The chiral phase transition with a chiral chemical potential in the framework of Dyson-Schwinger equations
Shu-Sheng Xu, Zhu-Fang Cui, Bin Wang, Yuan-Mei Shi, You-Chang Yang, Hong-Shi Zong
aa r X i v : . [ h e p - ph ] M a y The chiral phase transition with a chiral chemical potential in the framework ofDyson-Schwinger equations
Shu-Sheng Xu , , Zhu-Fang Cui , , Bin Wang , Yuan-Mei Shi , You-Chang Yang , , and Hong-Shi Zong , , , ∗ Key Laboratory of Modern Acoustics, MOE, Institute of Acoustics,and Department of Physics, Nanjing University, Nanjing 210093, China Department of Physics, Nanjing University, Nanjing 210093, China Department of Physics, Huazhong University of Science and Technology, Wuhan 430074, China Department of Physics and Electron ic Engineering,Nanjing Xiaozhuang University, Nanjing 211171, China School of Physics and Mechanical-Electrical Engineering, Zunyi Normal College, Zunyi 563002, China Joint Center for Particle, Nuclear Physics and Cosmology, Nanjing 210093, China and State Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, CAS, Beijing, 100190, China
Within the framework of Dyson-Schwinger equations (DSEs), we discuss the chiral phase transi-tion of QCD with a chiral chemical potential µ as an additional scale. We focus especially on theissues related to the widely accepted as well as interested critical end point (CEP). With the helpof a scalar susceptibility, we find that there might be no CEP in the T − µ plane, and the phasetransition in the T − µ plane might be totally crossover when µ <
50 MeV, which has apparentconsistency with the Lattice QCD calculation. Our study may also provide some useful hints tosome other studies related to µ .Key-words: chiral chemical potential, chiral phase transition, Dyson-Schwinger equationsPACS Number(s): 11.30.Rd, 25.75.Nq, 12.38.Mh, 12.39.-x I. INTRODUCTION
Quantum Chromodynamics (QCD), which describesthe interactions between quarks and gluons, is alreadycommonly accepted as an essential part of the StandardModel of particle physics. Dynamical chiral symmetrybreaking (DCSB) and quark color confinement are twofundamental features of QCD, and there are also manylaboratories and experiments on this field, such as the fa-mous Relativistic Heavy Ion Collider (RHIC) and LargeHadron Collider (LHC). However, thanks to the com-plicated non-Abelian feature of QCD itself, it is so dif-ficult to have a thorough understanding of the mecha-nisms of DCSB and confinement, especially in the in-teresting non-perturbative region, which means quarksand gluons are strongly coupled to each other and thenthe related processes have small momentum transfer (orequivalently, the coupling constant becomes large andrunning). In this case, nowadays people often and insome sense have to resort to various effective models tostudy them phenomenologically, such as the chiral per-turbation theory [1–5], the global color symmetry model(GCM) [6–10], the quasiparticle model [11–18], the QCDsum rules [19–22], the Nambu–Jona-Lasinio (NJL) modeland the related Polyakov-loop-extended Nambu–Jona-Lasinio (PNJL) model [23–34], Lattice QCD [35–37], andthe Dyson-Schwinger equations (DSEs) [38–45]. Throughthese studies, people hope to get profound insight of ournature as well as the early Universe. ∗ Email: [email protected]
Generally speaking, chiral symmetry is an exact globalsymmetry only when the current quark mass m is zero(the chiral limit case). In the low temperature ( T ) andlow chemical potential ( µ ) phase (the hadronic phase,often referred to as Nambu-Goldstone phase or Nambuphase), this symmetry is spontaneously broken, and asa consequence there exist N f − N f is the number offlavour) pseudoscalar Nambu-Goldstone bosons, mean-while the QCD vacuum hosts a chiral condensate (twoquark condensate) h ¯ qq i (which can actually act as an or-der parameter for chiral phase transition). At present,it is commonly accepted that when temperature and/orquark chemical potential are high enough, the stronglyinteracting hadronic matter will undergo a phase tran-sition to some new phase, where the chiral symmetryis restored for the chiral limit case or partially restoredfor the m = 0 case. This new phase is usually calledWigner phase, and in some sense is related to the fa-mous quark gluon plasma (QGP), which is expected toappear in the ultra-relativistic heavy ion collisions or theinner core of compact stars. As for the nature of thechiral phase transition when m = 0, a popular scenariofavors a crossover at small chemical potential, and thenturning into a first order chiral transition for larger chem-ical potential at a critical end point (CEP) [46]. Such apicture is consistent with most Lattice QCD simulationsand various QCD-inspired models, as listed in the lastparagraph, however, it is not yet clarified directly fromthe first principles of QCD. The search for such a CEP isalso one of the main goals in the high energy physics ex-periments, such as the beam energy scan (BES) programat RHIC [47–50]. Unfortunately, Lattice Monte Carlosimulations cannot be used to resolve this issue directlydue to the “sign problem” [51–56], and until now there isstill no firm theoretical evidence for the existence of sucha CEP, so the calculations based on some effective QCDmodels are also irreplaceable nowadays.In Ref. [57], K. Fukushima et al firstly introduce thechiral chemical potential (also called axial chemical po-tential in some other literatures, such as Ref. [58]), µ ,which is conjugated to chiral charge density; and inRef. [59], M. Ruggieri suggests that the CEP of the chiralphase diagram can be detected by means of Lattice QCDsimulations of grand-canonical ensembles with this chiralchemical potential. By concrete calculations within somechiral models, the author shows that a continuation of theCEP at finite temperature and finite chemical potential,to a possible “CEP ” in the T − µ plane is reachable,which is then helpful in the determining of the CEP inthe T − µ plane from Lattice QCD. The existence of sucha possible CEP is also confirmed in some other chiralmodel studies, for example, Refs. [60, 61]. In Ref. [58],the authors investigated the effect of the vector inter-action as well as the finite current quark mass on thelocation of the CEP. In this paper, we will discuss therelated topics within the framework of Dyson-Schwingerequations, which is widely used as well as has been provedto be successful in hadron physics and phase transitionsof strongly interacting matters. The following of this pa-per is organized in such a way: in Sect. II we give abasic introduction to the DSEs at finite temperature andnonzero chemical potential as well as an effective modelgluon propagator, and with the help of a scalar suscepti-bility we also discuss the nature of the chiral phase tran-sition within this framework; then in Sect. III, we discussthe influences of the chiral chemical potential on the chi-ral phase transition of QCD in detail, and mostly focuson the behaviours of the CEP, not only the algebra butalso the numerical results; at last, a brief summary isgiven in Sect. IV. II. DYSON SCHWINGER EQUATIONS ANDAN EFFECTIVE GLUON PROPAGATOR
In this section, we will briefly review the formula ofDyson Schwinger equations, which is not only widelyused in the non-perturbative region of QCD, but alsoin some other fields like the Quantum Electrodynamicsin (2+1) dimensions (QED ) [38, 62–65], etc. At zerotemperature and zero chemical potential, the DSE forthe quark propagator reads [38] (we will always work inEuclidean space and take the number of flavors N f = 2while the number of colors N c = 3 throughout this pa-per. Moreover, as we employ a ultra-violet finite model,renormalization is actually unnecessary) S ( p ) − = S ( p ) − + 43 Z d q (2 π ) g D µν ( p − q ) γ µ S ( q )Γ ν , (1) where S ( p ) is the dressed quark propagator, S ( p ) − = iγ · p + m, (2)is the inverse of the free quark propagator, g is the strongcoupling constant, D µν ( p − q ) is the dressed gluon prop-agator, and Γ ν = Γ ν ( p, q ) is the dressed quark-gluon ver-tex. According to the Lorentz structure analysis, we have S ( p ) − = i pA ( p ) + B ( p ) , (3)where A ( p ) and B ( p ) are scalar functions of p . Af-ter the gluon propagator together with the quark-gluonvertex are specified, people can then solve this equationnumerically.The extension of the above quark DSE to its nonzerotemperature and nonzero quark chemical potential ver-sion is systematically accomplished by transcription ofthe quark four-momentum via p → p k = ( ~p, ˜ ω k ), where˜ ω k = ω k + iµ with ω k = (2 k + 1) πT , k ∈ ZZ the fermionMatsubara frequencies, and no new parameters are in-troduced [39] S ( p k ) − = S ( p k ) − + 43 T ZX g D µν ( p k − q n ) γ µ S ( q n )Γ ν . (4)where S ( p k ) − = iγ · p k + m, (5)and RP denotes P + ∞ l = −∞ R d ~q (2 π ) . Nevertheless, its solutionnow should have four independent amplitudes due to thebreaking of O (4) symmetry down to O (3) symmetry [39] S ( p k ) − = i ~p A ( ~p , ˜ ω k ) + B ( ~p , ˜ ω k )+ iγ ˜ ω k C ( ~p , ˜ ω k )+ ~p γ ˜ ω k D ( ~p , ˜ ω k ) , (6)where ~p = ~γ · ~p , ~γ = ( γ , γ , γ ), and the four scalarfunctions F = A , B , C , D are complex and satisfy F ( ~p , ˜ ω k ) ∗ = F ( ~p , ˜ ω − k − ) . (7)But as discussed in Ref. [39], the dressing function D ispower-law suppressed in the ultra-violate region, so thatactually does not contribute in all cases investigated inour work. At zero temperature but nonzero chemical po-tential case, D vanishes exactly since the correspondingtensor structure has the wrong transformation proper-ties under time reversal [66]. For these reasons, in mostcases we can just neglect D , and get the commonly usedgeneral structure of the inverse of quark propagator as S ( p k ) − = i ~p A ( ~p , ˜ ω k ) + B ( ~p , ˜ ω k ) + iγ ˜ ω k C ( ~p , ˜ ω k ) . (8)For the dressed-gluon propagator, the general form islike this, g D µν ( k nl ) = P Tµν D T ( ~k , ω nl ) + P Lµν D L ( ~k , ω nl ) , (9)where k nl = ( ~k, ω nl ) = ( ~p − ~q, ω n − ω l ), P T,Lµν are trans-verse and longitudinal projection operators, respectively.And for the domain
T < . D T = D L is a good approximation. For the in-vacuum interaction, in this work we will adopt the fol-lowing form of Ansatz as in Ref. [68], D T = D L = D π σ k nl e − k nl /σ , (10)which is a simplified version of the famous as well aswidely used one in Refs. [69, 70]. It can be proved thatthis dressed gluon propagator at T = 0 violates the ax-iom of reflection positivity [71], and is therefore not ob-servable; i.e., the excitation it describes is confined. Thesame is true of the dressed quark propagator which is alsonot positive definite and hence is confined (Actually, wecan take the gluon propagator as input, and the quarkpropagator can then be solved numerically. The resultsshow that there is no singularity on the real, positive, i.e.timelike, p axis, which implies that quarks are confined).As concerning the quark-gluon vertex, in this work wewill take the rainbow truncation, which means a simplebut symmetry-preserving bare vertex is adopted,Γ ν ( p n , q l ) = γ ν . (11)The status of propagator and vertex studies can betracked from Ref. [72].Now let us fix the related parameters and then showsome of the numerical results. D and σ are usuallyfixed by fitting the observables, such as the two-quarkcondensate, the pion decay constant ( f π = 131 MeV)and the pion mass ( m π = 138 MeV). In this work weadopt the one from Ref. [70], that D = 9 . × MeV and σ = 400 MeV. For the current quark mass we willuse m =5 MeV. Then substituting Eqs. (5), (8), (10), and(11) into Eq. (4), we can solve the quark DSE for eachvalue of temperature and chemical potential by means ofnumerical iteration. As an example, we show B (0 , ˜ ω ) asa function of µ for different T in Fig. 1, and the corre-sponding chiral susceptibility with respect to m , which isdefined as χ m ( T, µ ) = ∂B (0 , ˜ ω ) ∂m , (12)in Fig. 2.In general, we can see from Fig. 1 that the scalar func-tion B (0 , ˜ ω ) will decrease when the chemical potential µ increase, this phenomenon holds to be true for the tem-perature T and momentum ~p too. It is known that thescalar part B ( ~p , ˜ ω k ) of the quark propagator Eq. (4) insome sense reflects the dressing effect of the quark, so theresults show that the dressing effect becomes weaker andweaker for higher T, µ and ~p . We can also see from Fig. 1that for different values of T , B (0 , ˜ ω ) may behave differ-ent: for T larger than a critical T c = 129 MeV, B (0 , ˜ ω )change gradually but continuously from the Nambu solu-tion to the Wigner solution; while for T smaller than T c ,there will appear a sudden discontinuity at some critical µ . è è è è è è è è è è è è è è è è è è è è è è è è è è è è è è è è è è è è è è è è è è è è è è è è è è èé é é é é é é é é é é é é é é é é é é é é é é é é é é é é é é é é é é é é é é é é é é é é é é é é é é÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷ Μ H MeV L B H , Ω Ž LH M e V L ÷ T =
139 MeV é T =
129 MeV è T =
120 MeV
FIG. 1. B (0 , ˜ ω ) as a function of µ for three different T . èèèèèèèèèèèèèèèèèèèèèèèèèèèèèèèèèèèèèèèèèèèèèèèèèèèèèèè èèèèèèèèèèèèèèèèèèèèèèèèèèèèèèèèèèèèèèèèèèèèèèéééééééééééééééééééééééééééééééééééé éééééééééééééééééééééééééééééééééééééééééééééééééééééééééééé÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷
50 100 150 200020406080100 Μ H MeV L Χ m H T , Μ L ÷ T =
139 MeV é T =
129 MeV è T =
120 MeV
FIG. 2. χ m ( T, µ ) as a function of µ for three different T . To study the nature of the chiral phase transition, es-pecially to determine the critical value of µ c at T c , peo-ple often employ various susceptibilities of QCD [73–77].We can see from Fig. 2 that, for T ≥ T c , the suscepti-bility χ m indicate a crossover from the Nambu phase tothe Wigner phase, and the peak grow higher and higherwhen T approaches T c . At T c , χ m shows a sharp andnarrow divergent peak, and the value of this peak turnsto be ∞ , which demonstrate that here is a second-orderphase transition, and corresponding to the CEP. And for T ≤ T c , an obvious first order phase transition will occur.According to these results, we can move on to study thechiral phase transition, especially the behavior of CEP.In the following Sec. III, we will focus on the variance ofCEP when the chiral chemical potential is considered asan additional scale. III. INFLUENCES OF THE CHIRALCHEMICAL POTENTIAL ON THE CHIRALPHASE TRANSITION
The concept of chiral chemical potential was first pro-posed by K. Fukushima et al. in a study related to theexternal magnetic field [57]. Since topological chargechanging transitions can induce an asymmetry betweenthe number of right- and left-handed quarks due to theaxial anomaly, they introduce the chiral chemical po-tential µ , which couples to the difference between thenumber of right- and left-handed fermions. The chiralityis also expected to be produced in the high tempera-ture phase of QCD [58]. Many researchers argue thatalthough µ is a mere mathematical artifice instead of atrue chemical potential , it has the advantage that canbe simulated on the Lattice QCD with N c = 3, hence islikely to provide some useful information for the studiesof the CEP, and even for inhomogeneous phases or the in-ner structure of compact stellar objects. One of the mostinteresting features of the introduction of µ is that, itmakes the continuation of the CEP to a possible “CEP ”in the T − µ plane possible, which is of course helpfulin the determining of the CEP from Lattice QCD. Someother researchers also confirm such a possible CEP inrelated chiral model studies [60, 61]. In this part, wewill discuss the related topics within the framework ofDyson-Schwinger equations.To be specific, in order to study the effects of µ , peopleshould add the following term to the Lagrangian density, µ ¯ ψγ γ ψ. (13)And in our work, the quark propagator and its inversenow can include at most eight components according tothe Lorentz structure analysis, namely,1 , ~p, γ , ~pγ , γ , ~pγ , γ γ , ~pγ γ . (14)So now the general inverse form of the dressed quarkpropagator is S ( p n , µ ) − = i ~pA + B + iγ ˜ ω n C + ~pγ ˜ ω n D +( i ~pA + B + iγ ˜ ω n C + ~pγ ˜ ω n D ) γ , (15)The eight scalar functions F = A , B , C , D , A , B , C , D denotes F = F ( ~p , ˜ ω n , µ ), which are all complex andsatisfy the following equation, F ( ~p , ˜ ω k , µ ) ∗ = F ( ~p , ˜ ω − k − , µ ) . (16)Now the quark DSE at nonzero temperature andnonzero chemical potential is then, S ( p n , µ ) − = S ( p n , µ ) − + 43 T ZX g D µν ( p n − q l ) γ µ S ( q l , µ )Γ ν ( p n , q l ) , (17)where S ( p n , µ ) − = i ~p + m + iγ ˜ ω n − µ γ γ . (18) The reason for this is easy to understand, since the difference indensities of the right- and left-handed quarks, n = n R − n L ,does not conserve. For the details of S ( q l , µ ), please see the Appendix part.Substituting Eqs. (15), (9), and (11) into Eq. (17), wefound that the solution is the following coupled integralequations (for the sake of concise, here all the notion F ( p )means F ( ~p , ˜ ω n , µ )) A ( p ) = 1 + c ( T ) ZX k nl e − k nl /σ × K A ,B ( p ) = m + c ( T ) ZX k nl e − k nl /σ × K B ,C ( p ) = 1 + c ( T ) ZX k nl e − k nl /σ × K C ,D ( p ) = c ( T ) ZX k nl e − k nl /σ × K D ,A ( p ) = c ( T ) ZX k nl e − k nl /σ × K A ,B ( p ) = c ( T ) ZX k nl e − k nl /σ × K B ,C ( p ) = iµ /ω n c ( T ) ZX k nl e − k nl /σ × K C ,D ( p ) = c ( T ) ZX k nl e − k nl /σ × K D , (19)in which c ( T ) = 16 π T σ , K A = − [( ~p · ~q k nl + 2 ~k · ~p ~k · ~q ) σ A + 2 ~k · ~p ω nl ω l σ C ] / ( ~p k nl ) , K B = 3 σ B , K C = − ω l σ C /ω n − ω nl ( ~k · ~q σ A + ω nl ω l σ C ) / ( ω n k nl ) , K D = [ ~p · ~q (2 ω nl − k nl ) + 2 ~k · ~p ~k · ~q ] ω l σ D / ( ~p ω n k nl ) , K A = [( ~p · ~q k nl + 2 ~k · ~p ~k · ~q ) σ A + 2 ~k · ~p ω nl ω l σ C ] / ( ~p k nl ) , K B = − σ B , K C = ω l σ C /ω n + 2 ω nl ( ~k · ~q σ A + ω nl ω l σ C ) / ( ω n k nl ) , K D = [ ~p · ~q (2 ω nl − k nl ) + 2 ~k · ~p ~k · ~q ] ω l σ D / ( ~p ω n k nl ) . (20)Then, we can solve Eq. (19) numerically for specific chi-ral chemical potential µ , as well as for the temperature T and normal chemical potential µ . The critical T c [ µ ]and µ c [ µ ], which are coordinates of the new “criticalend point” at a specific µ (CEP[ µ ]), are determinedby the scalar susceptibility that is defined in Eq. (12) .In this work, we will concentrate on the behavior of theCEP[ µ ], which is expected to be linked to a possibleCEP in the T − µ plane [59]. In Fig. 3 and Fig. 4, weplot the relations between µ and T c [ µ ] as well as µ and µ c [ µ ], respectively, which are obtained by seekingthe corresponding CEP[ µ ] for different µ . Therefore, Here we summarize that, in this work “CEP” means the criticalend point in the T − µ plane with µ = 0, while CEP denotesthe possible one in the T − µ plane with µ = 0 [59], and CEP[ µ ]is generally the similar critical end point in the T − µ plane fora specific µ . æ æ æ æ ææ æ æ æ æ æ
100 150 200 250 3000100200300400500 T c @ Μ D H
MeV L Μ H M e V L æ CEP @ Μ D FIG. 3. The relation between µ and the corresponding T c [ µ ]in the T − µ plane. æ æ ææææææææææ æ æ
40 50 60 70 80 90 1000100200300400500 Μ c @ Μ D H
MeV L Μ H M e V L æ CEP @ Μ D FIG. 4. The relation between µ and the corresponding µ c [ µ ]in the T − µ plane. each point in these two lines means a “CEP’ in the T − µ plane for the corresponding µ .We can see Fig. 3 that, T c [ µ ] increase slowly when µ is smaller than approximately 300 MeV, and turns toincrease quickly for larger µ . Nevertheless, the wholeincrease is smooth and monotonic. The most interest-ing thing is as shown in Fig. 4, that µ c [ µ ] will decreasefirstly when µ is smaller than about 400 MeV, and thenincrease for larger µ . The behavior of µ c [ µ ] for differ-ent µ is quite different with the previous results fromsome chiral models, such as Fig. 4 of Ref. [59], that atsome critical value of µ , µ c [ µ ] will decrease to 0, wherea CEP is expected to exist. When µ is not very large,the qualitative properties of the results in Fig. 4 are simi-lar to those from chiral models, but the decrease of µ c [ µ ]is much slower. Our results also indicate there might beno CEP in the T − µ plane, and the phase transition inthe T − µ plane might be totally crossover when µ < T − µ plane, they can not guarantee the ro-bustness when µ acts as an additional scale, and theintroducing of µ might make the applicability of thecalculations T + µ + µ < Λ (Λ is some cutoff scale).
IV. SUMMARY
Thanks to the complicated non-Abelian feature ofQuantum Chromodynamics (QCD) itself, its two fun-damental features, namely, dynamical chiral symmetrybreaking (DCSB) and quark color confinement, have tobe studied phenomenologically through various effectivemodels at present, especially in the most interesting non-perturbative region. In this work, we discuss the chiralphase transition of QCD within the framework of Dyson-Schwinger equations (DSEs), with a chiral chemical po-tential µ as an additional scale other than the normaltemperature T and quark chemical potential µ . We givea basic introduction to the DSEs at finite temperatureand nonzero chemical potential as well as an effectivemodel gluon propagator firstly, and then mainly focus onthe calculations related to the famous critical end point(CEP) in the T − µ plane, which is predicted by manymodel studies, and has caused much interests both in theexperimental side (one of the main goals in some highenergy physics experiments) and theocratical side. Withthe help of a scalar susceptibility, that often act as anorder parameter of chiral phase transition, we find thatthere might be no CEP in the T − µ plane, which isthought to exist by some chiral model calculations, andthe phase transition in the T − µ plane might be to-tally crossover when µ <
50 MeV, which has apparentconsistency with the Lattice QCD calculation. DSEs iswidely used as well as has been proved to be successfulin hadron physics and phase transitions of strongly inter-acting matters, so that our study may also provide someuseful hints to some other studies related to µ . Last butnot least, we’d like to say that the related issues deservefurther studies. ACKNOWLEDGMENTS
This work is supported in part by the NationalNatural Science Foundation of China (under GrantNos. 11275097, 11475085, 11265017, and 11247219),the National Basic Research Program of China (un-der Grant No. 2012CB921504), the Jiangsu PlannedProjects for Postdoctoral Research Funds (under GrantNo. 1402006C), the National Natural Science Founda-tion of Jiangsu Province of China (under Grant No.BK20130078), and Guizhou province outstanding youthscience and technology talent cultivation object specialfunds (under Grant No. QKHRZ(2013)28).
APPENDIX: STRUCTURE OF THE QUARKPROPAGATOR
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