Nature of chiral phase transition in QED 3 at zero density
aa r X i v : . [ h e p - ph ] A ug Nature of chiral phase transition in QED at zero density Hong-tao Feng , ∗ , Jian-Feng Li , , Yuan-mei Shi , , , and Hong-shi Zong , , † Department of Physics, Southeast University, Nanjing, 211189, China College of Mathematics and Physics, Nantong University, Nantong 226019, China Department of Physics, Nanjing Xiaozhuang College, Nanjing 211171, China Department of Physics, Nanjing University, Nanjing, 210093, China State Key Laboratory of Theoretical Physics, Institute of Theoretical Physics,CAS, Beijing 100190, Peoples Republic of China and Joint Center for Particle, Nuclear Physics and Cosmology, Nanjing 210093, China
Based on the feature of chiral susceptibility and thermal susceptibility at finite temperature, thenature of chiral phase transition around the critical number of fermion flavors ( N c ) and the criticaltemperature ( T c ) at a fixed fermion flavors number in massless QED are investigated. It is showedthat, at finite temperature the system exhibits a second-order phase transition at N c or T c and eachof the estimated critical exponents is less than 1, while it reveals a higher-order continuous phasetransition around N c at zero temperature.Keywords: QED , chiral phase transition, chiral susceptibility, thermal susceptibility. PACS numbers: 11.10.Kk, 11.15.Ex, 11.15.Tk, 11.30.Rd
I. INTRODUCTION
The study of chiral phase transition (CPT) in (2+1)-dimensional quantum electrodynamics (QED ) has beenan active subject for 30 years since Appelquist et al .found that CPT occurs when the flavor of masslessfermions reaches a critical number N c [1]. They arrivedat this conclusion by analytically and numerically solv-ing the Dyson-Schwinger equation (DSE) for the fermionself-energy in the lowest order approximation where theinvolved one-loop boson polarization is obtained by thefree form of the fermion propagator. To indicate the valueof N c , D. Nash adopted an improved scheme and gave alarger N c [2]. Later, several groups investigated the de-pendence of chiral symmetry breaking on N and somegroups doubted the existence of N c [3, 4]. This questionwas answered by P. Maris et al , who used the coupledDSEs for the photon and fermion propagator to inves-tigate the influence of the full vacuum polarization andvertex function on the fermion propagator and they foundthat the critical number of fermion flavors for dynamicalmass generation of massless QED lies between 3 and 4[5–7].Nevertheless, what is the order of CPT around N c might be an interesting question. To reveal that, theauthors of [1] studied the light scalar degrees of freedomand the order parameter of CPT near N c and found thatthe phase transition is not second-order and is also un-like conventional first-order transition [8]. In addition,the results from Cornwall-Jackiw-Tomboulis effective po-tential also gave the same conclusion [9]. Although theabove reveals the characteristic CPT, it is interesting to ∗ Email: [email protected] † Email:[email protected] adopt an alternative method to reanalyze the nature ofthis phase transition and see whether it is consistent withthose results.At finite temperature, the value of N c should alsovary and chiral symmetry restores as the temperatureincreases at a fixed N ( < N c ). In this case, the fermionpropagator at finite temperature T can be written as S − ( T, P ) = i~γ · ~P A k ( P ) + i̟ n γ A ( P ) + B ( P ) , (1)where ̟ n = (2 n + 1) πT and A, B are the fermionwave-renormalization factor and self-energy, respectively.Adopting the lowest-order approximation of DSE and us-ing Eq. (1), Dorey investigated the CPT of QED at fi-nite temperature and showed that QED with dynamicalchiral symmetry breaking (DCSB) undergoes CPT intochiral symmetric phase when the temperature reaches acritical value T c and the corresponding N c decreases withthe increasing temperature [10].The above conclusion holds in massless QED . Then,another natural question may be raised: how does onechart the phase diagram of thermal QED around T c and whether or not the nature of CPT around N c atfinite temperature is the same as that at zero temper-ature. At the involved temperature, since the externalfields are screened by thermal excitations and the bosongains a nonzero mass, the feature of CPT at N c mightbe changed. However, as far as we know, the nature ofCPT at N c in thermal QED has not been reported inthe existing literature. Therefore, it is very interestingto study this problem.In recent years, some works in lattice QCD [11–13]showed that the peak of chiral susceptibility should bean essential characteristic of CPT. Later, based on tech-niques of continuum field theory, several groups [14–19]also reached the same conclusion. Thus, chiral suscep-tibility is competent for studying the feature of phasetransition in this nonperturbative system. Mealwhile, thethermal susceptibility give other ideal parameter to inves-tigate the characteristic of CPT at finite temperature[20].In this paper, we shall adopt the chiral and thermal sus-ceptibilities to study the nature of chiral phase transitionof QED at finite temperature. II. FORMALISM FOR CHIRALSUSCEPTIBILITY
The Lagrangian of QED involving N fermion flavorsof 4 × L = N X j =0 ¯ ψ j ( ∂ + i e A − m ) ψ j + 14 F σν + 12 ξ ( ∂ ρ A ρ ) . (2)In the absence of the mass term m ¯ ψψ , QED has chiralsymmetry. There are several equivalent choices of theorder parameter for chiral symmetry breaking; here weuse the fermion chiral condensate h ¯ ψψ i m = Z d p (2 π ) Tr[ S ( m, p )] , (3)where S is the dressed fermion propagator and T r de-notes trace operation over Dirac indices of the fermionpropagator. Based on Lorentz structure analysis, the in-volved massive/massless fermion propagator can be writ-ten as S − ( m, p ) = iγ · pE ( p ) + F ( p ) , (4) S − ( p ) ≡ S − (0 , p ) = iγ · pA ( p ) + B ( p ) . (5)In the high energy limit, the fermion propagator reducesto the free one, i.e., S − ( p ) = iγ · p in the chiral limitand S − ( m, p ) = iγ · p + m beyond the chiral limit. Fromthis it can be seen that, with a small fermion mass m ,the integral in Eq. (3) is divergent. In this case weshould employ a renormalization procedure to deal withthis divergence. A natural approach is to subtract thecondensate of the free fermion field from the above value.That is to say, we define the renormalized fermion chiralcondensate by h ¯ ψψ i ≡ h ¯ ψψ i m − h ¯ ψψ i mf , (6)where h ¯ ψψ i mf is the condensate of the free fermion gas.Below, we shall determine the transition point via themaximum of chiral susceptibility ∂ h ¯ ψψ i ∂m (see, e.g., Refs.[11, 21]) which is defined as[14] χ c = ∂ h ¯ ψψ i ∂m (cid:12)(cid:12)(cid:12)(cid:12) m → . (7)This equation indicates that the chiral susceptibility mea-sures the response of the chiral condensate (the orderparameter) to an infinitesimal change of the fermionmass responsible for explicit breaking of chiral symme-try. Note here that we evaluate the chiral susceptibilityand fermion chiral condensate in the chiral limit. From Eqs. (3)-(5), we immediately arrive at the chiralsusceptibility of QED in chiral limit χ c = 4 N Z d p (2 π ) (cid:26) p A D − p ABC − B D [ p A + B ] − p (cid:27) , (8)where C ( p ) = ∂E ( p ) ∂m (cid:12)(cid:12)(cid:12)(cid:12) m → , D ( p ) = ∂F ( p ) ∂m (cid:12)(cid:12)(cid:12)(cid:12) m → . (9) III. ZERO TEMPERATURE
The next task is to obtain the four functions
A, B, C, D . These functions can be obtained by solvingthe DSE for the massive fermion propagator, S − ( m, p ) = S − ( m, p ) + Z d k (2 π ) × [ γ σ S ( m, k )Γ ν ( m ; p, k ) D σν ( m, q )] , (10)where Γ ν ( m ; p, k ) is the full fermion-photon vertex and q = p − k . The coupling constant α = e has dimensionone, and provides us with a mass scale. For simplicity,in this paper temperature, mass and momentum are allmeasured in unit of α , namely, we choose a kind of nat-ural units in which α = 1. Form Eq. (5) and Eq. (18),we obtain the equation satisfied by E ( p ) and F ( p ) E ( p ) = 1 − p Z d k (2 π ) T r [ i ( γp ) γ σ S ( m, k ) × Γ ν ( m ; p, k ) D σν ( m, q )] , (11) F ( p ) = 14 Z d k (2 π ) × T r [ γ σ S ( m, k )Γ ν ( m ; p, k ) D σν ( m, q )] . (12)Another involved function D σν ( q ) is the full gauge bosonpropagator which is given by [17] D σν ( m, q ) = δ σν − q σ q ν /q q [1 + Π( m, q )] + ξ q σ q ν q , (13)where ξ is the gauge parameter and Π( q ) is the vacuumpolarization for the gauge boson which is satisfied by thepolarization tensor for gauge boson and readsΠ σν ( m, q ) = − Z d k (2 π ) T r [ S ( m, k ) γ σ S ( m, q + k )Γ ν ( m, p, k )] . (14)Using the relation between the vacuum polarizationΠ( m, q ) and Π σν ( q ),Π σν ( m, q ) = ( q δ σν − q σ q ν )Π( m, q ) , (15)we can obtain an equation for Π( q ) which has ultravioletdivergence. Fortunately, it is present only in the longi-tudinal part and is proportional to δ σν . We can removethe divergence by the projection operator P σν = δ σν − q σ q ν q , (16)and obtain a finite vacuum polarization[18].Finally, we choose to work in the Landau gauge, sincethe Landau gauge is the most convenient and commonlyused one. Once the fermion-boson vertex is known, weimmediately obtain truncated DSEs for the propagatorsof the fermion and the gauge boson and then the chi-ral susceptibility near N c is obtained. Of course, just asmentioned in Ref. [22], N c occurs only in homogeneoussystem, i.e., all the involved functions in this issue for thefermion and boson propagators should satisfy homogene-ity degrees. A. Rainbow approximation
The simplest and most commonly used truncatedscheme for the DSEs is the rainbow approximation,Γ ν → γ ν , (17)since it gives us rainbow diagrams in the fermion DSEand ladder diagrams in the Bethe-Salpeter equation forthe fermion-antifermion bound state amplitude. In theframework of this approximation, the coupled equationfor massive fermion propagator reduces to S − ( m, p ) = S − ( m, p )+ Z d k (2 π ) γ σ S ( m, k ) γ ν D σν ( m, q ) . (18)From Eq. (5) and Eq. (18), we obtain the equationsatisfied by E ( p ) and F ( p ) E ( p ) = 1 − p Z d k (2 π ) T r [ i ( γp ) γ σ S ( m, k ) γ ν D σν ( m, q )] , (19) F ( p ) = 14 Z d k (2 π ) T r [ γ σ S ( m, k ) γ ν D σν ( m, q )] , (20)In order to obtain these two functions, we start fromthe propagators with massive fermion. From the abovetwo equations and some tricks proposed in Ref. [23], weobtain the three coupled equations for E ( p ), F ( p ) andΠ( m, q ), E ( p ) = 1 + 2 p Z d k (2 π ) E ( k )( pq )( kq ) / ( q ) G ( k )[1 + Π( m, q )] , (21) F ( p ) = m + 2 Z d k (2 π ) F ( k ) /q G ( k )[1 + Π( m, q )] , (22)Π( m, q ) = 2 Nq Z d k (2 π ) E ( k ) E ( p ) G ( k ) G ( p ) × [2 k − k · q ) − k · q ) /q ] , (23)with G ( k ) = E ( k ) k + F ( k ).Adopting Eqs. (9) and (21-23) and settingΠ ′ ( q ) = ∂ Π( m,q ) ∂m | m → , we get the coupled equationsfor C ( p ) , D ( p ) and Π ′ ( q ), C ( p ) = 2 p Z d k (2 π ) ( p · q )( k · q ) C / ( q ) H ( k ) [1 + Π( q )] , (24) D ( p ) = 1 + 2 Z d k (2 π ) D /q H ( k ) [1 + Π( q )] , (25)Π ′ ( q ) = 2 Nq Z d k (2 π ) (cid:2) k − k · q ) − k · q ) /q (cid:3) Π ′ H ( k ) H ( p ) , (26)with H ( k ) = A ( k ) k + B ( k ) and C ≡ [ B ( k ) C ( k ) − A ( k ) C ( k ) k − A ( k ) B ( k ) D ( k )] (cid:2) q ) (cid:3) − A ( k ) H ( k )Π ′ ( q ) ,D ≡ [ A ( k ) D ( k ) k − B ( k ) D ( k ) − A ( k ) B ( k ) C ( k ) k ] (cid:2) q ) (cid:3) − B ( k ) H ( k )Π ′ ( q ) , Π ′ ≡ (cid:2) A ( p ) C ( k ) + A ( k ) C ( p ) (cid:3) H ( k ) H ( p ) − A ( k ) A ( p ) (cid:2) A ( k ) C ( k ) k + B ( k ) D ( k ) (cid:3) H ( p ) − A ( k ) A ( p ) (cid:2) A ( p ) C ( p ) p + B ( p ) D ( p ) (cid:3) H ( k ) , where A, B,
Π are obtained by Eqs. (21-23) at m =0. By application of iterative methods, we can obtain A, B,
Π and the above functions for the scalar vertex.
B. Improved scheme for DSE
To improve the truncated scheme for DSE, there areseveral attempts to determine the functional form forthe full fermion-gauge-boson vertex [24–27], but noneof them completely resolve the problem. However, the Ward-Takahashi identity (WTI)( p − k ) ν Γ ν ( m ; p, k ) = S − ( m, p ) − S − ( m, k ) , (27)provides us an effectual tool to obtain a reasonableansatze for the full vertex [24]. The portion of the dressedvertex which is free of kinematic singularities, i.e. BCvertex, can be written as,Γ ν ( m, p, k ) = E ( p ) + E ( k )2 γ ν + F ( p ) − F ( k ) p − k ( p + k ) ν +( p + k ) E ( p ) − E ( k )2( p − k ) ( p + k ) ν . (28)Since the numerical results obtained using the first partof the vertex coincide very well with earlier investigations[6, 16], we choose this one as a reasonable ansatzeΓ BC ν ( m ; p, k ) . = 12 (cid:2) E ( p ) + E ( k ) (cid:3) γ ν (29) to be used in our calculation. Following the procedure inrainbow approximation, we also obtain the three coupledequations for E ( p ) , F ( p ) and Π( m, q ) in the improvedtruncated scheme for DSEs, E ( p ) = 1 + Z d k (2 π ) E ( k )[ E ( p ) + E ( k )]( pq )( kq ) / ( q ) p G ( k )[1 + Π( m, q )] , (30) F ( p ) = m + Z d k (2 π ) [ E ( p ) + E ( k )] F ( k ) /q G ( k )[1 + Π( m, q )] , (31)Π( m, q ) = Nq Z d k (2 π ) E ( k ) E ( p )[ E ( p ) + E ( k )] G ( k ) G ( p ) [2 k − k · q ) − k · q ) /q ] , (32)and the corresponding unknown functions for C ( p ) , D ( p ) , Π ′ ( q ) are, C ( p ) = 1 p Z d k (2 π ) ( pq )( kq ) /q [1 + Π( q )] (cid:26) [ C − C ][1 + Π( q )] H ( k ) − A ( k )[ A ( p ) + A ( k )]Π ′ ( q ) H ( k ) (cid:27) ,D ( p ) = 1 + Z d k (2 π ) q )] (cid:26) [ D − D ][1 + Π( q )] H ( k ) − B ( k )[ A ( p ) + A ( k )]Π ′ ( q ) H ( k ) (cid:27) , Π ′ ( q ) = Nq Z d k (2 π ) Π ′ Π ′ − ′ Π ′ H ( k ) H ( p ) [2 k − k · q ) − k · q ) /q ] , with C ≡ { A ( k ) C ( k ) + A ( p ) C ( k ) + A ( k ) C ( p ) } H ( k ) ,C ≡ A ( k )[ A ( p ) + A ( k )][ A ( k ) C ( k ) k + B ( k ) D ( k )] ,D ≡ { D ( k )[ A ( k ) + A ( p )] + B ( k )[ C ( k ) + C ( p )] } H ( k ) ,D ≡ B ( k )[ A ( p ) + A ( k )][ A ( k ) C ( k ) k + B ( k ) D ( k )] , Π ′ ≡ [ A ( k ) C ( p ) + A ( p ) C ( k )][ A ( k ) + A ( p )] + A ( k ) A ( p )[ C ( k ) + C ( p )] , Π ′ ≡ H ( k ) H ( p ) , Π ′ ≡ A ( k ) A ( p )[ A ( p ) + A ( k )] , Π ′ ≡ [ A ( k ) C ( k ) k + B ( k ) D ( k )] H ( p ) + [ A ( p ) C ( p ) p + B ( p ) D ( p )] H ( k ) , where A, B,
Π are obtained by Eqs. (30-32) in the chirallimit.
C. Chiral susceptibility around N c By application of numerical methods,
A, B,
Πand the functions for the scalar vertex can be ob-tained. The typical behaviors for the six functions A ( p ) , B ( p ) , C ( p ) , D ( p ) and Π( q ) , Π ′ ( q ) are shownin Fig. 1. From Fig. 1 it can be seen that, excepting that A ( p ) and D ( p ) approach 1, the other functions vanishin the large momentum limit and all the six functions arealmost constant in the infrared region. Substituting the above functions into Eq. (8), we im-mediately obtain the value of chiral susceptibility andfermion chiral condensate with a range of fermion flavors.The results are plotted in Fig. 2. From this figure, we seethat, with N increasing, the chiral susceptibility shows anobvious peak in the rainbow approximation and BC ver-tex approximation, while h ¯ ψψ i diminishes and vanishesat a critical number of fermion flavors where CPT occurs.Since each ansatze keeps different symmetry of the sys-tem, N c depends a little on the choice of the ansazte forthe dressed vertex. In addition, we also see that the sus-ceptibility around N c is apparently different from that athigh temperature and high density [17]. The peak showsa neither divergent nor discontinuous behavior which il- A ( p ) p B ( p ) p N=1N=2 Π ( q ) q C ( p ) p D ( p ) p -4 -3 -2 -1 − Π ’ ( q ) q FIG. 1: The behavior of A ( p ) , B ( p ) , C ( p ) , D ( p ) , Π( q )and − Π ′ ( q ) in BC vertex approximation at N = 1 , N N χ c ∆ FIG. 2: The dependence of chiral susceptibility and fermionchiral condensate at zero temperature on N in the rainbowapproximation (upper pannel) and BC vertex approximation(lower pannel), where ∆ = − lg h ¯ ψψ i N h ¯ ψψ i N =0 . lustrates that CPT at N c is neither of first-order nor ofsecond-order and thus is a higher-order continuous phasetransition, which is consistent with the previous works[8, 9]. IV. FINITE TEMPERATURE
With the involved temperature, O (3) symmetry of thesystem reduce to O (2) and the gauge boson acquires anonzero mass. The mass of the photon implies that ex-ternal electric fields are screened by thermal excitations[10] and so we expect that the feature of CPT may bechanged by the excitations. A. Truncated DSE
To give an insight of CPT, we shall adopt DSE for thefermion propagator and techniques of temperature fieldtheory to calculate the chiral and thermal susceptibilityat finite temperature with the increasing N and analyzethe transition of QED near N T c , and also reveal thenature of CPT at the critical temperature T Nc with afixed N .Now, let us give a short review of some studies onthe effect of the wave function renormalization factor A k and A . Just as mentioned in Sec. I, the chiral phasetransition (CPT) in QED was first studied in Ref. [1],where it is found that CPT occurs at N c ≈ .
24. Theyarrived at this conclusion by solving the lowest order DSEfor the fermion self-energy. Later, some groups adoptedimproved schemes for DSE to study this problem andobtained qualitatively similar results with N c ≈ . A, E and the fermion mass function
B, F do not undergo significant changes. From this, we alsoignore the frequency dependence of fermion self-energy B and then the corresponding DSE for the scalar part ofinverse fermion propagator reads [17] F ( P ) = m + 2 T Z d K (2 π ) ∞ X n = −∞ F ( K ) / [ Q + Π( Q )] ̟ n + K + F ( K )= Z d K (2 π ) F ( K ) tanh E k T E k [ Q + Π( Q )] , (33)where E k = p K + F ( K ) and the zero frequency bo-son polarizationΠ( Q ) = N Tπ Z d x ( ln (cid:18) M ( x )2 T (cid:19) − m tanh M ( x )2 T T M ( x ) ) , (34)with M ( x ) = m + x (1 − x ) Q .With the general equation for the chiral susceptibil-ity (8), we can obtain the chiral susceptibility at finitetemperature χ c = 4 N T X n Z d P (2 π ) × (cid:26) [ ̟ n + P − B ( P )] D ( P )[ ̟ n + P + B ( P )] − ̟ n + P (cid:27) = 2 N Z d P (2 π ) × ( D ( P ) E p (cid:20) P E p tanh E p T + B ( P )sech E p T T E p − E p " P E p tanh E p T ) , (35)where E p = √ P . The unknown function in the aboveequation, D ( P ), is obtained by F ( P ) D ( P ) = lim m → ∂F ( P ) ∂m = 1 + Z d K (2 π ) E k [ Q + Π( Q )] × (cid:26)(cid:20) D ( K ) K E k − B ( K )Π ′ ( Q ) Q + Π( Q ) (cid:21) tanh E k T + D ( K ) B ( K )2 T E k sech E k T (cid:27) , (36)with Π ′ ( Q ) = lim m → ∂ Π( Q ) ∂m . From Eq. (34), we easilyfind that Π ′ ( Q ) = 0.Similarly, thermal susceptibility measures the responseof the chiral condensate to an infinitesimal change of tem-perature χ T = ∂ h ¯ ψψ i ∂T = Z d P (2 π ) E p " B ′ ( P ) tanh E p T − B ( P ) B ′ ( P ) tanh E p T E p + B ( P )sech E p T (cid:18) B ′ ( P ) B ( P )2 T E p − E p T (cid:19)(cid:21) (37)with B ′ = ∂B∂T . B. Numerical results
After solving the above coupled DSEs by means of theiteration method, we can now calculate chiral fermion condensate and the above two susceptibilities, which canbe regarded as a function of N given by Eq. (35) and(37) with a range of temperature. The typical behaviorsof the susceptibilities and condensate are shown in Fig.3 and Fig. 4. The upper lines of Fig. 3 give the behavior χ cR χ TR < ΨΨ > N FIG. 3: The behaviors of chiral susceptibility and thermalsusceptibility around the critical fermion flavors with several T (where χ cR = χ c N , χ TR = − χ T , from left to right denote T = 2 . × − , − , − ). of chiral susceptibility and the lower lines in this figureshow the fermion chiral condensate, while the other linesbetween the two group denote that of thermal suscepti-bility. As is shown in Fig. 3, for any given temperature, χ c almost keeps a constant for small and large N , whileit shows an apparent peak at some critical number offermion flavors. This number depends on the tempera-ture and diminishes as the temperature increases. When N reaches a critical value N T c , the appearance of van-ishing fermion chiral condensate and divergence peak of χ T occur at the same point. This critical fermion flavorsalso decreases with the increase of T , which is similarto the results in the previous works [10, 28, 29]. By allappearances, at any temperature the peak of each sus-ceptibility lies at N T c . Moreover, near N c , the chiralsusceptibility at finite temperature shows a different be-havior from that at zero temperature. From Fig. 3, wealso see that the chiral and thermal susceptibilities ex-hibits a very narrow, pronounced and in fact divergentpeak at N T c , which is a typical characteristic of second-order phase transition driven by the restoration of chiralsymmetry at finite temperature.For a fixed N , with the increasing temperature, chiralsymmetry restores at a temperature T Nc and each sus-ceptibility exhibits the same behavior around the criticalpoint. From Fig. 4, we see that the chiral and thermalsusceptibility reveal their infinite value at T Nc which alsoillustrate the typical second-order phase transition. T χ cR χ TR < ΨΨ > FIG. 4: The behaviors of chiral and thermal susceptibilitiesaround the critical temperature with several N (from left toright denote N = 3 , , C. Critical exponents
Just as shown above, the chiral phase transition at fi-nite temperature is second order, a natural question is:what are the critical exponents? Now, let us try to an-swer this question. Around the critical points, the phasetransitions are characterized by the corresponding criti-cal exponents which are an important contemporary goalto exhibit the feature of CPT. We find that the fermionchiral condensate near the critical point reveals h ¯ ψψ i ∼ t α , N = const, h ¯ ψψ i ∼ n β , T = const, (38)with the reduced temperature t = 1 − T /T c and the re-duced fermion flavors number n = 1 − N/N c . The typicalbehavior of the condensate near the point of CPT can beseen in Fig. 5 and find that, in each figure, the slopeof the line of infrared fermion self-energy B (0) is sameto that of h ¯ ψψ i which indicates that B (0) and h ¯ ψψ i il-lustrate the same value of critical exponent in masslessQED .Numerically, for a fixed N , the estimated α at t → + and also the estimated β at n → + with several T aregiven as N α T β − − t, n → + reveals its criticalfeature as χ c ∼ t − γ c , N = const,χ T ∼ t − γ T , N = const,χ c ∼ n − δ c , T = const,χ T ∼ n − δ T , T = const, -5.5 -5.0 -4.5 -4.0 -3.5 -3.0-7-6-5-4-3 ln(B(0))ln(< ψψ >) ln(t) -11.5 -11.0 -10.5 -10.0 -9.5 -9.0 -8.5-9.0-8.5-8.0-7.5-7.0-6.5-6.0 ln(n) ln(B(0)) ln(< ψψ >) FIG. 5: The critical behavior of h ¯ ψψ i and B (0) near the pointof CPT with a range of t at N = 1 (top) and a range of n at T = 0 .
025 (bottom). and their critical behaviors can be found in Fig. 6.From the numerical results, we estimate the criticalexponents of the susceptibility with several N or T andgive γ, δ in the following table:N γ c γ T T δ c δ T − − χ c is larger than that of χ T . V. CONCLUSIONS
The primary goal of this paper is to investigate the na-ture of chiral phase transition of QED near the criticalvalue, including critical number of fermion flavors andcritical temperature through a continuum study of thechiral and thermal susceptibilities. Based on the suit-able approximation of truncated DSEs for the fermionpropagator and numerical model calculations, we studythe behavior of the two susceptibilities near the criticalpoint of CPT in QED . It is found that, with the riseof the number of fermion flavors, the appearance of the -6.0 -5.5 -5.0 -4.5 -4.0 -3.5 -3.0-5-4-3-2-10123 ln(t) -ln| χ Τ |-ln( χ c ) -6.0 -5.5 -5.0 -4.5 -4.0 -3.5 -3.0-6-5-4-3-2-10123 -ln| χ Τ |-ln( χ c ) ln(n) FIG. 6: The critical behavior of two susceptibilities near thepoint of CPT with a range of t at N = 1 (top) and a range of n at T = 0 .
025 (bottom). peak of chiral susceptibility and CPT occur at the samecritical point, but the peak reveals apparently differentbehavior at zero and finite temperature.At zero temperature the chiral susceptibility near thecritical number of fermion flavors reveals a finite and con-tinuous peak, which exhibits that CPT is neither of firstorder nor of second order, and thus it should be a continu-ous phase transition of higher order. However, apart fromzero temperature, each of chiral and thermal susceptibil-ity at either critical fermion flavors or chiral temperatureshows a large and in face divergent peak which illustratesa typical characteristic of second-order phase transitiondriven by chiral symmetry restoration in thermal QED .Finally, though the analysis for the critical exponents,it is found that the critical exponents of chiral/thermalsusceptibility which characterize the chiral phase tran-sition is between 0.2 and 1 and, in the same boundarycondition, the critical exponent of thermal susceptibilityis less than that of chiral susceptibility. VI. ACKNOWLEDGEMENTS
This work was supported in part by the National Nat-ural Science Foundation of China (under Grant Nos.11105029, 11275097, and 11347212) and the ResearchFund for the Doctoral Program of Higher Education (un-der Grant No 2012009111002) and by the FundamentalResearch Funds for the Central Universities (under GrantNo 2242014R30011). [1] T. Appelquist, D. Nash, and L.C.R. Wijewardhana,Phys. Rev. Lett. , 2575 (1988).[2] D. Nash, Phys. Rev. Lett. , 3024 (1989).[3] D.C. Curtis, M.R. Pennington and D. Walsh, Phys. Lett.B , 313 (1992).[4] M.R. Pennington and D. Walsh, Phys. Lett. B , 246(1991).[5] P. Maris, Phys. Rev. D , 4049 (1996).[6] C.S. Fischer, R. Alkofer, T. Dahm, and P. Maris, phys.Rev. D , 073007 (2004).[7] A. Bashir, C. Calcaneo-Roldan, L.X. Gutierrez-Guerrero,and M.E. Tejeda-Yeomans, Phys. Rev. D , 033003(2011).[8] T. Appelquist, J. Terning, and L.C.R. Wijewardhana,Phys. Rev. Lett. , 2081 (1995).[9] H.T. Feng, B. Wang, W.M. Sun, and H.S. Zong, Phys.Rev. D , 105042 (2012).[10] N. Dorey and N.E. Mavromatos, Phys. Lett. B , 163(1991).[11] F. Karsch and E. Laermann, Phys. Rev. D , 6954(1994) .[12] M. Cheng et al., Phys. Rev. D ,034505 (2007).[14] M. He, Y. Jiang, W. M. Sun, and H. S. Zong, Phys. Rev.D , 076008 (2008). [15] L. Chang, Y. X. Liu, C. D. Roberts, Y. M. Shi, W. M.Sun, and H. S. Zong, Phys. Rev. C , 035209 (2009).[16] M. He, F. Hu, W. M. Sun, and H. S. Zong, Phys. Lett.B , 32 (2009).[17] H.T. Feng, S. Shi, P.L. Yin, and H.S. Zong, Phys. Rev.D , 065002 (2012).[18] H.T. Feng, B. Wang, W.M. Sun, and H.S. Zong, Eur.Phys. J. C , 2444(2013).[19] Y. Zhao, L. Chang, W. Yuan, and Y.X. Liu, Eur. phys.J. C , 483 (2008).[20] A. H¨oll, P. Maris, and C.D. Roberts, Phys. Rev. C ,1751 (1999).[21] C.S. Fisher, J. Luecker, and J.A. Muller, Phys. lett. B , 438 (2011).[22] A. Bashir, A. Raya, I.C. Cloet, and C.D. Roberts, Phys.Rev. C , 055201 (2008).[23] H.T. Feng, W.M. Sun, F. Hu, and H.S. Zong, Inter. J.Mod. Phys. A20 , 2753 (2005).[24] J. S. Ball and T. W. Chiu, Phys. Rev. D , 4165(1990).[26] A. Ayala and A. Bashir, Phys. Rev. D , 025015 (2001).[27] A. Bashir and A. Raya, Phys. Rev. D , 105001 (2001).[28] N. Dorey and N.E. Mavromatos, Nucl. Phys. B386 , 614(1992).[29] I.J.R. Aitchison and M. Klein-Kreisler, Phys. Rev. D ,67