A chiral random matrix model with 2+1 flavors at finite temperature and density
aa r X i v : . [ h e p - ph ] D ec A chiral random matrix model with 2+1 flavors at finite temperature and density ∗ H. Fujii a and T. Sano a,b a Institute of Physics, University of Tokyo, Tokyo 153-8902, Japan b Department of Physics, University of Tokyo, Tokyo 113-0033, Japan (Dated: November 19, 2018)Phase diagram of a chiral random matrix model with the degenerate ud quarks and the s quarkat finite temperature and density is presented. The model exhibits a first-order transition at finitetemperature for three massless flavors, owing to the U A (1) breaking determinant term. We studythe order of the transition with changing the quark masses and the quark chemical potential, andshow that the first-order transition region expands as the chemical potential increases. We alsodiscuss the behavior of the meson masses and the susceptibilities near the critical point. Introduction.—
Study of the QCD critical point (CP)[1, 2, 3] is an intriguing fundamental issue since its ex-perimental confirmation will yield a strong evidence forthe QCD phase transition, and energy-scan experimentssearching for the QCD-CP are being performed at Rel-ativistic Heavy Ion Collider at BNL. Although the exis-tence of the critical point in the QCD phase diagram ispresumably inferred from the model studies and latticeQCD results[1, 2, 4], its absence is also a possibility[4, 5].In this paper, we adopt as a schematic model for QCDthe chiral random matrix (ChRM) model [6] which incor-porates the U A (1)-breaking determinant term[7, 8]. Wereport the phase diagram of this model with the degener-ate ud-quark mass m ud and the s-quark mass m s at finitetemperature T and quark chemical potential µ .The ChRM models have been successfully applied forqualitative study of chiral properties of QCD[9, 10]. Ina ChRM model the Dirac operator on gluon field back-ground is modeled by a matrix D in the space of constantmodes with small Dirac eigenvalues, retaining the chiralsymmetry { D, γ } = 0. The partition function of themodel is given as an average of det D over random en-semble of matrix elements, which mimics the complexityof the gluon dynamics. The finite T and µ effects aretreated schematically as non-random external parame-ters appearing in D . In Ref. [10], the phase diagram ofthe ChRM model has been explored in the T - µ plane anda tri-critical point (TCP) is found on the phase bound-ary in the massless limit. The TCP changes to a simpleCP when the quark mass is nonzero. This result is con-sistent with the phase structure obtained in other modelstudies with two quark flavors [1, 2, 11] implying the sce-nario that the CP exists in the QCD phase diagram asan endpoint of the first-order phase boundary.Nature of the chiral transition in QCD is sensitiveto the number of light quark flavors, especially to thevalue of the s-quark mass m s . Unfortunately, however,the phase structure of the conventional ChRM model[10]is independent of the number of flavors N f . In orderto remedy this problem, we have recently incorporated ∗ Work supported in part by Grant-in-Aid of MEXT, Japan(No. 19540269 and 19540273). the U A (1)-breaking determinant interaction [7, 8] in theChRM model [6] by extending the zero-mode space [12]with the instanton gas model picture in mind. This isthe first ChRM model which describes the N f depen-dence of the chiral transition allowing us to explore thephase structure varying the parameters m ud and m s inaddition to T and µ . Model with determinant interaction.—
The chiral sym-metry breaking manifests itself in the nonzero density ofthe zero Dirac eigenvalues through the Banks-Casher re-lation [13]. The origin of small Dirac eigenvalues maybe instanton configurations of background gauge fieldand other nonperturbative gluon dynamics. Here in ourmodel we divide these fermionic modes into two cate-gories, N + and N − topological zero modes associatedwith N + instantons and N − anti-instantons respectively,and 2 N near-zero modes generated by other complexdynamics [6, 12]. The Dirac operator D then approx-imated with a martix of 2 N + N + + N − dimensionswith N + , N − and N being of the order of the space-time volume O ( V ). The thermodynamic limit is takenas 2 N + N + + N − → ∞ . Note that N ± should varydepending on the instanton distribution.For fixed number of zero modes the model partitionfunction is written in the chiral basis as Z N + ,N − = Z dR e − N Σ tr RR † N f Y f =1 det( D + m f ) , (1)with D = (cid:18) R + C i R † + C T (cid:19) , (2)where R ∈ C ( N + N + ) × ( N + N − ) is a random matrix fol-lowing a Gaussian ensemble distribution with the vari-ance 1 / ( N Σ ) and C ∈ C ( N + N + ) × ( N + N − ) is a matrixrepresenting the effects of T and µ . The matrix D has | N + − N − | exact zero eigenvalues when R and C are rect-angular, which is interpreted as a realization of the indextheorem in the ChRM model. We adopt here the simplestform for C [10]: C = ( µ + i T ) N/ µ − i T ) N/
00 0 0 , (3)where T and µ are schematic representation for thetemperature and chemical potential effects, respectively.Note that D with µ = 0 is non-Hermitian whereas thepartition function (1) is still invariant under µ ↔ − µ .One should appreciate that the N + × N − right-bottomblock in C corresponding to the topological zero modesis set to zero. This seems a reasonable assumption if onenotice that the finite T and µ effects are introduced asa boundary condition in the Matsubara formalism andthat the localized topological zero modes will be insen-sitive to the boundary. This discrimination is importantindeed in reproducing the physical T dependence of thetopological susceptibility[6, 14].The complete partition function of the model is ob-tained by summing over N + and N − with a distributionfunction P ( N ± ): Z RM = X N + ,N − P ( N + ) P ( N − ) Z N + ,N − . (4)The P ( N ± ) reflects a modeling of the instanton distribu-tion in the QCD ground state. The authors of Ref. [12]adopted the Poisson distribution, which involves arbi-trarily large number for N ± and results in a model withno stable ground state. Inspired by the lattice gas modelwithin a finite box, we instead choose the binomial dis-tribtion [6], P ( N ± ) = (cid:18) γNN ± (cid:19) p N ± (1 − p ) γN − N ± , (5)where γ is a parameter of O ( V ) and p is interpreted asthe probability for a single instanton to occupy a unitvolume V / ( γN ). This distribution sets an upper bound γN for N ± and gives rise to a stable effective potentialas a function of order parameters. In fact, applying thestandard bosonization procedure to (4), we find Z RM = Z dS e − N Σ tr S † S × det N (cid:2) ( S + M )( S † + M † ) − ( µ + i T ) (cid:3) × det N (cid:2) ( S + M )( S † + M † ) − ( µ − i T ) (cid:3) × (cid:2) α det( S + M ) + 1 (cid:3) γN (cid:2) α det( S † + M † ) + 1 (cid:3) γN ≡ Z dS e − N Ω( S ; T, µ ) , (6)where we defined the effective potential Ω( S ; T, µ ) inthe last line. S ∈ C N f × N f is the order parameter matrix,and M is the mass matrix. The parameter α = p/ (1 − p ). Note that the integrand of Z RM is a polynomial of S except for the exponential factor originating from theGaussian ensemble distribution. Large values of S aresuppressed by this Gaussian weight.The determinant term with the coefficient α representsthe anomaly which breaks explicitly the U A (1) symmetryof the effective potential Ω( S ) even when M = 0. For S = φ N f ( φ ∈ R ) with M = 0, Ω simplifies toΩ N f = N f (cid:0) Σ φ − ln[ φ − ( µ + i T ) ][ φ − ( µ − i T ) ] (cid:1) − γ ln | αφ N f + 1 | . (7)We see that the anomaly term yields − αγφ N f whenexpanded. In Ref. [6] we studied this ChRM modelwith two and three equal-mass flavors at finite T with µ = 0, to show a second- (first-) order phase transitionfor N f = 2(3). Phase diagram and meson masses with 2+1 flavors.—
Choosing S = diag( φ ud , φ ud , φ s ) in the 2+1 flavor casewith M = diag( m ud , m ud , m s ), we haveΩ = Σ φ − (ln[ ϕ − ( µ + i T ) ] + ( T → − T ))+ (cid:2) Σ φ − (ln[ ϕ − ( µ + i T ) ] + ( T → − T )) (cid:3) − γ ln | αϕ ϕ s + 1 | , (8)where ϕ ud = φ ud + m ud and ϕ s = φ s + m s . The groundstate is determined by the saddle-point condition ∂ Ω ∂φ ud = 0 , ∂ Ω ∂φ s = 0 , (9)which becomes exact in the thermodynamic limit.Prior to the numerical analysis, we comment on themodel parameters Σ , α and γ . Setting Σ = 1 by redefi-nition of S and other parameters, we searched such a setof parameters α , γ , m ud and m s that reproduces quanti-tatively the (ratios of the) meson masses in the vacuum,but it was unsuccessful. However, the model can describethe mass hierarchy qualitatively as seen below in Fig. 3,and we wish to study the model phase diagram as anschematic model for QCD. Note that all the quantitiesare dimensionless in this work.We also remark that the anomaly term makes asymmetry-broken phase more stable. Indeed, no symme-try restoration occurs at finite T for αγ > Σ (= 1) with N f = 2, and the situation is similar even for N f = 3.Hence one must assume αγ . µ = 0 with T = 0 isfirst-order. It is seen in the simple case (7) because thesymmetric phase φ = 0 and the broken phase φ > µ are separated with the point φ = µ where Ω = ∞ or theintegrand of Z RM vanishes. This feature survives in moregeneral cases with 2+1 flavors.Let us study the phase diagram in the T - m ud - m s space.Since our model shows a first-order transition at finite T for m ud = m s = 0 and a crossover for large m ud and m s [6], there must be a line of a second-order transitionseparating these two regions in the m ud - m s plane. Thiscritical line is determined by the condition Ω ( n ) = 0 ( n =1 , ,
3) with Ω ( n ) ≡ ∂ n Ω /∂φ n ud , where Ω is a function ofa single order parameter φ ud with φ s eliminated by thesecond equation in (9). Note that Ω (2) = 0 is equivalentto det ∂ Ω( S ) /∂φ i ∂φ j = 0 ( i, j = ud , s), which impliesthe vanishing σ mass (see below). m s m ud (a) α =0.3 α =0.4 α =0.5 0 0.1 0.2 0.3m ud2/5 (b) γ =0.9 γ =1.0 γ =1.1 FIG. 1: The critical curves on the m ud - m s plane (a) for α = 0 . , . , . γ = 1 and (b) for γ = 0 . , . , . α = 0 .
5. The finite- T transition is first-order in the smaller-mass region and crossover in the larger-mass region. On the m s axis with m ud = 0, there is the N f = 2 chiral symmetry.The TCP is denoted by a dot for each parameter. We present in Fig. 1 the critical line projected ontothe m ud - m s plane for several values of α and γ . Whenwe increase the strength of the anomary term α and/or γ ,the region of the first-order transition expands. For eachparameter a TCP is found on the m s axis, where the N f = 2 chiral transition changes from a first to a second-order one. Near the TCP the critical line behaves as( m TCPs − m s ) ∝ m / as is expected from the mean-fieldLandau-Ginzburg analysis, which is clearly seen in Fig. 1(b). On the other hand, the line smoothly intersects the m ud axis with a finite slope [ ∗ ] . In fact, the model withthe anomaly term is symmetric under m ud ↔ − m ud butasymmetric under m s ↔ − m s .Next we extend our calculation to the finite µ case.Fig. 2 exhibits the phase diagram in the m ud - m s - µ space. We see that the region of the first-order transi-tion expands as µ is increased. This behavior indicatesthe existence of the CP in the T - µ plane with the phys-ical quark masses, provided that the finite- T transitionat µ = 0 is crossover. Varying the anomaly parameters α and γ , we have confirmed that the expansion of thefirst-order region with increasing µ is a robust result inour model as far as we keep α and γ constant.The (screening) mass matrices for the scalar andpseudo-scalar mesons are defined as the curvature of thepotential around ¯ S = diag( φ ud , φ ud , φ s ): M s 2 ab = ∂ Ω( S ) ∂σ a ∂σ b (cid:12)(cid:12)(cid:12)(cid:12) S = ¯ S , M ps 2 ab = ∂ Ω( S ) ∂π a ∂π b (cid:12)(cid:12)(cid:12)(cid:12) S = ¯ S , (10)where S = λ a ( σ a + i π a ) / √ σ a and π a , and with λ a being the Gell-Mann matrices and λ = q diag(1 , , M ab . Because the quark mass term m ud = m s breaks the SU(3) flavor symmetry to cause nonzero mix-ing M = M = 0 in both the scalar and pseudo-scalar [ ∗ ] We thank T. Hatsuda’s comment on this point.
0 0.02 0.04 0 0.1 0.2 0 0.1 0.2 µ m ud m s µ FIG. 2: Critical surface in the m ud - m s - µ space with theparameters γ = 1 and α = 0 .
5. A series of TCP’s is denotedby a thick line, and the second-order transition with N f = 2chiral symmetry occurs in the shaded area. channels, we diagonalize the matrices to get the masseigenvalues corresponding to σ and f for the scalars and η and η ′ for the pseudo-scalars. One can show that theflat direction of Ω( φ ud , φ s ) near the CP coinsides withthe σ fluctuation direction.In Fig. 3, we show the meson masses as a function of µ at T = T c with parameters γ = 1, α = 0 . m ud =0 .
05 and m s = 1 .
0. The model reproduces the empiricalhierarchy of the meson masses qualitatively in small µ region, thanks to the anomaly term. With increasing T and/or µ the pseudo-scalar meson masses remain nearlyconstant while the scalar ones, especially the σ mass,decrease. At the CP ( T = T c , µ = µ c ), the σ mesonbecomes massless. At higher µ > µ c , pairs of the masses M σ - M π , M κ - M K , and M δ - M η ′ get almost degenerate,reflecting the approximate N f = 2 chiral symmetry. M µσ , πκ , Kf , ηδ , η ’ FIG. 3: Mesonic masses as functions of µ with T = T c fixedfor parameters γ = 1, α = 0 . m ud = 0 .
05 and m s = 1 . T c , µ c ) = (0 . , . Generalizing the mass matrix to M = λ a ( s a + i p a ) / √ s a and p a , the scalar susceptibilities aredefined as χ s ab = − ∂ Ω( S ( M ); M ) ∂s a ∂s b , (11)where S ( M ) solves (9) for a fixed M , and the derivativesare evaluated at M = diag( m ud , m ud , m s ). The pseudo-scalar susceptibilities χ ps ab are defined similarly as the re-sponse to p a . We find the (pseudo-)scalar susceptibilities χ s ab and χ ps ab in a diagonal form χ s(ps) ab = δ ab Σ (cid:0) Σ /M s(ps) aa − (cid:1) (12)for a, b = 1 , . . . ,
7. There is an mixing term for a, b = 0 , χ s(ps) ab = Σ (cid:16) Σ ( M s(ps)2 ) − ab − δ ab (cid:17) , (13)which becomes diagonal when the mass matrix M s(ps)2 is transformed to be diagonal. Note that the scalar sus-ceptibility in the σ channel diverges when the (screening)mass M σ vanishes at the CP. So do the quark number sus-ceptibility χ q = − ∂ Ω /∂µ as well as the ‘specific heat’ χ T = − ∂ Ω /∂T , through the mode mixing generatedby the finite condensate ϕ ud and ϕ s [15]. Conclusion.—
As a schematic model for QCD, we haveanalyzed for the first time in the ChRM model the phasestructure with 2+1 flavors at finite T and µ , which be-comes possible by the inclusion of the U A (1) breakingterm[6]. We have drawn the critical curve separating thefirst-order transition region and the crossover region inthe m ud - m s plane with µ = 0, and we have shown thatthe first-order transition region expands as the strengthof the anomaly term is increased.Extending the model to the finite µ case, we haveshown the critical surface in the m ud - m s - µ space. Wehave found that the first-order transition region expandswith increasing µ , which is a supportive result for the ex-istence of QCD-CP on the T - µ plane: one encounters a CP as µ is increased from zero, provided that the finite- T transition is crossover at µ = 0. The meson mass hierar-chy in the vacuum is qualitatively reproduced with theU A (1) anomaly term and the SU(3) flavor breaking. Atthe CP the (screening) mass of σ vanishes. Although wetreat the model parameters independent of T and µ , pos-sible rapid quenching of these parameters at finite µ cangive rise to a shrinkage of the first-order transition region[16]. Furthermore, the role of the vector interaction be-tween the quarks[17] deserves further study. These arebeyond the scope of this schematic model. Appendix.—
The mass matrix M ab is diagnal exceptfor a, b = 0 and 8. For a = 1 , , M δ,π =Σ ± Re ϕ ± z ( ϕ − z ) ± γ αϕ s αϕ ϕ s + 1 , (14)and for a = 4 , . . . , M κ,K =Σ ± Re ϕ ud ϕ s ± z ( ϕ − z )( ϕ − z ) ± γ αϕ ud αϕ ϕ s + 1 , (15)where ϕ ud = φ ud + m ud , ϕ s = φ s + m s and z = µ +i T . Theupper (lower) sign corresponds to the (pseudo-) scalarmeson. The elements M ab for a, b = 0 , λ ud ≡ diag(1 , ,
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