Temperature Dependence of the Effective Bag Constant and the Radius of a Nucleon in the Global Color Symmetry Model of QCD
aa r X i v : . [ h e p - ph ] S e p Temperature Dependence of the Effective Bag Constant and the Radius of a Nucleonin the Global Color Symmetry Model of QCD
Yuan Mo, Si-xue Qin, and Yu-xin Liu
1, 2, ∗ Department of Physics and State Key Laboratory of Nuclear Physics and Technology, Peking University, Beijing 100871, China Center of Theoretical Nuclear Physics, National Laboratory of Heavy Ion Accelerator, Lanzhou 730000, China
We study the temperature dependence of the effective bag constant, the mass, and the radiusof a nucleon in the formalism of the simple global color symmetry model in the Dyson-Schwingerequation approach of QCD with a Gaussian-type effective gluon propagator. We obtain that, asthe temperature is lower than a critical value, the effective bag constant and the mass decreaseand the radius increases with the temperature increasing. As the critical temperature is reached,the effective bag constant and the mass vanish and the radius tends to infinity. At the same time,the chiral quark condensate disappears. These phenomena indicate that the deconfinement and thechiral symmetry restoration phase transitions can take place at high temperature. The dependenceof the critical temperature on the interaction strength parameter in the effective gluon propagatorof the approach is given.
PACS numbers: 14.20.Dh, 12.40.Yx, 11.15.Tk, 25.75.Nq
I. INTRODUCTION
The phase transitions of quantum chromodynamics(QCD), for example the evolution between chiral sym-metry breaking and its restoration, the color (or sim-ply quark ) confinement and deconfinement, have beenthe most active topics in nuclear and particle physics inrecent years [1]. Even though recent investigation hasprovided hints that the phase transitions can be drivenby the intrinsic characteristics, such as the running cou-pling strength and the current quark mass, of the system(see, for example, Refs. [2–6]), the more promising andmuch better believed is that the QCD may undergo phasetransitions into a chirally symmetric and color deconfinedphase at high temperature and/or density [7].To demonstrate the phase transitions, one usually im-plements the variation behaviors of not only the fea-tures of QCD vacuum and the strong interaction mat-ter but also the properties of hadrons at finite temper-ature and/or density. On theoretical side, one needsin principle QCD, which has been widely accepted asthe fundamental theory of strong interaction, to carryout the investigation. However, as a basic theory, QCDstill suffers from difficulties in the low energy region,which relates directly to strong interaction matter andhadrons. Then, besides the approaches of lattice simu-lations, QCD sum rules, instanton model(s) and Dyson-Schwinger equations and several models, such as the bagmodel [8], quark-meson coupling model (QMC) [9–11],Nambu-Jona-Lasinio (NJL) model [12], Polyakov-loopimproved NJL model [13], global color symmetry model(GCM) [14], and so on, have been developed. The NJLmodel has been widely used since it preserves the fea-ture of chiral symmetry and its dynamical breaking and ∗ Corresponding author: [email protected] is easy to carry out numerical calculation. Even though,with the Polyakov-loop improvement, the quark confine-ment effect is included at statistical level, the commonlyaccepted one still only takes into account the point (con-tact) interactions among quarks. The bag model is theone which handles hadrons as bubbles of perturbativevacuum immersed in the physical vacuum. However, allnonperturbative physics is included in a quantity—bagconstant, which is dealt with a phenomenological param-eter in the model. And the QMC model involves the sim-ilar problem. For the GCM, since it can take the resultof the Dyson-Schwinger equation (DSE) [15] approach asinput, it manifests well the properties of chiral symmetryand its dynamical breaking. Because the bag constant inthe model is taken as the difference between the energydensities of the perturbative and the physical vacuums,the color confinement effect is also handled well in somesense. The GCM is then believed to be a quite sophis-ticated model which involves as many characteristics ofQCD as possible. And the NJL model, the QMC model,and the bag model can be treated as the special cases ofthe GCM.Due to its solid QCD foundation, the GCM has beenwidely taken to study not only the properties of nucleonand some mesons [14, 16–18] but also the QCD vacuumstructure [19, 20]. It has also been extended to investi-gate the properties of strong interaction matter at finitetemperature and/or density and those of some hadronsin the matter [21–26]. For the property of nucleon infinite density matter, it has been discussed with vari-ous models for the effective gluon propagator in the DSEand the variation behaviors of the mass, the radius andthe bag constant of the nucleon have been given explic-itly [23]. However, in the case of finite temperature, onlythe changing feature of the bag constant has been dis-cussed with the Munczek-Nemirovsky model [27] for theeffective gluon propagator of the DSE [22]. We will then,in this paper, discuss some of the properties of a nucleonat finite temperature with a sophisticated effective gluonpropagator in the DSE.The paper is organized as follows. In Sec. II, we de-scribe briefly the formalism of the GCM soliton model.In Sec. III, we describe the algorithm to carry out thenumerical calculation of the GCM soliton at finite tem-perature and the obtained results. Finally we give a briefsummary and some remarks in Sec. IV.
II. BRIEF DESCRIPTION OF GCM
The original action in the global color symmetry model(GCM) defined in Euclidean metric is expressed as [14] S GCM (¯ q, q ) = Z d x ¯ q ( x )( iγ · p + m ) q ( x )+ g Z Z d xd yj aµ ( x ) D abµν ( x − y ) j bν ( y ) , (1)where j aµ ( x ) = ¯ q ( x ) λ µ γ µ q ( x ) is the local quark current, D abµν ( x − y ) is the full gluon propagator, m is the cur-rent quark mass, g is the quark-gluon coupling constant.The Euclidean metric is such that a · b = a µ b µ , and { γ µ , γ ν } = 2 δ µν . Taking the gluon propagator to be colordiagonal in the Feynman-like gauge, i.e., D ab µν ( x − y ) = δ ab δ µν D ( x − y ), and applying the Fierz transformationto reorder the quark fields, one can rewrite the action as S GCM [ B θ ( x, y ))] = Z Z d xd y ¯ q ( x )[( iγ · p + m ) δ ( x − y )+Λ θ B θ ( x, y )] q ( x )+ Z Z d xd y B θ ( x, y ) B θ ( y, x )2 g D ( x − y ) , (2)where { Λ θ } are direct products of Lorentz, flavor andcolor matrices of quarks which produce the scalar, vector, and pseudoscalar terms labeled by θ . B θ ( x, y ) are bilocalBose fields. Theoretically, it can be proved that the GCMis valid in any gauge even though one takes the Feynman-like gauge in deriving the above expression [28].By integrating the quark fields, one gets the action S GCM [ B θ ( x, y ))] = − T r ln G − [ B θ ( x, y )]+ Z Z d xd y B θ ( x, y ) B θ ( y, x )2 g D ( x − y ) , (3)where the inverse of the quark propagator can be writtenas G − ( x, y ) = ( iγ · p + m ) δ ( x − y ) + Λ θ B θ ( x, y ) . (4)Generally, the bilocal fields can be expanded as B θ ( x, y ) = B θ ( x, y ) + X i Γ θ ( x, y ) φ θi (cid:18) x + y (cid:19) , (5)where the first term is the translation invariant vacuumconfiguration. The second term stands for the fluctua-tions of the vacuum which can be identified as effectivemeson fields since the θ stands for the quantum num-ber of Bose fields. In the lowest order, one takes theGoldstone mode, φ θ = { σ, ~π } , which is thought of as themost important low energy degree of freedom. The vac-uum configuration can be determined by the saddle pointcondition δS GCM /δB θ = 0. One has then B θ ( x, y ) = g D ( x − y ) tr [ G ( y, x )Λ θ ] , (6)and the quark self-energy Σ( x − y ) = Λ θ B θ ( x, y ). Theequation of quark self-energy in momentum space coin-cides with that of the truncated Dyson-Schwinger equa-tion (DSE)Σ( p ) = Z d x Λ θ B θ ( x, y ) e iq · x = g Z d q (2 π ) t µν D ( p − q ) λ a γ µ iγ · ( q + m ) + Σ( q ) γ µ λ a , (7)where t µν = δ µν − k µ k ν /k , with k = p − q , γ µ is the colorSU(3) matrix. Generally, the quark self-energy functioncan be decomposed asΣ( p ) = S − ( p ) − S − ( p ) ,S − ( p ) = iγ · p A ( p ) + B ( p ) ,S − ( p ) = iγ · p + m , (8)and A ( p ), B ( p ) are scalar functions of p . Recalling the configuration of the bilocal fields inEq. (5) and considering the Bethe-Salpeter amplitude ofthe mesons and the partial conservation of axial-vectorcurrent, one can prove [29] that, when considering themost important low energy degree of freedom, i.e., theGoldstone mode φ θ = { σ, ~π } , Eq. (5) can be rewritten asΛ θ [ B θ ( x, y ) − B θ ( x, y )] = B ( x − y ) (cid:20) σ ( x + y iγ ~τ · ~π ( x + y (cid:21) , (9)where B is just the scalar part of the inverse of thequark propagator which can be determined by solvingthe quark’s DSE.With a nontopological-soliton ans¨atz[30], the action ofthe GCM soliton with quarks in chiral limit ( m = 0)can be given [14, 17, 18] as S GCM = q { iγ · p − α [ σ ( x ) + i~π ( x ) · ~τ γ ] } q + Z [ f σ ∂ µ σ ) + f π ∂ µ ~π ) − V ( σ, π )] d z , (10)with V ( σ, π ) ≈ − Z d p (2 π ) (cid:26) ln (cid:20) A ( p ) p + ( σ + ~π ) B ( p ) A ( p ) p + B ( p ) (cid:21) − ( σ + ~π − B ( p ) A ( p ) p + B ( p ) (cid:27) , (11)and the quark meson coupling constant α is given as α ( x ) = Z d p (2 π ) B ( p ) e − ip · x . It is evident that such a quark meson coupling constantis just the vacuum configuration of the bilocal fields andis independent of the meson fields.With the stationary condition of the soliton, one hasthe equations of motion for the quarks and mesons as { iγ · p − α [ σ ( x ) + i~π ( x ) · ~τ γ ] } q = 0 , (12) − ~ ∇ σ ( ~r ) + δVδσ ( ~r ) + Q σ ( ~r ) = 0 , (13) − ~ ∇ ~π ( ~r ) + δVδ~π ( ~r ) + Q ~π ( ~r ) = 0 , (14)where Q σ and Q ~π are the source terms contributed fromthe valence quarks, and can be written as Q σ ( ~R ) = X j =1 Z j Z d xd y ¯ u j ( ~x ) B ( ~x − ~y ) × δ ( ~x + ~y − ~R ) u j ( ~y ) , (15) Q ~π ( ~R ) = X j =1 Z j Z d xd y ¯ u j ( ~x ) B ( ~x − ~y ) iγ ~τδ ( ~x + ~y − ~R ) u j ( ~y ) , (16)with Z j being the renormalization constant [17] Z j = − Z d pd q ¯ u j ( ~p ) ∂G − ( iǫ j ; ~p, ~q ) ∂ǫ j u j ( ~q ) . (17)The quark field and σ , π meson fields can be deter-mined by solving Eqs. (12)-(14) self-consistently. As aconsequence, the corresponding eigenenergies can be ob-tained. It is apparent that the meson fields correspond-ing to the vacuum configuration can be simply taken as σ = 1, π = 0 due to the (normalized with f π ) restriction π + σ = 1. The vacuum configuration is a minimum of V ( σ, π ) and V (1 ,
0) = 0. In light of the nontopological-soliton ans¨atz[14, 30], one can approximate the soliton asa chiral bag with bag constant B = V ( σ, ~π ) − V (1 ,
0) = 12 Z d p (2 π ) (cid:26) ln (cid:20) A ( p ) p + B ( p ) A ( p ) p +( σ + ~π ) B ( p ) (cid:21) + ( σ + ~π − B ( p ) A ( p ) p + B ( p ) (cid:27) . (18)With the correction from the motion of center of mass,the zero-point effect, and the color-electronic and color-magnetic interactions being taken into account, the totalenergy of a bag (involving three valence quarks) is givenas E B ( R ) = 3 ε j ( R ) + 43 πR B − Z R , (19)where ε j ( R ) is the energy eigenvalue of the quark’s equa-tion of motion, R is the radius of the bag, Z /R denotes the contribution of the corrections of the motion of thecenter of mass, zero-point energy, and other effects with Z being a parameter. Just the same as that in Ref. [14],the bag is identified as a nucleon in the present work.With the equilibrium condition dE B ( R ) dR = 0 , we can obtain the radius of a nucleon. III. ALGORITHM AND NUMERICAL RESULTSA. Algorithm
From the description in last section, we know that theproperty of a nucleon (for instance, its mass, radius andbag constant) is determined by the solutions of the equa- tions of motion of the quarks and (chiral) mesons in thesoliton. To solve the equations of motion, one needsthe solutions A ( p ) and B ( p ) of the quark’s Dyson-Schwinger equation. Then after solving the quark’s DSEin Eq. (7), or more explicitly [with the help of the de-composition in Eq. (8)] the coupled equations( A ( p ) − p = g C F Z d q (2 π ) D ( p − q ) tr [ iγ · p t µν γ µ S ( q )Γ ν ( p, q )] ,B ( p ) − m = g C F Z d q (2 π ) D ( p − q ) tr [ t µν γ µ S ( q )Γ ν ( p, q )] , (20)where C F is the eigenvalue of the quadratic Casimir oper-ator in the fundamental representation of the color sym-metry group [for SU ( N c ), C F = ( N c − / N c , it reads4 / N c = 3].To investigate the temperature dependence of theproperty of a nucleon, one should at first discuss the formof the quark’s DSE at finite temperature.It has been well known that the appearance of(nonzero) temperature T in the QCD reduces the O (4)symmetry to O (3). Thus the quark’s four-momentum p should be rewritten as p = ( ~p, ω n ), where ω n =(2 n + 1) πT ( n ∈ Z ) are the discrete Matsubara frequen-cies of the quark, and the four-dimensional integral needsto be replaced by [31] Z d p (2 π ) → T ∞ X n = −∞ Z d p (2 π ) . (21)The decomposition of the dressed quark propagator needsto be rewritten as S − ( ~p, ω n ) = i~γ · ~pA ( | ~p | , ω n ) + iγ ω n C ( | ~p | , ω n ) + B ( | ~p | , ω n ) . (22) Furthermore, the gluon propagator at finite temperaturecan be generally expressed as [31] D µν ( ~k, ω n ) = D T ( ~k, ω n ) k P Tµν ( k ) + D L ( ~k, ω n ) k P Lµν ( k ) , (23)with the transverse and longitudinal projectors P Tµν ( k ) = (cid:26) (cid:0) δ ij − k i k j ~k (cid:1) δ iµ δ jν , µ, ν = 1 , , , , µ or/and ν = 4 ,P Lµν ( k ) = (cid:18) δ µν − k µ k ν k (cid:19) − P Tµν ( k ) . (24)Due to the lack of detailed information about the gluonpropagator at finite temperature, one usually na¨ıvely as-sumes that the transverse and longitudinal parts of thegluon propagator are equal and independent of temper-ature (see, for example, Ref. [32]), i.e., one has approxi-mately D T ( ~k, ω n ) = D L ( ~k, ω n ) = D ( k ). In this assump-tion and bare vertex approximation, we get three coupledintegral equations for the functions A ( ~p, ω n ), C ( ~p, ω n )and B ( ~p, ω n ) as A ( | ~p | , ω n ) = 1 + g T C F ~p X m Z d ~q (2 π ) D ( k ) k n [( ~p · ~q ) k + 2( ~p · ~k )( ~q · ~k )] A ( | ~q | , ω m ) + 2 ω m Ω k ( ~p · ~k ) C ( | ~q | , ω m ) o ,C ( | ~p | , ω n ) = 1 + g T C F ω n X m Z d ~q (2 π ) D ( k ) k n [ ω n ω m k + 2 ω n ω m Ω k ] C ( | ~q | , ω m ) + 2 ω n Ω k ( ~p · ~k ) A ( | ~q | , ω m ) o ,B ( | ~p | , ω n ) = m + g T C F X m Z d ~q (2 π ) D ( k )∆ 3 B ( | ~q | , ω m ) , (25)where k ≡ ~k + Ω k , Ω k ≡ ω n − ω m , and ∆ = ~q A + ω m C + B . As mentioned in last section, in view of the non-topological soliton ans¨atz, one can take a nucleon as asoliton bag. In the chiral limit ( m = 0), one has foundthat there exist two types of solutions for the quark’sDSE. One is the Nanmbu-Goldstone solution which cor-responds to the chiral symmetry spontaneously brokenphase. The other is the Wigner solution which represents the state with the chiral symmetry. One gets then theeffective bag constant as the pressure difference betweenthe Nambu-Goldstone solution and the Wigner solution,which reads B ( T ) ≡ P [ G NG ] − P [ G W ]= 4 N c X m Z d p (2 π ) (cid:26) ln (cid:20) ∆ NG ∆ W (cid:21) + ~p A NG + ω m C NG ∆ NG − ~p A W + ω m C W ∆ W (cid:27) , (26)where ∆ NG ≡ ~p A NG + ω m C NG + B NG , ∆ W ≡ ~p A W + ω m C W , and A NG , C NG , B NG , A W , C W denotes theNambu-Goldstone solution, the Wigner solution for theDSE, respectively. With the solutions of the DSE as input, we can havethe explicit expression of the quark’s equation of motionat finite temperature as[ i~γ · ~pA ( ~p, T )+ iγ ω n C ( ~p, T )+ B ( ~p, T )] u j ( ~p, T )+ Z d k (2 π ) B (cid:18) p + k , T (cid:19) h ˆ σ ( ~p − ~k ) + iγ ~τ · ~π ( ~p − ~k ) i u j ( ~k, T ) = 0 , (27)where ˆ σ ≡ σ −
1. After solving the related eigenequationwe can obtain the eigenenergy of the quark at finite tem-perature, ε j ( T ). One can in turn study the properties ofa nucleon at finite temperature by extending the GCMsoliton model described in last section.In the case of zero temperature, one usually takes onlythe lowest energy for the ε j ( R ) in Eq. (19). At finite-temperature, due to the influence of temperature, thecontribution of the excited states of the quarks must beincluded. Then the energy of the bag, i.e., the mass of anucleon, at finite temperature T should be written as M ( T ) = E B ( T ) = 3 ε j ( T ) − Z R + 43 πR B ( T ) , (28)where ε j ( T ) is the average of a quark’s energies at allpossible states.In the spirit of the most simple approximation of theGCM [14] (mentioned at the end of the last section),the quarks in the bag (GCM soliton) at finite tempera-ture can also be regarded as the free one which satisfiesthe Dirac equation, and the energy eigenvalue can be ex-pressed as ε j ( T ) = κ j R ( T ) , (29)where j denotes the quantum number labeling the energylevel. Since quarks are fermions, we take the Fermi-Diracstatistics to evaluate the average energy of each quark in the bag at finite temperature, and have ε j ( T ) = N ∞ X j =0 ε j ( T )1 + e ε j ( T ) /T , (30)where N is the degeneracy of quarks.Then, by solving the stability condition dM ( T ) dR = 0,i.e., dE B ( T ) dR = 3 dε j ( T ) dR + Z R + 4 πR B ( T ) = 0 , (31)we can obtain the (stable) nucleon radius R ( T ) and themass of a nucleon (the energy of the bag) M ( T ).It is apparent that after solving the quark’s DSE andin turn the equations of motion of the quark and mesonfields at zero and nonzero temperature, we can obtainthe bag constant, the mass and the radius of a nucleonin the corresponding circumstance and discuss the vari-ation characteristic of the property with respect to tem-perature. To solve the quark’s DSE, we take a simplifiedform of the effective gluon propagator in Ref. [33] g D ( k ) = 4 π D k ω exp (cid:18) − k ω (cid:19) , (32)where D and ω are dimensional parameters that canbe determined by fitting empirical date. Such an ef-fective gluon propagator is naturally an extension ofthe Munczek-Nemirovsky model [27] and consistent withthose given in lattice QCD calculations (see, for instance,Refs. [34, 35]) and solving the coupled DSEs of quark,gluon, and ghost (see, for instance, Ref. [36]). It has alsobeen shown to be successful in describing many hadronproperties [15, 33, 37–39]. B. Calculation and Numerical Results
We first solve the quark’s DSE at chiral limit ( m = 0)and zero temperature with parameters D = 1 . and ω = 0 . p [GeV ] S o l u t i on o f D SE A(p )B(p )M(p )10 −2 −4 FIG. 1: (Color online) Calculated result of the Nambu-Goldstone solution of the quark’s DSE at zero tempera-ture, with parameters in the effective gluon propagator ω =0 . D = 1 GeV . We then solve the quark’s DSE at nonzero temper-ature with the same effective gluon propagator. Fig-ure 2 illustrates the obtained results of the functions A ( | ~p | , T ), C ( | ~p | , T ), B ( | ~p | , T ) in the Nambu-Goldstonesolution and the mass function M ( | ~p | , T ) at a tempera-ture T = 30 MeV, as an example of those at nonzero tem-perature. We find from the figure that, just as expected,as the temperature is lower, the functions A ( | ~p | , T ), C ( | ~p | , T ), B ( | ~p | , T ) and M ( | ~p | , T ) have the correct zerotemperature limit. Hence, even though we have not in-cluded explicitly the temperature effect in the effectivegluon propagator, the calculated result can demonstratethe temperature dependence of quark propagator withquite high precision. In addition, it can be noticed fromFig. 2 that functions A ( | ~p | , ω ) and C ( | ~p | , ω ) have thesame behavior when the temperature is low, just as thatof the quark propagator at low chemical potential [40].To study the Matsubara frequency dependence of thequark propagator, we illustrate the variation behaviors S o l u t on o f D SE C(|p|, ω )A(|p|, ω )B(|p|, ω )M(|p|, ω )0 |p| [GeV] −2 −1 FIG. 2: (Color online) Calculated results of the functions A ( | ~p | , T ), C ( | ~p | , T ), B ( | ~p | , T ) and M ( | ~p | , T ) in the Nambu-Goldstone solution of the quark propagator at a temperature T = 30 MeV, with parameters in the effective gluon propaga-tor ω = 0 . D = 1 GeV . of the functions A ( | ~p | , ω n ) and B ( | ~p | , ω n ) at a tempera-ture T = 30 MeV in Fig. 3 [since Fig. 2 has shown thatthe functions A ( | ~p | , ω n ) and C ( | ~p | , ω n ) at T = 30 MeVare almost exactly equal to each other, we do not showthe function C ( | ~p | , ω n )] as a representative. From Fig. 3,we can find that, when the temperature is low, both func-tions A and B involve an approximate bilateral symme-try about n [since for a fixed temperature, ω n is propor-tional to n due to the definition ω n = (2 n + 1) πT ], andthe function B decreases rapidly as the Matsubara fre-quency increases. It gives us a posterior knowledge thatwhen we carry out the summation of the Matsubara fre-quencies ω n , it is not necessary to do that up to a verylarge number of n .To demonstrate the temperature dependence of thequark propagators explicitly, we illustrate the calculatedvariation behaviors of the Wigner solution at zero mo-mentum and zero mode Matsubara frequency [i.e., thefunctions A W ( | ~p | = 0 , ω ) and C W ( | ~p | = 0 , ω )] and theNambu-Goldstone solution under the same conditions[i.e., the functions A NG ( | ~p | = 0 , ω ), C NG ( | ~p | = 0 , ω ) and B NG ( | ~p | = 0 , ω )] with respect to temperature in Fig. 4as a representative. From Fig. 4, we can find that, whenthe temperature is low, the functions A NG and C NG co-incide with each other very well. So do the A W and C W except that the temperature for the deviation be-tween them to appear is lower. It indicates that only asthe temperature is quite high, the breaking from O(4)symmetry to O(3) symmetry (especially, for the physicalstate, i.e., the Nambu-Goldstone state) becomes obvious.Moreover, the decreasing feature of the function B NG is amanifestation of gradual restoration of chiral symmetry.With the solutions of the quark’s DSE as input, wesolve the equations of motion of the quark and the chiralmeson fields, and then obtain the property of a nucleon −20 −10 0 10 2011.21.41.5 n A ( , ω n ) −20 −10 0 10 2000.20.40.6 n B ( , ω n ) −20−10 0 10 2011.21.41.5 A ( | p | , ω n ) −20 0 20−10 1000.20.40.6 B ( | p | , ω n ) (d)(a) (b)(c) 10 −2 n n −2 |p| [GeV] |p| [GeV] FIG. 3: (Color online) Calculated results of the functions A ( | ~p | , ω n ) and B ( | ~p | , ω n ), where ω n = (2 n + 1) πT , in theNambu-Goldstone solution of the quark propagator in chirallimit and at a temperature T = 30 MeV [(a), (b), respectively]and the special case at | ~p | = 0 [(c), (d), respectively]. The cal-culations are also carried out with parameters ω = 0 . D = 1 GeV in the effective gluon propagator. at zero and nonzero temperature. The obtained prop-erty of a nucleon at zero temperature is bag constant B (0) = (162 MeV) , mass M (0) = 939 MeV (with theparameter Z is fixed as 3.08), and radius R (0) = 0 .
85 fm.It should be noted that the presently fixed value of theparameter Z , 3 .
08, is larger than the usually taken one,1 .
84. Such a large value arises from the fact that theterm − Z /R is a combination of the contributions fromnot only the zero-point energy but also those of the color-electronic and color-magnetic interactions, the correctionon the motion of center-of-mass and other effects. And itis consistent with the most recent result [41] and our pre-vious results [23]. The gained variation behaviors of thenucleon’s bag constant, mass, and radius with respectto temperature are illustrated in Figs. 5, 6, 7, respec-tively. From Figs. 5–7, one can notice that, with theincreasing of temperature if it is below a critical one,the bag constant and the mass of a nucleon decrease,and the radius of the nucleon increases. At critical tem-perature T = 133 MeV, the bag constant and the massof the nucleon vanish and the radius tends to be infi-nite. It manifests that the nucleon can no longer existas a bag soliton, so that the quark deconfinement hap-pens. Admittedly, such a obtained critical temperaturemay be model-dependent, however, the gradual variationfeatures of the nucleon’s property indicate that the de-confinement phase transition process is that, with the in-crease of temperature, the nucleons in the matter toucheswith each other at first due to the increase of the radius,then the fields of the ingredients of the nucleons mixedwith each other and the bound strength gets weaker si-multaneously. As the bound (the bag constant) vanishes,the deconfinement phase transition occurs. Therefore, T [GeV] S o l u t i on s o f D SE A W (0, ω )C W (0, ω )A NG (0, ω )C NG (0, ω )B NG (0, ω )(a)(b) FIG. 4: (Color online) Calculated results of the temperaturedependence of the Wigner solution of the quark’s DSE atzero momentum and zero mode Matsubara frequency [i.e.,the functions A W ( | ~p | = 0 , ω ) and C W ( | ~p | = 0 , ω )] (panel(a)) and the Nambu-Goldstone solution under the same con-ditions [i.e., the functions A NG ( | ~p | = 0 , ω ), C NG ( | ~p | = 0 , ω )and B NG ( | ~p | = 0 , ω )] (panel (b)). The calculations are alsocarried out with parameters ω = 0 . D = 1 GeV in theeffective gluon propagator. T [GeV] B c ( T ) / B c ( ) FIG. 5: Calculated result of the variation behavior of the bagconstant of a nucleon with respect to temperature. the temperature driven deconfinement process may be,in fact, a crossover but not a low order phase transition.When discussing the QCD phase transition, one usu-ally interests in the chiral symmetry and its dynamicalbreaking, too, and takes the chiral quark condensate asa order parameter in the case of chiral limit, which is
T [GeV] M [ G e V ] FIG. 6: Calculated result of the variation behavior of the massof a nucleon with respect to temperature.
T [GeV] R [f m ] FIG. 7: Calculated result of the variation behavior of theradius of a nucleon with respect to temperature. defined as − h ¯ qq i = N c T n =+ ∞ X n = −∞ Z d p (2 π ) tr [ S ( ~p, ω n )] . (33)We then calculate the temperature dependence of thechiral quark condensate. The obtained result is shownin Fig. 8. The figure displays evidently that the chiralquark condensate decreases gradually with the increaseof temperature and vanishes at a critical temperature. Itindicates that dynamical chiral symmetry breaking effectgets weaker and weaker with the increase of temperatureand the chiral symmetry can be restored as the tempera-ture reaches the critical one. In the case with parameters ω = 0 . D = 1 GeV in the effective gluon propa-gator, the critical temperature for the chiral symmetryto be restored is also approximately 133 MeV, the sameas that for the deconfinement. T [GeV] −< q q > / [ G e V ] FIG. 8: Calculated result of the variation behavior of thechiral quark condensate with respect to temperature. D’ T ’ c FIG. 9: Calculated result of the variation behavior of thescaled critical temperature with respect to the scaled strengthparameter D ′ in the effective gluon propagator, where T ′ c = T c /ω and D ′ = D/ω are the scaled quantities. As mentioned above, the critical temperature for theQCD phase transitions to happen may be parameter de-pendent. To demonstrate the parameter dependence ex-plicitly, we carry out a series calculations with variousvalues of the coupling strength parameter D and thescreening width parameter ω in the effective gluon prop-agator. Due to the good behavior of the Gaussian-typegluon propagator, we can scale the T c and D by ω withdefinition T ′ c ≡ T c /ω and D ′ ≡ D/ω . The obtainedvariation behavior of the T ′ c with respect to the D ′ isdisplayed in Fig. 9. One can find easily from the figurethat the critical temperature increases when the couplingstrength gets larger. Furthermore, there exists a criticalscaled coupling strength D ′ , below which the critical tem-perature for the deconfinement maintains zero. In fact,below the critical coupling strength, the quark’s DS equa-tion does not have a Nambu-Goldstone solution [4, 5]. Inother words, there exists only quarks in chiral symmetry.In turn, we have only deconfined quarks, but no nucleons(more general, hadrons) even if the temperature is zero. IV. SUMMARY AND REMARKS
In this paper we have calculated the temperature de-pendence of the quark propagator by solving the quarkDSE with a Gaussian-type effective gluon propagator.Based on the calculations, we investigated the tempera-ture dependence of the bag constant, the mass and theradius of a nucleon in the framework of the GCM soli-ton model. It shows that, as the temperature is lowerthan a critical value, the bag constant and the mass de-crease and the radius increases with the increasing of thetemperature. In the case with parameters ω = 0 . D = 1 GeV in the effective gluon propagator, the criticaltemperature is found to be about 133 MeV. At the criti-cal temperature, the bag constant and the mass decreaseto zero and the radius increases to infinity. It means thatthe nucleon can no longer exist as a bag soliton, so thatthe deconfinement phase happens. This indicates evi-dently that the quark deconfinement phase transition cantake place at high temperature. Moreover, we give thedependence of the critical temperature on the interactionstrength parameter D in the effective gluon propagator.It shows that, as the interaction strength parameter islarger than a critical value, the critical temperature in-creases with the increasing of the strength parameter.Even though the temperature dependence of some ofthe properties of nucleon is given with some approxi-mations and model parameters in the present work, thequalitative behavior would be universal and it is the firstone given with quite a sophisticated approximation ofQCD. Of course, there are various aspects to be im-proved. For example, we take the commonly used ef-fective gluon propagator, which is independent of tem-perature, to solve the quark DSE and make use of the preliminary GCM soliton model [14, 16, 23]. It is neces-sary to implement the real GCM soliton model [17, 18]with solving at first the coupled DSEs of the quark, gluonand ghost, and then the coupled equations of the quarkand the chiral fields, with the inclusion of the temper-ature effect. In more detail, for the quark gluon inter-action vertex, we take simply the bare vertex γ µ in ourpresent work, in fact more realistic vertex functions, suchas the BC vertex [42], which has been shown to be able toimprove the calculation of meson properties greatly [39],even the BC vertex together with the transverse partbeing included simultaneously [43–45], should be imple-mented to make the calculation with much more solidQCD foundation. It is also necessary to notice the dimen-sionless quantity Z which takes account of the contri-butions of the zero-point effect, the color-electronic andcolor-magnetic interactions, the motion of center-of-massand all the others. In our present work, we handle it, inthe commonly taken way, as a free parameter to be fixedby the property of a nucleon in free space. In fact, all theaspects of the Z are quite complicated and have recentlybeen paid great attentions (see for example, Refs. [46, 47]tried to evaluate the zero-point effect part from the gluonfield fluctuations directly). Furthermore, extending theresult of the thermal Casimir effect in ideal metal rect-angular boxes [48], we infer that the Z (at least, thezero-point effect part) may depend on temperature. Itwould then be interesting to study the temperature de-pendence of the parameter Z . The related investigationsare in progress. Acknowledgements
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