Continuum study of various susceptibilities within thermal QED 3
Pei-lin Yin, Yuan-mei Shi, Zhu-fang Cui, Hong-tao Feng, Hong-shi Zong
aa r X i v : . [ h e p - ph ] A ug Continuum study of various susceptibilities within thermal QED Pei-lin Yin , Yuan-mei Shi , , Zhu-fang Cui , , Hong-tao Feng , 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 and electronic engineering,Nanjing Xiaozhuang University, Nanjing 211171, China Department of Physics, Southeast University, Nanjing 211189, 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
In this paper, the relations of four different susceptibilities (i.e., the chiral susceptibility, thefermion number susceptibility, the thermal susceptibility and the staggered spin susceptibility) areinvestigated both in and beyond the chiral limit. To this end, we numerically solve the finitetemperature version of the truncated Dyson-Schwinger equations for fermion and boson propagator.It is found that, in the chiral limit, the four susceptibilities give the same critical temperature andsignal a typical second order phase transition. But the situation changes beyond the chiral limit:the critical temperatures from the chiral and the thermal susceptibilities are different, which showsthat to define a critical region instead of an exclusive point for crossover might be a more suitablechoice; meanwhile, both the fermion number and the staggered spin susceptibilities have no singularbehaviors any more, this may mean that they are no longer available to describe the crossoverproperties of the system.PACS Numbers: 11.10.Kk, 11.15.Tk, 11.30.Qc
I. INTRODUCTION
The Quantum Chromodynamics (QCD) vacuum ex-hibits many non-perturbative phenomena that don’tpresent in Quantum Electrodynamics (QED) vacuum.As the temperature and/or chemical potential increase,the QCD vacuum changes and the system undergoes aphase transition into another phase when the tempera-ture and/or chemical potential reach its critical values.There may be many different phases corresponding to dif-ferent regions of temperature and/or chemical potential.The researches for properties of strongly interacting mat-ter within different phases as well as behaviors of phasetransitions, so as to map the phase diagram for the sys-tem in the plane of temperature and chemical potential,are quite important in today’s basic physics theory andexperiment, so a great many of studies has been done onthis field [1–8].In drawing the phase diagram of strongly interact-ing matter, the researches of chiral symmetry breaking-restoration and confinement-deconfinement phase transi-tions are important aspect. Some quantities that char-acterize the above two kinds of phase transition are in-troduced both in Lattice QCD simulation and contin-uum model studies, for instance, the chiral fermion con-densate, the chiral susceptibility [9–12], the quark num-ber susceptibility [13–18], the thermal susceptibility [19–21] and so on. For the latter, the approach includeschiral perturbation theory [22], Dyson-Schwinger equa-tions (DSEs) and Bethe-Salpeter equation (BSE) [23, ∗ [email protected] ) has been studied quiteintensively in recent years. It has many features similarto QCD, such as dynamical chiral symmetry breaking(DCSB) [32–38] and confinement [39–42]. In addition,due to the coupling constant being dimensionful (its di-mension is √ mass ), QED is superrenormalizable, so itdoes not suffer from the ultraviolet divergence which ispresent in QED . Apart from these interesting features,QED with N f massless fermion flavors can be regardedas a possible low energy effective theory for strongly cor-related electronic systems [43–45].In order to see what will happen in the case of QED ,the chiral phase transition driven by the temperature inQED is investigated by analyzing the temperature de-pendence of susceptibility in the present paper. Althoughthere are some works on studying the chiral phase tran-sition by susceptibility in the past few years, such as thechiral susceptibility, the fermion number susceptibilityand so on, the research of the relations among them inthe same framework seems scant, as far as the present au-thors know. The motivation of present paper is to discussspecifically the behavior of four different susceptibilities(i.e., the chiral susceptibility, the fermion number sus-ceptibility, the thermal susceptibility and the staggeredspin susceptibility [46]) with the temperature varying inand beyond the chiral limit separately to compare thesimilarity and difference among them.This paper is organized as follows: In Sec. II, model-independent analytical expressions for the four suscep-tibilities are given, which express susceptibilities as in-tegrals of dressed propagators and dressed vertex. InSec. III, calculations of the four susceptibilities withinthe DSEs framework are performed. A brief summaryand discussions are given in Sec. IV. II. ANALYTICAL TREATMENT
Dynamical properties of a many-particle system can beinvestigated by measuring the response of the system toan external perturbation that disturbs the system onlyslightly in its equilibrium state. A noticeable measureis the susceptibilities that are defined as the first-orderderivative of the order parameter with respect to the ex-ternal field. The order parameter is radically different intwo phases and thus characterizes the phase transitionof the system. As a result, the divergent or some othersingular behaviors of susceptibilities are usually regardedas essential characteristics of phase transition.In this section, by means of the external field methodin Ref. [47], model-independent expressions for the foursusceptibilities are given.
A. Formalism of the chiral susceptibility
It is commonly accepted that with the temperatureand/or chemical potential increasing, strongly interact-ing matter will undergo a phase transition from theNambu-Goldstone phase (or Nambu phase, in which thecondensate of particle-antiparticle pairs, the order pa-rameter of the chiral phase transition, is non-zero dueto DCSB) to the Wigner phase (where chiral symmetryis partially restored and thus the condensate vanishes).The fluctuation of this order parameter is related to thechiral susceptibility which measures the response of thechiral condensate to a small perturbation of the currentmass of the fermion.The chiral susceptibility is defined as χ c = ∂ ( −h ¯ ψψ i ) ∂ m = TV ∂ ln Z ∂m , (1) where h ¯ ψψ i is the fermion chiral condensate in the pres-ence of current mass m , and Z denotes the partition func-tion of the system.Formally, we can express the chiral condensate bymeans of the dressed fermion propagator − h ¯ ψψ i = Z d p (2 π ) Tr[ S ( m , p )] , (2)where the notation Tr denotes trace over Dirac indices ofthe propagator, and S is the dressed fermion propagatorat finite current mass m .Substituting Eq. (2) into Eq. (1) and adopting theidentity ∂S ( m, p ) ∂m = − S ( m, p ) ∂S − ( m, p ) ∂m S ( m, p ) , (3)we immediately arrive at χ c = − Z d p (2 π ) Tr[ S ( m, p ) ∂S − ( m, p ) ∂m S ( m, p )] , (4)Analogizing the well-known Ward identity in QED, weconsider the current mass m as a constant backgroundscalar field coupled to the fermion fields by the term m ¯ ψψ . Then S ( m , p ) is the dressed fermion propagator inthe presence of such a background field and the deriva-tive of its inverse with respect to m yields the so-calledone-particle-irreducible (1PI) dressed scalar vertexΓ( m, , p ) = ∂S − ( m, p ) ∂m , (5)where p is the relative momentum, and the total momen-tum of the dressed scalar vertex vanishes because thebackground scalar field m is a coordinate-independentconstant.Substituting Eq. (5) into Eq. (4) gives χ c = − Z d p (2 π ) Tr[ S ( m, p )Γ( m, , p ) S ( m, p )] , (6)therefore, we obtain an integral formula for the chiral sus-ceptibility at zero temperature and chemical potential. Itexpresses the chiral susceptibility in terms of the dressedfermion propagator and the dressed scalar vertex, whichare just basic quantities in quantum field theory. TheDSEs-BSE approach provides us a desirable frameworkto calculate them, and hence the chiral susceptibility.Here, we note that there is a linear divergence in theabove integral. In order to obtain something meaningfulfrom the chiral susceptibility, we need to subtract thelinear divergence of the free chiral susceptibility. Theregularized chiral susceptibility is defined by χ cr = χ c − χ cf = − Z d p (2 π ) Tr[ S ( m, p )Γ( m, , p ) S ( m, p ) − S ( m, p ) S ( m, p )] , (7)This expression can be generalized to the case of fi-nite temperature. According to finite temperature fieldtheory, the corresponding finite temperature version ofthe chiral susceptibility can be obtained by replacing theintegration over the third component of the momentumwith summation over Matsubara frequencies χ cr ( T ) = − T X n Z d P (2 π ) Tr[ S ( m, p n )Γ( m, , p n ) S ( m, p n ) − S ( m, p n ) S ( m, p n )] , (8)where p µn = ( ω n , ~p ). The notation ω n denotes the fermionMatsubara frequencies, i.e. ω n = (2 n +1) π T , ~p representsthe spatial component of the momentum and its modulusis written as P . Therefore, we have obtained a model-independent integral formula for the chiral susceptibilityat finite temperature and vanishing chemical potential. B. Formalism of the fermion number susceptibility
In addition to the above chiral phase transition, astemperature and/or chemical potential increase, stronglyinteracting matter will also experience a phase transitionfrom the confinement phase (where the degree of free-dom for system is hadron and thus the baryon numberis an integer) to the deconfinement phase (in which thedegree of freedom is quark and gluon, and so the baryonnumber is a fraction). The fluctuation of fermion num-ber is theoretically constructed from measurement of thefermion number susceptibility, i.e the response of fermionnumber density to an infinitesimal change in the chemicalpotential.The fermion number susceptibility is defined as χ n = ∂ρ ( µ ) ∂µ (cid:12)(cid:12)(cid:12)(cid:12) µ =0 = TV ∂ ln Z ∂µ (cid:12)(cid:12)(cid:12)(cid:12) µ =0 , (9)where ρ represents the fermion number density, and µ isthe chemical potential of the fermion.Meanwhile, the fermion number density can be ex-pressed as [48] ρ ( µ ) = − Z d p (2 π ) Tr[ S ( µ, p ) γ ] , (10)where S ( µ, p ) is the dressed fermion propagator at finitecurrent mass and chemical potential.Substituting Eq. (10) into Eq. (9) and using the iden-tity ∂S ( µ, p ) ∂µ = − S ( µ, p ) ∂S − ( µ, p ) ∂µ S ( µ, p ) , (11)one easily arrive at χ n = Z d p (2 π ) Tr[ S ( µ, p ) ∂S − ( µ, p ) ∂µ S ( µ, p ) γ ] (cid:12)(cid:12)(cid:12)(cid:12) µ =0 , (12) In the same way as mentioned above, we consider A ν as a constant background vector field coupled to thefermion fields in the manner ¯ ψγ ν ψ A ν . Then S ( A , p ) isthe dressed fermion propagator in the presence of thisbackground field and the derivative of its inverse withrespect to A ν gives the so-called 1PI dressed vector ver-tex Γ ν ( A , , p ) = − ∂S − ( A , p ) ∂ A ν , (13)where p is the relative momentum and the total momen-tum of this vertex still vanishes due to the constant back-ground field. Putting A ν = δ ν µ in Eq. (13), we obtainΓ ( µ, , p ) = − ∂S − ( µ, p ) ∂µ , (14)Combining Eq. (12) with Eq. (14) gives χ n = − Z d p (2 π ) Tr[ S ( µ, p )Γ ( µ, , p ) S ( µ, p ) γ ] (cid:12)(cid:12)(cid:12)(cid:12) µ =0 , (15)thus, we have also got the expression for the fermionnumber susceptibility, in term of dressed fermion propa-gator and dressed vector vertex which can be calculatedby means of DSEs-BSE approach at zero temperatureand chemical potential.Finally, similar to Eq. (8), the above equation can begeneralized to the finite temperature version χ n ( T ) = − T X n Z d P (2 π ) Tr[ S ( µ, p n )Γ ( µ, , p n ) × S ( µ, p n ) γ ] (cid:12)(cid:12)(cid:12)(cid:12) µ =0 , (16)therefore, we have obtained a model-independent integralformula for the fermion number susceptibility at finitetemperature and vanishing chemical potential. C. Formalism of the thermal susceptibility
The chiral condensate is commonly used to character-ize the chiral phase transition of strongly interacting mat-ter. In the case of finite temperature, it is a function oftemperature and shows some singular behaviors near thephase transition point. In the issue, the thermal suscep-tibility, that is, the response of the chiral condensate toinfinitesimal change of temperature, has attracted quitea few interests over the years. Some works in Ref. [19–21]showed that the peak of the thermal susceptibility shouldbe a crucial character of chiral phase transition.As mentioned above, the thermal susceptibility is de-fined as χ T ( T ) = ∂ h ¯ ψψ i ∂T = − V ∂ ( T ln Z ) ∂T ∂m , (17)where h ¯ ψψ i denotes the finite-temperature fermion chiralcondensate at finite current mass, which can be obtainedfrom Eq. (2) by replacing the integration over the thirdcomponent of momentum with summation over Matsub-ara frequencies − h ¯ ψψ i = T X n Z d P (2 π ) Tr[ S ( m , p n )] . (18)Substituting Eq. (18) into Eq. (17), the thermal sus-ceptibility can be expressed as χ T ( T ) = − T X n Z d P (2 π ) Tr[ S ( m , p n ) T + ∂S ( m , p n ) ∂T ] , (19)therefore, once the finite-temperature dressed fermionpropagator is known, one can calculate the thermal sus-ceptibility. D. Formalism of the staggered spin susceptibility
Because of its success in interpreting the existenceof antiferromagnetic correlation in underdoped cuprates,the U(1) gauge fluctuation effect has aroused great in-terest and extensive attention both in theory and experi-ment in recent years. The response corresponding to thisfluctuation is the staggered spin susceptibility which canbe directly measured in experiments and so provides anideal tool to probe the characteristics of strongly corre-lated system. In recent work, based on functional anal-ysis, the general formula for the staggered spin suscep-tibility was given in term of dressed fermion propagatorand dressed pseudoscalar vertex, and thus it can be cal-culated within the framework of DSEs-BSE approach.The general expression for the low-energy behavior ofthe regularized staggered spin susceptibility was given byRef. [49] χ s = Z d p (2 π ) Tr[ S ( p )Γp( p ) S ( p ) − S ( p ) S ( p )] , (20)where the notation Γp represents pseudoscalar vertexthat satisfies the corresponding inhomogeneous BSE.The corresponding finite temperature version of thestaggered spin susceptibility can be obtained χ s ( T ) = T X n Z d P (2 π ) Tr[ S ( p n )Γp( p n ) S ( p n ) − S ( p n ) S ( p n )] , (21)where Γp satisfies the finite temperature version of theinhomogeneous BSE. III. NUMERICAL RESULTSA. Dyson-Schwinger equations in QED Given the chiral symmetry and parity transformation,we will employ the four-dimension matrix representationand four-component spinors as in four space-time dimen-sions. In Euclidean space, the Lagrangian density ofQED with N f massless fermion flavors reads L = N f X f =1 ¯ ψ f ( −6 ∂ − m + ie A ) ψ f − F µν − ξ ( ∂ · A ) , (22)where the subscript f is a flavour label; f = 1, 2, ..., N f fora theory with N f distinct types or flavours of electricallyactive fermions. We will only work with one flavor in thepresent paper. Using this Lagrangian density, one can de-rive in the standard way, for instance through functionalanalysis, the DSEs for propagators.For the fermion propagator, we obtain the finite tem-perature version of DSEs S − ( m, p n ) = S − ( m, p n ) + Σ( m, p n ) , (23)Σ( m, p n ) = T X n Z d K (2 π ) γ µ S ( m, k n )Γ ν ( p n , k n ) D µν ( q n ) , (24)where S − = i~γ · ~p + iγ ω n + m is just inverse of thefree fermion propagator, Σ is the fermion self-energy, Γ ν is the full fermion-boson vertex, and D µν is the dressedboson propagator.Other than zero temperature, the O(3) symmetry ofthe system reduces to O(2), and based on the Lorentzstructure analysis, the inverse of fermion propagator canbe written as S − ( m , p n ) = i ~γ · ~ pA k ( m, p n ) + i γ ω n A ( m, p n )+ B ( m, p n ) , (25)where A k and A are familiar wave function renormal-ization factors; B is fermion self-energy function, and atensor term proportional to σ µν is ruled out by PT in-variance.At zero temperature, the results in Ref. [36] show thatwhen the 1/N order contribution to the renormaliza-tion factor is included, the critical number of fermionflavors take almost the same value as the case where A = 1. At finite temperature, the comparison of studiesin Ref. [50, 51] also suggests that the 1/N order contri-bution to the factor only changes the results slightly. Sowe expect that the 1/N order contribution to A k and A is not important and we will take A k = A =1 in thepresent paper. In addition, the conclusions in Ref. [52]indicate that by summing over the frequency modes andtaking suitable simplifications, the qualitative aspects ofthe result obtained under the zero frequency approxi-mation for fermion self-energy don’t undergo significantchanges. From this, we will also ignore the frequencydependence of the self-energy.For the boson propagator, we will follow Ref. [50] inretaining only the µ = ν = 3 component of the bosonpropagator, and ignore all but the zero-frequency com-ponent, that is to say, we employ the boson propagator D µν ( m, T, Q ) = 2 δ µ δ ν Q + Π( m, T, Q ) , (26)where Q = ~q = ( ~p − ~k ) . The zero frequency bosonpolarization with current mass readsΠ( m, T, Q ) = Tπ Z d x (cid:26) ln(4cosh p m + x (1 − x ) Q T ) − m tanh √ m + x (1 − x ) Q T T p m + x (1 − x ) Q (cid:27) , (27)Substituting Eq. (24), Eq. (25) and Eq. (26) into Eq.(23), we can immediately obtain (to be concise, hereafterwe use B to represent B ( m, T, P ), and Π to representΠ( m, T, Q ) in the right sides of equations) B ( m, T, P ) = m + T X n Z d K (2 π ) B/ ( Q + Π)( ω n + K + B ) , (28)with the help of the identity X n ω n + x = tanh x T xT , (29)the above equation can be reduced to B ( m, T, P ) = m + Z d K (2 π ) B tanh √ K + B T √ K + B ( Q + Π) . (30) B. Chiral symmetry breaking-restoration phasetransition in the chiral limit
In this section, within the framework of DSEs ap-proach, we will investigate the behavior of the four sus-ceptibilities with varying temperature. All equations in-volved in the section are in the chiral limit.
1. The chiral susceptibility χ c For the chiral susceptibility, substituting Eq. (5) andEq. (25) into Eq. (8) gives χ cr ( T ) = 2 Z d P (2 π ) (cid:26) B m [ P tanh √ P + B T ( √ P + B ) + B sech √ P + B T T ( P + B ) ] − P tanh P T (cid:27) , (31) c T FIG. 1. Dependence of the chiral susceptibility χ c on temper-ature T . where B is the function that satisfies Eq. (30) in theabsence of current mass. The function B m is just thederivative of B with respect to current mass and can bewritten as B m = 1 + Z d K (2 π ) B m (cid:26) K tanh √ K + B T ( √ K + B ) + B sech √ K + B T T ( K + B ) (cid:27) Q + Π , (32)Using the iterative method for the above two equa-tions, we can immediately arrive at the typical behaviorof them, and thus the chiral susceptibility. As a result, weplot the behavior of the chiral susceptibility with varyingtemperature in Fig. 1 .From Fig. 1, it can be seen that the chiral susceptibil-ity exhibits a very narrow, pronounced, and, in fact, di-vergent peak at the critical temperature T c = 2 . × − ,which is a typical characteristic of the second order phasetransition. This conclusion is in good agreement with theresult based on the continuum model studies of two-flavorQCD [54].
2. The fermion number susceptibility χ n For the fermion number susceptibility, substituting Eq.(14) and Eq. (25) into Eq. (16), and employing thesimilar approximation in Ref. [38] for the dressed vectorvertex, we can obtain χ n ( T ) = 4 T Z d P (2 π ) exp √ P + B T (exp √ P + B T + 1) , (33) In the studies within QED , the coupling constant α = e hasdimension one and provides us with a mass scale. Accordingly,a kind of natural unit e = 1 is often used (for example, seeRef. [53]). For simplicity, in this paper the temperature and themass are both measured in this unit. n T FIG. 2. Dependence of the fermion number susceptibility χ n /χ nf on temperature T , where χ nf is the free fermion numbersusceptibility at finite temperature. Once the fermion self-energy function is obtained, wecan calculate the fermion number susceptibility. The be-havior of χ n ( T ) is shown in Fig. 2.As is shown in Fig. 2, where χ n is normalized by thefree fermion number susceptibility χ nf and is hence di-mensionless, the fermion number susceptibility rises astemperature increases, then shows an apparent inflexionat the critical temperature T c = 2 . × − , and finallyis almost constant.
3. The thermal susceptibility χ T For the thermal susceptibility, substituting Eq. (25)into Eq. (19), then the latter can be reduced to χ T ( T ) = 2 Z d P (2 π ) (cid:26) B T [ P tanh √ P + B T ( √ P + B ) + B sech √ P + B T T ( P + B ) ] − B sech √ P + B T T (cid:27) , (34)where the function B T represents the derivative offermion self-energy function with respect to temperature,and can be expressed as B T = Z d K (2 π ) (cid:26) B T [ K tanh √ K + B T ( √ K + B ) + B sech √ K + B T T ( K + B ) ] − B sech √ K + B T T − B Π T tanh √ K + B T √ K + B ( Q + Π) (cid:27) Q + Π , (35)while the function Π T denotes the derivative of bosonpolarization function with respect to the temperature,which is written asΠ T = 1 π Z (cid:26) ln(4cosh p x (1 − x ) Q T ) T T FIG. 3. Dependence of the thermal susceptibility χ T on tem-perature T . − p x (1 − x ) Q tanh √ x (1 − x ) Q T T (cid:27) , (36)from the two functions B and B T , one can obtain thedependence of the thermal susceptibility on temperature.As a result, the behavior of this susceptibility is plottedin Fig. 3.From Fig. 3, it is found that the thermal susceptibilityrises with temperature increasing, then reaches its max-imum at the critical temperature T c = 2 . × − , andvanishes when temperature is above T c
4. The staggered spin susceptibility χ s For the staggered spin susceptibility, we focus on itslow-energy behavior, and so the dressed pseudoscalar ver-tex Γp can be written asΓp( p n ) = γ C ( p n ) + i~γ · ~pγ D k ( p n ) + iγ w n γ D ( p n ) , (37)Substituting Eq. (37) into Eq. (21), we immediatelyarrive at χ s ( T ) = 2 Z d P (2 π ) ( C tanh √ P + B T √ P + B − tanh P T P ) . (38)Meanwhile, the dressed pseudoscalar vertex satisfiesthe finite temperature version of the inhomogeneous BSEΓp( p n ) = γ − T X n Z d K (2 π ) γ µ S ( k n )Γp( k n ) S ( k n ) γ ν × D µν ( q n ) , (39)Substituting Eq. (37) into Eq. (39) gives C ( P ) = 1 + Z d K (2 π ) C tanh √ K + B T √ K + B ( Q + Π) . (40)By solving the functions B and C , we can obtain thestaggered spin susceptibility with a range of temperature,and the results are shown in Fig. 4. s T FIG. 4. Dependence of the staggered spin susceptibility χ s on temperature T . As can be seen from Fig. 4, the staggered spin suscepti-bility decreases with temperature increasing, then dropsrapidly at the critical temperature T c = 2 . × − , andis almost constant when temperature is higher.From above, we may safely draw the conclusion thatthese four susceptibilities exhibit the singular behaviorsat the same critical temperature T c = 2 . × − , wherethe chiral phase transition occurs. In addition, the chi-ral susceptibility exhibits a divergent peak at the criticaltemperature, which is a typical characteristic of the sec-ond order phase transition. C. Chiral symmetry breaking-restoration phasetransition beyond the chiral limit
In this section, we will recalculate the four susceptibil-ities to analyze the effect of current mass on the chiralphase transition driven by temperature.
1. The chiral susceptibility χ c Regarding the chiral susceptibility, following the samestep as Eq. (31), the chiral susceptibility beyond thechiral limit is obtained χ cr ( m, T ) = 2 Z d P (2 π ) (cid:26) B m [ P tanh √ P + B T ( √ P + B ) + B sech √ P + B T T ( P + B ) ] − [ P tanh √ P + m T ( √ P + m ) + m sech √ P + m T T ( P + m ) ] (cid:27) , (41)where the self-energy function B satisfies Eq. (30) andthus its derivative with respect to current mass can beexpressed as B m = Z d K (2 π ) (cid:26) B m [ K tanh √ K + B T ( √ K + B ) + B sech √ K + B T T ( K + B ) ] c T m=1.0x10 -4 m=2.0x10 -4 m=1.0x10 -3 FIG. 5. Dependence of the chiral susceptibility χ c on temper-ature T for several m . − B Π m tanh √ K + B T √ K + B ( Q + Π) (cid:27) Q + Π + 1 , (42)the function Π m is the derivative of polarization functionwith respect to current mass, which satisfiesΠ m = − mπ Z [ x (1 − x ) Q tanh √ m + x (1 − x ) Q T ( p m + x (1 − x ) Q ) + m sech √ m + x (1 − x ) Q T T ( m + x (1 − x ) Q ) ] , (43)From Eq. (30), Eq. (41) and Eq. (42), we can obtainthe dependence of the chiral susceptibility on tempera-ture and current mass. The behaviors of χ c with regardto T for several m are plotted in Fig. 5.As is shown in Fig. 5, the chiral susceptibility exhibitsa quite different behavior in the presence of current mass.The peak of the chiral susceptibility becomes not so sharpand pronounced as in the chiral limit and its height isgreatly suppressed and evidently finite, which is a typicalcharacter of a crossover. In addition, with the currentmass increasing, the critical temperature where the chiralsusceptibility takes its maximum also rises, but the valueof the peak falls monotonously.
2. The fermion number susceptibility χ n Similar to Eq. (33), the fermion number susceptibilitybeyond the chiral limit can be written as χ n ( m, T ) = 4 T Z d P (2 π ) exp √ P + B T (exp √ P + B T + 1) , (44)where the fermion self-energy function B satisfies Eq.(30).From Eq. (27), Eq. (30) and Eq. (44), the dependenceof the fermion number susceptibility on temperature andcurrent mass is immediately obtained. We show the be-haviors of χ n with respect to T for several m in Fig. 6. m=1.0x10 -4 m=2.0x10 -4 m=1.0x10 -3 n T FIG. 6. Dependence of the fermion number susceptibility χ n /χ nf on temperature T for several m , where χ nf is the freefermion number susceptibility at finite temperature and cur-rent mass of the fermion. In Fig. 6, the fermion number susceptibility at fi-nite current mass also reveals a different picture fromthe case of chiral limit. As temperature increases, thefermion number susceptibility also rises monotonouslyand almost keeps a constant at last. It is notewor-thy that χ n becomes smooth and no singular behaviorsemerge in a range of temperature, which may show thatthe fermion number susceptibility can not describe thecrossover properties of the system well. It is consistentwith the result obtained using the NJL model [30] .
3. The thermal susceptibility χ T For the thermal susceptibility, analogizing with Eq.(34), we finally arrive at χ T ( m, T ) = 2 Z d P (2 π ) (cid:26) B T [ P tanh √ P + B T ( √ P + B ) + B sech √ P + B T T ( P + B ) ] − B sech √ P + B T T (cid:27) , (45)where the two functions B and B T satisfy, separately,Eq. (30) and Eq. (35). The function Π T involved here isa little different from Eq. (36), and is written asΠ T = 1 π Z (cid:26) ln(4cosh p m + x (1 − x ) Q T ) − p m + x (1 − x ) Q tanh √ m + x (1 − x ) Q T T + m sech √ m + x (1 − x ) Q T T (cid:27) , (46)According to the equation above, we obtain the de-pendence of the thermal susceptibility on temperatureand current masses. As a result, the behaviors of χ T with regard to temperature for several current massesare plotted in Fig. 7. m=1.0x10 -4 m=2.0x10 -4 m=1.0x10 -3 T T FIG. 7. Dependence of the thermal susceptibility χ T on tem-perature T for several m . From Fig. 7, we can evidently see that the thermalsusceptibility shows an apparent peak. With tempera-ture increasing, the thermal susceptibility also rises, thentakes its maximum and decreases slowly when the tem-perature is higher. Similar to the chiral susceptibility,as current mass increases, the critical temperature atwhich the thermal susceptibility is maximal rises, whilethe value of the peak falls slowly.
4. The staggered spin susceptibility χ s Following the same step as Eq. (38), the staggered spinsusceptibility in the presence of current mass is expressedas χ s ( m, T ) = 2 Z d p (2 π ) (cid:26) C tanh √ P + B T √ P + B − [ P tanh √ P + m T ( √ P + m ) + m sech √ P + m T T ( P + m ) ] (cid:27) , (47)By solving Eq. (30) and Eq. (40), the dependenceof the staggered spin susceptibility on temperature andcurrent mass can be obtained. We plot the behaviors of χ s with respect to temperature at several current massesin Fig. 8.From Fig. 8, it can be seen that the staggered spinsusceptibility in this case shows a quite different picturefrom the chiral limit case. With the temperature increas-ing, the staggered spin susceptibility decreases slowly andis almost a constant in the end. Similar to the fermionnumber susceptibility, the staggered spin susceptibility issmooth and no singular behaviors occur in the tempera-ture range we studied.From what we have mentioned above, we can concludethat the four susceptibilities in the presence of currentmass have intrinsic differences from the cases of chirallimit. Both the chiral and the thermal susceptibilitiesreveal an apparent but not divergent peak signalling atypical crossover behavior, meanwhile the critical tem-peratures from these two susceptibilities are different, m=1.0x10 -4 m=2.0x10 -4 m=1.0x10 -3 s T FIG. 8. Dependence of the staggered spin susceptibility χ s on temperature T for several m . which shows that to define a critical region instead ofan exclusive point for crossover might be a more suitablechoice. Moreover, the fermion number and the staggeredspin susceptibilities are smooth with varying tempera-ture and no singular behaviors arise, this may mean thatthese two susceptibilities can not describe the crossoverproperties of the system well. IV. SUMMARY AND CONCLUSIONS
In this paper, we study the relations of the four dif-ferent susceptibilities (viz., the chiral susceptibility, thefermion number susceptibility, the thermal susceptibilityand the staggered spin susceptibility) both in and beyondthe chiral limit. We first give the general integral formula for the four different susceptibilities by means of the ex-ternal field method, and then investigate the temperaturedependence of them in the framework of DSEs.Our model study reveals that, in the chiral limit, thefour susceptibilities give the same critical temperature T c = 2 . × − , where the chiral phase transition oc-curs. In addition, the chiral susceptibility shows that thisis a second order phase transition at finite temperatureand vanishing chemical potential. On the other hand, inthe presence of current mass, the results are quite dif-ferent: the critical temperatures from the chiral and thethermal susceptibilities are different, which shows that todefine a critical region instead of an exclusive point forcrossover might be a more suitable choice. In addition,both the fermion number and the staggered spin suscep-tibilities have no singular behaviors any more, this maymean that they are no longer available to describe thecrossover properties of the system.Of course, the model adopted in this work is schematic,to further confirm these observations, we need to studythis problem in some more realistic models in the future. ACKNOWLEDGMENTS
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