµ.It 0 as tEucl →±∞. In this way, the overall worldline configuration introduces a Wilson loop in the path describes the propagation of the perturbative gluon modes in the presence of the background gauge field a 1)Its incorporation into the worldline form of the propagator modes Bµ and, in this sense, it is expected to incor- serves to signify the spin of the accomodated modes. porate confinement effects associated with the SVM. PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 5 2005 864 KARANIKAS, KTORIDES Generally speaking, one would expect that, in a Setting ui = x0 + αix(t), i =1, 2,onewrites study of a physically relevant process with the full 1 (and proper) inclusion of nonperturbative effects, no c c gFµ2 ν2 (u2)gFµ1 ν1 (u1) B (10) need would arise for the introduction of an infrared 2Nc cutoff to regulate the various expressions entering 1 = TrF φ(x0,u2)gFµ2 ν2 (u2) the computation at long distances. An infrared scale Nc should, in other words, naturally arise suppressing × φ(u ,x )φ(x ,u )gF (u )φ(u ,x ) contributions from very large distances (|l|→∞). 2 0 0 1 µ1ν1 1 1 0 B ≡ (2) − Given that the object of the present study is to in- ∆µ2ν2,µ1ν1 (u2 u1), vestigate perturbative/nonperturbative interference, where long-distance effects in the (nonphysical) process of quark–quark high-energy “collision” in the forward x0 direction, it becomes necessary to regulate infrared φ(x0,ui)=P exp ig dv · B(v) divergences associated with the upper limit of the T u integral. Our choice of introducing the infrared cutoff i is via the replacement and is unity in the FS gauge. Its insertion serves to ∞ ∞ underline the gauge invariance of the field strength − 2 correlator. (···)dT → dT e Tλ (···). With the above in place and upon making in 0 0 Eq. (8) the redefinition ti → Tti, i =1, 2,onedeter- On a simple dimensional basis and given the length mines scales entering the problem, one could make the as- 1 sociation λ ∝ σ|l|. v · v 2N 2 I(l)= 1 2 − c v · v dα α (11) 2| |2 2 − 1 2 2 2 Upon expanding the exponential term in Eq. (7), 4π l Nc 1 one obtains 0 1 1 1 ∞ v · v 2N 2 1 2 c 2 I(l)= − v2µv1ν (8) 2 −Tλ 2| |2 2 − × dα1α1 dt2 dt1θ(t2 − t1) dT T e 4π l Nc 1 ∞ T T 0 0 0 0 × −Tλ2 − 1 dT e dt2 dt1θ(t2 t1) 1 × Dx(t)exp− dtx˙ 2(t) 0 0 0 4 T x(0) = 0 0 x(T )=l × D −1 2 x(t)exp dtx˙ (t) v v 4 × 16 2µ 1ν ∆(2) (x(t ) − x(t )) x(0) = 0 0 µρ,ρν 2 1 v2 · v1 x(T )=l 1 1 (2) × · + x˙ µ2 (t2)xν2 (t2)˙xµ1 (t1)xν1 (t1)∆µ ν ,µ ν δµν TrF gx˙(t2) B(x(t2))gx˙(t1) T 2 2 2 1 1 Nc × − O 4 4 × 2 c [α2x(t2) α1x(t1)] + ( g F B). B(x(t1)) B + TrF gFµρ(x(t2)) Nc × c O 4 4 On a kinematic basis, the correlator can be repre- gFρν (x(t1)) B + ( g F B). sented as follows [6, 10]: Observe now that, in the FS gauge, the following (2) − ∆µ2ν2,µ1ν1 (z2 z1) (12) relation holds, 2 =(δµ µ δν ν − δµ ν δν µ )D(z ) Tr gB (x(t ))gB (x(t )) (9) 2 1 2 1 2 1 2 1 F µ2 2 µ1 1 B 1 ∂ 1 1 − 2 + (zµ2 δν2ν1 zν2 δµ2ν1 )D1(z ) 2 ∂zµ =(x − x ) (x − x ) dα α dα α 1 2 0 ν2 1 0 ν1 2 2 1 1 ∂ + (z δ − z δ )D (z2) . 0 0 ∂z ν2 µ2µ1 µ2 ν2µ1 1 × − ν1 TrF gFµ2ν2 (x0 + α2(x2 x0)) × gF (x + α (x − x )) , One notices [10, 11] that the first term enters as a dis- µ1ν1 0 1 1 0 B tinct feature of the non-Abelian nature of the gauge which brings into play the field strength correlator. symmetry (it is not present, e.g., in QED). According PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 5 2005 THEORETICAL EVIDENCE FOR A TACHYONIC GHOST-STATE CONTRIBUTION 865 to the premises of the SVM, it is associated with the tain 4 2 (QCD) string tension σ by [10, 11] d k · 1 µ I(l)=v · v eik l + . (16) ∞ 1 2 (2π)4 k2 k4 1 2 dz2D(z2)= d2zD(z2) ≡ σ. (13) π π The integral is infrared divergent and should require 0 the introduction of a corresponding cutoff.Alterna- tively, one could restrict the validity of the replace- Now, the central objective the present analysis is 2 2 to determine first-order contributions to the ampli- ment, according to Eq. (15), to the region k >µ . Then, one would determine tude coming, via the SVM, from the nonperurba- 4 2 tive/confining sector of QCD. The corresponding d k · 1 µ I(l)=v · v eik l + (17) lowest order correction is expected, on dimensional 1 2 (2π)2 k2 k4 2 2 grounds, to be of the order of the string tension σ. k >µ This implies, as already pointed out by Simonov [25], 2 2 v1 · v2 1 µ |l| 4 that we shall set aside the D term entering the 1+ ln 1 |l|2 (2π)4 4 eµ2|l|2 kinematical analysis of the correlator, according to Eq. (12), which cannot furnish contributing terms + O(µ4|l|4) for µ2|l|2 < 1. of dimension [m]2. In the Appendix, the following expression for the quantity I(l) is established: Comparing the above result with that of Eq. (14), one 1 v · v deduces I(l)= 1 2 (14) 4π2 |l|2 3 Nc µ2 =4ασ = (1 − κ)σ (18) σ π N 2 − 1 × 1+ασ|l|2ln C + O(σ2|l|4) , c λ2 2.15σ 0.4 GeV2, where in full accord with the phenomenologically deter- 3 Nc mined estimate for the tachyonic “pole” [20–22]. One α ≡ (1 − κ) 2 − ∼|| π Nc 1 also observes that λ l σ, as per our original suppo- sition. with the constant parameter estimated to be κ 0.5. Returning to the original full expression, which As pointed out by Simonov [21], the first—and most provides the full dynamical input for the amplitude, we important—term contributing to α comes from the write paramagnetic, attractive interaction between the spin +∞ of the gluons with the nonperturbative background 2 a a · field [cf. Eq. (5)]. It should also be noted that the Eii jj g tii tjj v1 v2 ds2 (19) constant C entering the argument of the logarithm −∞ is connected with the choice of parametrization of +∞ 2 1 σ D(z ) (see, e.g., [7]). In the context of a correspond- × ds 1+σ|l|2αln C ing result having to do with an amplitude for a phys- 1 4π2|l|2 λ2 ically relevant, hence protected from infrared diver- −∞ gences, process, then any such parameter would dis- +∞ +∞ 2 a a 1 appear. −g t t v · v ds ds ii jj 1 2 2 1 4π2|l|2 Suppose the following problem is now posed: −∞ −∞ Given the above results, which pertain to the gluon 2| |2 µ l 4 2 a a propagation in a confining environment, look for an × 1+ ln −g t t v · v 2| |2 ii jj 1 2 equivalent effective particle-like mode propagation, 4 eµ l summarizing their full content. In this spirit, we shall +∞ +∞ d4k 1 µ2 proceed to assess the possibility that the gist of all we × ds ds eik·l + . 2 1 (2π)2 k2 k4 have done up to here can be reproduced via the intro- −∞ −∞ duction of an effective propagator. Following [20], we k2>µ2 make the substitution Concerning the logarithmic factors entering the 2 above result, it is useful to remark that the various 1 → 1 µ 2 2 + 4 , (15) constants appearing in the arguments are not of any k k k particular importance—at least in the approximation whose additional term signifies the presence of a lin- we have been working—given that they would dis- ear term in the static potential. Then, one would ob- appear with the appropriate choice for the infrared PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 5 2005 866 KARANIKAS, KTORIDES cutoff. More importantly, thinking in terms of the sig- 4. SUMMATION OF LARGE LOGARITHMS nificance of the above results if they became part of an ∼ g2 1/b2µ2 amplitude corresponding to a physically consistent, The presence of terms ln ,entering 2 2 hence protected from infrared divergences, process, through the function f(b µ ), imposes the need of then any dependence from these constants should be their summation in the perturbative series. In the absent. absence of the background field, i.e., in the framework Going over to Minkowski space, the previous re- of pQCD, it is well known that such a summation can lation assumes the form be accomplished by employing the renormalization group strategies, which, for the quark–quark scat- − 2 a a · Eii jj g tii tjj iv1 v2 (20) tering process under consideration, can be justified +∞ +∞ 4 2 onthebasisthat1/b plays the role of an ultraviolet d k −ik·l 1 µ × ds2 ds1 e − + . cutoff. As Simonov has shown [11], the presence (2π)2 k2 k4 of the background field does not alter the Callan– −∞ −∞ k2>µ2 Symanzyk (CS) equation, relevant for the summa- The integration leads to the result tion. The physical basis on which this is so can be articulated by the following two arguments: +∞ +∞ − · − · (i) Contributions from the nonperturbative sector · ik v1s1 ik v2s2 v1 v2 ds2 ds1e (21) do not introduce additional divergences, given that −∞ −∞ they are finite at short distances. 2 =(2π) v1 · v2δ(k · v1)δ(k · v2) (ii) Dimension-carrying quantities arising from 2 the nonperturbative sector (correlators) are struc- =(2π) coth γδ(k )δ(k−), + tured in terms of combinations of renormalization where γ is determined by group invariant quantities gB; i.e., they behave as external momenta, as opposed to masses which are v · v s s/m2 1 cosh γ ≡ 1 2 = ⇒ γ (22) subject to renormalization. |v ||v | 2m2 1 2 Consequently, the called-for renormalization group ln(s/m2) ⇒ coth γ 1. evolution follows the footsteps of the procedure employed in the purely perturbative analysis of the It follows same process [5]. In this connection, it is recalled [2, g2 − a a 2 2 26] that the Wilson contour configuration associated Eii jj tii tjj i coth γf(b µ ), (23) 4π with Eiijj mixes with one corresponding to a pair where of closed loops resulting from an alternative way of identifying the points at infinity [5]. It is associated 1 1 µ2 2 2 2 ik⊥·b with f(b µ )= d k⊥e 2 + 4 . (24) π k⊥ k⊥ αS 2 2 E¯ = − c (γ coth γ − 1)δ δ (28) k⊥>µ ij ji π F ij ji 2 2 2 2 αS a One immediately notices that, if µ b 1,then × f(b µ )+ c (γ coth γ − 1 − iπ coth γ)t π F ij 4e f(µ2b2) ln , (25) × a 2 2 O 2 µ2b2 tji i coth γf(b µ )+ (αS). which recovers the known perturbative result—with Accordingly, and upon introducing, in shorthand ≡ ≡ an infrared cutoff given by λ2 ≡ µ2/(4e). notation, W1 δii δjj + Eii jj and W2 δij δji + 2 2 E¯ , the CS equation assumes the form As b grows, while remaining in the region µ b < ij ji 1,onefinds ∂ ∂ M + β(g) W = −Γ W , (29) 4e µ2b2 1 ∂M ∂g a ab b f(µ2b2) ln 1 − + µ2b2. (26) µ2b2 4 2 a, b =1, 2, In turn, this gives with M playing the role of the ultraviolet cutoff whose g2 running takes place between some lower scale (at − a a Eii jj tii tjj i coth γ (27) which corresponding initial conditons are set) and 4π 2 2 an upper scale set by 1/b. The anomalous dimension 4e µ b 1 2 2 × ln 1 − + µ b . matrix Γab, computed in the context of perturbation µ2b2 4 2 theory (see [2, 3, 5]), reads PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 5 2005 THEORETICAL EVIDENCE FOR A TACHYONIC GHOST-STATE CONTRIBUTION 867 iπ − coth γiπcoth γ Nc αS O 2 (Γab)= + (αS). (30) π iπ −γ coth γ +1+iπ coth γ +1 Nc(coth γ − 1) − coth γ Nc What does change, with respect to the perturbative 1 /b2 analysis, on account of the presence of nonpertur- × 2 2 dτ a a O 2 f(b0µ ) αB(τ)tii tjj + (αB). bative background contributions is the dependence τ 1/b2 of the Wa on the B-field correlators; i.e., one has 0 (n) Wa = Wa[{∆ },M,g]. Given that the computation It follows that has taken into account only the two-point correlator, 1 /b2 the extra dependence of the Wa will involve the string dτ α (τ)=α (1/b2)f(b2µ2) (34) tension [cf. Eq. (13)]. The inclusion of this additional τ B S dimensional parameter will have its effects on the 2 1/b0 running coupling constant. − 2 2 2 O 2 αS(1/b0)f(b0µ )+ (αS). With this in mind, let us recast Eq. (29) in integral Upon comparing Eqs. (27) and (33), one obtains form: µ2 4τ Wa[σ,M2,gB(M2)] (31) α (τ)=α (τ) 1+ ln − 2 + O(α2 ). B S 4τ µ2 S M2 dM (35) = P exp − Γ(gB(M)) M It should be noted that the validity of the above results M1 ab holds for τ/µ2 > 1 and × W [σ,M,g (M )] b 1 B 1 4π 1 αS(τ)= 2 < 1. with the path ordering becoming necessary because β0 ln(τ/Λ ) the anomalous dimension matrices do not commute An indicative estimate, on the basis of Eq. (35), is with each other. Concerning the integration limits, a that, if α 0.5,thenα 0.5(1 + 0.05). Follow- consistent choice, given the premises of the present S B ing [3, 5], one surmises that the amplitude A for the calculation, is to take M =1/b and set M =1/b 2 1 0 process under consideration behaves as with b2σ<1. It is observed that the nonperturba- 0 2 W 1 /b tive input enters a not only through their explicit ∼ −Nc s dτ O 2 dependence on the string constant, but also—which A exp ln αB(τ)+ (αS) is the most important—through a running coupling 2π m2 τ 1/b2 constant gB, which obeys the equation 0 (36) ∂ αS s M gB(M)=β(gB(M)). (32) ∝ exp − N ln f(b2µ2) , ∂M 2π c m2 The solution of the latter calls for initial conditions from which one reads a Reggeized behavior for the which are influenced by the presence of the non- gluon. The difference from the usual purely perturba- perturbative background and specifically by σ.Such tive result is that the function f(b2µ2) is now con- matters have been studied by Simonov in [25]. nected with the modified propagator, as per Eq. (24). Turning our attention to the “deformation” (to the In conclusion, we have demonstrated that the one-loop order) of the running coupling constant, on nonperturbative input, through the SVM, to the account of its additional dependence on the back- analysis of a hypothetical quark–quark scattering ground field, we proceed as follows. Knowing the process in the Regge kinematical region produces a result which, in a phenomenological context, has perturbative result to order αS, we go to Eq. (31) and present its solution in the form been argued to be extremely attractive in repro- ducing low-energy hadron phenomenology. In a W1[σ, 1/b, αB(1/b)] = δii δjj − i coth γ (33) sense, this investigation could be considered as a PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 5 2005 868 KARANIKAS, KTORIDES special example, which justifies Simonov’s more and ∞ general argumentation [21] according to which the 2 π − 2 perturbative–nonperturbative interference in static R(p; t ,t ,α ,α ) ≡ dT e Tλ (A.4) QCD interactions at small distances implies the 2 1 2 1 p presence of a linear term in the potential. 0 4 d q 2 × e−q /(4p)(δ δ − δ δ ) (2π)4 µ2ν2 µ1ν1 µ2ν1 µ1ν2 ACKNOWLEDGMENTS T We are grateful to our colleague Prof. F.K. Di- 1 × Dx(t)exp− dtx˙ 2(t) x˙ (t ) akonos for programming the computations for the 4 µ2 2 constants ci. x(0) = 0 0 We would also like to acknowledge financial x(T )=l × support through the research program “Pythagoras” xν2 (t2)˙xµ1 (t1)xν1 (t1) (grant no. 016) and by the General Secretariat of × exp[iq · [α2x(t2) − α1x(t1)]] Research and Technology of the University of Athens. with |l|,asdefined in the text. APPENDIX The above path integrals can be executed by employing standard techniques, given that “parti- cle” action functionals are quadradic (plus a linear Given the set of defining worldline formulas given term) [16–18]. Ignoring terms giving contributions by Eqs. (11)–(15) in Section 3, we proceed to derive O(b2) and using condensed notation from hereon, one Eq. (16). In the course of the derivation, the various determines quantities and parameters appearing in the last part ∞ 4 of the section are specified. 1 1 − 2 d q Q = dT e Tλ (A.5) Writing 16 p2 (2π)4 ∞ 0 2 2 −pz2 q D(z )= dpD˜(p)e , (A.1) × exp − − Tq2G [1 + O(l2q2)] 4p 12 0 one determines and ∞ 2 4 1 v · v 2N 1 1 − 2 d q I(l)= 1 2 c v · v (A.2) R = dT e Tλ (A.6) 2 | |2 2 − 1 2 2 4 4π l Nc 1 16 p (2π) 0 1 1 1 1 2 q 2 × dα2α2 dα1α1 dt2 dt1θ(t2 − t1) × exp − − Tq K 4p 12 0 0 0 0 2 2 4 2 2 ∞ × [c0 + Tq c1 + T q c2 + O(l q )], × dpD˜(p)[48Q(p; t2,t1) where the following one-dimensional particle-pro- 0 pagator-type quantities have been introduced: 4 4 ≡ − − − R(p; t2,t1,α2,α1)] + O( g F B), ∆12 =∆(t1,t2) t1(1 t2)θ(t2 t1) (A.7) + t2(1 − t1)θ(t1 − t2), where we have made the change ti → Tti, i =1, 2, and have introduced the quantities G = G(t ,t )=∆(t ,t )+∆(t ,t ) (A.8) ∞ 12 2 1 2 2 1 1 2 − | − | −| − | π −Tλ2 2∆(t1,t2)= t2 t1 (1 t2 t1 ) Q(p; t2,t1) ≡ dT e (A.3) p and 0 2 4 K12 = K(t2,t1)=α ∆(t2,t2) (A.9) × d q −q2/(4p) D 2 4 e x(t) 2 − (2π) + α1∆(t1,t1) 2α1α2∆(t1,t2). x(0) = 0 x(1) = l The coefficients entering Eq. (A.6) are given by the 1 expressions 1 2 × exp − dtx˙ (t) exp[iq · (x(t2) − x(t1))] ∂ ∂ 4T c0 = −72∆12 ∆12 ∆12, (A.10) 0 ∂t1 ∂t2 PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 5 2005 THEORETICAL EVIDENCE FOR A TACHYONIC GHOST-STATE CONTRIBUTION 869 c1 = 48∆12∂2∆12∂1K12 (A.11) REFERENCES +24α1∂1∆12∂2∆12(α1∆11 − α2∆12) 1.H.ChengandT.T.Wu,ExpandingProtons: Scat- − tering at High Energies (M.I.T.Press,Cambridge, +24α2∂1∆12∂2∆12(α2∆22 α1∆12) 1987). − 2 2 12(α1∆12∂1∆11∂2∆12 + α2∆12∂2∆22∂1∆12 2. L. N. Lipatov, Nucl. Phys. B 309, 379 (1988). − α α ∆ ∂ ∆ ∂ ∆ − α α ∆ ∂ ∆ ∂ ∆ ) 3. G. P. Korchemsky, Phys. Lett. B 325, 459 (1994); 1 2 12 1 11 2 22 1 2 12 1 12 2 12 I. A. Korchemskaya and G. P. Korchemsky, Nucl. − 12(α2∆22 − α1∆12)(α1∂1∆11∂2∆12 Phys. B 437, 127 (1995). 4. D. Kabat, Comments Nucl. Part. Phys. 20, 325 − α2∂1∆12∂2∆12) − 12(α1∆11 − α2∆12) (1992). × − (α2∂2∆22∂1∆12 α1∂1∆12∂2∆12) 5. A. I. Karanikas and C. N. Ktorides, Phys. Rev. D 59, and 016003 (1999). 205 − 6. H.G.DoschandYu.A.Simonov,Phys.Lett.B , c2 = 24(α2∆22 α1∆12) (A.12) 339 (1988). × (α1∆11 − α2∆12)∂1K12∂2∆12. 7. O. Nachtmann, Lectures Given at the 35th Inter- national University School of Nuclear and Par- Given the above, the “paramagnetic” contribution ticle Physics “Perturbative and Nonperturbative to Eq. (A.2) becomes Aspects of Quantum Field Theory”,Schladming, ∞ 1 Austria, 1996, HD-THEO-96-38. 2N 8. G. K. Savvidi, Phys. Lett. B 71B, 133 (1977). I =12 c v · v dpD˜(p) dt (A.13) p 2 − 1 2 2 9. N. G. Van Kampen, Physica 74, 215, 239 (1974); Nc 1 0 0 Phys. Rep. 24, 172 (1976). 1 10. A. Di Giacomo, H. G. Dosch, V. I. Shevchenko, and Yu. A. Simonov, Phys. Rep. 372 (2002). × − dt1θ(t2 t1)Q(p; t2,t1) 11. Yu. A. Simonov, Lect. Notes Phys. 479, 144 (1995); 0 Yad. Fiz. 65, 140 (2002) [Phys. At. Nucl. 65, 135 ∞ (2002)]. 2 · 12 2Nc v1 v2 dp ˜ 12. G. ’t Hooft, The Background Field Method in = 2 D(p) Gauge Field Theories, in “Karpacz 1975, Proceed- 16 Nc − 1 4π p 0 ings” (Acta Universitatis Wratislaviensis, Wroclaw, − p 1976), No. 368, Vol. 1, p. 345. × ln 4e γE + O(λ2/p) , λ2 13. V. A. Fock, Izv. Akad. Nauk USSR, OMEN, 557 (1937). and since 14. R. P.Feynman, Phys. Rev. 80, 440 (1950). ∞ ∞ dp 15. J. Schwinger, Phys. Rev. 82, 664 (1951). D˜(p)= dz2D˜(z2) (A.14) 16. M. J. Strassler, Nucl. Phys. B 385, 145 (1992). p 17. M. Reuter, M. G. Schmidt, and C. Schubert, Ann. 0 0 ∞ Phys. (N.Y.) 259, 313 (1997). 18. A. I. Karanikas and C. N. Ktorides, J. High Energy 1 2 2 2 = d zD˜(z ) ≡ σ, Phys. 9911, 033 (1999). π π 0 19. Yu. A Simonov and J. A. Tjon, Ann. Phys. (N.Y.) 300, 54 (2002). Eq. (A.13) gives 20. K. G. Chetyrkin, S. Narison, and V.I. Zakharov, Nucl. 3N v · v σ Phys. B 550, 353 (1999). I = c 1 2 σ ln C . (A.15) p N 2 − 1 4π2 λ2 21. Yu. A. Simonov, Phys. Rep. 320, 265 (1999). c 22.S.L.Huber,M.Reuter,andM.G.Schmidt,Phys. 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Original English Text Copyright c 2005 by Ilgenfritz, Martemyanov, M¨uller-Preussker, Veselov. Instantons or Monopoles: Dyons at Finite Temperature* E.-M. Ilgenfritz1), B.V.Martemyanov2)**,M.Muller-Preussker¨ 1),andA.I.Veselov2) Received May 21, 2004; in final form, September 22, 2004 Abstract—This article in honor of Yurii Antonovich Simonov’s 70th birthday reviews some recent work related to the semiclassical approach to QCD at finite temperature based on classical solutions with non- trivial holonomy. By cooling Monte Carlo generated lattice SU(2) gauge fields, we investigate approximate solutions of the classical field equations as a possible starting point for rethinking the semiclassical approxi- mation of the path integral. We show that old findings of cooling have to be reinterpreted in terms of Kraan– van Baal solutions with generically nontrivial holonomy instead of Harrington–Shepard caloron solutions with trivial holonomy. The latter represent only a subclass of possible topological configurations and for T 1. INTRODUCTION vacuum was mainly ignored by the physicists work- ing with the instanton model, while instantons were It is a great honor for us to be invited to give a mainly ignored by the confinement community, which tribute to Yurii Antonovich Simonov on the occasion concentrated instead on monopoles and vortices [10, of his 70th birthday. All of us, in one or another 11] (in the context of projecting Yang–Mills the- context, had the opportunity to work together with ory to U(1)N−1 and Z(N) Abelian gauge theories, him and gratefully remember this. More than that, his correspondingly). There have been claims for some ideas had an important impact on our thinking on the time [12] that random instantons could create a string vacuum structure of gluodynamics and QCD. tension of the right magnitude. Finally, among other not realistic features of the model pointed out, it be- For long time, the vacuum was imagined as a came clear that one of the main counterarguments random medium (dilute gas or less dilute liquid) of in- − raised by Yurii Antonovich against the model [13], stantons [1–5] described by a density of about 1 fm 4 the violation of approximate Casimir scaling in the and a radius ρ =0.3−0.5 fm, and this picture was instanton model, cannot be circumvented [14]. easily extended to finite temperature with the instan- Yurii Antonovich, who has designed a model- tons being replaced by Harrington–Shepard (HS) independent scheme for dealing with confinement, calorons [6, 7] and where the temperature acted to spectroscopy, and other hadronic features in terms regulate the infrared explosion with growing ρ (for of n-point field strength cumulants [15], with only the recent reviews, see [8, 9]). The starting point for the two-point correlator getting involved in the leading T =0model was the assumption of a repulsion be- (so-called Gaussian) approximation, from this point 2 tween instantons at distance d (d > const · ρ1ρ2)[4] of view has always been skeptical that instantons which seemed to solve many puzzles at once. This alone could give a description of all these aspects. idea then has opened the way to the instanton liquid Instead, he has pointed out how instantons would be picture of the vacuum [5]. This picture was rather modified on top of other, more infrared (“confining successful for doing hadron phenomenology [8], i.e., background”), fields without losing the role that they for understanding the behavior of light quarks. have to play for chiral symmetry breaking. This different starting point did not preclude that The importance of the existence of instantons for we together once estimated the strength and correla- the understanding of gluodynamics, however, is still tion length of the field strength correlator in the dilute under debate, and this makes us uncomfortable about instanton gas model [16], surprisingly with parame- the status of instantons. The confining aspect of the ters in the above-mentioned ballpark. In a follow-up ∗ paper to this work, we have found, however, that the This article was submitted by the authors in English. field strength correlator would not be defined correctly 1)Institut fur¨ Physik, Humboldt-Universitat¨ zu Berlin, Ger- many. without taking the interplay of instantons (instanton 2)Institute of Theoretical and Experimental Physics, Bol’shaya correlations) into account [17]. Cheremushkinskaya ul. 25, Moscow, 117259 Russia. During the 1990s, there were numerous attempts **E-mail: [email protected] to identify instanton-like structures, the density and 1063-7788/05/6805-0870$26.00 c 2005 Pleiades Publishing, Inc. INSTANTONS OR MONOPOLES 871 size (or, better, size distribution) of instantons, from monopole structure of the true configurations [33] ab initio lattice gauge field samples. This task turned nor the mesonic two-point functions [34] could be out to be much more difficult than expected. Apart reproduced on the reconstructed configurations based from the very last time, all these studies relied on on the (anti-)instanton sizes and loci alone. very subjective tools of smoothing the UV gauge field There were two ideas about what could be missing. fluctuations (cooling [18–20], blocking and inverse One of them focused on the eventual role of field blocking [21–23], APE smearing [24]). Whereas the strength correlations between the hot spots which existence of hot spots (very localized regions of large were not recordable by the cycling method. This could field strength, where it turns out self-dual or anti- have been measured by Yurii Antonovich’s gauge- self-dual) immediately caught the eye, the number of invariant field strength correlator, restricted to the these objects and the sizes were strongly dependent loci of instantons. However, smoothing and other on details and prejudices. Only recently, thanks to methods (“restricted overimproved cooling”)have the mastering of chirally improved [25] or chirally failed [35, 36] to detect these correlations. Thus, it perfect (so-called overlap) Dirac operators [26], the is fair to say that the uncorrelated instanton model hope emerged to reach the goal of an objective iden- correctly represents the result of cooling, provided the tification of lumps of topological charge density. But cooling is stopped at the right (average) multiplicity of still, the correct interpretation of the lumps is a task hot spots. The other idea was that a nontrivial holon- which poses new problems. omy in one of the four Euclidean directions might Yurii Antonovich, in the mid-1990s, proposed have escaped attention and therefore a corresponding his own model of the vacuum structure which was constant background Aµ should be recorded and used thought to replace the instanton model. One could to model the lattice field [34]. This construction was say that it started from a certain limit of the HS helpful for reconstruction of the mesonic correlators caloron which itself can be understood as a chain just empirically. At this stage, realizing a possible of instantons, enforcing the Euclidean (thermal) correlation between holonomy and the topological periodicity b =1/T , looked at for time t in the interval lump structure was out of scope, while the message 0 PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 5 2005 872 ILGENFRITZ et al. higher topological charge (even more so in generically Tc, we describe the search for (approximate) solutions non(anti-)self-dual fields), various nonlinear super- of the lattice equation of motion. We shall show that positions will be realized. Constituents of one caloron the nontrivial holonomy plays an important role and approaching each other lead to configurations which shall see solutions carrying all attributes of KvB solu- are also not static and resemble more instantons tions. But we shall also discuss some findings related despite the nontrivial holonomy. This is typically to by-products (“artifacts”) of the cooling process (for what happens if the inverse temperature becomes big instance, dyon–antidyon pairs, left over from partial compared to the distance between calorons. caloron–anticaloron annihilations), as well as of the Very recently, Diakonov [9] discussed the possible finiteness of the volume with torus topology (Dirac consequences of the KvB solutions as background sheets). In Section 6, we demonstrate under what cir- fields in the semiclassical approximation of the Yang– cumstances KvB solutions become dissociated into Mills path integral. Together with Petrov and young dyon pairs or recombine into calorons, the latter still coworkers [40], he went through the difficult task of having the internal nontrivial holonomy substructure. computing the one-caloron functional determinant We shall see that, even on a symmetric torus, instan- as a prerequisite for establishing a KvB caloron gas tons with nontrivial holonomy show up, for which the or liquid approximation. The hope is that the path analytic form is not yet known. Section 7 presents integral could finally be represented in terms of the the first results from ensembles obtained at higher dyon constituents instead of the full calorons. action plateaus. Section 8 presents the conclusion In the present review, we will describe the results and possible directions of future investigations. of investigations of calorons with nontrivial holonomy and of constituent dyons obtained by cooling of 2. CALORONS WITH NONTRIVIAL Monte Carlo lattice gauge field configurations [41– HOLONOMY IN CONTINUUM SU(2) 47]. While the new type of classical solutions with 2 2 YANG–MILLS THEORY an action S =8π /g0 does remember the physical (equilibrium) situation only through the nontrivial In the following, we consider the case of SU(2) holonomy it inherited from the confined phase, the pure gauge theory in continuum 4D Euclidean space. dependence on inverse temperature b and volume We assume periodicity of the gauge fields in the has important consequences for the rebuilding of fourth, i.e., imaginary time, direction, as it is required the caloron model, in particular, for its working near for the path-integral representation of the partition the confinement–deconfinement phase transition. function in the nonzero temperature case. Classical Higher action plateaus reflect more about the struc- instanton solutions periodic in x4 are therefore called ture of the equilibrium configurations, but this is not calorons [6] and have been constructed generalizing yet completely understood. It seems more important the one-instanton solution [1] in the singular gauge to study the Polyakov loop structure of lumps of with the help of the so-called ’t Hooft ansatz [51] action which become visible after APE smearing [24]. 1 τa a Aµ(x)= η¯µν ∂ν log φ(x), (1) Another way to reveal the dissociation into dyon con- g0 2 stituents is provided by the specific localization be- µ, ν =1,...,4, havior of zero modes of the Dirac operator depending on the boundary conditions imposed on the fermion with an infinite chain of equidistant singularities fields [48, 49]. Gattringer and his coworkers [50] have ∞ + ρ2 recently made a first attempt to identify such dyon φ(x)=1+ (2) − 2 − 2 constituents even in equilibrium lattice fields with the −∞ (x x0) +(t nb) help of the zero modes of a chirally improved lattice n= πρ2 sinh(2πr/b) Dirac operator. =1+ . This paper, in which we shall restrict ourselves br cosh(2πr/b) − cos(2πt/b) to the case of SU(2) lattice gauge fields, is orga- Here, η¯a denotes the anti-self-dual ’t Hooft tensor; nized as follows. In Section 2, we will give a the- µν oretical introduction to the objects under consider- τa/2, a =1, 2, 3, are the generators of the SU(2) | − | − 0 ation: instantons, calorons, calorons with nontrivial group; r = x x0 and t = x4 x4 determine the holonomy, etc. In Section 3, we shall brieflydiscuss spatial and temporal distance to the caloron center, the recent work by Diakonov and coworkers [40]. whereas b gives the time period to be identified with − Section 4 provides all necessary lattice definitions, the inverse temperature T 1; for simplicity, mostly in particular, the observables considered in order to b =1is assumed in what follows; ρ denotes the scale identify calorons with nontrivial holonomy. Then, in size of the solution. Section 5, at lowest action plateaus for temperatures The action and the topological charge of the near but mostly below the deconfining temperature caloron solution—integrated over one periodicity PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 5 2005 INSTANTONS OR MONOPOLES 873 strip of width b—correspond to the one-instanton previousonebyanangle4πω in color space, ω being ≡ 2 2 values S = Sinst 8π /g0 and Qt =1, respectively. the parameter of holonomy, as we will see below. The The anti-instanton solution with the opposite charge rotation axis is arbitrary; for definiteness, the third Qt = −1 is easily obtained by replacing the ’t Hooft axis is taken. The solution is made periodic by a a a tensor η¯µν by the self-dual one ηµν [2]. nonperiodic (in time) gauge transformation g(x)= −2πitωτ3 The gauge potentials of the (anti)caloron solutions e . The explicit expression of the KvB caloron fall off at spatial infinity, providing a trivial asymptotic solution looks as follows: holonomy, i.e., per 1 3 Aµ = η¯µν τ3∂ν log φ (7) b 2 1 1 2 P(x)=P exp i A (x,t)dt (3) + φRe((¯η − iη¯ )(τ1 + iτ2)(∂ν +4πiωδν,4)˜χ) 4 2 µν µν 0 + δµ,4 · 2πωτ3, →P∞ = ±1 ∈ Z for |x|→∞, 2 where whereas φ(x)=ψ(x)/ψˆ(x), (8) b ψ(x)=− cos(2πt)+cosh(4πrω¯)cosh(4πsω) P(x)=P exp i A (x,t)dt →∓1 (4) 4 r2 + s2 + π2ρ4 + sinh(4πrω¯) sinh(4πsω) 0 2rs for x → x − 0 + πρ2(s 1 sinh(4πsω)cosh(4πrω¯) holds complementary at the center of the (anti)calo- + r−1 sinh(4πrω¯)cosh(4πsω)), ron. The HS chain of instantons is aligned along the ψˆ(x)=− cos(2πt)+cosh(4πrω¯)cosh(4πsω) temporal direction with an identical orientation in r2 + s2 − π2ρ4 color space. In the large-scale limit, ρ →∞,and + sinh(4πrω¯) sinh(4πsω), after applying an appropriate gauge transformation, 2rs πρ2 it turns into a static solution which can be identified χ˜ = e−2πits−1 sinh(4πsω)+r−1 sinh(4πrω¯) with a BPS monopole [52] in Euclidean space, where ψ the fourth component A plays the role of the Higgs 4 ω ω¯ field in the adjoint representation. The solution has instead of Eq. (6). The holonomy parameters and are related to each other: ω¯ =1/2 − ω,0 ≤ ω ≤ 1/2; both electric and magnetic charge. This motivates its | − | | − | name “dyon” (D). r = x x1 and s = x x2 are the 3D distances to the locations of the two centers of the new caloron In order to generalize the solution to the case of solution. The distance between the centers d ≡|x − nontrivial holonomy, it is useful to rewrite the HS 1 x2| is connected with the scale size and the width of caloron (1), (2) in terms of separate “isospin” com- the time periodicity strip through ponents 2 1 πρ /b = d. (9) A (x)= η¯3 τ ∂ log φ(x) (5) µ 2 µν 3 ν The time component of the caloron potential becomes 1 1 − 2 nonzero at spatial infinity, providing the nontrivial + φ(x)Re (¯ηµν iη¯µν )(τ1 + iτ2)∂ν χ(x) asymptotic holonomy 2 with (b =1) b P(x)=P exp i A (x,t)dt (10) φ(x)=ψ(x)/ψˆ(x), (6) 4 πρ2 0 2πiωτ ψ(x)=cosh(2πr) − cos(2πt)+ sinh(2πr), →P∞ = e 3 for |x|→∞, r ˆ − ψ(x)=cosh(2πr) cos(2πt), whereas b 1 πρ2 sinh(2πr) χ(x)=1− = . P →± φ ψr (x)=P exp i A4(x,t)dt 1 (11) 0 A caloron with nontrivial holonomy [38, 39] is a for x → x solution with two spatial centers. It can be viewed— 1,2 in the limit (ρ/b 1)—as an instanton chain, where holds simultaneously at the two spatial centers x1, x2 each of the instantons is rotated compared to the of the KvB caloron [38, 53]. PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 5 2005 874 ILGENFRITZ et al. In terms of ω, the normalized trace of the holon- constant-time slice and watching its evolution over omy, the Polyakov loop, which we shall take as a the periodicity interval b [44]. direct measure of the holonomy, at spatial infinity Finally, let us comment on the zero-mode eigen- becomes functions of the fermionic massless Dirac operator in 1 1 the background of the KvB solutions. They have been L(x) ≡ trP(x) → L∞ ≡ trP∞ =cos(2πω), 2 2 studied analytically in [48] and [49]. One finds closed (12) solutions depending on the type of (anti)periodic boundary conditions (b.c.) imposed on the fermion whereas it shows a dipole structure with opposite fields in the imaginary time direction. In the case of ± peak and dip values 1 at the centers of the solution. well-separated dyon pairs, i.e., for d = πρ2/b 1, From Eqs. (8), it can be easily seen that, for trivial the zero-eigenmode densities become very simple asymptotic holonomy, i.e., for ω → 0 or ω¯ → 0,and expressions, by sending one of the positions of the centers to | − |2 − 1 2 infinity, the “old” HS caloron emerges as described ψ (x) = ∂µ [tanh(2πrω¯)/r] (14) by Eqs. (5), (6). 4π for antiperiodic b.c., On the other hand, for nontrivial asymptotic holonomy, we can distinguish two limiting cases. | + |2 − 1 2 ψ (x) = ∂µ [tanh(2πsω)/s] First, when the separation d between the centers 4π becomes sufficiently large, two well-separated con- for periodic b.c. stituents emerge which are static in time. The “mass” This means that the zero-mode eigenfunctions are ratio of these dissociated constituents is equal to always localized around one of the constituents of the ω/ω¯ . Since the full solution is self-dual, the ratio KvB solution, for antiperiodic b.c. at that constituent isthesamefortheactionasfortheequal-sign which has L(x1)=−1 at its center x1.Underthe topological charge carried by the constituents, the switching to periodic b.c. for the fermion fields, the latter summing up to one unit of topological charge, zero-mode localization jumps to the other constituent Qt =1. The separated constituents form a pair of monopole of the gauge field. Therefore, the fermionic BPS monopoles (or dyons) with opposite magnetic zero modes provide a convenient way to identify a charges. In the following, we will call this limiting monopole-pair structure in the gauge fields. case a DD pair. Second, in the small-distance limit, we shall observe just one lump of action and integer topological charge which looks very similar to the 3. A SEMICLASSICAL APPROACH HS caloron. Thus, we denote this nondissociated TO CONFINEMENT BASED ON THE KvB KvB solution by caloron. Still, this one-lump solution SOLUTIONS contains the typical internal dipole structure for the For two decades, the only known generalization of local Polyakov loop variable as mentioned above. the instanton solution to finite temperature was the The action density in all three cases described HS caloron [6]. In [7], alternatives were discussed above can be expressed by a simple formula [38] but soon abandoned. The argument against solu- tions with nontrivial asymptotic holonomy was the 1 s(x)=− ∂2∂2 log ψ(x). (13) following. The one-loop effective action obtained by 2 µ ν integrating out short-wavelength fluctuations on top of constant A4 (in the absence of a nonvanishing For well-separated dyons, when the functions electric or magnetic field) was known to be [55] φ(x) and ψ(x) are almost time-independent, the strongest time dependence comes from the first part 1/T 3 of the function χ˜(x). This dependence is represented Seff(A4)= dt d xP (φ) (15) −2πit by the phase e and is nothing else but the 0 homogeneous rotation of the first dyon, which has a a L(x1)=−1, in color space around the third axis with φ = A4A4 +1and with angle 2π over the period. The second dyon 1 P (φ)= φ2φ¯2. (16) with L(x2)=+1 is static. Such a relative rotation 3T (2π)2 of two dyons (which form a monopole–antimonopole pair) gives the so-called Taubes winding necessary to Here, φ is related to the holonomy parameter ω in- produce a unit topological charge from a monopole– troduced in the previous section by L =cos(2πω)= antimonopole pair [54]. One can detect this rotation cos(φ/(2T )). The notation φ¯ is shorthand to denote in a gauge-invariant fashion by looking at the gauge- 2πT − φ. For a single dyon or a caloron with asymp- invariant field strength correlator defined on each totically nontrivial holonomy ω, the consequence PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 5 2005 INSTANTONS OR MONOPOLES 875 22 |2πT−φ| would be a positive free-energy density fpert = P (φ) 3 2πT 3 Λ of the background, compared to the cases of trivial + |2πT − φ| , πT holonomy, ω =0 or ω =1/2. For the topological object under discussion, embedded in an infinite volume, this would mean that its contribution to the which becomes minimal at maximally nontrivial partition function is infinitely suppressed. holonomy. For sufficiently low temperature, it could This argument was questioned first by Diakonov be strong enough to overcompensate the positive in [9]. He pointed out that, independent of details perturbative free-energy density fpert. of the decomposition of the caloron amplitude into Very recently, the SU(2)-caloron amplitude was dyon amplitudes, the total free-energy density of a evaluated by Diakonov and coworkers [40]. Based dyon gas would eventually reverse the argument, on this knowledge, the conditions have been clarified i.e., overcompensate the positive free-energy density under which the dissociation into dyonic constituents in the region of nontrivial holonomy. A sketchy actually occurs. This is a necessary prerequisite for way to construct such a mechanism, in the case of the free-energy density eventually to acquire a form SU(2) gluodynamics, goes as follows [9]. Realize resembling (19). In [40], the temperature Tc was esti- that the asymptotic holonomy finally determines how mated at which trivial holonomy becomes unstable. the action (and topological charge) of the caloron The fluctuation determinants have been evaluated is distributed among the two dyonic constituents. by integrating the derivatives with respect to φ,be- Let us label them M and L, carrying magnetic − − ginning from φ =0(trivial caloron). The prefactor has and electric charges M =(+, +) and L =( , ). been fixed also by connecting the result to the limiting They are sources of self-dual fields. Correspondingly, ¯ − case of the trivial caloron dealt with in [7]. A closed an anticaloron is composed of M =(+, ) and expression has been obtained for the single-caloron ¯ L =(−, +), which are sources of anti-self-dual fields. amplitude written in terms of the dyon coordinates S = S The respective share of the total action inst is z1 and z2 in the limit of dyon separation r12 = |z1 − 2ωS and (1 − 2ω)S . Equal sharing is possi- inst inst z2|1/T : ble only in the background of maximally nontrivial 4 holonomy, ω =1/4. Assuming the factorization of 8π2 Z = d3z d3z T 6C (20) the caloron amplitude into that of a dyon pair, the KvB 1 2 g2 partition function of the dyon and antidyon gases γ 22/3 5/3 would be Λe E 1 φφ¯ × 2π + r12 1 4πT Tr12 T Zdyon = (17) 4φ¯ NM !NM¯ !NL!NL¯ ! 4φ −1 −1 NM ,NM¯ ,NL,NL¯ × (φr +1)3πT φr¯ +1 3πT 12 12 N +N ¯ 8π2 |φ| M M × − (3) − × d3z|φ|3 exp − exp V P (φ) 2πr12P (φ) , g2 2πT with a constant C ≈ 1 and P (φ) as in Eq. (16). The 3 3 × d z|2πT − φ| second derivative of the latter, 1 1 8π2 |2πT − φ| NL+NL¯ P (φ)= πT 1 − √ − φ (21) × exp − . π2T 3 g2 2πT 1 Then the dyon pressure × φ¯ − πT 1 − √ , 3 2 log Zdyon 8π |φ| T =2|φ|3 exp − (18) determines the leading dyon–dyon interaction, which V g2 2πT is a pure quantum effect. Λ is the scale parameter of 8π2 |2πT − φ| the Pauli–Villars regularization scheme. When the +2|2πT − φ|3 exp − g2 2πT dyon separation is much larger than their core sizes, /φ > /T r /φ>¯ /T would give a negative contribution to the free-energy r12 1 1 , 12 1 1 , the expression simplifies further to density. Invoking a running coupling depending on temperature T in the usual way would lead to a dyon 8π2 4 free-energy density Z = d3z d3z T 6(2π)8/3C (22) KvB 1 2 g2 22 |φ| Λ 3 2πT 22 4φ 4φ¯ f = −cT |φ|3 (19) ΛeγE 3 φ 3πT φ¯ 3πT dyon πT × 4πT 2πT 2πT PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 5 2005 876 ILGENFRITZ et al. (3) × exp −2πr12P (φ) exp −V P (φ) , to undergo the deconfinement phase transition at a critical coupling βc 2.299 [56]. where the terms (φ/(2πT))4φ/3πT and We generate the quantum gauge field ensemble 4φ/¯ 3πT { } φ/¯ (2πT) can be understood as fugacities Ux,µ by simulating the canonical partition function using the standard heat-bath Monte Carlo method. of M and L dyons, respectively. The derivation The equilibrium field configurations will be cooled by allows one to estimate a linear-plus-Coulomb-like iteratively minimizing the action S. Usually, cooling interaction between the constituents and relates the (leading) linear one to the sign of P (φ).For|L| > in one form or another is used in order to smooth out short-range fluctuations, while (initially) leaving 0.787597, it is attractive and the integral over r 12 some long-range physics intact. The cooling method converges and permits one to define the fugacity of applied here is the standard relaxation method de- the compound caloron. Based on this knowledge, scribed a long time ago in [18]. in [40], the critical temperature Tc =1.125Λ has been estimated as the temperature below which trivial This method, if applied without any further limita- holonomy becomes unstable. At this temperature, tion, easily finds approximate solutions of the lattice the gluodynamic system starts to relax to larger φ. field equations as shoulders (plateaus) of action in the For |L| < 0.787597, the dyon interaction becomes relaxation history. Under certain circumstances, this repulsive, and the caloron dissociates into separate defines and preserves the total topological charge of dyons. aconfiguration. However, the short-range structure of the vacuum fields is changed. Still, the type of This fully dissociated case, however, is not covered classical solutions which are being selected depends by the derivation given in [40]. The statistical me- on the phase which the quantum ensemble {U } is chanics of the fully dissociated dyon gas still needs x,µ meant to describe. We want to investigate smoothed to be developed more in depth in order to justify the sketchy scenario leading to a free-energy density (19) fields at different stages of cooling by using a stopping criterion which selects the plateaus in a given interval with holonomy becoming stabilized at ω =1/4 (con- of action. finement). The smoothed fields have been analyzed according to the spatial distributions of the following observ- 4. LATTICE OBSERVABLES ables: USED TO DETECT DYONS AND CALORONS Action density computed from the local plaquette Now we are going to describe what lattice gauge values: theory can tell about the holonomy of semiclassical s(x,t)= s(x,t; µ, ν). (24) background fields. We concentrate on SU(2) gauge µ<ν fields discretized on an asymmetric lattice with peri- odic boundary conditions (p.b.c.) in all four directions. Topological density computed with the standard In a very few cases, also fixed holonomy boundary discretization and averaged over the time variable: conditions (f.h.b.c.) have been implemented. For the 1 1 latter, all timelike links at the spatial boundary Ω q (x)=− (25) t N 24 · 32π2 are fixed to equal Abelian values such that L(x)= t 0 for x ∈ Ω in accordance with the vanishing order ±4 parameter in the confinement phase. × /µνρσtr [Ux,µνUx,ρσ] . The respective ensembles of configurations have t µ,ν,ρ,σ=±1 been created by heat-bath Monte Carlo using the 2 Polyakovloop defined as standard Wilson plaquette action (β =4/g0 ), N 1 t S = s(x,t)= s(x,t; µ, ν), (23) L(x)= tr U . (26) 2 x,t,4 x,t x,t µ<ν t=1 1 s(x,t; µ, ν)=β 1 − trUx,µν , Abelian magnetic fluxes and monopole charges 2 defined within the maximally Abelian gauge (MAG). † † Ux,µν = Ux,µUx+ˆµ,ν Ux+ˆν,µUx,ν. The latter is found by maximizing the gauge func- tional The lattice size will be N 3 × N with varying spatial s t 1 g g† −1 ≡ A[g]= tr(U τ3U τ3) (27) extension Ns =8to 24 and with T Nt =4, 5,or 2 x,µ x,µ x,µ 6.ForNt =4,forinstance,themodeliswellknown PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 5 2005 INSTANTONS OR MONOPOLES 877 g with respect to gauge transformations Ux,µ → Ux,µ = Mean violation of the equations of motion † g(x)Ux,µg (x +ˆµ). Abelian link angles θx,µ are then 1 1 † ∆= tr (U − U˜ )(U − U˜ ) , defined by Abelian projection onto the diagonal U(1) 4N 3N 2 x,µ x,µ x,µ x,µ s t x,µ part of the link variables Ux,µ ∈ SU(2). According to (30) the DeGrand–Toussaint prescription [57], a gauge- ¯ † invariant magnetic flux Θp through an oriented pla- U˜ = c U U U ≡ x,µ x,ν x+ˆν,µ x+ˆµ,ν quette p (x, µν) is defined by splitting the plaquette ν= µ − − ¯ Θp = θx,µ + θx+ˆµ,ν θx+ˆν,µ θx,ν into Θp = Θp + † + U U − U − 2πnp, np =0, ±1, ±2,suchthatΘ¯ p ∈ (−π,+π].The x−ν,νˆ x ν,µˆ x+ˆµ ν,νˆ (quantized) magnetic charge of an elementary 3-cube is the local link variable representing the solution of c is then the lattice equation of motion, with all degrees of 1 m = Θ¯ , freedom coupled to it held fixed. The factor c is just c 2π p p∈∂c a normalization of the staple sum such that U˜x,µ ∈ → ˜ where ∂c is the boundary of this cube. SU(2). The replacement Ux,µ Ux,µ is exactly the local cooling step as applied throughout this paper. Eigenvalues and eigenmode densities of the non-Hermitian standard Wilson–Dirac operator D[U] ψ (y)=λψ (x), (28) 5. SIMPLEST SOLUTIONS OF THE SU(2) xrα,ysβ sβ rα LATTICE FIELD EQUATIONS y,s,β In the first part of our investigation, we have D[U] = δxyδrsδ xrα,ysβ αβ searched for topologically nontrivial objects with the − { − µ κ δx+ˆµ,y(1D γ )rs(Ux,µ)αβ lowest possible action, i.e., at S Sinst and below, µ late in the cooling history, in order to find systematic µ † dependences of the selected solutions on the spatial + δy+ˆµ,x(1D + γ )rs(Uy,µ)αβ , boundary conditions and on the temperature of the original Monte Carlo ensemble. First, we started from are studied with both time-antiperiodic and time- equilibrium configurations generated in the neighbor- periodic b.c. For our purposes, i.e., for demonstrating hood of the deconfinement phase transition. Later on, the qualitative properties of a caloron solution, on we lowered the temperature by increasing the lattice smooth lattice fields it will be sufficient to consider extent in the time direction. Cooling was always this operator which explicitly breaks chiral invariance. stopped on plateau values of the action S<1.5S We find the λ spectrum and the eigenfunctions with inst as a function of the number of iteration steps. We the help of the implicitly restarted Arnoldi method have selected those configurations for which the mean and use the standard ARPACK code package for this violation of the equations of motion ∆ according to aim [58, 59]. Eq. (30) passed a low-lying minimum. Nonstaticity δt,isdefined as For each β,wehavescannedthetopological Nt content of several hundred cooled configurations. At δt = |s(x,t+1)− s(x,t)|/S. (29) 4 Nt =4 and β =2.2, i.e., in the confinement phase x,t (T 0.8Tc), for both kinds of boundary conditions The normalization factor Nt/4 has been chosen such (p.b.c.andf.h.b.c),wehavemostfrequentlyfound that δt will not change for a given KvB solution dis- objects which can be identified as KvB solutions cretized on lattices with a varying time extent Nt.It in a nondissociated form or dissolved into a dyon allows one to discriminate between caloron and DD pair. Below, we shall demonstrate that they indeed solutions taking into account that δt depends mono- show all known features of KvB solutions despite tonically on the spatial distance d between the two the fact that, on a 4-torus, solutions of topological centers of the solution. The “bifurcation” value for charge Qt = ±1 do not exist in a mathematically the recombination of dyon pairs into single calorons strict sense [60]. The frequency of appearance of DD ∗ is δt 0.27 for analytic, lattice-discretized KvB so- vs. that of calorons in the ensemble of the detected lutions of maximal nontrivial asymptotic holonomy KvB solutions will be discussed later. Compared with L∞ =0 for which the constituent dyons get equal the confinement case, we have seen self-dual KvB ∗ masses. If δ<δ , two dyons can be distinguished by solutions to become strongly suppressed at Nt =4, t ∗ the two maxima of the action density. For δ>δt ,the β =2.4, i.e., at T 1.3Tc. dyons will appear recombined into a caloron with only Moreover, for T ≤ Tc at the level of one-instanton one action density maximum. action values, we have detected configurations which PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 5 2005 878 ILGENFRITZ et al. have vanishing total topological charge, Qt =0,and content of the field configurations under inspection. consist of two distinct topological lumps of opposite For static DD solutions, we always find a pair of static charge. They are static in time and can be easily (anti)monopoles with world lines coinciding with the interpreted as DD¯ pairs. By tracing back the cooling centers of the dyons. history, we have convinced ourselves that such pairs (ii) Caloron configurations: In Fig. 2, we show can occur through a partial annihilation of D with D¯ a typical caloron solution, with an approximately in superpositions of DD pairs with D¯D¯ pairs. These 4D rotationally invariant action distribution, also non-self-dual DD¯ configurations will be discarded obtained at β =2.2 from cooling with p.b.c. Again, from our further discussion in this section. we plot 2D cuts for the topological charge density and for the Polyakov loop and present the plot for Definitely, below the one-instanton action level, fermionic eigenvalues together with the 2D cuts we have also detected very stable but purely Abelian of eigenmode density for the distinct real eigen- objects which in an early work by Laursen and value. The full topological charge Qt is unity. But Schierholz [61] were interpreted as ’t Hooft–Polyakov most important for us is the observation of narrow monopoles thought to characterize dominantly the opposite-sign peaks of the Polyakov loop (the “dipole deconfinement phase. We shall show below that such structure”), which shows that this caloron possesses configurations are more likely extended Dirac sheets a nontrivial asymptotic holonomy different from “old” (DS), which, in the thermodynamic limit, should not HS calorons, i.e., intrinsic dyon–dyon constituents. play a role in the Euclidean path integral. The fermionic zero modes for time-periodic and time- (i) DD pairs: For a special DD solution found with antiperiodic b.c. for this configuration are only slightly p.b.c., we show in Figs. 1a and 1b 2D cuts of the shifted relative to each other. A reasonable fit with the topological charge density qt(x) and of the Polyakov analytic solution can be obtained, showing that this loop distribution L(x), respectively. We clearly see caloron is nothing but a limiting case of the generic the opposite-sign peaks of the Polyakov loop vari- KvB solutions. The typical caloron configurations able correlated with the equal-sign maxima of the show, after putting them into MAG, a closed Abelian topological charge density. The boundary values of monopole loop circulating around the maximum of the Polyakov loop vary slightly because they are not the action density in the 4D space. fixed here to a well-defined value. This is the only At that point, we may conclude that cooling, even difference observed between the two types of b.c. In with nonfixed holonomy at the spatial boundary, yields principle, for p.b.c., the holonomy can be arbitrary. As almost-classical solutions which show all character- a consequence, the ratio of the action carried by the istics of the KvB calorons. well-separated dyon constituents can take any value. (iii) Dirac sheets: Finally, by cooling with both For the same DD solution, Figures 1c and 1d show kinds of spatial b.c., we have found objects becoming the scatter plot of the 70 lowest complex eigenvalues very stable against cooling at even lower action, i.e., of the Wilson–Dirac fermion operator according to S PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 5 2005 INSTANTONS OR MONOPOLES 879 (a) (b) ×10–8 0.6 6 4 0.2 2 –0.2 0 15 10 15 10 15 5 10 10 0 5 0 5 0.2 (c) 0.2 (d) 0 0 –0.2 –0.2 –0.10 –0.08 –0.06 –0.08 –0.07 –0.06 –0.05 (e) (f) ×10–3 ×10–3 2 2 1 1 0 0 10 15 10 15 10 10 0 5 0 5 Fig. 1. Various portraits of a self-dual DD pair obtained by cooling under periodic gluonic boundary conditions. The subpanels show appropriate 2D cuts of the topological charge density in lattice units (a) and of the Polyakov loop (b), the plot of lowest fermionic eigenvalues (Re, Im) (c, d), and the 2D cut of the real-mode fermion densities in lattice units (e, f) for the cases of time-periodic (c, e) and time-antiperiodic (d, f) fermionic boundary conditions, respectively (β =2.2 and lattice size 163 × 4). 2 ≈ 1 3 2 3 8π Nt Nt follows, we shall concentrate on (anti-)self-dual con- 2 (Bx) Ns Nt = 2 = Sinst . 2g0 g0 Ns Ns figurations of the KvB type for action values S Sinst. All DS events found in the confined phase for Ns =20 (when put into MAG) show exactly such an Abelian 6. DYON PAIRS AND THEIR magnetic field. For smaller Ns in the confined phase, RECOMBINATION INTO CALORONS the fluxes are not always completely homogeneous The possibility of observing the dyonic con- and absolutely stable, whereas the fluxes are definitely stituents of a KvB caloron as separate lumps of action unstable for all Ns in the deconfined phase. As we depends on the value of (πρ/b)2.InSU(2), lattice have shown in [47], this stability pattern [63, 64] is gauge theory (LGT) at β =2.2 on a lattice 163 × 4 directly related to the nontrivial holonomy observed fits KvB solutions provided ρ ≈ 2.5a (a is the lattice within the confinement√ phase (stability is guaran- spacing) [42]. With b =4a,wehave teed for |L| < cos( πN /N )) and to the trivial one t s 2 ≈ (L∞ = ±1), which naturally occurs in the deconfine- (πρ/b) 4 1. ment phase. This means that dyons become well-separated. On a Anyway, the DS configurations should not con- lattice 163 × 6 (with b =6a) and at the same β =2.2 tribute in the thermodynamic limit. Therefore, in what (i.e., at a temperature 1.5 times lower), the parameter PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 5 2005 880 ILGENFRITZ et al. (a) (b) ×10–7 1.5 –0.6 1.0 0.5 –0.8 0 –1.0 10 15 10 15 10 10 0 5 0 5 (c) 0.2 (d) 0.2 0 0 –0.2 –0.2 –0.05 0 0.05 –0.10 –0.08 –0.06 –0.04 (e) (f) ×10–3 ×10–4 4 6 4 2 2 0 0 10 15 10 15 10 10 0 5 0 5 Fig. 2. As Fig. 1, but for self-dual caloron configuration. (πρ/b)2 would be of the order O(1). Then, from this The decrease in action has slowed down to simple arithmetic, one would expect that calorons are |∆S|/Sinst < 0.05. not dissociated anymore into separate dyonic lumps. The action fits into the window 0.5 PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 5 2005 INSTANTONS OR MONOPOLES 881 SDS/Sinst 0.5 0.6 Nt/Ns 0.4 0.4 0.3 Nt = 4, F = 0.162 = 5, = 0.153 0.2 0.2 Nt F Nt = 6, F = 0.152 2 3 4 5 Ns/Nt 0 Fig. 3. Action of DS events in the confined phase (trian- –1.0 –0.5 0 0.5 1.0 gles, β =2.2)andinthedeconfined one (circles, β =2.4) Las for Nt =4as a function of the ratio Ns/Nt.Thecurve presents Nt/Ns. Fig. 4. The (normalized) distribution of the asymptotic holonomy L∞ ≡ Las for cooled configurations at one- line L∞ =0. If one performs cooling without spe- instanton action plateaus corresponding to three temper- cial restrictions concerning the holonomy, there is atures below deconfinement at Nt =4, 5,and6 (β =2.2, no guarantee that the asymptotic holonomy of the Ns =16). caloron configurations still coincides with the average Polyakov line of equilibrium configurations in con- finement. In order to define an approximate asymp- a considerable fraction of cooled configurations have totic holonomy L∞ for each cooled configuration, developed an asymptotic holonomy |L∞| > 1/6.In we have determined the average of L(x) over a 3D Fig. 5a, we show the probability distribution over δt subvolume where the local 3D action density s(x) for cooled configurations (with an action at the one- is low; for definiteness, s(x) < 10−4.InFig.4,we instanton plateau) without the cut with respect to the | | present the distributions of the cooled configurations asymptotic holonomy L∞ . One can see that a rel- over L∞ for the three cases Nt =4, 5,and6.In atively high fraction of configurations, obtained from the Monte Carlo equilibrium with Nt =4,haveδt < the legend, we show the respective volume fraction ∗ (F ≈ 0.15) of the 3D volume (i.e., far from the lumps δt =0.27. This means that they would be identifiable of action and topological charge) over which the as consisting of two constituents by looking for the “asymptotic” value L∞ is defined as an average. 3D action density on the lattice. For Nt =5,there are only a minority of cooled configurations which As explained above, the nonstaticity δt isamea- sure which describes the distance from a perfectly fall below the threshold δt =0.27. No static config- (Euclidean) time-independent configuration. In other urations (according to the nonstaticity criterion) have words, the distributions of nonstaticity of caloron been found among cooling products at Nt =6.We events obtained by cooling of equilibrium lattice con- have repeated the same analysis after applying the | | figurations can be considered as a substitute for the cut with respect to the asymptotic holonomy, L∞ < distribution in dyon distances d. This quantity can 1/6.Thenwegetmodified histograms in δt for the be directly measured for cooled lattice gauge field three temperatures. This is shown in Fig. 5b. The his- tograms got more pronounced peaks in δt,whichare configurations. We show the δt distributions for all ∗ our cooling products obtained at β =2.2 on 163 × positioned around 0.125, exactly around δt =0.27, andaround0.5forN =4, 5,and6, respectively. Nt lattices in Fig. 5. In an attempt to make a fair t comparison with calorons with nontrivial holonomy There are other criteria which could be used to and to correct for the possible evolution of the asymp- characterize a more or less static configuration, for totic holonomy away from L∞ =0during the cooling example, the presence of static Abelian monopoles process, we defined a subsample by the requirement emerging in the maximal Abelian projection. By |L∞| < 1/6. One can see that the cut with respect selecting the subsample of cooled configurations to the asymptotic holonomy selects cooled config- containing a static Abelian monopole–antimonopole urations from the flat central part of the histogram pair, we can determine the distance of the caloron showninFig.4.Ontheotherhand,wenoticethat constituents d through the distance R between these PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 5 2005 882 ILGENFRITZ et al. Nt = 4, Nt = 5, Nt = 6, (a) (b) 6 6 4 4 2 2 δ δ 0 0.4 0.8 1.2 t 0 0.4 0.8 1.2 t Fig. 5. Histograms (of nonnormalized distribution) for the nonstaticity δt after cooling, for three temperatures below the deconfinement transition with Nt =4, 5,and6 (β =2.2,Ns =16): (a) without any cut; (b)whenacut|Las| < 1/6 is applied ∗ (Las ≡ L∞). The thick vertical line marks the value δt where the caloron recombines. (anti)monopoles. We make sure that there is a clear amount of presence of opposite-sign Polyakov lines anticorrelation between R and δt as long as static representing eventually two different constituents in- Abelian monopoles are found. Practically all cooled side the same lump of action. Figure 6a shows how ∗ configurations with δt <δt possess static Abelian |Ltot| changes with δt. For the temperature nearest to monopole–antimonopole pairs, the locations of which the transition, at Nt =4, we see that |Ltot| falls from fall close to the opposite peaks of the Polyakov loop ≈4.0 to ≈1.0 at δt ≥ 0.5. We interpret this such that, ± ∗ L = 1. Above δt , however, the fraction, as well as in the region where constituents can be well separated the distance R, rapidly goes to zero. For Nt =6, i.e., according to the action density (at small δt), they are at lower temperature, at the peak value around δt = characterized by a relatively smooth change of the 0.5,only20% of the solutions are still characterized by Polyakov line inside. In the region of large δt where a static pair of Abelian monopoles, whereas at higher they are not separable according to the action density, nonstaticity this is never the case. the Polyakov line changes rapidly in the neighbor- In order to represent how the Polyakov line be- hood of the absolute maximum of action density. For haves inside a lump of action, we have explored the lower temperatures, Nt =5 and 6, the relationship neighborhood of the absolute maximum of the 3D between these properties of an action cluster and δt action density (denoted as central point x0). This is the same; however, separable lumps of action (at position could either be one of the two dyonic lumps low δt) become very rare. Figure 6b shows how the (as long as they are separable) or the center of a re- “dipole moment” |Mtot| of the Polyakov line around combined caloron. For this purpose, we have defined a maximum of action density rises with increasing a locally summed-up Polyakov line Ltot (including the nonstaticity δt. In the region where one can sepa- central point xi (i =0) and its six nearest neighbors rate the constituents according to the action density, xi (i =1,...,6)), the “dipole moment” is small, emphasizing again the 6 homogenity of the Polyakov line around the central point. In the region beyond δ∗, the dipole moment Ltot = L(xi), (32) t i=0 gradually stabilizes around a value of 1.5. and a Polyakov line “dipole moment” over the same In the same way as described so far, we have set of 3D lattice points with respect to the central analyzed configurations obtained by cooling at higher point, action plateaus. Although the separation between static and nonstatic does not have the clear meaning 6 as for the one-caloron case, the trend is the same: − Mtot = L(xi)(xi x0). (33) at lower temperature, the lumps of action tend to be i=1 more localized also in Euclidean time (“instanton- like”). The absolute value |Ltot| of the firstquantityteststhe amount of local coherence of the Polyakov line. The We have asked whether our findings at lowest ac- absolute value |Mtot| of the second quantity tests the tion cooling plateaus do really scale with the physical PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 5 2005 INSTANTONS OR MONOPOLES 883 | | Nt = 4, Nt = 5, Nt = 6, Las < 1/6 | | | | Ltot Mtot 2.0 5 (a) (b) 4 1.5 3 1.0 2 0.5 1 δ δ 0 0.4 0.8 1.2 t 0 0.4 0.8 1.2 t Fig. 6. The Polyakov line in the neighborhood of maxima of the action density: (a) modulus of the average summed-up Polyakov line |Ltot| (Eq. (32)) and (b) modulus of the corresponding “dipole moment” |Mtot| (Eq. (33)) vs. nonstaticity δt ∗ for the subsample with asymptotic holonomy near zero. The thick vertical line marks the nonstaticity δt where the caloron recombines (β =2.2, Ns =16). temperature. Therefore, at fixed lattice size, we have intersecting the lump through the center. The latter is varied β within the confinement case (e.g., for Nt =6, defined as the maximum of the 4D action density. For we compared β =2.2 with 2.36). We did not find a all types of Polyakov lines, the characteristic double significant change in the δt distributions as long as structure is seen exactly when the “asymptotic” value we stayed within the same phase. This shows that our of the respective Polyakov line is not close to ±1. Still, cooling results for the one-instanton action plateaus it is not clear how to construct analytically an instan- do not really depend on the physical temperature of ton solution which behaves nontrivially with respect the original Monte Carlo equilibrium gauge fields but to holonomy in all four directions. A KvB solution rather on the nonsymmetric geometry of the lattice. discretized on a symmetric lattice, for comparison, This will certainly change when we investigate higher shows nontrivial holonomy only for the Polyakov loop action plateaus with improved methods to remove ul- in one, i.e., the “time,” direction. traviolet fluctuations (improved cooling, smoothing, or smearing methods). 7. ENSEMBLES OF KvB SOLUTIONS We have also cooled equilibrium configurations AT HIGHER ACTION PLATEAUS generated with β =2.2 on symmetric lattices (164 ≤ representing “zero” temperature). In this case, we Within the confinement phase, for 0 PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 5 2005 884 ILGENFRITZ et al. s q 0.04 0.02 15 15 0.02 0.01 0 10 0 10 y y 5 5 5 5 x 10 x 10 15 0 15 0 plt plz 0.6 –0.2 0.4 15 –0.6 15 0.2 10 –1.0 0 10 y 0 y 5 5 5 5 x 10 x 10 15 0 15 0 plx ply 0 0.5 15 0 15 –0.5 10 –0.5 10 0 z 0 z 5 5 5 5 y 10 x 10 15 0 15 0 Fig. 7. Profiles of the action density (s), the topological charge density (q) (all in lattice units), and the Polyakov lines (plt, plz, plx, ply) calculated along all straight line paths parallel to the four axes for a 164 lattice caloron found by cooling a Monte Carlo generated equilibrium gauge field (β =2.2) down to the one-instanton action plateau. The center of the caloron (at the maximum of its action density) was found at the site (x, y, z, t)=(7, 8, 8, 14).Theplanesshowninthefigures cross just at this point. subsequent action windows. We have been monitor- violation of the classical lattice field equations per link ing the landscape of topological density and of the [see Eq. (30)]. Polyakov line operator as well as the localization On the first visible plateau, we find a gas of lo- of the fermionic zero modes in the semiclassical calized lumps carrying topological charge, where an candidate configurations. Correspondingly, we have identification in terms of dyons D and/or antidyons D¯ been triggering now by the condition is still difficult. (m − 1/2)Sinst PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 5 2005 INSTANTONS OR MONOPOLES 885 A A' 0.06 0.2 0.03 0 0 –0.2 0 0 10 10 20 20 20 10 20 10 B B' 0.06 0.2 0.03 0 0 –0.2 0 0 10 10 20 20 20 10 20 10 C C' 0.06 0.2 0.03 0 0 –0.2 0 0 10 10 20 20 20 10 20 10 Fig. 8. Configurations after 800 (A, A ), 1650 (B, B ), and 7000 (C, C ) cooling steps. A, B, C show 2D projections of the topological charge density qt(x) (in lattice units) and A ,B,C of the Polyakov loop L(x), respectively. Cooling has been employed with f.h.b.c., L(x)=0for x ∈ Ω (Ω is the 3D volume boundary) (β =2.2, lattice size 243 × 4). plateaus, we have stopped at iteration steps A (for strongly violating the equations of motion (as can be m =4), B (for m =3), and C (for m =2), respec- monitored by following ∆). The average violation of tively, where the mean violation ∆ of the lattice field the equations of motion peaks immediately before the equations passed through local minima. The corre- configuration drops to the next plateau. This example sponding semiclassical field configurations are then shows that we have superpositions of noninteger Qt displayed in Fig. 8 by means of the 2D-projected lumps, which can be interpreted as described in the (summed over the third coordinate) spatial topolog- previous section. To make sure that this is really the ical charge density and the 2D-projected Polyakov case, we also provide the eigenvalue scatter plots loop distribution. One can recognize in these figures and the density (ψψ¯ )(x) for the lowest modes of the that, at stage A, we have a superposition of six dyons Wilson–Dirac operator for stage B (see Fig. 9). and two antidyons. The topological charge sector has We have also studied the Abelian (anti)monopole DD¯ been independently determined to be Qt =2.A structure after MAG fixing and Abelian projection pair decays or annihilates from A to B such that we and have seen a strong correlation of the peaks of the have five dyons and one antidyon at the next stage. Polyakov loop with the positions of the (anti)monopo- The topological charge did not change. Finally, stage les. The pair structure in terms of Abelian monopoles, C exhibits a superposition of four dyons, again with occurring on all action plateaus, as well as the annihi- Q =2. The latter configuration is very stable. While t lation of single (monopole–antimonopole) pairs dur- it stays at the same action over thousands of cool- ing further relaxation, provides an additional signal for ing sweeps, the nonstationarity δ gradually rises. A t the topological content as superpositions of nontrivial closer look then shows that the scale size of one of the ¯ ¯ ¯ dyon pairs shrinks, transforming this pair into a small holonomy caloron, DD (DD), or DD pairs. undissociated and nonstatic caloron, which finally In order to understand this from the point of view disappears after having turned into a tiny dislocation, of single-caloron solutions with nontrivial holonomy, PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 5 2005 886 ILGENFRITZ et al. B 0.1 0 –0.1 –0.10 –0.08 –0.06 0.8 B1 0.8 B2 0.4 0.4 0 0 0 0 10 10 20 20 20 10 20 10 0.8 B3 0.8 B4 0.4 0.4 0 0 0 0 10 10 20 20 20 10 20 10 Fig. 9. The lattice field configuration depicted in Fig. 8 (B, B ) for 1650 cooling steps (f.h.b.c.). Here, B plots the eigenvalues of the Wilson–Dirac operator in the complex plane for κ =0.140 and the case of time-periodic fermionic b.c.; B1, B2 show 2D projections of the fermionic mode densities (in lattice units) related to the two distinct almost real eigenvalues, whereas B3, B4 present the densities (in lattice units) related to the two real eigenvalues. we have to find out whether (approximately) those the real distribution becomes flat. Therefore, at the asymptotic holonomy boundary conditions as typical lowest plateaus, a cut with respect to the asymptotic for the dyonic (antidyonic) semiclassical background holonomy L∞ is necessary to focus on the most static excitations are actually present during the cooling (most nontrivial) calorons. process when periodic boundary conditions (and not particular holonomy boundary conditions!) are ap- Concluding, we can state that multicaloron so- plied to the full volume. To answer this question, the lutions with nontrivial holonomy naturally occur in asymptotic holonomy L∞ as defined in Section 6 semiclassical configurations of higher action. From needs to be followed over the process of cooling. In the analytic point of view, at present, only special Fig. 10, we show histograms of L∞ obtained at dif- classes of multicaloron solutions with nontrivial ferent plateaus during the cooling histories of an en- holonomy are known [65]. The study of these solu- semble of O(200) configurations produced at β =2.2 tions shows that the calorons consist of dyon con- on a 163 × 4 lattice with standard p.b.c. We see a clear stituents which cannot be easily associated with sin- peak at L∞ =0for higher action plateaus. The distri- gle calorons. They constitute more complicated ob- bution is narrower than the pure Haar measure would jects which hopefully can be described as multidyon tell us. However, approaching lower lying plateaus, events. This should then allow one to parametrize PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 5 2005 INSTANTONS OR MONOPOLES 887 40 (a) 20 0 –1.0 –0.5 0 0.5 1.0 30 30 (b) (c) 20 20 10 10 0 0 –1.0 –0.5 0 0.5 1.0 –1.0 –0.5 0 0.5 1.0 30 40 (d) (e) 30 20 20 10 10 0 0 –1.0 –0.5 0 0.5 1.0 –1.0 –0.5 0 0.5 1.0 Fig. 10. Histogram P (L∞) of the values of the Polyakov loop at “infinity” (as explained in the text) seen on the first plateau (a) and at plateaus with√ m 4 (b), 3 (c), 2 (d), and 1 (e). For comparison, the distribution expected from the pure Haar 2 measure PHaar(L) ∼ 1 − L is shown with the same normalization (dashed curves). The equilibrium ensemble of O(200) configurations was generated at β =2.2 (for lattice size 163 × 4, p.b.c.). the path integral in terms of monopole degrees of always been interpreted in terms of standard instan- freedom, as has been discussed in Section 3. tons [1] or calorons [6], since the observables were mostly the action or topological charge densities, as well as the fermionic densities obtained from the 8. CONCLUSIONS AND OUTLOOK low-lying spectrum of the (chirally improved) Dirac In this review, we tried to demonstrate that the operator. Over many years, one has overlooked that semiclassical treatment of QCD at finite tempera- these solutions in general have an intrinsic holonomy ture [7] has to be reconsidered taking into account structure and a nontrivial asymptotic behavior related (anti-)self-dual solutions of the field equations with to nonvanishing potentials far away from the lumps nontrivial holonomy. Such solutions were analytically of topological charge. In this paper, we have reviewed described by van Baal and his coworkers [38, 39, 65]. attempts to reveal this structure by carefully inves- A first attempt to treat the path integral with these tigating the spatial Polyakov loop distributions and solutions was recently put forward by Diakonov and the localization of fermionic zero modes depending on collaborators [40]. whether time-periodic or time-antiperiodic boundary In order to show that KvB solutions are really conditions for the fermion fields are imposed. The relevant, we have used the lattice representation of the result is surprising. We clearly see a dyonic substruc- Yang–Mills path integral and generated statistical ture even though the calorons or instantons produced ensembles of finite-temperature fields with the help of at lower temperature or on a symmetric torus look the Monte Carlo method. Then, by iteratively mini- like integer charged (topological charge is assumed) mizing the action, we have searched for semiclassical pseudoparticles in the 4D Euclidean world. background configurations. The method is standard and has been known for almost twenty years [18–20], One is tempted to ask whether these observations and has been refined and improved in different ways. are only an artifact of the cooling method. Having in Anyway, over the years, the outcome, i.e., the (ap- mind that the low-lying fermionic spectrum is rela- proximate) solutions of the equations of motion, has tively robust against smoothing or smearing of the PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 5 2005 888 ILGENFRITZ et al. gauge fields, Gattringer and his coworkers [50] stud- 4. E.-M. Ilgenfritz and M. Muller-Preussker,¨ Nucl. ied the “mode hopping” of the fermionic zero modes Phys. B 184, 443 (1981). of the equlibrium lattice gauge fields when changing 5. E. V. Shuryak, Nucl. Phys. 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This is a nontriv- in Proceedings of the EPS International Con- ial generalization of the “old” caloron which by con- ference “High-EnergyPhysics ”, Palermo, 1975, Ed. by A. Zichichi (Editrice Compositori, Bologna, struction is an SU(2) object embedded into SU(3). 1976); S. Mandelstam, Phys. Rep. 23, 245 (1976); Still, it is not clear whether the new and nontrivial A.S.Kronfeld,M.L.Laursen,G.Schierholz, structures discussed in this paper will contribute to andU.J.Wiese,Phys.Lett.B198, 516 (1987); a better understanding of the mechanism driving the V. B o r n y a k o v et al., Phys. Lett. B 261, 116 (1991); deconfinement transition. How far a certain working G. Bali, V. Bornyakov, M. Muller-Preussker,¨ and picture of a dilute gas of these configurations can K. Schilling, Phys. Rev. D 54, 2863 (1996). be developed and whether it will improve our un- 11. G. ’t Hooft, Nucl. Phys. B 138, 1 (1978); L. Del Deb- derstanding of quark confinement itself remains an bio, M. Faber, J. Greensite, and S. Olejnik, Phys. Rev. open question to which we hope to come back in the D 55, 2298 (1997); L. Del Debbio et al., Phys. Rev. D 58, 094501 (1998); K. Langfeld, H. Reinhardt, and future. Nevertheless, we feel that some drawbacks O. Tennert, Phys. Lett. B 419, 317 (1998); M. En- found in interpreting lattice instantons [32–34] can gelhardt, K. Langfeld, H. Reinhardt, and O. Tennert, be cured by classical solutions which properly take Phys. Lett. B 452, 301 (1999); Phys. Rev. D 61, the nontrivial holonomy into account. 054504 (2000). 12. M. Fukushima et al., Phys. Lett. B 399, 141 (1997); Nucl. Phys. B (Proc. Suppl.) 63, 513 (1998); ACKNOWLEDGMENTS M. Fukushima, H. Suganuma, and H. Toki, Phys. We gratefully acknowledge many stimulating Rev. D 60, 094504 (1999). discussions with Yu.A. Simonov at ITEP, Moscow. 13. V.I. Shevchenko and Yu. A. Simonov, Phys. Rev. Lett. Moreover, we thank Pierre van Baal, Falk Bruck- 85, 1811 (2000). mann, Christof Gattringer, and Dirk Peschka for 14. M. Fukushima, E.-M. Ilgenfritz, and H. Toki, Phys. cooperation and useful discussions. Rev. D 64, 034503 (2001). 15. H. G. Dosch, Phys. Lett. B 190, 177 (1987); This work was partly supported by the Russian H.G.DoschandYu.A.Simonov,Phys.Lett.B Foundation for Basic Research (project nos. 02-02- 205, 339 (1988); A. Di Giacomo, H. G. Dosch, 17308, 03-02-19491, and 04-02-16079), by DFG V. I. Shevchenko, and Yu. A. Simonov, Phys. Rep. grant no. 436 RUS 113/739/0 and RFBR-DFG 372, 319 (2002). grant no. 03-02-04016, by the Federal Program 16. E.-M. Ilgenfritz, B. V.Martemyanov, S. V.Molodtsov, of the Russian Ministry of Industry, Science, and et al., Phys. Rev. D 58, 114508 (1998). Technology no. 40.052.1.1.1112, and by INTAS-00- 17. E.-M. Ilgenfritz, B. V. Martemyanov, and M. Muller-¨ 00111. Two of us (B.V.M. and A.I.V.) gratefully ap- Preussker, Phys. Rev. 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B (Proc. Suppl.) Phys. J. C (in press). 129-130, 653 (2004). PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 5 2005 Physics of Atomic Nuclei, Vol. 68, No. 5, 2005, pp. 890–891. From Yadernaya Fizika, Vol. 68, No. 5, 2005, pp. 925–926. Original English Text Copyright c 2005 by Kerbikov. Klein–Gordon Equation for Quark Pairs in Color Superconductor* B. O. Kerbikov** Institute of Theoretical and Experimental Physics, Bol’shaya Cheremushkinskaya ul. 25, Moscow, 117259 Russia Received July 23, 2004 Abstract—The wave equation is derived for quark pairs in color superconductor in the regime of low density/strong coupling. c 2005 Pleiades Publishing, Inc. During the last five years, color superconductiv- In the present paper, we shall derive an equa- ity became a compelling topic in QCD—see the re- tion for the quark pairs in the low-density/stong- view papers [1].Broadly speaking, we have a fair coupling limit.This equation is obtained directly from understanding of color superconductivity physics in the mean-field gap equation.Our derivation follows the high-density/weak-coupling regime.In the low- the work by Nozieres and Schmitt-Rink [4], who density/strong-coupling region, the situation is dif- obtained the Schrodinger¨ equation starting from the ferent.Here, the theory faces the well-known di ffi- BCS solution.Quarks in color superconductor are culties of nonperturbative QCD and use is made of relativistic particles, and therefore we shall arrive at models like NJL or instanton gas.By low density, the Klein–Gordon equation. we mean quark densities 3–4 times larger than that Our starting point is the expression for the ther- in normal nuclear matter.Model calculations show modynamic potential for the two-flavor superconduc- (see [2] and references in [1]) that, at such densities, tor (2SC).In the 2SC phase [1], pairing takes place the gap equation acquires a nontrivial solution.This between the u and d quarks, while the s quark is out was interpreted [1] as the onset of the Bardeen– of the game until the density increases, so that the Cooper–Schrieffer (BCS) regime, i.e., the formation chemical potential becomes substantially larger than of the condensate of Cooper pairs made of u and d the mass of the s quark.Here, we consider only the T =0 quarks.It is known, however, that a nonzero value case.The possible coexistence of the chiral and diquark condensates is neglected on the basis of the gap is only a signal of the presence of fermion of the Anderson theorem [9].The expression for the pairs.Depending on the dynamics of the system, on thermodynamic potential reads [2, 9] the fermion density, and on the temperature, such pairs may be either stable or fluctuating in time, may ∆2 Ω(T =0,µ;∆)= (1) form a BCS condensate or dilute Bose gas, or un- 4g dergo a Bose condensation.The continuous evolu- 2 − dqq2 (E − µ)2 +∆2 tion from the BCS regime to the regime of Bose– 2 q π Einstein condensation (BEC) is called the BCS– 2 2 | − | BEC crossover.Such a transition takes place either + (Eq + µ) +∆ + µ Eq + Eq , by increasing the strength of the interaction or by 2 2 decreasing the carrier density.The fact that the BCS where Eq = q + m . wave function may undergo a smooth evolution and The four-fermion interaction constant g has di- describe the tightly bound fermion pairs was first no- mension m−2.From (1), one obtains the following ticed long ago [3–6].According to [5], the remark that gap equation: “there exists a transformation that carries the BCS into BE state” was originally made by F.J. Dyson in 4g 2 ∆ ∆ ∆= 2 dqq + , (2) 1957 (i.e., the same year that the BCS paper [7] was π Eq E¯q published).The BCS –BEC crossover for quarks was where first discussed in [8].The general description of the 2 2 crossover for quarks will be given elsewhere. Eq = (Eq − µ) +∆ , ∗ This article was submitted by the author in English. 2 2 E¯q = (Eq + µ) +∆ . **e-mail: [email protected] 1063-7788/05/6805-0890$26.00 c 2005 Pleiades Publishing, Inc. KLEIN–GORDON EQUATION 891 An important remark concerning the structure of equation.The point µ = m −EB corresponds to a Eqs.(1), (2) is due here.We have tacitly assumed negative nonrelativistic chemical potential, typical for that the four-fermion interaction between quarks is the “molecular” limit of the BCS–BEC crossover. pointlike.In the general case, instead of (2), one The phase diagram in the (np/n¯p,µ) plane has two should write symmetric branches corresponding to quarks and antiquarks.Note also that the system described by 4g 2 ∆q ∆q ∆p = dqq Vpq + . (3) Eqs.(4), (5), and (7) possesses an “exciton-like” π2 E E¯ q q instability. However, the crude approximation (1), (2) is suf- I am grateful for discussions and remarks from ficient to obtain the general structure of the Klein– N.O. Agasian, T.D. Lee, E.V. Shuryak, D.T. Son, Gordon equation.Next, making use of the standard M.A. Stephanov, and A.M. Tsvelik. Special thanks go Bogolyubov functional, we introduce the wave func- to T.Yu. Matveeva for the help in preparing the article. tions of the quark–quark and antiquark–antiquark Financial support from BNL and grant no.Ssc- pairs 1774-2003 is gratefully acknowledged.We thank INT (Seattle) for its hospitality and the Department ∆ ∆ of Energy for the support during the Workshop INT- ϕp = ,χp = ¯ . Ep Ep 04-1. Then, with a little juggling of (2), (3), we obtain It is a pleasure and a honor to submit this paper to the following set of coupled equations for ϕ and χ : the Yurii Simonov Festschrift. p p 2 2 − ( p + m µ)ϕp (4) 4g REFERENCES = (1 − 2n ) dqq2(ϕ + χ ), π2 p q q 1.K.Rajagopal and F.Wilczek, in At the Frontier of Particle Physics. Handbook of QCD,Ed.by M.Shifman (World Sci.,Singapore, 2001), Vol.3, 2 2 ( p + m + µ)χp (5) p.2061; M.Alford, Annu.Rev.Nucl.Part.Sci. 51, 131 4g = (1 − 2¯n ) dqq2(χ + ϕ ), (2001); G.Nardulli, Riv.Nuovo Cimento 25, 1 (2002). π2 p q q 2. J.Berges and K.Rajagopal, Nucl.Phys.B 538, 215 (1999). where 3.D.M.Eagles, Phys.Rev. 186, 456 (1969); − − Ep µ − Ep + µ A.J.Leggett, in Modern Trends in the Theory 1 2np = , 1 2¯np = . (6) of Condensed Matter (Springer-Verlag, Berlin, Ep E¯p 1980), p.13. These two equations may be recasted into a single 4. P.Nozieres and S.Schmitt-Rink, J.Low Temp.Phys. Klein–Gordon equation following the standard pro- 59, 195 (1985). cedure [10].Let us de fine ψp = ϕp + χp and consider 5.P.W.Anderson, Basic Notions of Condensed the dilute limit np 1, n¯p 1. Matter Physics (Benjamin/Cummings Publ., Menlo Park, Cal., 1983), p. 105. One then gets 6. P.W.Anderson and W.F.Brinkman, in Proceedings 8g of the 15th Scottish University Summer School in (p2 + m2 − µ2)ψ = p2 + m2 dqq2ψ . (7) p π2 q Physics (Acad.Press, New York, 1975), p.315. 7. J.Bardeen, L.N.Cooper, and J.R.Schrie ffer, Phys. We recall that nonlocality in (7) is of a symbolic Rev. 108, 1175 (1957). character as soon as we use the pointlike four-fermion 8. B.O.Kerbikov, hep-ph/0110197; Yad.Fiz. 65, 1972 interaction. (2002) [Phys.At.Nucl. 65, 1918 (2002)]. From (4), (5), and (7), we see that, in the di- 9. B.O.Kerbikov, hep-ph/0106324. lute strong-coupling limit, the chemical potential µ 10.G. Baym, Lectures on Quantum Mechanics plays the role of the eigenvalue of the Klein–Gordon (W.A.Bejamin, New York, 1969), p.54. PHYSICS OF ATOMIC NUCLEI Vol.68 No.5 2005 Physics of Atomic Nuclei, Vol. 68, No. 5, 2005, pp. 892–898. From Yadernaya Fizika, Vol. 68, No. 5, 2005, pp. 927–934. Original English Text Copyright c 2005 by Palumbo. Path Integrals and Degrees of Freedom in Many-Body Systems and Relativistic Field Theories* F. P a l u m b o ** INFN, Laboratori Nazionali di Frascati, Italy Received June 2, 2004; in final form, October 29, 2004 Abstract—The identification of physical degrees of freedom is sometimes obscured in the path-integral formalism, and this makes it difficult to impose some constraints or to do some approximations. I review a number of cases where the difficulty is overcome by deriving the path integral from the operator form of the partition function after such identification has been made. c 2005 Pleiades Publishing, Inc. 1. INTRODUCTION Now the path-integral formalism is widely used because of its flexibility and the possibility of numer- It is a great pleasure for me to contribute to this ical applications, but the identification of degrees of volume in honor of Prof. Yu.A. Simonov. This is an freedom is not always easy, while it can be conve- occasion for me to remember again the origin of our niently achieved in the operator form of the partition friendship, which has extended over the years well function. Therefore, in all of the above problems, I beyond what it would appear from our joint papers. will first identify the relevant degrees of freedom in The subject of my contribution is focused on my the Fockspace, where the partition function is de- recent interests in several problems which have a fined; then I will introduce the appropriate constraints or approximations; and finally I will derive the con- common feature: the identification of degrees of free- strained or approximated path integral. dom in a path integral. Indeed, there are many situa- tions in which such identification is helpful or neces- As I said, all the subjects I mentioned have a sary. An old example is the thermodynamics of gauge common feature in the role played by the identification theories [1], but recently I met with many others. The of degrees of freedom, but are otherwise very different. first I will discuss here is how to find actions exactly Therefore, the motivations for their investigation are equivalent to the standard ones but closer to the con- given separately in the relevant sections. In Section 2, tinuum at finite lattice spacing [2]. This includes the I will report the results I will use later about the definition of the couplings of the chemical potential, standard derivation of the path integral from the op- an issue of particular importance in QCD [3]. The erator form of the partition function. In Section 3, chemical potential is used to fix the expectation value I will show how to derive an action different from of some charge operator. Fixing instead the value of the standard one and closer to the continuum in the nonrelativistic case. In Section 4, I will carry out the the charge, namely, selecting a specific charge sector, corresponding derivation for relativistic field theories, is somewhat more difficult, but interesting pieces of confining myself to the couplings of the chemical information can be obtained by an expansion in a potential. In Section 5, I will discuss the case of a given charge sector of the fermion determinant in given charge sector. I will present a general method series of the number of fermions [4]. The last prob- of bosonization valid for relativistic and nonrelativistic lem I will consider is how to describe the low-lying theories in Section 6 and my conclusions in Section 7. excitations of fermionic systems, both relativistic and nonrelativistic, by means of effective bosons, in short, how to bosonize them [5]. All these problems require 2. THE STANDARD DERIVATION the identification of the relevant degrees of freedom— OF THE EUCLIDEAN PATH INTEGRAL in the last case, to determine the structure of the com- FROM THE PARTITION FUNCTION posite bosons in terms of the fermionic constituents; in the first case, to show the equivalence of different actions. Let me introduce some definitions. I denote by τ the temporal lattice spacing, by N0 the number of ∗ This article was submitted by the author in English. temporal sites, by x0 or t the temporal component of **e-mail: [email protected] the site position vector x,byT the temperature, by µ 1063-7788/05/6805-0892$26.00 c 2005 Pleiades Publishing, Inc. PATH INTEGRALS AND DEGREES OF FREEDOM 893 the chemical potential, by Qˆ the(electric,baryon,...) order and introduce between the factors in Eq. (5) the Tˆ identity charge operator, and by (x0) the transfer matrix. T In the nonrelativistic case, is expressed in terms I = [dc+dc dd+dd]exp(−c+c − d+d)|cdcd|, of the Hamiltonian, (6) T =exp(−τH) , (1) where the basis vectors are coherent states and the Hamiltonian is the generator of continuous † † time translations. In relativistic theories, instead, only |cd = | exp(−ccˆ − ddˆ ). (7) the transfer matrix is known in general, and the above c+ c d+ d equation can be used to define a Hamiltonian, but The , , , are holomorphic/Grassmann vari- only as the generator of discrete translations by the ables and satisfy periodic/antiperiodic boundary con- time spacing τ. Another important difference is that ditions in time for bosons/fermions, respectively [6]. the nonrelativistic interactions are generally quartic They have the label of the time slice, in which the in the fields, while the relativistic ones are quadratic. identity operator is introduced. For the other indices, Both features contribute to make the derivation of the they are subject to the same convention as the cre- Euclidean path integral different in the two cases. ation and annihilation operators. The main property of coherent states is that they are eigenstates of the Tˆ is defined in terms of particle–antiparticle annihilation operators † ˆ† ˆ creation–annihilation operators cˆ , d , c,ˆ d acting in cˆ|cd = c|cd. (8) a Fockspace. It depends on the time coordinate x0 only through the dependence on it of other fields To get the functional form of the partition function, (for instance, gauge fields). In fact, the creation and it is only necessary to evaluate the matrix elements annihilation operators do not depend on x0.Theyde- c1d1|T |cd. This, as anticipated in the Introduction, pend on the spatial coordinates x and on the internal must be done in different ways for nonrelativistic and quantum numbers (Dirac, color, and flavor indices in relativistic theories. the case of QCD), comprehensively represented by The first case appears more difficult because of I,J,.... the quartic interactions, but since in general there In the transfer matrix formalism, often one has to are no ultraviolet divergencies (a detailed discussion deal with quantities at a given (Euclidean) time x0. of the departure from the standard form of the path For this reason, we adopt a summation convention integral in the presence of singular potentials can over spatial coordinates and intrinsic indices at fixed be found in [8]), we can make without any error the time. So, for instance, for an arbitrary matrix M,we approximation will write exp(−τHˆ ) ∼ 1 − τH.ˆ (9) † † cˆ M(x )ˆc = cˆ M (x )ˆc . (2) 0 x,I x,I;y,J 0 y,J Let us then consider a many-body system with the x,y,I,J Hamiltonian written in normal form Qˆ ˆ † † † † In this notation, the charge operator has the form H(ˆa , aˆ)= aˆxhx,yaˆy +ˆaxaˆyvx,yaˆyaˆx . (10) † † Qˆ =ˆc cˆ − dˆ d.ˆ (3) x,y The standard expression of the Euclidean path inte- The grand canonical partition function can be gral associated to this Hamiltonian is written as a time-ordered product ∗ Z = [da da]exp(−S), (11) µ Z = Tr exp Qˆ Tˆ(x0) , (4) T where x0 N0 which, using the relation T −1 = τN and assuming ∗ ∇ ∗ 0 S = τ at+1 tat + H(at+1,at) (12) the conservation of Qˆ, is conveniently rewritten as t=1 is the action and I denoted by Z = Tˆ(x )exp µτQˆ . Tr 0 (5) 1 x0 ∇ f = (f − f ) (13) t t τ t+1 t The standard way [6, 7] to obtain the path-integral the right time discrete derivative. Notice the time form of Z is to write all the operators in normal splitting between the fields and their conjugates, PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 5 2005 894 PALUMBO which implies a departure from the classical ex- time derivative remains unchanged, but in the other pression not only in derivative but also in potential terms, all the fields appear at the same time, terms. Needless to say, neglecting this time splitting ∗ ∇ − ∗ introduces finite errors [6]. S = τ h0 + at+1 tat +(h h)at at (19) In the relativistic case, the transfer matrix can be t ∗ } written as [7] + H(at ,at) . T † We can easily checkon solvable models that this x0 = Tx Tx0 , (14) 0 action gives the right results and that the terms aris- where ing from the rearrangement in antinormal ordering † † T =exp cˆ Mcˆ + dˆ M T dˆ exp cNˆ dˆ (15) cannot be neglected. In the case of fermions, the time splitting between (here, the superscript T means transposed and I do the fields and their conjugates in the potential terms not need to specify the matrices M, N). The matrix can be avoided in the same way as for bosons, but elements of T can be evaluated exactly [7], yielding also in a simpler way. In fact, the Grassmann fields, the standard form of the Euclidean path integral. unlike the holomorphic variables, are independent of their conjugates, so that the simple transformation ∗ → ∗ 3. ANTINORMAL ORDERING: A DIFFERENT at+1 at (20) ACTION IN THE NONRELATIVISTIC CASE eliminates the time splitting everywhere, with the ob- vious exception of the term with the time derivative, In the standard form of the action, the time is split which is changed into the left one. in the fields and their conjugates. This is an artifact Because in this case we have two different deriva- which makes the equations unnecessarily different tions of the path integral, by their comparison, we can from continuous ones, somewhat more complicated, get nontrivial identities. and somewhat confusing. In gauge theories, for in- stance, such time splitting introduces a coupling of the chemical potential to the temporal gauge fields 4. ANTINORMAL ORDERING: A DIFFERENT which has been considered of physical significance, COUPLING OF THE CHEMICAL POTENTIAL while it can be altogether avoided in a different deriva- IN RELATIVISTIC FIELD THEORIES tion of the path integral. I will first illustrate such a derivation in the nonrelativistic case. In relativistic field theories, the artificial time Instead of the normal order, I write the Hamilto- splitting affects only the coupling of the chemical nian in antinormal order (all the annihilation opera- potential. In the Hasenfratz–Karsch–Kogut formu- tors to the left of the creation ones) lation [9], which is the standard one, for Wilson ˆ † † † fermions such coupling takes the form H = h0 +ˆaxhx,yaˆy +ˆaxaˆyvx,yaˆyaˆx , (16) x,y − (+) (+) δS =2K q [exp µ 1] P0 U0T0 (21) where x − − 3 − − ( ) ( ) (+) h0 = N (−σhx,x + w) , (17) +[exp( µ) 1] P0 T0 U0 q, − hx,y = σhx,y wδx,y. where q is the quark field, K is the hopping parameter, Intheaboveequations, U0 the temporal linkvariable, and ± 1 w = σv + v , (18) P ( ) = (1l ± γ ) , (22) x,x x,z 0 2 0 z ± T ( )f(x )=f(x ± 1). N 3 is the number of spatial sites, and σ = −1 for 0 0 0 fermions and +1 for bosons. I assume that the po- Because of gauge invariance, in the presence of the tential is sufficiently regular for w to exist, otherwise a time splitting, a coupling with the temporal links regularization or a more drastic change in our proce- is needed, and this led to the conclusion that a dure is needed. Then I expand the transfer matrix and nonvanishing contribution of the chemical potential in each term I insert the identity between the creation must necessarily involve a Polyakov loop. But Creutz and annihilation operators. For the rightmost factor showed in a toy model [10] that this is not true, and before taking the trace, I move the creation operators I will report [4] in the sequel how to avoid such an to the leftmost position. In this way, the term with the artifact in the full-fledged QCD. PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 5 2005 PATH INTEGRALS AND DEGREES OF FREEDOM 895 I write the exponential of the charge in the follow- the transfer matrix acting in the quarkFockspace, ing way: impose the restriction to a given baryonic sector, and thenrewritethetraceasthedeterminantofamod- + + exp(µa0Qˆ)= [dc dc dd dd] (23) ified quarkmatrix. The round-trip is done by map- ping the Grassmann algebra generated by the quark × exp(δS − c+c − d+d)|cdcd|, fields into the Fockspace following the construction of Luscher¨ [7]. But while his paper is based on the where mere existence of the map, in order to enforce the + + δS =(1− cosh(µa0)) (c c + d d) (24) projection, I will make use of a concrete realization + + by means of coherent states. +sinh(µa0)(c c − d d). The first step is then to write the unconstrained The expression of δS is obtained by expanding the determinant as a trace in the Fockspace, exponential of the charge operator, putting all the terms in antinormal form, inserting in each mono- det Q = Tr Tˆ. (27) mial the unity between the set of annihilation and The second step is to impose the restriction to a sector the set of creation operators, and replacing them by with baryon number n by inserting in the trace the their Grassmannian eigenvalues. For the rightmost B appropriate projection operator Pˆn , exponential of the charge, before taking the trace, one B has to move the creation operators to the left of all the | Tˆ ˆ + + operators appearing under the trace. det Q nB = Tr PnB [dx dx dy dy] (28) After this, the construction of the path integral × exp −x+x − y+y x, y|TˆPˆ |−x, −y, proceeds in the standard way, and we get the stan- nB dard action with the exception of the coupling of the which will be expressed in terms of the determi- chemical potential, where all the fields appear at the nant of a modified quarkmatrix. The kernel same time: x, y|TˆPˆ |−x, −y has the integral form nB δS = q (1 − cosh(µa0)) + sinh(µa0)γ0 Bq. |Tˆ ˆ |− − + + x x, y PnB x, y = [dz dzdw dw] (29) (25) × exp −z+z − w+w x, y|T|ˆ z,w Here, I only need to say that the matrix B does not × | ˆ |− − depend on U0. What is important is to notice that the z,w PnB x, y . quark field and its conjugate are at the same time, and then the temporal Wilson variable disappears. The expression of the kernel x, y|T|ˆ z,w will not ˆ be reported here, while that of PnB can easily be 5. EXPANSION OF THE FERMION derived, DETERMINANT IN THE NUMBER ∞ | ˆ |− − − nB OF FERMIONS IN A GIVEN CHARGE z,w PnB x, y = ( 1) (30) SECTOR r=0 ×(ˆyw+)r(ˆxz+)(nB +r)(xxˆ+)nB+r(yyˆ+)r The use of the chemical potential is a way to 1 × . impose a given expectation value for some conserved ((n + r)!r!)2 charge. The alternative option of selecting a given B sector of the charge in the path integral presents Since xˆ+ and yˆ+ are creation operators of quarks and additional difficulties, but something can be learned antiquarks, respectively, we see that the rth term of by a series expansion of the fermion determinant [5]. this series gives the gauge-invariant contribution of To be concrete, we will refer to the case of QCD, nB valencequarksplusr quark–antiquarkpairs. but the method can be applied to other cases with | Needless to say, for nB =0, det Q nB does not appropriate modifications. reduce to the unconstrained determinant: Indeed, also In the absence of any condition on the baryon baryonic states are present in the unconstrained de- number, the quarkdeterminant is terminant, while they are absent in det Q| .In nB =0 QCD at nonvanishing temperature, it makes a differ- det Q = [dqdq]expSq, (26) ence whether we impose or not the condition nB = 0. In view of the relatively low value of the critical where Sq is the quarkaction and Q is the quark temperature with respect to the nucleon mass, how- matrix. My strategy is to write det Q as the trace of ever, we do not expect significant effects from the PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 5 2005 896 PALUMBO restriction to a given baryon sector unless we go to there are several recipes for specific cases which are exceedingly high temperatures. reviewed in [13]. A more flexible approach is based Let me now proceed to derive our final result. By on the Hubbard–Stratonovich transformation which evaluating the vacuum expectation values appearing linearizes the fermionic interaction by introducing in the last equation, we express the kernel of the bosonic auxiliary fields which are then promoted to projection operator in terms of Grassmann variables physical life. The typical resulting structure is that of only, chiral theories [14]. But in such an approach, an en- ergy scale emerges naturally, and only excitations of | ˆ |− − z,w PnB x, y (31) lower energy can be described by the auxiliary fields. ∞ 1 Moreover, in renormalizable relativistic field theories = (−1)nB (z+x)nB +r(w+y)r. like QCD, the fermion Lagrangian is quadratic to (n + r)!r! r=0 B start with, so that the Hubbard–Stratonovich trans- formation cannot be used. One can add quartic in- To evaluate the integral of Eq. (29), I rewrite the above teractions as irrelevant operators, and this can help equation in exponential form in numerical simulations, but has not led so far to ∞ 1 a formulation of low-energy QCD in terms of chiral z,w|Pˆ |x, y = (32) mesons. nB (n + r)!r! r=0 B I present a new approach [5] to bosonization which nB +r r does not suffer from the above limitations and can be ∂ ∂ + + × exp(−j1z x − j2w y)|j =j =0. applied to theories with quartic and quadratic inter- nB +r ∂jr 1 2 ∂j1 2 actions as well. It is based on the evaluation of the The integrals of Eqs. (28) and (29) are Gaussian and partition function restricted to the bosonic compos- we get the constrained determinant in terms of the ites of interest. By rewriting the partition function so determinant of a modified quarkmatrix obtained in functional form, we get the Euclidean ac- tion of the composite bosons from which in the non- ∞ 1 ∂nB +r relativistic theories we can derive the Hamiltonian. In det Q|n = (33) B nB +r this way, I derived the interacting boson model from (nB + r)!r! ∂j r=0 1 a nuclear Hamiltonian. In the case of pure pairing, I r × ∂ | reproduce the well-known results for the excitations r det (Q + δQ1 + δQ2) j1=j2=0. ∂j2 corresponding to the addition and removal of pairs of fermions, as well as for the seniority excitations The explicit form of the variations δQ1 and δQ2 of the which are inaccessible by the Hubbard–Stratonovich quarkmatrix is not important here, but we warn the method. Indeed, at least in this example, this theory reader that there is an error in their expression in [14]. does not have the structure of a chiral expansion. For the relativistic case, an investigation is in 6. BOSONIZATION IN MANY-BODY progress [15]. SYSTEMS AND RELATIVISTIC FIELD Let me start by defining the composites in terms of THEORIES the fermion operators cˆ: ˆ† 1 † † † 1 † † † bJ = cˆ BJ cˆ = cˆm1 BJ cˆm2 . (34) The low-energy collective excitations of many- 2 2 m1,m2 m1,m2 fermion systems can be described by effective bosons. Well-known examples are the Cooper pairs of su- Intheaboveequation,m represents all the fermion perconductivity, the bosons of the interacting boson intrinsic quantum numbers and position coordinates model of nuclear physics, the chiral mesons, and and J represents the quantum numbers of the com- the quarkpairs of color superconductivity in QCD. posites. I assume all the structure matrices BJ to In all these cases, the effective bosons are gener- have one and the same dimension, which I denote by ated by attractive interactions, but effective bosons 2Ω. The fermionic operators have canonical anticom- can arise also in the presence of repulsive forces, mutation relations, while for the composites as in the Hubbard model [11]. Some of the effec- † 1 † † † [ˆbJ ,ˆb ]= Tr(BJ B ) − cˆ B BJ c.ˆ (35) tive bosons are Goldstone bosons, and then there 1 J2 2 1 J2 J2 1 is a general theory which tells that they live in the coset space of the group which is spontaneously bro- It is then natural to require the normalization † ken and dictates how they are related to the orig- Tr(B ,BJ )=2δJ ,J . (36) inal fields [12]. But there is no general procedure J1 2 1 2 to reformulate the fermionic theory in terms of the A convenient way to get the Euclidean path in- effective bosonic degrees of freedom, even though tegral from the trace of the transfer matrix is to use PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 5 2005 PATH INTEGRALS AND DEGREES OF FREEDOM 897 coherent states of composites. Therefore, I introduce are interested in states with n = n + ν for an arbitrary the operator reference state n,weredefine P according to 2 ∗ − (Ω − n) P = db db(b|b) 1|bb|, (37) P = P . (46) n Ω2 0 We then have where † −1 | | · ˆ † ν b = exp(b b ) . (38) P |(ˆb )n = 1 − (47) n Ω − n I adopted the convention −1 ν +1 † · ˆ† ˆ† × 1 − |(ˆb )n, b b = bJ bJ . (39) Ω − n J which shows that Pn behaves like the identity in the If ˆb were operators of elementary bosons, P would neighborhood of the reference state up to an error be the identity in the boson Fockspace. I would of order ν/(Ω − n), namely, the measure b|b−1 is like P to be the identity in the fermion subspace of essentially uniform. P the composites. To see the action of on composite In the general case of many composites, we have operators, let us first consider the case where there is 1/2 only one composite with structure function satisfying b1|b =[det(1l+β1β)] , (48) the equation where the matrix β is † 1 † B B = 1l . (40) β = b · B (49) Ω Then we find and n n † † ∂ 0 ∂ i −Ω ˆ n0 ˆ ni 1 b1|(b ) ...(b ) = ... (50) | −1 ∗ I0 Ii ∂xn0 ∂xni b b = 1+ b1b (41) 0 i Ω × 1 · † ∗ · and exp Tr ln[1l+ (x B )(b1 B)] . 2 x=0 | ˆ† n ∗ n b1 (b ) = Cn(b ) , (42) We must now make an assumption which replaces Eq. (40), namely, that all the eigenvalues of the ma- where † Ω! trices BJ BJ are much smaller than Ω.Thenwefind C = (43) P n (Ω − n)!Ωn again that approximates the identity with an error of order 1/Ω. 1 2 n − 1 = 1 − 1 − ... 1 − . Now we are equipped to carry out the program Ω Ω Ω outlined at the beginning. The first step is the Z Now we can determine the action of P on the com- evaluation of the partition function C restricted posites to fermionic composites. To this end, we divide the inverse temperature in N0 intervals of spacing τ, − −1 1 ˆ† n n n +1 ˆ† n 1 P|(b ) = 1 − 1 − |(b ) , = N τ, (51) Ω Ω T 0 (44) and write P N0 which shows that behaves approximately like the Zc = Tr (PT ) , (52) identity with an error of the order of n/Ω.Itisperhaps worthwhile noticing that, in the limit of infinite Ω,we where T is the transfer matrix. In the nonrelativistic case, T is expressed in terms of the Hamiltonian recover exactly the expressions valid for elementary bosons, in particular, T − ˆ =exp τH , (53) Ω | 1 ∗ → ∗ b1 b = 1+ b1b exp(b1b), (45) and the Hamiltonian is the generator of continuous Ω time translations. In relativistic field theories, instead, Ω →∞. only the transfer matrix is known in general, and the above equation can be used to define a Hamiltonian, It might appear that the treatment of states with but only as the generator of discrete translations by n ∼ Ω is precluded, but this is not true. Indeed, if we the time spacing τ. PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 5 2005 898 PALUMBO At this point, we must evaluate the matrix element The bosonization of the system that we consid- b1|T |b and to do this we must distinguish between ered has thus been accomplished. In particular, the relativistic field theories and many-body systems. In fermionic interactions with external fields can be ex- the first case, the transfer matrix is a product of ex- pressed in terms of the bosonic terms which involve ponentials of quadratic forms in the fermion opera- the matrix M (appearing in h), and the dynamical tors [7], and the matrix element can be directly and problem of the interacting (composite) bosons can be exactly evaluated [15]. In the second case, one must solved within the path-integral formalism. Part of this at an intermediate stage expand with respect to the problem is the determination of the structure matrices time spacing τ. This does not introduce any error BJ . This can be done by expressing the energies in because one can retain all the terms which give finite terms of them and applying a variational principle contributions in the limit τ → 0. We will report here which gives rise to an eigenvalue equation. only the nonrelativistic calculation. The most general The Hamiltonian can be derived by standard pro- Hamiltonian can be written cedures. † 1 † † † 1 Hˆ =ˆc h cˆ − g cˆ F cˆ cFˆ c,ˆ (54) 0 K 2 K 2 K K 7. CONCLUSION where K represents all the necessary quantum num- bers. The single-particle term includes the single- particle energy with matrix e, any single-particle in- I showed that there are a number of problems teraction with external fields described by the matrix which can be easily dealt with in the operator form of M,andthechemicalpotentialµ, the partition function. Only afterwards can it be given h = e + M−µ. (55) the functional form of the path integral which is more 0 convenient for many purposes. Therefore, we will be able to solve the problem of There are other examples that I left out for different fermion–boson mapping by determining the inter- reasons. Among these, I would like to mention the action of the composite bosons with external fields. problem of the restriction of gauge theories to phys- Assuming for the potential form factors the normal- ical states, which has not yet found a general and ization † satisfactory solution. Tr(FK FK )=2Ω (56) and setting ∗ −1 REFERENCES Γ =(1l+β β − ) , (57) t t t 1 1. L. Dolan and R. Jackiw, Phys. Rev. D 9, 3320 (1974). we get the Euclidean action 2. F. Palumbo (in press). 1 ∗ 3. F.Palumbo,Phys.Rev.D69, 074508 (2004). S = τ Tr[ln(1l+ β β ) − ln Γ ] (58) 2τ t t t 4. F. Palumbo, Nucl. Phys. B 643, 391 (2002); hep- t lat/0202021. 1 ∗ † 5. F. Palumbo, nucl-th/0405045. − H + g Tr Tr(Γ β F )Tr(Γ F β − ) 1 4 K t t K t K t 1 6. J. W. Negele and H. Orland, Quantum Many- K Particle Systems (Addison-Wesley, New York, − † − ∗ † 1988). 2Tr ΓtF FK Tr[Γtβt F , ΓtFK βt−1]+ K K 7. M. Luscher,¨ Commun. Math. Phys. 54, 283 (1977). 1 ∗ T 8. H. Kleinert, Path Integrals in Quantum Mechanics, + Tr β (β − h + hβ − ) , 2 t t 1 t 1 Statistics and Polymer Physics (World Sci., Singa- pore, 1990). where [...,...]+ is an anticommutator. This action 9. P. Hasenfratz and F. Karsch, Phys. Lett. B 125B, 308 differs from that of elementary bosons because (1983); J. Kogut, M. Matsuoka, M. Stone, et al., (i) the time derivative terms (contained in the first Nucl. Phys. B 225, 93 (1983). line) are noncanonical; 10. M. Creutz, Found. Phys. 30, 487 (2000). (ii) the coupling of the chemical potential (which 11. M. Cini and G. Stefanucci, cond-mat/0204311 v. 1. appears in h) is also noncanonical, since it is not 12. S. Weinberg, The Quantum Theory of Fields (Cam- quadratic in the boson fields; bridge Univ. Press, Cambridge, 1996). (iii) the function Γ becomes singular when the 13. A. Klein and E. R. Marshalek, Rev. Mod. Phys. 63, number of bosons is of order Ω,whichreflects the 375 (1991). Pauli principle. 14. V. A. Miransky, Dynamical Symmetry Breaking in We remind the reader that the only approximation Quantum Field Theories (World Sci., Singapore, done concerns the operator P. Therefore, these are to 1993). be regarded as true features of compositeness. 15. S. Caracciolo and F. Palumbo (in press). PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 5 2005 Physics of Atomic Nuclei, Vol. 68, No. 5, 2005, pp. 899–908. From Yadernaya Fizika, Vol. 68, No. 5, 2005, pp. 935–944. Original English Text Copyright c 2005 by Mariani, Calogero. Isochronous PDEs* M. Mariani1) and F. Calogero2) Received May 13, 2004 Abstract—A number of well-known evolution PDEsare modi fied so that they then possess many solutions which are isochronous, i.e., completely periodic, with a fixed period that doesnot depend on the initial data (for large sets of such data). c 2005 Pleiades Publishing, Inc. 1. INTRODUCTION the remarkable phenomenology featured by their so- lutions. The following section is devoted to certain, Recently, a simple “trick”—amounting essentially mainly notational, preliminaries, the main purpose to a change of independent and dependent variables— of which is to minimize subsequent repetitions. The wasintroduced [1 –3] that associates to a given nonlinear evolution PDEsare then listed, with mini- evolution equation a family of deformed equations mal comments, in Section 3, both in their unmodified parametrized by a real constant ω to which (for avatars(with appropriate —if inevitably incomplete— positive ω)theperiod references) and in their ω-modified versions. Occa- 2π sionally, some solutions are also reported and tersely T = (1.1) discussed. The last section contains some final com- ω ments. is associated. For vanishing ω, ω =0, the original equation isrecovered; for positive ω, ω>0, to which 2. NOTATION AND PRELIMINARIES attention ishereafter con fined without loss of gener- ality, these “ω-modified” equations possess many so- The independent variablesof the unmodi fied evo- lutionswhich are isochronous —namely, completely lution PDE are denoted as ξ ≡ (ξ1,...,ξN ) and τ; periodic with period T or possibly with a period that the dependent variable of the unmodified evolution isa simple multiple or fraction of T .Inthecontext PDE isdenoted as w ξ; τ ≡ w (ξ1,...,ξN ; τ),and of the initial-value problem, these solutions generally if a second dependent variable also enters, it is emerge out of open sets of initial data having the denoted as w˜ ξ; τ ≡ w˜ (ξ ,...,ξ ; τ); upper case full dimensionality of the phase space underlining 1 N ˜ the problem at hand. In several recent papers, this letters, W ξ; τ ≡ W (ξ1,...,ξN ; τ), W ξ; τ ≡ phenomenology hasbeen explored and exploited, or W˜ (ξ1,...,ξN ; τ), are used for matrices. The in- just mentioned, in an ODE context (mainly, though dependent variablesof the ω-modified evolution not exclusively, in a many-body problem context) [1– PDE are denoted as x ≡ (x1,...,xN ) and t;the 21] and in a few recent papersin a PDE context [3, dependent variable of the ω-modified evolution PDE 17, 21–23]. The purpose and scope of this paper isdenoted as u (x; t) ≡ u (x1,...,xN ; t),andifa isto apply the trick to a number of well-known, second dependent variable also enters, it is de- autonomous, evolution PDEs which thereby yield, noted as u˜ (x; t) ≡ u˜ (x1,...,xN ; t); and again upper also autonomous (at least as regards the dependence case letters, U (x; t) ≡ U (x1,...,xN ; t), U˜ (x; t) ≡ onthetimevariable),ω-modified equationsthat, we ˜ believe, deserve to be exhibited and—at least some U (x1,...,xN ; t), are used for matrices. The relations of them—shall warrant further investigation and are among the (independent and dependent) variablesof likely to become useful theoretical tools, in view of the unmodified evolution PDE and the ω-modified evolution PDE are given by the following formulas ∗Thisarticle wassubmitted by the authorsin English. (the trick): 1) Dottorato in Matematica, Universitadi` exp (iωt) − 1 Roma “La Sapienza,” Roma, Italy; e-mail: τ = , (2.1) [email protected] iω 2)Dipartimento di Fisica, UniversitadiRoma` “La Sapienza,” and Istituto Nazionale di Fisica ξn = ξn(t)=xn exp (iµnωt) , (2.2) Nucleare, Sezione di Roma, Roma, Italy; e-mail: [email protected] n =1,...,N, 1063-7788/05/6805-0899$26.00 c 2005 PleiadesPublishing,Inc. 900 MARIANI, CALOGERO u ξ; t =exp(iλωt) w ξ; τ , (2.3a) below, in a few cases, we also exhibit the system of real evolution PDEsobtained in thismanner. u˜ ξ; t =exp iλωt˜ w˜ ξ; τ , (2.3b) 3. ISOCHRONOUS PDEs with analogousformulas[see(2.3)] in the matrix case. Here and hereafter, constants such as µn, λ, In this section, we display, with minimal com- λ˜, α, β (Greek letters) denote rational numbers (not mentary, a list of nonlinear evolution PDEs, each necessarily positive), which whenever necessary shall of them first in its unmodified avatar, then in its be properly assigned, while Latin letters such as a, ω-modified version. It is remarkable that so many b, c denote complex (or, asthe casemay be, real) well-known autonomous evolution PDEs possess ω- constants (sometimes, we keep such constants even modified versionswhich are aswell autonomous,at when they could be eliminated by trivial rescaling leastasregardstheir time dependence; in several transformations; and, of course, by such transforma- cases, this appears to be due to some minor miracles, tions, additional such constants might instead be in- inasmuch as the number of relevant parameters λ, µ, troduced). When N =1, we drop the index n; namely, ν, λ˜, ... issmaller than the equationsthey are re- we write ξ instead of ξ1, x instead of x1,andµ instead quired to satisfy in order to guarantee the autonomous of µ1;andforN =2, we also, to simplify the notation, character of the ω-modified equations, yet nontrivial write η instead of ξ2, y instead of x2,andν instead of parameters satisfying these conditions do exist. All µ2. Note that thistransformation (2.1) –(2.3) entails the ω-modified evolution PDEs displayed below pos- that, at the “initial” time, τ = t =0, the change of sess many isochronous solutions, but only in some variablesdisappearsaltogether: (rare) casesdowe exhibit below examplesof these ξ(0) = x,u(x;0)=w ξ;0 , (2.4) solutions. A discussion of each of these nonlinear evolution PDEswould indeed require much more u˜ ξ;0 =˜w ξ;0 . space. Let us also emphasize that the list reported below includes only some kind of representative in- Hereafter, subscripted variables denote partial differ- stances of this phenomenology; obviously, many more ≡ ≡ entiations, wτ ∂w ξ; τ /∂τ, uxn ∂u(x; t)/∂xn, examplescould be added. The listisarranged in a and so on. Let us emphasize—obviousasthismay user-friendly manner, being ordered according to the be—that, since the transition from an (unmodified) following taxonomic rules: of primary importance is PDE satisfied by w ξ; τ to the corresponding (ω- the number of independent variables; next, the num- modified) PDE satisfied by u (x; t) isperformed via ber of dependent variables; next, the order of the dif- the explicit change of variables(2.1) –(2.3) (the trick), ferential equation, but with primary attention to the properties such as integrability or solvability, if pos- time variable; then, the type of nonlinearity [except sessed by the unmodified PDE satisfied by w ξ; τ , when an equation is presented as a special case of a more general equation—see, for instance, (3.9) and carry over to the corresponding (ω-modified) PDE (3.34)]. satisfied by u (x; t), which generally hasin addition the property of isochronicity, as defined above. Let us The following unmodified (1 + 1)-dimensional moreover note that the property of isochronicity of the “generalized shock-type” PDE isintegrable, indeed ω-modified evolution PDE, which doesnot require solvable (see, for instance, [24]): that the original unmodified PDE from which it has α wτ = aw wξ. (3.1a) emerged be itself integrable, implies that, in some open set of its phase space, the ω-modified equation By setting µ =1− αλ, one getsthe corresponding ω- is generally integrable indeed, in some sense, super- modified evolution PDE integrable (for a discussion of this question in the α ut − iλωu + i (αλ − 1) ωxux = au ux. (3.1b) ODE context, where the conceptsof integrability and indeed superintegrability can be given a more precise The general solution to the initial-value problem for meaning, see[17] and [19]). Let usend thissection thisPDE (3.1b) isgiven, in implicit form, by the by pointing out that, in most cases, the ω-modified following formula: equationsare complex; they can of coursebe rewrit- iλωt ten in real form by introducing the real and imagi- u(x; t)=e u0 exp [i(1 − αλ)ωt] (3.1c) nary parts (or, instead, the amplitudes and phases) of all the quantitiesthat enter in thesePDEsand 1 − e−iωt × x + a [u(x; t)]α , by then considering the two, generally coupled, real iω PDEsthat are obtained from each complex PDE by considering separately its real and imaginary parts; where of course u0(x)=u(x;0). PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 5 2005 ISOCHRONOUS PDEs901 The following unmodified (1 + 1)-dimensional Here, we assume of course that the two dependent “generalized Burgers–Hopf” PDE (see [25]) reads variables u1 ≡ u1(x, t) and u2 ≡ u2(x, t) are real and α that the four arbitrary constants c1, c2, c3,andc4 are wτ = awwξ + b (w wξ) . (3.2a) ξ real aswell. By setting λ =1/(2 − α) and µ =(1− α)/(2 − α), The following unmodified (1 + 1)-dimensional onegetsthecorrespondingω-modified evolution “Schwarzian KdV” equation isintegrable (see,for PDE instance, [28]): 1 1 − α u + i ωu + i ωxu (3.2b) 2 t − − x wξξ α 2 α 2 wτ = wξξξ + a . (3.5a) α wξ = auux + b (u ux)x . Note that, for α =1, thisPDE becomesautonomous By setting µ =1/3, one getsthe corresponding also with respect to the space variable x, while for α = ω-modified evolution PDE 0 the PDE (3.2a) becomes the standard (solvable) 2 ω uxx Burgers–Hopf equation. ut − iλωu − i xux = uxxx + a . (3.5b) 3 w The following unmodified (1 + 1)-dimensional x dispersive KdV-like PDE is integrable, indeed solv- The following unmodified (1 + 1)-dimensional able (see, for instance, [26]): “Cavalcante–Tenenblat” equation isintegrable (see, 2 2 4 for instance, [29]): wτ = wξξξ +3 wξξw +3w (wξ) +3wξw . (3.3a) 1 3/2 wτ = a √ + b (wξ) . (3.6a) wξ By setting λ =1/6, µ =1/3, and (for notational ξξ simplicity) Ω=ω/6, one getsthe corresponding Ω- By setting λ =0, µ =2/3, one getsthe correspond- modified evolution PDE ing ω-modified evolution PDE u − iΩu − 2iΩxu = u (3.3b) t x xxx 2ω 2 2 4 ut − i xux (3.6b) +3 uxxu +3uux +3uxu . 3 1 3/2 The symmetry properties, and some explicit solu- = a √ + b (ux) . tions, of the following unmodified (1 + 1)-dimensional ux xx “generalized KdV” equation have been investigated recently [27]: The KdV class of unmodified (1 + 1)-dimensional integrable evolution PDEs (see, for instance, [30]) α β wτ = a (w )ξξξ + b w . (3.4a) reads ξ m wτ =Λ wξ,m=1, 2,..., (3.7a) By setting λ =2/(3β − α − 2) and µ =(β − α)/(3β − α − 2), one getsthe ω-modified evolution PDE where Λ isthe integro-di fferential operator (depend- − ing on the dependent variable w(ξ; τ))thatactsona − 2iω − i(β α)ω ut − − u − − xux (3.4b) generic (twice-differentiable and integrable at infin- 3β α 2 3β α 2 ity) function φ(ξ) asfollows: α β = a (u )xxx + b u . − x Λφ (ξ)=φξξ(ξ) 4w(ξ; τ)φ(ξ) (3.7b) ∞ − − We assume here of course that 3β α 2 =0.Par- ticularly interesting is the case with α = β, when this +2wξ(ξ; τ) φ ξ dξ . ω-modified PDE becomesautonomousalsoin the ξ space variable x. In the even more special case with α = β =2and by setting u = u1 + iu2, a = c1 + ic2, By setting λ =2/(2m +1), µ =1/(2m +1),and(for and b = c3 + ic4, we rewrite thisevolution PDE asa notational simplicity) Ωm = ω/(2m +1),onegets system of two coupled PDEs: the corresponding class of Ωm-modified evolution 2 − 2 − PDEs u1t + ωu2 = c1 u1 u2 2c2u1u2 (3.4c) x − m 2 − 2 − ut iΩm (2u + xux)=L ux, (3.7c) + c3 u1 u2 2c4u1u2 , xxx m =1, 2,..., − 2 − 2 u2t ωu1 = c2 u1 u2 +2c1u1u2 x where L isthe integro-di fferential operator (depend- + c u2 − u2 +2c u u . 4 1 2 3 1 2 xxx ing on the dependent variable u(x; t)) analogousto PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 5 2005 902 MARIANI, CALOGERO Λ, namely, the operator that actson a generic (twice- The following unmodified (1 + 1)-dimensional differentiable and integrable at infinity) function f(x) solvable PDE reads asfollows: wττwξ − wτξwτ =0. (3.10a) − Lf (x)=fxx(x) 4u(x; t)f(x) (3.7d) The corresponding ω-modified PDE isin thiscase t- ∞ autonomousfor any choice of λ and µ +2u (x; t) f x dx . x uttux − utxut + iω[(−λ + µ +1)utux (3.10b) x + λuutx − µxutxux + µxutuxx] For m =1, the PDE (3.7a) becomesthe well-known + ω2[λ(µ − 1)uu +(−λ + µ − 2)µx(u )2 KdV equation x x + λµxuuxx]=0. wτ + wξξξ =6wwξ, (3.8a) The general solution to this PDE (3.10b) reads and the corresponding ω-modified equation reads iλωt iµωt iωt ω u(x; t)=e f g xe + e , (3.10c) ut + uxxx − i (2u + ux)=6uux. (3.8b) 3 where f(z) and g(z) are two arbitrary analytic func- The unmodified (1 + 1)-dimensional “Monge– tions. Ampere” integrable PDE (see, for instance, [24, 31]) The following unmodified (1 + 1)-dimensional reads “Boussinesq” equation isintegrable (see,for − 2 instance, [32]): wττwξξ wξτ =0. (3.9a) wττ =(wξξξ + wwξ)ξ . (3.11a) The corresponding ω-modified PDE isin thiscase t- autonomousfor any choice of λ and µ By setting λ =1and µ =1/2, one getsthe corre- sponding ω-modified evolution PDE u u − (u )2 + iω[−(2λ +1)u u (3.9b) tt xx tx t xx iω 2 − utt − iωu − xux =(uxxx + uux) . (3.11b) +2(λ + µ)utxux]+ω [ λ(λ +1)uuxx 2 x − 2 2 + µ(µ 1)xuxuxx +(λ + µ) (ux) ]=0. The following unmodified (1 + 1)-dimensional The general solution to thisPDE (3.9b) isgiven in “nonlinear wave” equation (see, for instance, [24]) two steps: first, for any arbitrary function F (r), find reads α the function r(x; t) from the nondifferential equation: wττ =(w wξ)ξ . (3.12a) − − r(x; t)=xei(µ 1)ωt − e iωtF [r(x; t)] ; (3.9c) By setting µ =1− αλ/2, one getsthe corresponding ω-modified evolution PDE then, for any arbitrary function G(r) and constant a, u − i(2λ +1)ωu − iω(2 − αλ)xu (3.12b) evaluate the solution tt t tx t αλ − λ(λ +1)ω2u − (2 − αλ) λ +1− ω2xu iλωt iωt 4 x u(x; t)=e dt e G r(x; t ) . (3.9d) 2 a αλ 2 2 α − 1 − ω x uxx =(u ux) . A class of explicit solutions to this PDE (3.9b) is 2 x sin (ωt/2) Two special cases of this nonlinear PDE warrant u(x; t)=ei(λ+µ)ωtxf , i(µ−1/2)ωt (3.9e) explicit display: xe ω 2 ux where f(z) isan arbitrary function. Three particularly utt + u = 4 (3.12c) neat cases of the ω-modified evolution PDE (3.9b) are 2 u x worth explicit display: corresponding to α = −4, λ = −1/2; for λ = µ =0, − − 2 utt 5ωut 6ω u =(uux)x (3.12d) − 2 − uttuxx utx iωutuxx =0; (3.9f) corresponding to α =1, λ =2(some solutions to this for λ = −1, µ =1, ω-modified PDEsare exhibited in [21]). Another class (out of many possible ones) of un- − 2 uttuxx utx + iωutuxx =0; (3.9g) modified (1 + 1)-dimensional “nonlinear wave” equa- − tionsreads for λ = 1/2, µ =1/2, α a ∂pk w k ω 2 k 2 wττ = 3+α p , (3.13a) uttuxx − u + (u − xux) uxx =0. (3.9h) w k ∂ξ k tx 2 k PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 5 2005 ISOCHRONOUS PDEs903 ˜ where the numbers pk are nonnegative integers, or By setting λ = −λ and µ =1/2, one getsthe corre- possibly just integers: here and hereafter, we indicate sponding ω-modified system: ξ ξp ξ2 ω u − √ ∂ p ut − iλωu − i xux = a , (3.17b) f(ξ)= dξ dξ − ... f(ξ )dξ . 2 1+uu˜ ∂ξ−p p p 1 1 1 xx ω √ u˜ − u˜t + iλωu˜ − i xu˜x = b . By setting λ = 1/2 and µ =0, one getsthe corre- 2 1+uu˜ sponding ω-modified evolution PDE xx p αk The following unmodified (1 + 1)-dimensional ω 2 ak ∂ k u utt + u = . (3.13b) “Landau–Lifshitz” system of two coupled PDEs is 2 u3+αk ∂xpk k integrable (see, for instance, [38, 33]): w = − sin(w)˜w − 2cos(w)w w˜ (3.18a) The following unmodified (1 + 1)-dimensional τ ξξ ξ ξ system of two coupled PDEs is integrable (see, for +(a − b)sin(w)cos(˜w)sin(˜w), instance, [33]): wξξ 2 w˜τ = − cos(w)(˜wξ) 2 sin(w) wτ = wξξ +˜w , w˜τ = wξξ. (3.14a) +cos(w) a cos2(˜w)+b sin2(˜w)+c . By setting λ = λ˜ =1and µ =1/2, one getsthe cor- responding ω-modified system: By setting λ = λ˜ =0and µ =1/2, one getsthe cor- responding ω-modified system − − iω 2 ut iωu xux = uxx +˜u , (3.14b) iω 2 ut − xux = − sin(u)˜uxx − 2cos(u)uxu˜x (3.18b) iω 2 u˜t − iωu˜ − xu˜x = uxx. − 2 +(a b)sin(u)cos(˜u)sin(˜u), iω uxx The following unmodified (1 + 1)-dimensional u˜ − xu˜ = − cos(u)˜u2 t 2 x [sin(u)] x system of two coupled PDEs is integrable (see, for instance, [34, 33]): +cos(u) a cos2(˜u)+b sin2(˜u)+c . 2 wτ = a (ww˜)ξ , w˜τ = bw + cw˜ ξ . (3.15a) Note the simplification if a = b and, moreover, if c = −a. By setting λ =2(1− µ) and λ˜ =1− µ, one getsthe The following unmodified (2 + 1)-dimensional in- corresponding ω-modified system: tegrable PDE (see, for instance, [24]) reads u − 2i(1 − µ)ωu − iµωxu = a (uu˜) , (3.15b) wτ = awη + bww . (3.19a) t x x ξ − − − 2 u˜t i(1 µ)ωu˜ iµωxu˜x = bu + cu˜ x . By setting λ =0and µ = ν =1, one getsthe corre- sponding ω-modified evolution PDE The following unmodified (1 + 1)-dimensional u − iω (xu + yu )=au + buu . (3.19b) “Zakharov–Shabat” system of two coupled PDEs t x y y x isintegrable (see[35, 36]): The following unmodified (2 + 1)-dimensional in- 2 2 tegrable PDE (see, for instance, [24]) reads wτ + wξξ = w w,˜ w˜τ − w˜ξξ = −w˜ w. (3.16a) 2 wτ = awη + b (wξ) . (3.20a) By setting λ˜ =1− λ and µ =1/2, one getsthe cor- responding ω-modified system: By setting λ =0, µ =1/2,andν =1, one getsthe ω corresponding ω-modified evolution PDE u − iλωu − i xu + u = u2u,˜ (3.16b) t x xx x 2 2 ut − iω ux + yuy = auy + b (ux) . (3.20b) ω 2 2 u˜t − i(1 − λ)ωu˜ − i xu˜x − u˜xx = −uu˜ . 2 The following unmodified (2 + 1)-dimensional The following unmodified (1 + 1)-dimensional PDE reads α “Wadati–Konno–Ichikawa” system of two coupled wτ = a (wξwη − wwξη) . (3.21a) PDEsisintegrable (see[37]): By setting λ =1/(2α − 1) and µ = ν =0,onegets w w˜ the corresponding ω-modified evolution PDE wτ = a √ , w˜τ = b √ . 1+ww˜ 1+ww˜ iω ξξ ξξ u − u = a (u u − uu )α . (3.21b) (3.17a) t 2α − 1 x y xy PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 5 2005 904 MARIANI, CALOGERO A (rather trivial) separable solution of this PDE reads − 3iνΩyuy =(uu˜) , u˜y = ux. x iωt u(x, y; t)=exp f(x)g(y), (3.21c) 2α − 1 The following unmodified (2 + 1)-dimensional “long-wave equation” system of two coupled PDEs where f(x) and g(y) are two arbitrary functions. is integrable (see, for instance, [40, 41]): The following unmodified (2 + 1)-dimensional 1 2 PDE reads wτη +˜wξξ = w ξη , (3.24a) α 2 wττ = a(wξwη − wwξη) . (3.22a) w˜τ + wξξ =(ww˜ + wξη)ξ . By setting λ =2/(2α − 1) and µ = ν =0,onegets the corresponding ω-modified evolution PDE By setting λ =1/2, λ˜ =0, µ =1/2, ν = −1/2,and, Ω=ω/2 2α +3 2(2α +1) for notational convenience, , one getsthe u − iωu − ω2u (3.22b) corresponding ω-modified system: tt 2α − 1 t (2α − 1)2 α 1 2 = a(uxuy − uuxy) . u − iΩu − iΩ(xu − yu )+˜u = u , ty y xy yy xx 2 xy A (rather trivial) separable solution of this PDE reads (3.24b) 2iωt u˜t − iΩ(xu˜x − yu˜y)+uxx =(uu˜ + uxy)x. u(x, y; t)= b exp (3.22c) 2α − 1 2α +1 The following unmodified (2 + 1)-dimensional + c exp iωt f(x)g(y), system of two coupled PDEs is integrable (see, for 2α − 1 instance, [42]): where a and b are two arbitrary constants and f(x) 2 and g(y) are two arbitrary functions. wτ + wξξη =(w )η + wξw,˜ w˜ξ = wη. (3.25a) The following unmodified (2 + 1)-dimensional system of two coupled PDEs is integrable (see, for By setting λ˜ =1− λ/2, µ = λ/2,andν =1− λ,one instance, [39]): getsthe corresponding ω-modified system: wτ + wξξξ =(ww˜)ξ , w˜η = wξ. (3.23a) iλω ut − iλωu − xux − i(1 − λ)ωyuy (3.25b) ˜ 2 By setting λ = ν +1/3, λ =2/3, µ =1/3,and,for 2 notational convenience, Ω=ω/3, one getsthe cor- + uxxy =(u )y + uxu,˜ u˜x = uy. responding Ω-modified system: A nontrivial family of solutions of this system (3.25b) ut + uxxx − i(3ν +1)Ωu − iΩxux (3.23b) readsasfollows: −4a2eiλωt u(x, y; t)= , (3.25c) cosh axeiλωt/2 − f(t) − g yei(1−λ)ωt − beiωt 2 eiλωt/2 u˜(x, y; t)= − f (t)+eiωtg yei(1−λ)ωt − beiωt iωb − 4a2 a 2 2a + , cosh axeiλωt/2 − f(t) − g(yei(1−λ)ωt − beiωt) where a and b are two arbitrary constants and integrable (see, for instance, [43]): f(t) and g(z) are arbitrary functions(and of course − ˜ 2 f (t) and g (z) denote their derivatives). Conditions Wτ + Wξξξ 3Wη =3 W (3.26a) ξ sufficient to guarantee that thissolutionwill be ˜ ˜ isochronous are given in [21]. +3i W, W , Wξ = Wη. The following unmodified (2 + 1)-dimensional Here, W ≡ W (ξ,η; τ) and W˜ ≡ W˜ ≡ W˜ (ξ,η; τ) are “matrix KP” system of two coupled matrix PDEs is matrices (of course, of the same rank), and the no- PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 5 2005 ISOCHRONOUS PDEs905 tation W, W˜ denotestheir commutator. By setting where m isan integer and f(w) isan analytic func- − ˜ tion. By setting λ =0and µ = 2/m, one getsthe λ =2/3, λ =1, µ =1/3,andν =1, one getsthe corresponding ω-modified evolution PDE corresponding ω-modified system: ∂m+2u ∂m+1u ∂m+2u 2 +3iω +4iωx (3.29b) U + U − 3U˜ − iωU (3.26b) 2 m m m+1 t xxx y 3 ∂t ∂x ∂t∂x ∂t∂x m m+1 1 2 ∂ u 2 2 ∂ u − iωxU − iωyU =3 U 2 +3i U, U˜ , − 2ω − 3+ ωx x y x ∂xm m m ∂xm+1 3 U˜ = U . 2 2 ∂m+2u x y − ω2x2 = f (u) . m ∂xm+2 The following class of unmodified (N +1)-dimen- sional PDEs, The following unmodified (N +1)-dimensional ∂m+1w PDE reads = f (w) , (3.27a) ∂τ∂ξm N w = w [a (w )αn ] (3.30a) where m is a positive integer (or possibly just an ττ τ n ξn integer) and f(w) isan arbitrary analytic function, n=1 N gets transformed, by setting λ =0and µ = −1/m, − + b (w )2(βαn 1) w . into the corresponding ω-modified evolution PDE n ξn ξnξn n=1 ∂m+1u ∂mu ω ∂m+1u + iω + i x = f (u) . (3.27b) By setting λ = β − 1 and µ =1− β +1/α ,one ∂t∂xm ∂xm m ∂xm+1 n n getsthe corresponding ω-modified evolution PDE For instance, for m =1 and f(w)=exp(aw), Eq. (3.27a) becomesthe Liouville equation utt − iω (2β − 1)ut (3.30b) wτξ =exp(aw), (3.28a) and the corresponding ω-modified evolution PDE N 1 + 1 − β + x u (3.27b) reads α n txn n=1 n u + iωu + iωxu =exp(au) . (3.28b) tx x xx N − 2 − − 1 The general solution to this ω-modified Liouville ω β(β 1)u + β 1 β + xnuxn αn equation reads n=1 − u(x; t)=g(xe iωt)+f (t) (3.28c) = ut − iω (β − 1)u x 2 − − − ln be iωt dy exp ag ye iωt a N − 1 x0 + 1 β + xnuxn αn t n=1 a N N − ds exp [iωs − af (s)] , − × αn 2(βαn 1) 2b [an (uxn ) ]+ bn (uxn ) uxnxn . 0 n=1 n=1 where b isan arbitrary (nonvanishing) complex con- stant, x0 isan arbitrary real constant, g(x) isan The following unmodified (N +1)-dimensional arbitrary analytic function, and f(t) isaswell an “nonlinear diffusion” PDE (see, for instance, [24]) arbitrary function of the real time variable t,butit reads must of course be periodic with period T [see (1.1)] for N α the isochronicity property to hold (other conditions on wτ = w wξnξn . (3.31a) this general solution sufficient to guarantee thisare n=1 given in [21]). By setting µn =(1− αλ)/2, one getsthe corre- The following unmodified (N +1)-dimensional sponding ω-modified evolution PDE PDE reads N ∂m+2w 1 − αλ = f (w) , (3.29a) u − iλωu − i ω x u (3.31b) ∂ξm∂τ2 t 2 n xn n=1 PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 5 2005 906 MARIANI, CALOGERO N u ≡ u(x; t) α The general solution to thisPDE (3.33b) = u uxnxn . isgiven by the implicit formula n=1 n iωt − Note the simplification for λ =1/α. (e 1)f0(u)+ xkfk(u)=c, (3.33c) k=1 The following unmodified (N +1)-dimensional where the N +1 functions fk(z), k =0, 1,...,N, “nonlinear heat equation with a source” (see, for are arbitrary. For N =1,theunmodified (1 + 1)- instance, [24]) reads dimensional Bateman equation reads N 2 2 − wττ(wξ) + wξξwτ 2wξwτ wξτ =0, (3.34a) αn β wτ = an (w wξ ) + bw . (3.32a) n ξn and the ω-modified version of this equation reads n=1 2 2 uttu + uxxu − 2uxtuxut (3.34b) By setting λ =1/(β − 1) and µ =(β − α )/[2 × x t n n − − 2 (β − 1)], one getsthe corresponding ω-modified evo- + iω[2λu(uxuxt utuxx)+(2µ 1)(ux) ut] 2 2 2 2 2 2 lution PDE + ω [λ u(ux) − λ u uxx + λ(2µ − 1)u(ux) N − 3 iω iω (β − α ) + µ(µ 1)x(ux) ]=0. u − u − n x u (3.32b) t β − 1 2 (β − 1) n xn The general solution to this equation (3.34b) reads (in n=1 implicit form) N = a (uαn u ) + buβ. iωt − −iλωt n xn xn (e 1)f e u(x; t) (3.34c) n=1 + xeiµωtg e−iλωtu(x; t) = c, (N +1) The following unmodified -dimensional with f(z) and g(z) two arbitrary functionswhich solvable “Bateman” PDE (see, for instance, [44]) can be easily determined in terms of the initial data, reads say, u0(x)=u(x;0)and u1(x)=ut(x;0). By setting ··· λ =0, µ =1/2, and (for notational convenience) Ω= 0 wτ wξ1 wξn ω/2,theω-modified (1 + 1)-dimensional Bateman ··· equation (3.34b) takesthe simple(real) form wτ wττ wξ1τ wξnτ 2 2 − 2 3 det ··· =0. (3.33a) uttux + uxxut 2uxtuxut =Ω xux. (3.34d) wξ1 wτξ1 wξ1ξ1 wξnξ1 . . . . . Note that, if u(x; t) isa solutionto thisPDE, v(x; t)= ...... f [u(ax; t − b)] is also a solution, with f(z) an arbi- ··· a b wξn wτξn wξ1ξn wξnξn trary function and and two arbitrary constants. The initial-value problem for thisequation issolved The corresponding ω-modified PDE isin thiscase by the implicit formula t-autonomousfor any choice of λ and µn.Forλ = u(x; t) (3.34e) µ =0, it reads n (inv) u1 u [u(x; t)] tan(Ωt) 0 x ··· = u0 + , 0 ut ux1 uxn (inv) Ω u u [u(x; t)] cos[Ωt] ··· 0 0 ut utt ux1t uxnt where of course u(inv)(z) (respectively, u (z))arethe det u u u ··· u (3.33b) 0 0 x1 x1t x1x1 xnx1 inverse (respectively, the derivative) of the function . . . . . ...... (inv) u0(z) (namely, u0 [u0(x)] = x, u0(x)=du0(x)/dx). ··· And two explicit solutionsof thisequation (3.34d) uxn utxn ux1xn uxnxn read asfollows: ··· c1x + c2 cos [Ω(t − c3)] 0 ux1 uxn u(x; t)=f , (3.34f) c4x + c5 cos [Ω(t − c6)] u u ··· u − x1 x1x1 xnx1 iωutdet =0. − . . .. . c1 cos [Ω(t c2)] . . . . u(x; t)=f x cos [2Ω(t − c3)] + cos [2Ω(c2 − c3)] u u ··· u xn x1xn xnxn (3.34g) PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 5 2005 ISOCHRONOUS PDEs907 + c tan [Ω(t + c − 2c )] . and the general solution, in implicit form, to this PDE 4 2 3 reads 1/α Here, f(z) denotesan arbitrary (twice di fferentiable) ak − u(x; t)=eiωt/α 1+ 1 − e iωt [u(x; t)]α function, and the constants ck are arbitrary. Clearly, ω these solutions are real if the function f(z) and the (4.2b) constants ck are themselves real, and the conditions i ak −iωt α that the function f(z) and the constants ck must × u0 x − log 1+ 1 − e [u(x; t)] , satisfy in order to guarantee the isochronicity of these k ω solutions are rather obvious. where of course the initial datum, u0(x)=u(x;0), The following unmodified (N +1)-dimensional should itself be periodic with period L (or some ap- PDE reads propriate integer multiple, or fraction, of L), in order 2 wττ = wτ (3.35a) for thissolutionto be periodic for all time with period × f (w, w ,...,w ,w ...,w ,w ...) , L (or some appropriate integer multiple, or fraction, ξ1 ξN ξ1ξ1, ξN ξN ξ1ξ1ξ1 of L)—in addition of course to being periodic in t with where f(w,wx1,...,wξN,wξ1ξ1,...,wξN ξN,wξ1ξ1ξ1 ...) period T [see (1.1)] or with some integer multiple of isan arbitrary analytic function. By setting λ = µn = T , depending on the analyticity propertiesof u0(z) 0, one getsthe corresponding ω-modified evolution asa function of the complex variable z. An explicit PDE special case of this implicit equation (corresponding − 2 iqx utt iωu = ut (3.35b) to u0(x)=e ) reads × iωt/α iqx f (u, ux1 ,...,uxN ,ux1x1, ...,uxN xN ,ux1x1x1 ...) . u(x; t)=be e (4.2c) 1/α+q/k ak −iωt α 4. OUTLOOK × 1+ 1 − e [u(x; t)] , ω Asmentioned in the introductory Section 1, each yielding, for q = −k/α, the trivial solution to (4.2a) of the ω-modified equationsreported above would require a separate investigation to exhibit the exis- (ωt − kx) u(x; t)=b exp i . (4.2d) tence and characterize the propertiesof isochronous α solutions. This shall eventually be done, probably mainly in the context of possible applications of these For q = k/α, one obtainsinsteadfrom (4.2c) the equations. Throughout this paper, we focused on ω- explicit solution to (4.2a) modified equationsthat feature many isochronous − ω solutions. By a variant of the trick (2.1)–(2.3), it is u(x; t)=ei(ωt kx)/α (4.2e) 2 2 iωt − 2 in some cases possible to generate equations that 2a bk (e 1) feature many solutions which are completely periodic ikx iωt not only in the time variable t but also in the space × ω − 2abe (e − 1) variable x. We only exhibit here a single example, obtained from the unmodified PDE (3.1a) via the 1/α − 2 − ikx iωt − following change of variables: ω 4abkωe (e 1) . eiωt − 1 τ = , (4.1a) Of course, many other explicit solutions could be iω exhibited in specific cases, for instance, in the special eikx − 1 case with q = −k and α =2 ξ = , (4.1b) ik u(x; t) (4.2f) iλωt iρkx u(x; t)=e e w(ξ; τ), (4.1c) ω ± 4ab2kωe−2ikx(eiωt − 1) + ω2 = ± . where, to apply thistransformationto the unmodi fied 2ak (1 − eiωt) PDE (3.1a), we set λ =1/α and ρ = −1/α, while ω and k are two real (indeed, without loss of generality, Finally, we also rewrite below the evolution PDE positive) constants that determine the basic period (4.2a) with α =1in real form, setting u = u1 + iu2 in the (real) time variable t [see (1.1)] and the basic and a = c1 + ic2, where the two dependent variables ≡ ≡ period, L =2π/k, in the (also real) space variable x. u1 u1(x; t) and u2 u2(x; t), aswell asthe two The modified PDE corresponding to (3.1a) then reads constants c1 and c2, are of course now real: u1 + ωu2 = c1 [u1u1 + u2u2 − 2ku1u2] (4.3) − iω α ik t x x ut u = au ux + u , (4.2a) − 2 − 2 α α c2 u1u2x + u2u1x + k u1 u2 , PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 5 2005 908 MARIANI, CALOGERO − − u2t ωu1 = c2 [u1u1x + u2u2x 2ku1u2] 24. 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