Influence of the isospin and hypercharge chemical potentials on the location of the CEP in the mu_B-T phase diagram of the SU(3)_L x SU(3)_R chiral quark model
aa r X i v : . [ h e p - ph ] F e b In(cid:29)uen e of the isospin and hyper harge hemi al potentials on the lo ation of theCEP in the µ B − T phase diagram of the SU (3) L × SU (3) R hiral quark modelP. Ková s ∗ Resear h Group for Statisti al and Biologi al Physi s of the Hungarian A ademy of S ien es, H-1117 Budapest, HungaryZs. Szép † Resear h Institute for Solid State Physi s and Opti s of the Hungarian A ademy of S ien es, H-1525 Budapest, HungaryWe investigate the in(cid:29)uen e of the asymmetri quark matter ( ρ u = ρ d = ρ s ) on the mass of thequasiparti les and the phase diagram of the hiral quark model parametrized at one-loop level of therenormalized theory, using the optimized perturbation theory for the resummation of the perturba-tive series. The e(cid:27)e t of various hemi al potentials introdu ed in the grand anoni al ensemble isinvestigated with the method of relativisti many-body theory. The temperature dependen e of thetopologi al sus eptibility is estimated with the help of the Witten(cid:21)Veneziano mass formula.PACS numbers: 11.10.Wx, 11.30.Rd, 12.39.FeI. INTRODUCTIONThe study of a system of parti les at (cid:28)nite density and temperature is phenomenologi ally interesting be ause inheavy ion ollision experiments the initial state is su h that the hemi al potentials µ B , µ I , µ Y ( onjugate to the baryon harge, third omponent of the isospin and hyper harge, respe tively) are non vanishing, although the last two are mu hsmaller than the (cid:28)rst one. Assuming thermal equilibrium, thermal models show that the strangeness hemi al potentialin entral Si + Au ollisions at the Brookhaven AGS experiment was 20-25% of the baryoni hemi al potential forwhi h the best (cid:28)t gives µ B = 540 MeV [1℄. For entral Pb + Pb ollisions at CERN SPS experiments the value of thestrangeness hemi al potential was ∼ − and that of the isospin hemi al potential ∼ − of the value of µ B estimated to be around − MeV [2, 3℄. The Compressed Baryoni Matter (CBM) experiment at FAIR inDarmstadt will explore regions of the QCD phase diagram with moderate temperature up to su h high values of thebaryoni density whi h are omparable with those in the ore of neutron stars [4℄.In many-body theory hemi al potential is introdu ed to any onserved harge. In heavy ion ollision experimentsthe baryon number, isospin and hyper harge an be onsidered onserved due to the short time elapsed betweenthe formation of the (cid:28)reball and its freeze-out, during whi h only the strong intera tions play important role, theele troweak intera tions being negligible. It is expe ted that in the very early stage of the (cid:28)reball's evolution strangenessis abundantly produ ed in the de on(cid:28)ned phase through gluon-gluon fusion [5℄, while in the hadroni phase in the vi inityof the transition multi-mesoni rea tions will play an important role in the fast redistribution of strange quarks [6℄.The in(cid:29)uen e of the isospin hemi al potential on the hiral phase transition is urrently a tively investigated, be ausethis e(cid:27)e t an in prin iple be tested experimentally. As noti ed in [7℄, using di(cid:27)erent isotopes of an element in heavyion ollision experiments will vary µ I keeping µ B onstant. Moreover, the system with real µ I represents no extradi(cid:30) ulty in latti e (cid:28)eld theory ompared to the introdu tion of µ B . For two (cid:29)avors the simulations at µ B = 0 and µ I = 0 is not even a(cid:27)e ted by the sign problem [8℄. For µ I = 0 a generi result oming from e(cid:27)e tive models ofthe strongly intera ting matter without the U (1) A anomaly appeared to be the splitting in the µ B − T plane of the(cid:28)rst order transition line into two transition lines. This e(cid:27)e t was observed in random matrix model [9℄, NJL model[10℄, strong oupling limit of the staggered latti e QCD [11℄, all with two (cid:29)avors and in the three (cid:29)avors ladder QCD[12℄. This would imply the existen e of not only the two phases having h ¯ uu i 6 = 0 , h ¯ dd i 6 = 0 and h ¯ uu i = h ¯ dd i = 0 ,respe tively, but also of a phase with h ¯ uu i = 0 and h ¯ dd i 6 = 0 . It was shown in [13, 14℄ that the stru ture with twotransition lines and riti al end points eases to exist for a su(cid:30) iently strong U (1) A breaking, above whi h the twostrongly oupled ondensates vanish simultaneously. In a hadron resonan e gas model it was found that at (cid:28)xed baryon hemi al potential the pseudo riti al temperature of the transition between the hadroni and the quark-gluon plasmaphases is lowered as either the isospin or the strangeness hemi al potential is in reased [15℄.Be ause of their phenomenologi al impli ations, it is natural to study to what extent these results are present inanother low energy e(cid:27)e tive model, the hiral quark model, widely used for studying the hiral behavior of stronglyintera ting matter. In the past few years we have investigated the thermodynami s of this model for two and threequark (cid:29)avors at µ B = 0 and µ B = 0 , while leaving µ I = µ Y = 0 [16(cid:21)19℄. As a ontinuation of these previous studies,in this paper we onsider the in(cid:29)uen e of the hemi al potentials on the hiral phase transition up to su h high values ∗ Ele troni address: kpeti leopatra.elte.hu † Ele troni address: szepzsa hilles.elte.huof the isospin hemi al potential above whi h the ondensation of pseudos alar mesons o urs. The pion and kaon ondensation phase, whi h is beyond the s ope of our present investigation, was studied both with latti e methods andusing e(cid:27)e tive theories [20(cid:21)26℄.The paper is organized as follows. In se tion II we present the model, its one-loop parametrization and the intro-du tion of the hemi al potentials. The variation of the lo ation of the riti al end point in presen e of µ I and µ Y isstudied in se tion III. There we investigate also the temperature and density dependen e of the one-loop pole massesof the pseudos alar mesons. We on lude in se tion IV.II. THE SU (3) L × SU (3) R SYMMETRIC CHIRAL QUARK MODELThe Lagrangian of the model ontaining expli it symmetry breaking terms is L = 12 Tr ( ∂ µ M † ∂ µ M + m M † M ) − f (cid:0) Tr ( M † M ) (cid:1) − f Tr ( M † M ) − g (cid:0) det( M ) + det( M † ) (cid:1) + ǫ σ + ǫ σ + ǫ σ + ¯ ψ ( i /∂ − g F M ) ψ. (1)The onstituent quarks are ontained in the (cid:28)eld ψ : ¯ ψ = (¯ u, ¯ d, ¯ s ) . The two × omplex matri es are de(cid:28)ned interms of the s alar σ i and pseudos alar π i (cid:28)elds as M = √ P i =0 ( σ i + iπ i ) λ i and M = P i =0 ( σ i + iγ π i ) λ i , with λ i : i = 1 . . . the Gell(cid:21)Mann matri es and λ := q . The (cid:28)elds with well de(cid:28)ned quantum numbers are obtainedwith a blo k-diagonal transformation f α = T αi f i , f ∈ { σ, π } , where T = diag (1 , τ, , τ, τ, and τ = √ (cid:18) − i i (cid:19) . As α goes from to , the omponents of the s alar and pseudos alar (cid:28)elds go trough σ , a +0 , a − , σ , κ + , κ − , κ , ¯ κ , σ and π , π + , π − , π , K + , K − , K , ¯ K , π , respe tively. The physi al (cid:28)elds π (neutral pion), η and η ′ mesons in thepseudos alar se tor and a (neutral a ), σ and f in the s alar se tor are obtained as linear ombinations of the orresponding (cid:28)elds in the two mixing 0,3,8 se tors.In this paper we investigate the pattern of symmetry breaking realized in nature, with the SU (3) A × U (1) A × SU (3) V symmetry ompletely broken, that is the isospin SU (2) V is also broken. In addition to the spontaneous symmetrybreaking, expli it breaking is also onsidered with the introdu tion of external (cid:28)elds for all the diagonal generators ofthe s alar se tor. This results in having three non-vanishing ondensates in the broken symmetry phase: v δ = h σ δ i , for δ = 0 , , . The ondensates determines the tree-level s alar and pseudos alar masses: m S,αβ = m δ αβ − G αβγ v γ + 4 ˜ F αβγδ v γ v δ ,m P,αβ = m δ αβ + 6 ˜ G αβγ v γ + 4 ˜ H αβ,γδ v γ v δ . (2)The tensors appearing above arise after the evaluation of the tra e in (1) and the transformation of the (cid:28)elds to thebasis with good quantum numbers. The onne tion between these oupling tensors and the original ones appearing in(1) whi h an be found in [27, 28℄ is given by: ˜ G αβγ = X i,j,k =0 G ijk T − iα T − jβ T − kγ , ˜ H αβ,γδ = X i,j,k,l =0 H ij,kl T − iα T − jβ T − kγ T − lγ , ˜ F αβγδ = X i,j,k,l =0 F ijkl T − iα T − jβ T − kγ T − lγ . (3)The transformations preserve the symmetry stru ture of the tensors, that is ˜ G αβγ and ˜ F αβγδ are ompletely symmetri and ˜ H αβ,γδ is symmetri upon the inter hange of two indi es whi h are on the same side of the omma.The tree-level mass square matri es are not diagonal in the , , subspa e, but sin e they are real and symmetri diagonalization is a hieved with an orthogonal transformation. The tree-level orthogonal matri es in the s alar andpseudos alar se tors are denoted with O S and O P , respe tively. Denoting the eigenvalues of the pseudos alar and thes alar × mass matri es in the , , se tor with λ P, { min,mid,max } and λ S, { min,mid,max } the tree-level masses of themesons are as follows m π + = m π − = m P, , m a +0 = m a − = m S, ,m π = λ P, min , m a = λ S, mid ,m K + = m K − = m P, , m κ + = m κ − = m S, ,m K = m K = m P, , m κ = m κ = m S, ,m η = λ P, mid , m σ = λ S, min ,m η ′ = λ P, max , m f = λ S, max . (4)Note, that some of the tree-level masses of s alars and pseudos alars oin ide. As we will see, the introdu tion of α t γσ,α α t γπ,α a − G γ − H γ δ v δ π − G γ − F γ δ v δ a +0 G γ − H γ δ v δ π + G γ − F γ δ v δ κ − G γ − H γ δ v δ K − G γ − F γ δ v δ κ + G γ − H γ δ v δ K + G γ − F γ δ v δ ¯ κ G γ − H γ δ v δ ¯ K G γ − F γ δ v δ κ G γ − H γ δ v δ K G γ − F γ δ v δ . TABLE I: The t γf,α oe(cid:30) ients appearing in the equations of state (6), (7), (8). The summation index δ goes over , , .isospin and hyper harge hemi al potentials distinguish between the parti les and as a result all the one-loop polemasses will be di(cid:27)erent for µ I , µ Y = 0 .The tree-level fermion masses are: M u = g F √
12 ( √ v + √ v + v ) , M d = g F √
12 ( √ v − √ v + v ) , M s = g F √
12 ( √ v − v ) . (5)The evolution of the ondensates with the temperature or/and the hemi al potentials is determined by the threeequations of state h ∂L∂σ i = m v − c √ v − v − v ) + 13 (3 g + g ) v + ( g + g )( v + v ) v + g √ v − v ) v − ε − X f ∈{ σ,π } α =1 , , ... t f,α h f † α f α i − X γ ∈{ , , } h(cid:0) O TS S O S (cid:1) γγ h σ γ σ γ i + (cid:0) O TP P O P (cid:1) γγ h π γ π γ i i + g F √ N c ( h ¯ uu i + h ¯ dd i + h ¯ ss i ) , (6) h ∂L∂σ i = (cid:18) m − c √ v + c √ v + ( g + g v + v ) + ( g + g ) v + √ g v v (cid:19) v − ǫ − X f ∈{ σ,π } α =1 , , ... t f,α h f † α f α i − X γ ∈{ , , } h(cid:0) O TS S O S (cid:1) γγ h σ γ σ γ i + (cid:0) O TP P O P (cid:1) γγ h π γ π γ i i + g F N c ( h ¯ uu i − h ¯ dd i ) , (7) h ∂L∂σ i = m v + c √ v v + c √ v − v ) + ( g + g v + v ) v + g √ v − v ) v + ( g + g ) v v − ε − X f ∈{ σ,π } α =1 , , ... t f,α h f † α f α i − X γ ∈{ , , } h(cid:0) O TS S O S (cid:1) γγ h σ γ σ γ i + (cid:0) O TP P O P (cid:1) γγ h π γ π γ i i + g F √ N c ( h ¯ uu i + h ¯ dd i − h ¯ ss i ) , (8)where in the mixing se tor σ γ stands for σ, a , f and similarly π γ denotes π , η, η ′ as γ = 0 , , , respe tively. f † α denotes the antiparti le of f α , that is e.g. for f = σ and α = 1 one has σ = a +0 and σ † = a − . In this notation h f + α f α i = T βB ( m f α ) , h ¯ qq i = − m q T βF ( m q ) , where T βB ( m f α ) and T βF ( m q ) stands for the bosoni , and the fermioni tadpole integrals, respe tively. These integrals are given in Appendix B of [19℄. The oe(cid:30) ients t γf,α are listed inTable I. In the mixing se tor, that is for γ = 0 , , , the × matri es read: S γ = ˜ G − v ˜ F γ − v ˜ F γ − v ˜ F γ ,P γ = ˜ G + 43 v ˜ H γ + 43 v ˜ H γ + 43 v ˜ H γ , (9)with the de(cid:28)nition: ( ˜ G γ ) αβ ≡ ˜ G γαβ , ( ˜ F γδ ) αβ ≡ ˜ F αβγδ , and ( ˜ H γδ ) αβ ≡ ˜ H αβ,γδ . All the indi es run through 0, 3, or 8.A. One-loop parametrization of the model at zero temperature and densityOne has some freedom in hoosing the set of equations whi h determines the 13 parameters of the model, namely m , f , f , g, g F , v , v , v , ǫ , ǫ , ǫ and l f , l b . These latter two parameters are the fermioni and bosoni renormalizations ales. For the parametrization we follow the method des ribed in [19℄ where the renormalization of the model wasalso dis ussed. The only di(cid:27)eren e in the present ase is the appearan e of v and ǫ . Sin e at zero temperature − i Σ π + = X f ∈ ( σ,π ) α =0 ... (cid:1) f α π + π + + X δ =0 , , (cid:2) π + σ δ π + π + + (cid:3) π + ¯ κ K + π + + (cid:4) π + κ + ¯ K π + + X δ =0 , , (cid:5) π + a +0 π δ π + + (cid:6) π + u ¯ d π + − i Σ γγ ′ , , = X f ∈ ( σ,π ) α =0 ... (cid:7) f α π γ π γ + X δ =0 , , (cid:8) π γ a − π + π γ ′ + (cid:9) π γ a +0 π − π γ ′ + (cid:10) π γ κ − K + π γ ′ + (cid:11) π γ κ + K − π γ ′ + (cid:12) π γ ¯ κ K π γ ′ + (cid:13) π γ κ ¯ K π γ ′ + X δ,δ ′ =0 , , Æ π γ σ δ π δ ′ π γ ′ + X q = u,d,s (cid:15) π γ q ¯ q π γ ′ − i Σ K + = X f ∈ ( σ,π ) α =0 ... (cid:16) f α K + K + + (cid:17) K + κ π + K + + X δ =0 , , (cid:18) K + σ δ K + K + + (cid:19) K + a +0 K K + + X δ =0 , , (cid:20) K + κ + π δ K + + (cid:21) K + u ¯ s K + FIG. 1: Diagrammati representation of the one-loop pseudos alar self-energies used for the parametrization. The label asso iatedto the line denotes the propagating parti le.and densities the e(cid:27)e t of isospin breaking is small we use the same values for l f and l b as in [19℄ where these weredetermined by minimizing the deviation of the predi ted mass spe trum from the physi al one. The external (cid:28)elds aredetermined from the equations of state (6), (7), (8) on e the remaining 8 parameters are known.In order to avoid the appearan e of negative propagator mass squares in the one-loop (cid:28)nite temperature al ulationsin the broken symmetry phase we use the Optimized Perturbation Theory (OPT) of Ref. [29℄. This amounts to repla ethe mass parameter − m in the Lagrangian with an e(cid:27)e tive, eventually temperature-dependent, mass parameter m : L mass = 12 m Tr M † M −
12 ( m + m ) Tr M † M ≡ m Tr M † M −
12 ∆ m Tr M † M. (10)The ounterterm ∆ m is taken into a ount (cid:28)rst at one-loop level, while m will repla e m in all the tree-level massesand is determined using the riterion of fastest apparent onvergen e (FAC). We have hosen to implement this riterionby requiring that for π + the one-loop mass al ulated at vanishing external momentum stays equal to the tree-level mass( M π + = m π + ). We have he ked that imposing this equation for the neutral pion rather than the harged one results inno signi(cid:28) ant hanges in the parameters. We note here that we were for ed to use the de(cid:28)nition M π + = − iG − ( p = 0) instead of de(cid:28)ning the one-loop mass as the pole of the propagator be ause in this latter ase the solution to the gapequation, to be presented below, eases to exist above a ertain temperature, in a ordan e with previous investigationsusing the OPT [18, 29℄.As des ribed in details in [19℄, with the appli ation of FAC one an eliminate the e(cid:27)e tive mass parameter m infavor of the tree-level pion mass m π + in all the other tree-level masses of the propagators used to al ulate the one-loopself-energies: m = m π + + c √ v − c √ v − √ g v v − (cid:16) g + g (cid:17) v − (cid:18) g + 3 g (cid:19) v − (cid:16) g g (cid:17) v . (11)In this way one obtains the following gap equation m π + = − m − c √ v + c √ v + √ g v v + (cid:16) g + g (cid:17) v + (cid:18) g + 3 g (cid:19) v + (cid:16) g + g (cid:17) v + Re Σ π + ( p = 0 , m i ( m π + )) , (12)where Σ π + denotes the self-energy of π + shown diagrammati ally in Fig. 1. Equation (12) is the (cid:28)rst from a set of four oupled non-linear equations whi h determines m , f , f , g , if one knows g F , v , v , v . Two further equations of theset are given by the one-loop equation for the η and K + pole masses M η = − m + h ˜ O TP (cid:16) M tree + Re Σ , , ( p = M η ) (cid:17) ˜ O P i , (13) M K + = − m − c √ v + c v − c √ v + g √ v v − g √ v v + 2 g √ v v + (cid:16) g + g (cid:17) v + (cid:16) g + g (cid:17) v + (cid:18) g + 7 g (cid:19) v + Re Σ K + ( p = M K + ) , (14)where M tree is the tree-level mass squared matrix of the mixing se tor without the mass parameter m , the orthogonalmatrix ˜ O P diagonalizes the expression in the round bra ket, and Σ , , is the self-energy matrix of the pseudos alarmixing se tor. This matrix is determined numeri ally. The last equation in the set is the FAC riterion for the kaon,whi h requires Re Σ K + ( p = M K + ) − ∆ m = 0 .The parameters g F , v , v , v are determined as follows. A linear ombination of v and v is determined by thetree-level PCAC relation for the pion de ay onstant (see Appendix of [28℄) f π := d a v a = r v + 1 √ v . (15)One an see from (5) that the same linear ombination enters the expression of the average mass of the two light onstituent quarks, so that the Yukawa oupling is given by g F = ( M u + M d ) /f π . Another linear ombination of v and v appears in the expression of M s in (5) whi h together with the PCAC relation (15) determines v and v : v = r f π (cid:18) M s M u + M d (cid:19) , v = 1 √ f π (cid:18) − M s M u + M d (cid:19) . (16)The remaining parameter, v is obtained by requiring that the di(cid:27)eren e between the tree-level masses of π + and π equals the physi al value ( ∆ m π ): − (cid:2) O TP M tree O P (cid:3) = (∆ m π ) . (17)This equation has two roots for v , a negative and a positive one. The positive root would give m K < m K + for thekaon masses. Sin e the opposite relation holds in nature, we hoose the negative solution whi h is the physi ally validone.We use the following values for the physi al quantities: m π + = 139 . MeV, ∆ m π = 4 . MeV, M K + = 493 . MeV, M η = 547 . MeV, f π = 93 MeV, ( M u + M d ) / MeV, M s = 530 MeV and in addition l b = 520 MeV and l f = 1210 MeV for the two renormalization s ales.B. Introdu tion of the hemi al potentialsThe introdu tion of the hemi al potential for a system with a set of onserved harge operators is reviewed below.For vanishing external (cid:28)elds the Lagrangian (1) is invariant under the following global ve tor transformations M → e − iα G G M e iα G G = M − iα G [ G, M ] + O ( α G ) ,ψ → e − iα G G ψ = ψ − iα G ψ + O ( α G ) , (18)where G denotes the representation of the baryon (B), third omponent of the isospin (I) and hyper harge (Y) operatorswhi h are related to the diagonal generators as B = q λ , I = λ and Y = √ λ . The oe(cid:30) ients in front of thediagonal matri es are hosen su h as to obtain the right quantum numbers when applying the operators on the quark(cid:28)elds. The onsequen e of this symmetry is the existen e of onserved Noether ve tor- urrents J Gµ = − δLδ ( ∂ µ M ) ij i [ G, M ] ji − δLδ ( ∂ µ M † ) ij i [ G, M † ] ji − δLδ ( ∂ µ ψ i ) iG ij ψ j . (19)The onserved harge is de(cid:28)ned as Q G = R d xJ G ( x ) . In terms of parti le number operators the onserved baryon,isospin and hyper harges read as Q B = 13 ( N u + N d + N s − N ¯ u − N ¯ d − N ¯ s ) , (20) Q I = 12 ( N u − N ¯ u − N d + N ¯ d + N κ + − N κ − + N ¯ κ − N κ + N K + − N K − + N ¯ K − N K )+ N a +0 − N a − + N π + − N π − , (21) Q Y = 13 ( N u − N ¯ u + N d − N ¯ d − N s + 2 N ¯ s ) + N κ + − N κ − + N κ − N ¯ κ + N K + − N K − + N K − N ¯ K . (22)Note the di(cid:27)erent sign of N K , N ¯ K , N κ , N ¯ κ in Q I relative to Q Y . This is be ause the parti les K , ¯ K , κ , ¯ κ fall intodi(cid:27)erent doublets from the point of view of the I and Y quantum numbers: K , K + and K − , ¯ K form a I doubletwhile K , K − and K + , ¯ K form a Y doublet (likewise for s alars).The statisti al density matrix of the system is given by ρ = exp[ − β ( H − µ G Q G )] , (23)with G going over B, I, Y in the summation over this index. Using (20), (21), (22) one an rewrite (23) by regroupingthe terms in the exponent a ording to di(cid:27)erent number operators and obtain ρ = exp[ − β ( H − µ i N i )] , where i goesover all the non-singlet parti les to whi h the following hemi al potentials were introdu ed in terms of µ B , µ I , µ Y : µ u = − µ ¯ u = 13 µ B + 12 µ I + 13 µ Y ,µ d = − µ ¯ d = 13 µ B − µ I + 13 µ Y ,µ s = − µ ¯ s = 13 µ B − µ Y ,µ a +0 = µ π + = − µ a − = − µ π − = µ I ,µ κ + = µ K + = − µ κ − = − µ K − = 12 µ I + µ Y ,µ κ = µ K = − µ ¯ κ = − µ ¯ K = − µ I + µ Y . (24)The singlet parti les ( π , η, η ′ , a , σ, and f ) do not ontribute to the onserved harges and in onsequen e no hemi al potential is introdu ed for them. By looking at (24) one an see that di(cid:27)erent members of a given multiplet(e.g. π + and π − ) a quire a di(cid:27)erent ombination of the baryon, isospin and hyper harge hemi al potentials, whi hmeans that the hemi al potentials remove ompletely the degenera y between the members of the multiplets whi h weobserve in the va uum, both at tree and one-loop level. We have to keep tra k of the e(cid:27)e t of 21 individually di(cid:27)erentparti les, whi h makes things more ompli ated than in previous studies of this model.The e(cid:27)e t of the hemi al potentials is taken into a ount through the propagators whi h are introdu ed usingthe de(cid:28)nition familiar from the theory of many-body systems. The relativisti formalism was developed in [30℄ andis reviewed in Appendix A, where the al ulation of the self-energy using the (cid:28)nite-density Green's fun tion is alsosket hed.In order to see expli itly that the parti le and its antiparti le re(cid:29)e t di(cid:27)erently the presen e of a (cid:28)nite density mediumwe give here the tree-level propagators of K + and K − : G K + ( k ) = i E k (cid:20) n K + ( E k ) k − E k + iǫ − n K + ( E k ) k − E k − iǫ − n K − ( E k ) k + E k − iǫ + n K − ( E k ) k + E k + iǫ (cid:21) ,G K − ( k ) = i E k (cid:20) n K − ( E k ) k − E k + iǫ − n K − ( E k ) k − E k − iǫ − n K + ( E k ) k + E k − iǫ + n K + ( E k ) k + E k + iǫ (cid:21) , (25)where n K ± ( E p ) = 1 e β ( E p − µ K ± ) − and E p = q p + m K ± . The interpretation of the terms on the right hand side of(25) is as follows (from left to right): addition of a parti le, removal of a parti le, addition of an antiparti le, removalof an antiparti le. Note, that in the propagator of the K + the parti le is K + and the antiparti le is K − , while in thepropagator of the K − the parti le is K − and the antiparti le is K + .For all the other s alar and pseudos alar (cid:28)elds the propagators an be written analogously using the hemi alpotentials de(cid:28)ned in (24). For the fermions the propagators are given in Appendix A.III. THERMODYNAMICS OF THE MODEL AT FINITE DENSITYA. The in(cid:29)uen e of µ I and µ Y on the CEPWith the parameters (cid:28)xed in the previous se tion, we an solve the model at (cid:28)nite temperature and density usingthe formalism des ribed in Se tion II B and in Appendix A. One al ulates the 1-loop integrals entering the (cid:28)nitetemperature and density version of the equations whi h determine the state of the system: the three equations ofstate (6), (7), (8) and the gap-equation for m π + (12). The relevant integrals are given in Appendix A. An observedsmooth variation of the order parameters with the intensive parameter ( T , or µ B,I,Y ) indi ates analyti rossover typetransition. A (cid:28)rst order phase transition is signaled by the multivaluedness of either one of the three ondensates in agiven range of variation of the intensive parameter. The point where by varying some parameter(s) the nature of the -0.1-0.095-0.09-0.085-0.08-0.075-0.07-0.065-0.06-0.055-0.05 0 50 100 150 200 250 v [ M e V ] T[MeV] µ I =30 MeV µ B -0.09-0.085-0.08-0.075-0.07-0.065-0.06-0.055-0.05 0 50 100 150 200 250 v [ M e V ] T[MeV] µ B =500 MeV µ I FIG. 2: The generi temperature dependen e of v for a rossover transition: on the l.h.s. µ B hanges and µ I = 70 MeV, on ther.h.s. µ I hanges while µ B = 500 MeV. On both panels the referen e urve (ref. urve) refers to the ase µ B = µ I = µ Y = 0 .phase transition hanges from rossover to (cid:28)rst order one orresponds to a se ond order phase transition.The riti al end point (CEP) is a se ond order phase transition point on the µ B − T plane where by in reasing µ B thephase transition as a fun tion of T hanges from rossover to (cid:28)rst order ( µ I and µ Y are kept onstant). At vanishing µ I and µ Y the CEP is lo ated in the point ( T, µ B ) CEP = (63 . , . MeV. The pseudo- riti al temperature at vanishing hemi al potentials is T c ( µ B,I,Y = 0) = 157 . MeV.Here it is important to note that, with the expli it isospin breaking taken into a ount, these values have signi(cid:28) antly hanged with respe t to those obtained without isospin breaking at all (neither expli it nor spontaneous): ( T, µ B ) CEP =(74 . , . MeV and T c ( µ B,I,Y = 0) = 154 . MeV [19℄. At (cid:28)rst sight this is surprising sin e we have seen that at T = µ = 0 the e(cid:27)e t of the expli it symmetry breaking is minimal. The di(cid:27)eren e is due to the behavior of the v withthe temperature. Without expli it isospin symmetry breaking v is identi ally zero for µ I = 0 . When ǫ = 0 one an seeby looking at the referen e urve of Fig. 2 that with in reasing temperature v is de reasing signi(cid:28) antly ompared to its T = 0 value and rea hes a minimum around the phase transition point where the in(cid:29)uen e of v be omes the strongest.The l.h.s. panel of Fig. 2 shows that the baryo hemi al potential magni(cid:28)es this e(cid:27)e t, implying that approa hing theCEP the in(cid:29)uen e of v is even stronger. A ording to our onje ture made in [19℄ that a smoother rossover at µ B = 0 will require a larger value of µ B to turn the phase-transition in T into a (cid:28)rst order one, implying a larger value of µ B, CEP, we an expe t that the larger value of µ B, CEP in the ase of the expli it isospin breaking ompared to the asein whi h the isospin breaking is absent orresponds to a higher value of the width of the hiral sus eptibility ∆ T c ( xχ ) .Indeed, by looking at Fig. 3 one an see, that in the ase with expli it isospin symmetry breaking ∆ T c ( xχ ) in reasedby ∼ , approa hing the value of ∆ T c ( χ ¯ ψψ ) = 28(5)(1) MeV at at µ I = 0 . This value was obtained on the latti ein Ref. [31℄ after the extrapolation in the ontinuum limit was done, though in this latti e investigation the e(cid:27)e t of
15 16 17 18 19 20 21 22 23 0 10 20 30 40 50 60 70 ∆ T c ( x χ ) [ M e V ] µ I [MeV] ε ≠ ε = FIG. 3: The width of the peak of the hiral sus eptibility ∆ T c ( xχ ) as fun tion of the isospin hemi al potential with (without)expli it symmetry breaking external (cid:28)eld ǫ = 0 ( ǫ = 0 ). In the hiral quark model χ = dx/dǫ x where x = p / v − v ) is thenon-strange ondensate, ǫ x = p / ǫ − ǫ ) and as shown in [19℄ χ ¯ ψψ ∼ xχ .
0 10 20 30 40 50 60 70 0 50 100 150 200 250 30 35 40 45 50 55 60 65T
CEP [MeV] µ I [MeV] µ Y [MeV] T CEP [MeV] µ B,CEP [MeV] µ I [MeV] µ Y [MeV] µ B,CEP [MeV]
FIG. 4: The surfa es swept by the oordinates T CEP and µ B, CE P of the riti al end point as fun tion of µ I and µ Y .isospin breaking was not taken into a ount. It would be interesting to see whether similar e(cid:27)e t is produ ed on latti ewhen m u = m d .Varying µ I and µ Y the lo ation of the CEP in the µ B − T plane hanges. Fig. 4 shows the surfa es swept by thetwo oordinates of the CEP as fun tions of µ I and µ Y . One an see that µ Y has pra ti ally no in(cid:29)uen e on T CEP,whi h de reases very slowly, while with its in rease µ B, CEP signi(cid:28) antly de reases. The in rease of µ I pushes the CEPtowards higher values of µ B, CEP and lower values of T CEP. This behavior is in on ordan e to what was previouslywritten on the in(cid:29)uen e of v on the CEP at µ I = 0 , sin e by looking at the left hand side of Fig. 2 one sees that at(cid:28)nite µ I the isospin ondensate v in reases even more with the temperature.One an gain intuition on the way the hemi al potentials µ I and µ Y in(cid:29)uen e the oordinates of the CEP byattempting a simple interpretation of our results in terms of generalized Clausius-Clapeyron equations applied to oursystem. The parti le number and entropy densities of the two oexisting phases will be determined assuming anideal gas of the quasiparti le degrees of freedom, whi h di(cid:27)er only in their respe tive masses on the two sides of thephase oexisting urves. The Clausius-Clapeyron equation su essfully des ribe the slopes of phase oexisten e urvesof strong matter as fun tions of various hemi al potentials and quark masses [11, 32, 33℄. They are derived fromthe Gibbs-Duhem relation whi h onne t the variation of the intensive thermodynami al parameters of a ma ros opi system: dp = sdT + n B dµ B + n I dµ I + n Y dµ Y . (26)Here n B , n Y , n I are the parti le number densities and s is the entropy density. Keeping the pressure plus any othertwo intensive parameters onstant one (cid:28)nds the following set of onditions for the phase oexisten e when one variesthe remaining two intensive parameters along the oexisten e (cid:16)surfa e(cid:17): dTdµ B (cid:12)(cid:12)(cid:12)(cid:12) µ Y ,µ I = − ∆ n B ∆ s , dTdµ Y (cid:12)(cid:12)(cid:12)(cid:12) µ B ,µ I = − ∆ n Y ∆ s , dTdµ I (cid:12)(cid:12)(cid:12)(cid:12) µ B ,µ Y = − ∆ n I ∆ s ,dµ B dµ Y (cid:12)(cid:12)(cid:12)(cid:12) T,µ I = − ∆ n Y ∆ n B , dµ B dµ I (cid:12)(cid:12)(cid:12)(cid:12) T,µ Y = − ∆ n I ∆ n B . (27)On the right hand side of the equations above ∆ refers to the di(cid:27)eren e of the values of a given extensive quantityin the symmetri and broken symmetry phase. In the two oexisting phases the relevant parti le number and/orentropy densities ( n G , G = B, I, Y, and s ) an be al ulated from the partition fun tion using the formulas n G = T V − ∂ ln Z/∂µ G and s = V − ∂ ( T ln Z ) /∂T. Our simpli(cid:28)ed pi ture of the omposition of the two phases in terms ofnon-intera ting mixtures of 15 quasiparti les is given by ln Z = V X i γ i (2 s i + 1) Z d p (2 π ) h βω i + ln(1 + α i e − β ( ω i − µ i ) ) + ln(1 + α i e − β ( ω i + µ i ) ) i , (28)where i ∈ π ± , π , K ± , K , η, η ′ , a ± , a , κ ± , κ , σ, f , u, d, s , γ i = N c , α i = 1 , s i = 1 / for fermions and γ i = α i = − , s i = 0 for bosons, respe tively. The energies ω i = p p + m i are al ulated with help of the tree-level mass expressions(4) after substituting into them the order parameter values determined in our (cid:28)eld theoreti al treatment for the twophases, that is by solving (6), (7), (8), and (12).The simple model predi ts that ∆ n B , ∆ n Y and ∆ s is always positive, while ∆ n I is always negative. Moreover, thefollowing relations are obtained: ∆ n B ≈ ∆ n Y , ∆ s > ∆ n B and ∆ s > | ∆ n I | . The dis ontinuity of the parti le numberdensities is determined by the ontributions of essentially three quasiparti les: u, d, and π ± . From our simple andtransparent model we get the sign and even the magnitude of the shifts of the CEP in agreement with Fig. 4 with thesingle ex eption of dµ I /dT . The ideal gas model does not reprodu e the value of this derivative obtained by solvingour model. We interpret this dis repan y as a result of the strong oupling between the h ¯ uu i ∼ p / v + p / v + v and h ¯ dd i ∼ p / v + p / v − v ondensates not aptured by the ideal gas approximation. As one an he k alsoin [13℄ the strong oupling between these ondensates redu e the temperature of the CEP when a (cid:28)nite µ I is swit hedon. This is the same tenden y we found in our (cid:28)eld theoreti al al ulation. For the other three shifts it is the massdi(cid:27)eren es of the lightest quasiparti les of the e(cid:27)e tive model whi h exert the strongest in(cid:29)uen e on the variation ofCEP position. B. Quasi-parti le massesWe turn to the study of the dependen e of the tree-level masses and the one-loop pole masses on the temperatureand the hemi al potentials. The one-loop pole masses are determined as the zeros of the real part of the orrespondingone-loop inverse propagators at vanishing spatial momentum. For example, the equation determining the one-loop π + mass reads: M π + = Re G − π + ( p = M π + , p = 0) . If there are more than one solutions of this type of equations, then wefollow that solution whi h in the va uum lies loser to the physi al mass. Usually this solution is lost as the temperaturein reases and some other solution is found.
100 200 300 400 500 600 700 800 900 1000 0 50 100 150 200 250 m t r ee [ M e V ] T [MeV] (b) π π ± ησ a ± a m t r ee [ M e V ] (a)K ± K , − K η′κ ± κ , − κ f m - l oop [ M e V ] µ I [MeV] (d) π + π − T=0, µ B =0T=15, µ B =0T=30, µ B =0T=30, µ B =600 π + π m - l oop [ M e V ] π − π T=0, µ B =0T=0, µ B =0T=0, µ B =600T=0, µ B =650T=0, µ B =700 FIG. 5: The temperature dependen e of the tree-level masses is shown in panels (a) and (b). The µ I dependen e of the one-looppion masses for di(cid:27)erent values of µ B at T = 0 (panel ( )) and T = 0 (panel (d)).In Fig. 5(a) and Fig. 5(b) we see that the tree-level masses of π ± , π and σ learly re(cid:29)e t the restoration of the SU (2) symmetry at high temperature. This is not shown by the masses of a ± , a and η . We annot go to higher values of thetemperature be ause at T ≃ MeV the non-strange ondensate x de reases below the value of v and the tree-levelmass of the u quark turns into negative. At this temperature there is still no sign for the tenden y of the SU (3) hiralpartners to be ome degenerate.In Fig. 5( ) we an see the dependen e of the harged and neutral pion masses on the isospin hemi al potential.The harged pions have by far the most signi(cid:28) ant dependen e on µ I from all of the harged pseudos alar mesons. At T = µ B = 0 the splitting between π + and π − is ontrolled by the bubble diagram involving π + and π − respe tively(see Fig. 1) and the splitting point is at µ I ≃ m π . One an see that at T = µ B = 0 the mass of π depends mildly on0 µ I . This dependen e intensi(cid:28)es with the in rease of T and µ B , but it remains true that the dependen e on µ I is lessstrong than for the ase of m π ± . It is interesting to note that for large values of µ B , when µ u/d > m u/d and the fermionbubble ontributes to the one-loop self-energies, the shape of the m π ± ( µ I ) urves hanges: m π + starts to in rease with µ I and the in rease of m π − with µ I is slowed down and eventually turned over into a de rease in a given interval of µ I .Panel (d) shows that the in rease of the temperature has a similar e(cid:27)e t as µ B in that it turns over the µ I -dependen eof m π ± with respe t to the behavior at T = µ B = 0 starting at a low value of µ I . [ M e V ] T/T c ∆ ∆ x ∆ ∆ x
100 150 200 250 300 350 1.410.60.2 ∆ /f π [ M e V ] T/T c tree1-loop FIG. 6: Estimation of the topologi al sus eptibility through the Witten(cid:21)Veneziano mass formula using one-loop and tree-levelmasses. The T -dependent pion de ay onstant ( f π ( T ) ) is approximated with the non-strange ondensate x . The value of thepseudo riti al temperature is T c = 157 . MeV. In the inserted (cid:28)gure ∆ /f π is plotted based on the tree and one-loop masses.In Fig. 6 we plot, both at tree and at one-loop level a ombination of the masses and the pion de ay onstant ∆ = ( m η + m η ′ − m K ) f π / , whi h through the Witten(cid:21)Veneziano mass formula [34, 35℄ N f f π χ T = m η + m η ′ − m K (29)with N f = 3 , an be onsidered as an estimation of the topologi al sus eptibility χ T ( T ) , whi h plays a ru ial role in thephenomenology of the U (1) A anomaly (see e.g. [36, 37℄ for re ent studies in terms of e(cid:27)e tive des riptions). In prin iple χ T ( T ) an also be omputed dire tly in our model if the quantity orresponding to the topologi al harge density Q T ofthe QCD is extra ted. This an be done by omparing the four-divergen e of the singlet axial ve tor urrent, whi h inQCD involves the U (1) A anomaly term with the orresponding urrent of the hiral quark model. Sin e the determinantterm of Eq. (1) breaks the U (1) A symmetry, the orresponden e is Q T ∼ g (det( M ) − det( M † )) = g Im det M .The de rease with T of the estimated χ T ( T ) seen in Fig. 6 doesn't mean the restoration of the U (1) A symmetry, sin ethrough f π ( T ) , χ T ( T ) is dominated by the restoration of the hiral symmetry. In view of (29) this an be also seen onthe inserted (cid:28)gure of Fig. 6. However, the fa t that at T = 0 the estimated χ T ( T ) is so lose to the value obtained onthe latti e in [38℄ and the urve itself stays within of the latti e points, ould imply that the e(cid:27)e tive restorationof the U (1) A symmetry, if ontained in the latti e data 1, ould be implemented in an e(cid:27)e tive des ription based on the hiral quark model. In the NJL model the latti e result on χ T ( T ) [38℄ is onverted into the temperature-dependen eof the strength of the determinant term, by (cid:28)tting it with the expli it formula of the sus eptibility al ulated in [39℄.IV. CONCLUSIONSIn this paper we studied the in(cid:29)uen e of the isospin and hyper harge hemi al potentials on the µ B − T hiral phasediagram of the three (cid:29)avored hiral onstituent quark model with expli itly broken SU (3) L × SU (3) R symmetry. Themodel was parametrized at one-loop level and optimized perturbation theory was used for the resummation of theperturbative series. Only one riti al end point (CEP) is found for both spontaneous and expli it isospin breaking. Inthe latter ase, based on the width of the peak of the hiral sus eptibility, the rossover transition at µ B,I,Y = 0 isfound to be weaker than in the former ase. Compared to the ase without isospin breaking, in the ase with expli itisospin breaking, the lo ation of the CEP moves to a higher value of µ B and a lower value of T . For µ I = µ Y = 0 U (1) A symmetry requires that χ T ( T ) de reases faster than f π ( T ) with the in rease of T so that χ T ( T ) /f π ( T ) → .1the oordinates of the CEP are: ( T, µ B ) CEP = (63 . , . MeV. This value of µ B, CEP is about three times largerthan the value found on the latti e [40℄ and in reases (de reases) linearly with µ I ( µ Y ), while T CEP is two (cid:28)fth of thelatti e value and de reases slightly with the in rease of µ Y and signi(cid:28) antly with the in rease of µ I . Using an idealgas pi ture and the generalized Clausius-Clapeyron equations we ould interpret semiquantitatively with one ex eptionthe in(cid:29)uen e of µ Y and µ I hemi al potentials on the CEP as resulting from the quasiparti le masses. We also studiedthe dependen e of the harged and neutral one-loop pion masses on the isospin hemi al potential at di(cid:27)erent valuesof the temperature and the baryon hemi al potential. As a ontinuation of the present study, it would be interestingto investigate at what value of the µ I do the harged pions ondensate.A knowledgmentWork supported by the Hungarian S ienti(cid:28) Resear h Fund (OTKA) under ontra t number T046129, NI68228. Zs.Sz. is supported by OTKA Postdo toral Grant no. PD 050015. We thank A. Patkós for dis ussion and suggestions,espe ially on the orre t way of introdu ing the various hemi al potentials, and for areful reading of the manus ript.APPENDIX A: THE FORMALISM OF RELATIVISTIC MANY-BODY THEORY FOR A SYSTEM ATFINITE DENSITY AND TEMPERATUREWe review below the method of relativisti many-body theory developed in [30℄ for the perturbative al ulation ofthe self-energy at (cid:28)nite temperature and density.First we present the derivation of the tree-level Green's fun tions for K − , K + , whi h depend both on the isospinand hyper harge hemi al potentials. The (cid:28)eld operators K − ( x ) and K + ( x ) are written in terms of reation andannihilation operators a + ( p ) , b + ( p ) and a ( p ) , b ( p ) , respe tively, as K − ( x ) = Z d p (2 π ) p E p (cid:0) a + ( p ) e ip · x + b ( p ) e − ip · x (cid:1) (cid:12)(cid:12)(cid:12) p = E p ,K + ( x ) = Z d p (2 π ) p E p (cid:0) b + ( p ) e ip · x + a ( p ) e − ip · x (cid:1) (cid:12)(cid:12)(cid:12) p = E p , (A1)where E p = q p + m K ± . This means that a + ( p ) reates a K + parti le, b + ( p ) reates a K − parti le, et . Theoperators have the usual non-zero ommutators [ a ( p ) , a + ( k )] = [ b ( p ) , b + ( k )] = δ ( p − k ) . (A2)The two point fun tions for K − and K + are de(cid:28)ned as G K − ( y − x ) := h T K − ( y ) K + ( x ) i β = Θ( y − x ) h K − ( y ) K + ( x ) i β + Θ( x − y ) h K + ( x ) K − ( y ) i β ,G K + ( y − x ) := h T K + ( y ) K − ( x ) i β = Θ( y − x ) h K + ( y ) K − ( x ) i β + Θ( x − y ) h K − ( x ) K + ( y ) i β , (A3)where the average is to be taken over a grand anoni al ensemble, that is for an operator O one has h O i β = Tr [ e − β H O ] Tr e − β H , (A4)with H = H − µ i Q i . We make this distin tion between K + and K − propagators be ause the parti le and its antiparti lefeel di(cid:27)erently the presen e of the dense medium, resulting in a di(cid:27)erent mass dependen e on the hemi al potential.In our ase this di(cid:27)eren e in the mass manifests itself (cid:28)rst at one-loop level.Substituting (A1) into (A3), taking only the non-intera ting part of the Hamiltonian H , with the help of the ommutator relations given in (A2) and the Campbell(cid:21)Baker(cid:21)Hausdor(cid:27) relation one evaluates the expe tation valuesobtaining h a + ( p ) a ( q ) i β = δ ( p − q ) n K + ( E p ) , h b + ( p ) b ( q ) i β = δ ( p − q )¯ n K + ( E p ) , (A5)where n K + ( E p ) = 1 e β ( E p − µ K + ) − , ¯ n K + ( E p ) = 1 e β ( E p + µ K + ) − . Note, that ¯ n K + ( E p ) = n K − ( E p ) . Using (A5) andthe Fourier representation of Θ( t ) in (A3) one obtains in momentum spa e the K + and K − propagators given in (25).Next, we al ulate a one-loop bosoni bubble appearing in Fig. 1. With the standard rules of the perturbation theory,using the onventions of [41℄ the π + self-energy is given by − i Σ π + ( y, x ) = − G βγ + 4 ˜ H β,γδ v δ )(3 ˜ G β ′ γ ′ + 4 ˜ H β ′ ,γ ′ δ ′ v δ ′ ) G π β ′ π β ( y, x ) G σ γ ′ σ γ ( y, x ) . (A6)2The (cid:28)rst non-mixing bubble graph in the diagrammati representation of Σ π + given in Fig. 1 is obtained with the hoi e β = 4 , β ′ = 5 implying γ = 7 , γ ′ = 6 . Using that ˜ G = ˜ G and ˜ H , δ = ˜ H , δ the ontribution of this graphis − i Σ K + ¯ κ π + ( y, x ) = (cid:1) π + κ xK − π − ¯ κ yK + π − π + = − h G + 4 ˜ H , δ v δ i G K + K − ( y, x ) G ¯ κ κ ( y, x ) . (A7)The labels in the graph denote the (cid:28)eld operators, e.g. on the left hand side π − reates a π + parti le.Going to momentum spa e one has Σ K + ¯ κ π + ( p ) = − iV π + K − κ Z d k (2 π ) G K + ( k ) G ¯ κ ( p − k ) = 4 V π + K − κ I βB ( p, m K + , µ K + , m ¯ κ , µ ¯ κ ) , (A8)where the vertex is V π + K − κ = 4 h c √ + g √ v − √ g (cid:16) v √ − v (cid:17)i , and G K + ( k ) ≡ G K + K − ( k ) .Generally, at (cid:28)nite hemi al potentials and temperature for a bosoni bubble diagram one al ulates at vanishingspatial external momentum ( p = 0 ) an integral of the form: I βB ( p , m , µ , m , µ ) = − i Z d k (2 π ) G ( k ) G ( p − k ) (cid:12)(cid:12)(cid:12)(cid:12) p =0 = Z d k (2 π ) E E (cid:20) n + n p − E − E − n − ¯ n p − E + E + ¯ n − n p + E − E − n + ¯ n p + E + E (cid:21) , (A9)where for the propagators one uses a form similar to that in (25) and to arrive at the se ond equality one performs a ontour integration in the omplex energy plane. The distribution fun tions n i ≡ n i ( E i ) with E i = p k + m i ontainthe hemi al potential for parti le or antiparti le whi h is reated by the (cid:28)elds of the vertex in the left hand side.We rewrite the integral (A9) as: I βB ( p , m , µ , m , µ ) = I µ,T =0 B ( p , m , m )+ 18 π p X i =1 P Z ∞ m i dE q E − m i (cid:20) n i ( E ) p a i − E + ¯ n i ( E ) p a i + E (cid:21) , (A10)where the remaining integral is evaluated numeri ally, P stands for prin ipal value. The va uum integral I µ,T =0 B ( p , m , m ) is given by the expression (B4) of [19℄, n i = 1 / (exp( β ( E − µ i )) − is the Bose-Einstein distributionand a i = [1 + ( − i − ( m − m ) /p ] / .For fermions the method is identi al to that used for the bosons. The fermion propagators for the onstituent quarks u, ¯ u are de(cid:28)ned as D u ( y − x ) := h T u ( y )¯ u ( x ) i β = Θ( y − x ) h u ( y )¯ u ( x ) i β − Θ( x − y ) h ¯ u ( x ) u ( y ) i β ,D ¯ u ( y − x ) := h T ¯ u ( y ) u ( x ) i β = Θ( y − x ) h ¯ u ( y ) u ( x ) i β − Θ( x − y ) h u ( x )¯ u ( y ) i β , (A11)whi h in the momentum spa e read D u ( k ) = i ( / k + m u )2 E k (cid:20) − f + u ( E k ) k − E k + iǫ + f + u ( E k ) k − E k − iǫ − − f − u ( E k ) k + E k − iǫ − f − u ( E k ) k + E k + iǫ (cid:21) ,D ¯ u ( k ) = i ( / k + m u )2 E k (cid:20) − f − u ( E k ) k − E k + iǫ + f − u ( E k ) k − E k − iǫ − − f + u ( E k ) k + E k − iǫ − f + u ( E k ) k + E k + iǫ (cid:21) , (A12)where f + u ( E p ) = 1 e β ( E p − µ u ) + 1 and f − u ( E p ) = 1 e β ( E p + µ u ) + 1 are the distribution fun tions for u type quarks andantiquarks.Then for the fermioni bubble appearing in the π + self-energy (see Fig. 1) one has Σ u ¯ dπ + ( p ) = − g F N c i Tr Z d k (2 π ) γ D ¯ d ( k ) D u ( k + p ) = g F N c I βF ( p, m d , µ ¯ d , m u , µ u ) . (A13)Similarly to equation (A9) in ase of fermions we use the integral: I βF ( p , m , µ , µ , m ) = − i Tr Z k γ D ( k ) γ D ( k + p ) (cid:12)(cid:12)(cid:12)(cid:12) p =0 = Z d k (2 π ) (cid:20) E ( f +1 + f − −
1) + 1 E ( f +2 + f − − (cid:21) +2( p − ( m − m ) ) Z d k (2 π ) E E (cid:20) − f +1 − f +2 p − E − E + f +1 − f − p − E + E − f − − f +2 p + E − E − − f − − f − p + E + E (cid:21) = − T µ,T =0 F ( m ) − T µ,T =0 F ( m ) + 2( p − ( m − m ) ) I µ,T =0 B ( p , m , m )+ T T =0 F ( m )+ T T =0 F ( m ) − p − ( m − m ) π p X i =1 P Z ∞ m i dE q E − m i (cid:20) f + i ( E ) p a i − E + f − i ( E ) p a i + E (cid:21) (A14)where f ± i = 1 / (exp( β ( E ∓ µ i )) + 1))) + 1)