Chiral crossover transition from the Dyson-Schwinger equations in a sphere
CChiral crossover transition from the Dyson-Schwinger equations in a sphere
Yin-Zhen Xu, ∗ Chao Shi, † Xiao-Tao He, ‡ and Hong-Shi Zong
1, 3, 4, 5, § Department of physics, Nanjing University, Nanjing 210093, China Department of Nuclear Science and Technology,Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China Department of physics, Anhui Normal University, Wuhu 241000, China Nanjing Proton Source Research and Design Center, Nanjing 210093, China Joint Center for Particle, Nuclear Physics and Cosmology, Nanjing 210093, China (Dated: September 28, 2020)Within the framework of DysonSchwinger equations of QCD, we study the effect of finite volumeon the chiral phase transition in a sphere with the MIT boundary condition. We find that thechiral quark condensate (cid:104) ¯ ψψ (cid:105) and pseudotransition temperature T pc of the crossover decreases asthe volume decreases, until there is no chiral crossover transition at last. We find that the systemfor R = ∞ fm is indistinguishable from R = 10 fm and there is a significant decrease in T pc with R as R <
R < . I. INTRODUCTION
The quantum chromodynamics (QCD) as a underlyingtheory describing strong interactions exhibits two fasci-nating aspects: confinement and dynamical chiral sym-metry breaking (DCSB). As the temperature increases,the strongly-interacting matter will undergo a phasetransition from hadronic matter to quark-gluon plasma(QGP) with deconfinement and chiral restoration. QCDphase transitions are experimental and theoretical fron-tiers, which have been studied in relativistic heavy-ioncollisions (HICs) at CERN (France/Switzerland), BNL(USA), and GSI (Germany) [1, 2]. However, most of thetheoretical calculations are based on the thermodynamiclimit (namely, the volume of the system V → ∞ ). It isworth bearing in mind that the QGP system produced inHICs always has a finite volume, depending on the colli-sion nuclei, the center of mass energy and the centrality.According to the the UrQMD transport approach [3], thevolume of Au-Au and Pb-Pb collisions before freeze-outis about 50 ∼
250 fm [4]. It is believed that the radiiof possible quark gluon plasma are estimated to be 2 to10 fm. Therefore, there’s a problem we need to consider:does the size and shape of QGP system produced in HICsaffect the phase transition?There has been a lot of theoretical studies for the ef-fect of finite volume on QGP phase transition, withinthe Nambu-Jona-Lasinio (NJL) model [5–7], Polyakov-Nambu-Jona-Lasinio (PNJL) model [8–10], quark-mesonmodel [11–13], and Dyson-Schwinger equations (DSEs)[14–19]. In most existing theoretical studies, for the sakeof convenience, the systems are usually treated as a cube,and anti-periodic boundary condition (APBC) is used.However, when the volume of the fireball produced in ∗ [email protected] † [email protected] ‡ [email protected] § [email protected] collision is small enough, not only its size but also itsshape have an non-negligible effect on the QCD phasetransition. In order to simulate more realistic conditionsuch as the fireball expected to arise in HICs, the au-thors of Ref. [20] consider a sphere with the MIT bound-ary condition under the framework of NJL model for thefirst time. However, it should be pointed out that theNJL model is a non-renormalizable theory, in which theconfinement property is not preserved. Meanwhile, thegauge sector of QCD, i.e., the gluon degrees of freedom,is lost. This led us to consider a more realistic approachto study QGP phase transitions in a sphere with the MITboundary condition.In this work, we employ the framework of Dyson-Schwinger equations to deal with the finite size effects ina sphere. DSEs has been widely used in studying stronglyinteracting phenomena in vacuum and in heat bath [21–32]. It is capable of simultaneously implementing colorconfinement and expressing DCSB [30, 33–38]. Recently,as mentioned before, it is used to study the effects offinite volume on QGP crossover transition. However,those studies were all for cubic systems i.e. APBC. Inthis work, we study finite size effects with MIT bound-ary condition for the first time.This paper is organized as follows: In Section II, we in-troduce the DSEs at finite temperature within the MITboundary condition. In Section III, chiral quark con-densation and chiral susceptibility of a spherical systemat different radii are defined and calculated. On thisbasis, the influence of system volume and shape on chi-ral crossover transition temperature is discussed, and theresults are compared with those obtained by traditionalAPBC based on cubic systems. In Section IV, we presenta brief summary. a r X i v : . [ nu c l - t h ] S e p II. DYSONSCHWINGER EQUATIONS IN AFINITE VOLUME
The formulation of DSEs at nonzero temperature aredescribed in Refs. [21, 36]. The T (cid:54) = 0 dressed-quarkpropagator is obtained from the following gap equation: S (˜ ω n , (cid:126)p ) − = Z ( i(cid:126)γ · (cid:126)p + iγ ˜ ω n + Z m m )+ Z T (cid:88) l (cid:90) d p (2 π ) g D µν ( k Ω ) λ a γ µ S (˜ ω l , (cid:126)q ) λ a γ ν , (1)where ˜ ω n = (2 n + 1) πT are the fermionic Matsubarafrequencies; m is the current-quark mass and m = 0 de-fines the chiral limit. Z , ,m are the vertex, quark field,and mass renormalization constants, respectively. As weemploy an ultraviolet-finite model, renormalization is un-necessary, i.e. Z , ,m = 1. D µν ( k Ω ) is the dressed-gluonpropagator which has the form: g D µν ( k Ω ) = P Tµν ( k Ω ) D (cid:0) k (cid:1) + P Lµν ( k Ω ) D (cid:0) k + m g (cid:1) , (2)where k Ω = (˜ ω n − ˜ ω l , (cid:126)p − (cid:126)q ), and m g = (16 / π T isthe gluon Debye mass. Since the temperature breaks theLorentz symmetry, the tensor structure of gluon has bothtransverse and longitudinal parts, where P T,Lµν are trans-verse and longitudinal projection operators respectively, P Tµν ( k Ω ) := (cid:26) , µ and/or ν = 4 δ ij − k i k j k , µ, ν = i, j = 1 , , , (3) P Lµν ( k Ω ) + P Tµν ( k Ω k ) = δ µν − k µ k ν k . (4)The choice of interaction kernel is not unique. In thiswork, we use a simplified form of Maris-Tandy model[32]: D (cid:0) k (cid:1) = D π ω k e − k /ω . (5)The parameters D and ω are not independent: A changein D can be compensated by an alteration of ω [38]. Inthis paper we choose a typical value ω = 0 . D ω = (0 . [32].The gap equation’s solution can be generally expressedas S (˜ ω n , (cid:126)p ) − = i(cid:126)γ · (cid:126)pA (cid:0) (cid:126)p , ˜ ω n (cid:1) + B (cid:0) (cid:126)p , ˜ ω n (cid:1) + iγ ˜ ω n C (cid:0) (cid:126)p , ˜ ω n (cid:1) + (cid:126)γ · (cid:126)pγ ˜ ω n D (cid:0) (cid:126)p , ˜ ω n (cid:1) , (6)with the four scalar dressing functions A, B, C, D. Thedressing function D, however, is power-law suppressed inthe UV [21] and does not contribute in all cases investi-gated here. The mass function of quarks can be defined as [39, 40] M (cid:0) ˜ ω n , (cid:126)p (cid:1) = B (cid:0) ˜ ω n , (cid:126)p (cid:1) C (˜ ω n , (cid:126)p ) (7)and the Euclidean constituent mass M E := { p | p > , p = M (cid:0) ˜ ω , (cid:126)p (cid:1) } , which can be seen as a free particlewith mass M E .For finite size system, Eq. (1) should be modified.Three momenta will be discretized by the boundary con-dition. (cid:90) d (cid:126)p (2 π ) → V (cid:88) p k . (8)The allowed values of momentum modes depend onthe selection of boundary conditions. For anti-periodicboundary condition (APBC), we have (cid:126)p k = (cid:88) k i = ± , ± ,... k i πL ˆ e i (9)with L is the size of cubic box. Another boundary condi-tion is multiple reflection expansion (MRE), which intro-duces a IR cutoff in the momentum space and modifiesthe density of states [19].However, APBC works on a cubic box and MRE be-comes invalid for very small volume. In this work, we useMIT boundary condition. Under spherical MIT bound-ary condition, the allowed momentum values are givenby the following eigen-equations j l κ ( pR ) = − sgn( κ ) pE + M j ¯ l κ ( pR ) (10)where l κ = (cid:26) − κ − κ < κ for κ > l κ = (cid:26) − κ for κ < κ − κ > κ = ± , ± , . . . and j l ( x ) is the l -th ordered sphericalBessel function. R is the radius of sphere, p is the allowedmomentum value. The M is the Euclidian constituentquark mass M E . III. FINITE VOLUME EFFECTS ON THECHIRAL PHASE TRANSITION
Solving the DSEs at finite temperature and finite size,we obtain the numerical results of fully dressed quarkpropagator. We then study the chiral phase transitiontemperature T c . The corresponding order parameter isthe chiral condensate. In the chiral limit, we have −(cid:104) ¯ ψψ (cid:105) = N c T ∞ (cid:88) n = −∞ tr D (cid:90) d p (2 π ) S ( (cid:126)p, ˜ ω n ) (11)When m (cid:54) = 0, Eq. (11) diverges, we do not have a well-defined chiral condensate. Hence, we employ the renor-malized chiral condensate defined as [23, 41] −(cid:104) ¯ ψψ (cid:105) = N c T ∞ (cid:88) n = −∞ tr D (cid:90) d p (2 π ) [ S ( (cid:126)p, ˜ ω n ) − S ( (cid:126)p, ˜ ω n )] , (12)For finite size system it becomes −(cid:104) ¯ ψψ (cid:105) V = N c TV (cid:88) k,n tr D [ S ( (cid:126)p k , ω n ) − S ( (cid:126)p k , ω n )] (13)with S ( (cid:126)p, ˜ ω n ) being free quark propagator and S ( (cid:126)p k , ω n ) being the fully dressed quark propagator.A direct computation of finite volume DSEs, i.e., Eqs.(1, 8) is quite difficult. The reason is obvious: In general,for the simplified Maris-Tandy model, the ultraviolet cut-off of the three momentum integral is O (10) GeV and thesummation of Matsubara frequency should be consistentwith it. As the volume increases and the temperaturedecreases, the number of allowed values of three momen-tum and Matsubara frequency is so large that it is diffi-cult to calculate numerically. Some people approximatesthe momentum modes summation using an integral withan infrared cutoff [8, 15]. However, this approximation isknown to get worse as the system size decreases. In thiswork we adopt a new approximation, i.e., we rewrite Eq.(8) as (cid:90) d (cid:126)p (2 π ) → V (cid:88) | p k | <λ + (cid:90) | p k | >λ d (cid:126)p (2 π ) (14)As the high momentum modes are generally denser thanlow momentum modes, we therefore approximate thesummation by an integral for high momentum modes.Here λ is an adjustable parameter, depending on volumeand temperature. We can keep increasing it until thenumerical result is stable.For the summation of Matsubara frequency, symmetryof quark propagator can help us reduce the computationeffort, F (˜ ω n , (cid:126)p k ; T ) = F ∗ (˜ ω − n , (cid:126)p k ; T ) (15)with F = A, B or C . At low temperature, the num-ber of Matsubara frequency is of ∼ O (10 ). For thisreason, the numerical calculation is very difficult withinDSEs framework when T < . n is large enough, F is very smooth, so in the iteration wecan reduce computing complexity by interpolation. Thetechnique goes as follows: We take all the low Matsub-ara frequencies and keep only a few high frequencies bymeans of a mapping n (cid:48) = Int[ n γ · a + n · (1 − a )] , (16)where n = 1 , , , ...N ; a = − N (cid:48) + N − N (cid:48) +( N (cid:48) ) γ and N ( N (cid:48) ) is thenumber of elements in array n ( n (cid:48) ). By controlling the R = = = = = = = = = ∞ fm [ GeV ] -< ψψ > R ( G e V ) R = = = = = = = = = ∞ fm [ GeV ] χ m ( T ) FIG. 1. Upper panel: chiral condensate at different volumes.Lower panel: chiral susceptibility at different volumes. Fora spherical system, when
R >
10 fm, the size of the systemcan be regarded as infinite. The volume change of the systemstarts to have a significant effect when the
R <
R < . scaling factor γ , we can map the evenly distributed array n to an array n (cid:48) that gets sparse as n (cid:48) enlarges. We thensolve the scalar functions F at frequencies n (cid:48) , while at therest frequencies the F are obtained by the cubic splineinterpolation.When the chiral condensate is obtained, we can furtherstudy the chiral susceptibility [42–45] χ mV ( T ) = − ∂∂m (cid:104) ¯ ψψ (cid:105) V . (17)In the chiral limit, chiral symmetry is restored via asecond-order transition at T c in which chiral susceptibil-ity diverges. At nonzero current mass, the chiral symme-try restoration transition is replaced by a crossover. Thepseudocritical temperature T pc is obtained as the maximaof the chiral susceptibility with respect to temperature.In this work, we choose the current-quark mass m =4 . m π = 0 .
135 GeV) and decay constant ( f π = 0 .
095 GeV)with the Bethe-Salpeter equation. When the volume ofthe system is infinite, we have T pc = 137 MeV, close tothe recent lattice QCD simulation value T pc = 156 . ± . T pc graduallydecreases and the curve of chiral susceptibility χ mV ( T ) be-comes flat while the system size dwindling (see Fig. 1).When the volume is small enough, the influence of theshape of the system on T c cannot be ignored. Our earlier MITAnti - periodic =( / π ) / L [ fm ] T p c [ G e V ] FIG. 2. This figure shows the effect of the system shape onthe T pc . For a spherical system, the chiral crossover transitionis more sensitive to the size of the system. study shows that in a cubic box, when L is greater than 3fm ( V ∼
27 fm ), the size of the system can be regardedas infinite [15–17, 20]. As Fischer at al pointed out in Ref.[14], at zero temperature, the finite size will have largeeffects when cubic side length goes below L = 1 . V ∼ ),the finite volume effect of the system emerges. The sizechange of the system starts to have a significant effect on T pc when the radius is decreased to 4 fm ( V ∼
270 fm ).We argue that there is no chiral crossover transition in thesystem when the radius is less than 1.5 fm ( V ∼
14 fm ).In other words, it is meaningless to discuss the chiralcrossover transition in a very small space size. Considerthat DCSB can only occur in infinitely large systems inprinciple, chiral symmetry has restoratd when the size issmall enough.We note that similar results were found in Ref. [20].Therein the NJL model study with MIT boundary con-dition shows that a system whose radius is above 14 fmcan be regarded as an infinitely large system, while in acubic the marginal size is L = 3 fm. Hence, our resultsare closer to Ref. [20] than to DSEs with APBC [15–17]. This indicates that the boundary condition potentiallyplays an important role when we want to simulate therealistic fireball produced in HICs. IV. SUMMARY AND PERSPECTIVE
Based on the DSEs formalism, we consider the in-fluence of the finite volume on the chiral transition ofQCD at finite temperature in a cubic and in a spherical.For the cubic volume, we use the widely adopted anti-periodic boundary condition and for the spherical vol-ume we choose MIT boundary condition which had beenused in NJL model [20]. While a cubic system withinAPBC could be regarded as an infinite system if the vol-ume was greater than 27 fm ( L ∼ ( R ∼ R < . ACKNOWLEDGEMENTS
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