Extending the truncated Dyson-Schwinger equation to finite temperatures
aa r X i v : . [ nu c l - t h ] O c t Extending the truncated Dyson-Schwinger equation to finitetemperatures
S. M. Dorkin,
1, 2
M. Viebach ∗ , L. P. Kaptari,
1, 4 and B. K¨ampfer
4, 3 Bogoliubov Lab. Theor. Phys., 141980, JINR, Dubna, Russia International University Dubna, Dubna, Russia Institut f¨ur Theoretische Physik, TU Dresden, 01062 Dresden, Germany Helmholtz-Zentrum Dresden-Rossendorf,PF 510119, 01314 Dresden, Germany ∗ Now at Institute of Power Engineering, TU Dresden, 01062 Dresden, Germany bstract In view of the properties of mesons in hot strongly interacting matter the properties of the solutionsof the truncated Dyson-Schwinger equation for the quark propagator at finite temperatures within therainbow-ladder approximation are analysed in some detail. In Euclidean space within the Matsubaraimaginary time formalism the quark propagator is not longer a O(4) symmetric function and possesses adiscrete spectrum of the fourth component of the momentum. This makes the treatment of the Dyson-Schwinger and Bethe-Salpeter equations conceptually different from the vacuum and technically muchmore involved. The question whether the interaction kernel known from vacuum calculations canbe applied at finite temperatures remains still open. We find that, at low temperatures, the modelinteraction with vacuum parameters provides a reasonable description of the quark propagator, whileat temperatures higher than a certain critical value T c the interaction requires stringent modifications.The general properties of the quark propagator at finite temperatures can be inferred from lattice QCD(LQCD) calculations. We argue that, to achieve a reasonable agreement of the model calculationswith that from LQCD, the kernel is to be modified in such a way as to screen the infra-red part of theinteraction at temperatures larger than T c . For this, we analyse the solutions of the truncated Dyson-Schwinger equation with existing interaction kernels in a large temperature range with particularattention on high temperatures in order to find hints to an adequate temperature dependence of theinteraction kernel to be further implemented in the Bethe-Salpeter equation for mesons. This will allowto investigate the possible in medium modifications of the meson properties as well as the conditionsof quark deconfinement in hot matter. PACS numbers: . INTRODUCTION The description of mesons as quark-antiquark bound states within the framework of theBethe-Salpeter (BS) equation with momentum dependent quark mass functions, determinedby the Dyson-Schwinger (DS) equation, is able to explain successfully many spectroscopicdata, such as meson masses [1–7], electromagnetic properties of pseudoscalar mesons and theirradial excitations [8–10] and other observables [10–17]. Contrary to purely phenomenologicalmodels, like the quark bag model, such a formalism maintains important features of QCD,such as dynamical chiral symmetry breaking, dynamical quark dressing, requirements of therenormalization group theory etc., cf. Ref. [18]. The main ingredients here are the full quark-gluon vertex function and the dressed gluon propagator, which are entirely determined by therunning coupling and the bare quark mass parameters. In principle, if one were able to solvethe Dyson-Schwinger equation, the approach would not depend on any additional parameters.However, due to known technical problems, one restricts oneself to calculations within effectivemodels which specify the dressed vertex function Γ µ and interaction kernel D µν . The rainbow-ladder approximation [2] is a model with rainbow truncation of the vertex function Γ µ → γ µ in the quark DS equation and a specification of the dressed quark-quark interaction kernel as g D µν ( k ) → G ( k ) D freeµν ( k ). (Here, γ µ is a Dirac gamma matrix and D µν stands for the gluonpropagator; g is the coupling strength and k denotes a momentum.)The model is completely specified once a form is chosen for the effective coupling G ( k ).The ultraviolet behavior is chosen to be that of the QCD running coupling α ( k ); the ladder-rainbow truncation then generates the correct perturbative QCD structure of the DS and BSequations. Moreover, the ladder-rainbow truncation preserves such an important feature ofthe theory as the maintenance of the Nambu-Goldstone theorem in the chiral limit, accordingto which the spontaneous chiral symmetry breaking results in an appearance of a (otherwiseabsent) scalar term in the quark propagator of the DS equation. As a consequence, in the BSequation a massless pseudoscalar bound state should appear . By using the Ward identities,it has been proven (see, e.g. Refs. [19–21]) that in the chiral limit the DS equation for thequark propagator and the BS equation for a massless pseudo-scalar in ladder approximationare completely equivalent. It implies that such a massless bound state (pion) can be interpretedas a Goldstone boson. This results in a straightforward understanding of the pion as both aGoldstone boson and quark-antiquark bound state.3nother important property of the DS and BS equations is their explicit Poincar´e invariance.This frame-independency of the approach provides a useful tool in studying processes when arest frame for mesons cannot or needs not be defined.The merit of the approach is that, once the effective parameters are fixed (usually theeffective parameters of the kernel are chosen, cf. Ref. [22, 23], to reproduce the known datafrom lattice calculations, such as the quark mass function and/or quark condensate), the wholespectrum of known mesons is supposed to be described, on the same footing, including alsoexcited states. The achieved amazingly good description of the mass spectrum with only feweffective parameters encourages one to employ the same approximations to the truncated Dyson-Schwinger (tDS) and truncated Bethe-Salpeter (tBS) equations also at finite temperatures withthe hope that, once an adequate description of the quark propagators at non-zero temperature( T ) is accomplished, the corresponding solution can be implemented in to the BS equation formesons to investigate the meson properties in hot and dense matter.At low temperatures the properties of hadrons in nuclear matter are expected to changein comparison with the vacuum ones, however the main quantum numbers, such as spin andorbital momenta, space and inner parities etc. are maintained. The hot environment maymodify the hadron masses, life time (decay constant) etc. Contrarily, at sufficiently large tem-perature in hot and dense strongly interacting matter, phase transitions may occur, related toquark deconfinement phenomena, as e.g. dissociation of hadrons in to quark degrees of free-dom. Therefore, these temperature regions are of great interest, both from a theoretical andexperimental point of view. Hitherto, the truncated DS and BS formalism has been mostlyused at large temperatures to investigate the critical phenomena near and above the pseudo-critical and (phase) transition values predicted by lattice simulation data (cf. Refs. [24–27] andreferences therein quoted). It has been found that, in order to achieve an agreement of themodel results with lattice data, a modification of the vacuum interaction kernel is required.Namely, the infra-red term has to vanish abruptly in this region. Accordingly, it has beensuggested [24, 25] to employ a kernel with a Heavyside step-like behaviour in the vicinity ofthe (pseudo-) critical temperature T c . Then, it becomes possible to achieve a rather reliabledescription of such quantities as the quark spectral function, plasmino modes, thermal massesetc., see also Ref. [28]. However, a use of such a discontinuously modified interaction in theBS equation in the whole temperature range becomes hindered. Another strategy of solvingthe DS equation in a larger interval of temperatures is to utilize directly the available LQCD4esults to fit, point by point, the interaction kernel at given temperatures. In such a way oneachieves a good description of the quark mass function and condensate for different tempera-tures, including the region beyond T c [29, 30]. The success of such approaches demonstratesthat the rainbow approximation to the DS equation with a proper choice of the interactionkernel is quite adequate in understanding the properties of quarks in hot environment. Never-theless, for systematic studies of quarks and hadrons within the BS equation, on needs a smoothparametrization of the kernel in the whole interval of the considered temperatures. In view ofstill scarce LQCD data, such a direct parametrization from ”experimental” data is problem-atic. An alternative method is to solve simultaneously a (truncated) set of Dyson-Schwingerequations for the quark and gluon propagators within some additional approximations [31].This approach also provides good description of quarks in vicinity of T c , however it becomestoo cumbersome in attempts to solve the BS equation, since in this case one should solve a toolarge system of equations. It should also be noted that there are other investigations of thequark propagator within the rainbow truncated DS equation, which employ solely the vacuumparameters in calculations of T -dependencies of quarks [32] without further attempts to ac-commodate the kernel to LQCD results. As a result one finds that the critical behaviour of thepropagators (e.g. chiral symmetry restoration) starts at temperatures much smaller than theones expected from LQCD.In the present paper we are interested in a detailed investigation of the quark propagator inthe whole range of temperatures, from zero temperatures up to above T c , and find a reliablesmooth parametrization of the kernel. We start with the interaction kernel known at T = 0and extend it, step by step, to larger temperatures by finding the prerequisites to meet therequirements of the LQCD and to be able to implement the kernel into the BS equation insubsequent studies of the hadron bound states at finite temperatures.In quantum field theory, a system embedded in a heat bath can be described within theimaginary-time formalism, known also as the Matsubara approach [33–35]. Due to finitenessof the heat bath temperature T the Fourier transform to Euclidean momentum space becomesdiscrete, resulting in a discrete spectrum of the energy, known as the Matsubara frequencies.Consequently, the interaction kernel and the DS solution become also discrete with respectto these frequencies. Moreover, since the heat bath already fixes a particular frame, the cor-responding DS and BS equations are not longer O (4) symmetric. This requires a separatetreatment of the transversal and longitudinal parts of the kernel with the need of an additional5unction in parametrizing the quark propagators. All this makes the consideration of the DSequation different from the vacuum case. However, here is the hope that the phenomenologicalinteraction kernel defined at T = 0 can be, to some extent, applied for finite temperatures aswell.In the present paper we investigate the prerequisites to the interaction kernel of the DS for-malism at finite temperatures to be able to investigate, in a subsequent step, different processeswith the challenging problem of changes of meson characteristics at finite temperatures. Ourgoal is to determine with what extend the rainbow truncation of the DS equation is applicable ina large interval of temperature, starting from low values, with the effective parameters, known toaccomplish an excellent description of the hadron properties in vacuum, towards temperaturesabove the critical values predicted by lattice calculations. We try to find a proper modificationof the kernel at higher temperatures to be able to describe the properties of the quark prop-agator in the whole temperature range. A reliable parametrization of the T -dependence willallow to implemented it directly into the BS equation in the same manner as at T = 0 andto investigate, e.g. in-medium changes of mesons in hot environment. This is crucial, e.g. inunderstanding the di-lepton yields in nucleus-nucleus collisions. Our future goal is to investi-gate to what extend the effective parameters, known to accomplish an excellent description ofthe hadron properties in vacuum, can be utilized in the BS equation to investigate the hadronmodifications in hot and dense matter below and above the critical or cross-over temperature.For this we consider the quark propagators from the DS equation in a large temperature rangeand investigate their properties and compare qualitatively with other approaches, such as theLQCD calculations.Our paper is organized as follows. In Sec. II, we recall the truncated BS and DS equations invacuum and at finite temperatures. The rainbow approximation for the DS equation kernel invacuum is introduced and the system of equations for the quark propagator, to be solved at finitetemperature, is presented. Numerical solution for the chirally symmetric case is discussed inSec. III, where the chiral quark condensate and spectral representation for the quark propagatorare introduced. It is found that, to achieve a reasonable behaviour of the spectral functionsabove the critical temperature, a modification of the interaction kernel is needed. In Sec. IVwe consider the solution of the truncated DS equation for finite bare masses. The inflectionpoints of the quark condensate and the mass function are considered as a definition of thepseudo-critical temperature at finite quark masses. The procedure of regularization of integrals6n calculating the quark condensate from the solution of the DS equation is discussed in somedetail. It is shown that, for finite quark masses, the inflection method determines the pseudocritical temperatures by ∼
50% smaller than the ones obtained by other approaches, e.g. bylattice QCD calculations. The possibility to reconcile the model and lattice QCD results isconsidered too. The impact of the infrared term in the interaction kernel in the vicinity andabove the critical temperature is also briefly discussed. Summary and conclusions are collectedin Sec. VI. A brief explanation of the meaning of the rainbow-ladder approxiamtion is presentedin the Appendix.
II. BASIC FORMULAEA. Dyson-Schwinger and Bethe-Salpeter equations in vacuum
To determine the bound-state mass of a quark-antiquark pair one needs to solve the DS andthe homogeneous BS equations, which in the rainbow ladder approximation and in Euclideanspace read S − ( p ) = S − ( p ) + 43 Z d k (2 π ) (cid:2) g D µν ( p − k ) (cid:3) γ µ S ( k ) γ ν , (1)Γ( P, p ) = − Z d k (2 π ) γ µ S ( η P + k )Γ( P, k ) S ( − η P + k )) γ ν (cid:2) g D µν ( p − k ) (cid:3) , (2)where η and η = 1 − η are the partitioning parameters defining the quark momenta as p , = k ± η , P with P and k denoting the total and relative momenta of the bound system, respec-tively; Γ( P, k ) stands for the BS vertex function being a 4 × S ( p ) = ( iγ · p + m ) − and S ( p ) = ( iγ · pA ( p ) + B ( p )) − are the propagators of bare and dressed quarks, respectivelywith mass parameter m and the dressing functions A ( p ) and B ( p ). In Euclidean space weuse the Hermitian matrices γ = γ , γ E = − i γ M which obey the anti-commutation relation { γ µ , γ ν } = 2 δ µ,ν ; for the four-product one has ( a · b ) = P i =1 a i b i . The masses M of mesonsas bound states of a m -quark and m -antiquark follow from the solution of BS equation, P = M , in specific J P C channels, with the solution of the DS equation (1) as input intothe calculations in Eq. (2). The interaction between quarks in the pair is encoded in g D µν Usually, for quarks of masses m , the partitioning parameters are chosen as η , = m , / ( m + m ). How-ever, in general the BS solution is independent of the choice of η , . S , the gluon propagator D µν andthe quark-gluon vertex function Γ µ , all with full dressing (and, if needed, supplemented byghosts and their respective vertices), are considered as an integral formulation being equiva-lent to QCD. In practice, due to numerical problems, the finding of the exact solution of thesystem of coupled equations for S − D µν − Γ µ can hardly be accomplished and therefore someapproximations [5, 17, 18] are appropriate. Note that we must know the form of D µν ( k ) andΓ µ ( k, p ), not only in the ultraviolet range, where perturbation theory is applicable, but also inthe infra-red range, where perturbation theory fails and lattice simulations are to be correctedfor finite-volume effects. D µν ( k ) and Γ µ )( k, p ) satisfy DS equations. However, studies of theseequations in QCD are rudimentary and are presently used only to suggest qualitatively reliable ans¨atze for these functions. That is why the quantitative studies of the quark DS equation todate have employed model forms of the gluon propagator and quark-gluon vertex. Leaving adetailed discussion of the variety of approaches in dressing of the gluon propagator and vertexfunction in DS equations (see e.g. Refs. [40, 41] and references therein quoted) we mention onlythat in solving the DS equation for the quark propagator one usually employs truncations ofthe exact interactions and replaces the gluon propagator combined with the vertex function byan effective interaction kernel [ g D µν ]. This leads to the truncated Dyson-Schwinger equationfor the quark propagator which may be referred to as the gap equation. In explorative calcula-tions, the choice of the form of the effective interaction is inspired by results from calculationsof Feynman diagrams within pQCD maintaining requirements of symmetry and asymptoticbehaviour already implemented, cf. Refs. [5, 10, 18, 41]. The results of such calculations, evenin the simplest case of accounting only for one-loop diagrams with proper regularization andrenormalization procedures, are rather cumbersome for further use in numerical calculations,e.g. in the framework of BS or Faddeev equations. Consequently, for practical purposes, thewanted exact results are replaced by suitable parametrizations of the vertex and the gluonpropagator. Often, one employs an approximation which corresponds to one-loop calculationsof diagrams with the full vertex function Γ ν , substituted by the free one, Γ ν ( p, k ) → γ ν (wesuppress the color structure and account cumulatively for the strong coupling later on). To em-phasize the replacement of combined gluon propagator and vertex we use the notation [ g D µν ],where an additional power of g from the second undressed vertex is included.8 . Choosing an interaction kernel Note that the nonperturbative behaviour of the kernel [ g D µν ] at small momenta, i.e. inthe infra-red (IR) region, nowadays is not uniquely determined and, consequently, suitablemodels are needed. In principle, constraints on the infra-red form of the kernel can be soughtfrom studies of the DS equations with the fully dressed gluon propagator, D µν ( k ), and thedressed gluon-quark vertex Γ ν ( p, k ). However, there is almost no information available from DSequation studies; the gluon propagator itself has been often studied via the gap equation, andfrom such studies one can merely qualitatively conclude that the gluon propagator is enhancedin the infra-red. There are several ans¨atze in the literature for the IR kernel, which can beformally classified in the two groups: (i) the IR part is parametrized by two terms - a deltadistribution at zero momenta and an exponential, i.e. Gaussian term, and (ii) only the Gaussianterm is considered. In principle, the IR term must be supplemented by a ultraviolet (UV) one,which assures the correct asymptotics at large momenta. A detailed investigation [7, 42] of theinterplay of these two terms has shown that, for bound states, the IR part is dominant for light( u , d and s ) quarks with a decreasing role for heavier ( c and b ) quark masses for which theUV part may be quite important in forming mesons with masses M q ¯ q > − M q ¯ q ≤ − T ) where one canexpect that at sufficiently large T some phase transition can occur and/or quark dissociationof mesons into quark degrees of freedom in hot matter. In such a temperature range, the IRterm is expected to be screened [24, 25] and, consequently, the perturbative UV behaviour canbecome important even for light mesons.Following examples in the literature [2, 5, 9, 10, 12, 16] the interaction kernel in the rainbowapproximation in the Landau gauge is chosen as g ( k ) D µν ( k ) = (cid:0) D IR ( k ) + D UV ( k ) (cid:1) (cid:18) δ µν − k µ k ν k (cid:19) ,D IR ( k ) = 4 π Dk ω e − k /ω , D UV ( k ) = 8 π γ m F ( k )ln[ τ + (1 + k Λ QCD ) ] , (3)where the first term originates from the effective IR part of the interaction determined by soft,non-perturbative effects, while the second one ensures the correct UV asymptotic behaviourof the QCD running coupling. In what follows we restrict ourselves to two models. (i) Theinteraction consists of both the IR and UV terms: Such an interaction is known as the Maris-9andy (MT) model [2]. (ii) The UV term is ignored at all: This interaction is known asAlkofer-Watson-Weigel [16] kernel, referred to as the AWW model. It should be noted thatat zero temperatures these models, with only a few adjustable parameters - the IR strength D , the slope parameter ω and quark mass parameter m in the AWW model and additionally τ , Λ QCD , γ m and m t in the formfactor F ( k ) = (1 − exp {− k / [4 m t ] } ) /k in the MT model -provide a good description of the pseudoscalar, vector and tensor meson mass spectra [3, 4, 6, 7].Therefore, at finite temperatures a tempting choice of the interactions is to keep them the sameas in vacuum. C. Finite temperatures
The theoretical treatment of systems at non-zero temperatures differs from the case of zerotemperatures. In this case, a preferred frame is determined by the local rest system of thethermal bath. This means that the O (4) symmetry is broken and, consequently, the dependenceof the quark propagator on p and p requires a separate treatment. To describe the propagatorin this case a third function C is needed, besides the functions A and B introduced above forvacuum. Yet, the theoretical formulation of the field theory at finite temperatures can beperformed in at least two, quite different, frameworks which treat fields either with ordinarytime variable t ( −∞ < t < ∞ ), e.g. the termo-field dynamics (cf. [37]) and path-integralformalism (cf. [38, 39]), or with imaginary time it = τ (0 < τ < /T ) which is known as theMatsubara formalism [33–36]. In this paper we utilize the imaginary-time formalism withinwhich the partition function is defined and all calculations are performed in Euclidean space.Since at T = 0 the (imaginary) time evolution is restricted to the interval [0 . . . /T ], the quarkfields become anti-periodic in time with the period 1 /T . In such a case the Fourier transform isnot longer continuous and the energies p of particles become discrete [33–35] which are knownas the Matsubara frequencies, i.e. p = ω n = πT (2 n + 1) for Fermions ( n is an integer, runningfrom minus to plus infinity). The inverse quark propagator is now parametrized as S − ( p , ω n ) = i γp A ( p , ω n ) + iγ ω n C ( p , ω n ) + B ( p , ω n ) . (4)Accordingly, the interaction kernel is decomposed in to a transversal and longitudinal part[ g D µν ( q , Ω mn )] = P Tµν D T ( q , Ω mn ,
0) + P Lµν D L ( q , Ω mn , m g ) , (5)10here Ω mn = ω m − ω n and the gluon screening (Debye) mass m g is introduced in the longitudinalpart of the propagator, where q = q +Ω mn + m g enters. The scalar coefficients D L,T are definedbelow. The projection operators P L,Tµν can be written as P Tµν = , µ, ν = 4 ,δ αβ − q α q β q ; µ, ν = α, β = 1 , , , P Lµν = δ µν − q µ q ν q − P Tµν . (6)The gap equation has the same form as in case of T = 0, Eq. (1), except that within the Mat-subara formalism the integration over k is replaced by the summation over the correspondingfrequencies, formally Z d p (2 π ) −→ T ∞ X n = −∞ Z d p (2 π ) . (7)Then the system of equations for A, B and C to be solved reads (cf. also Ref. [29]) A ( p , ω n ) = 1 + 43 T ∞ X m = −∞ Z d k (2 π ) (cid:26) qkq (cid:18) − pkp (cid:19) σ A ( k , ω m ) D T ( q , Ω nm ) + (cid:20) pkp σ A ( k , ω m ) + 2 Ω mn q (cid:18) − pkp (cid:19) ω m σ C ( k , ω m ) −− mn q qkq (cid:18) − pkp (cid:19) σ A ( k , ω m ) (cid:21) D L ( q , Ω nm , m g ) (cid:27) , (8) B ( p , ω n ) = m q + 43 T ∞ X m = −∞ Z d k (2 π ) (cid:20) D L ( q , Ω nm , m g ) + 2 D T ( q , Ω nm , (cid:21) σ B ( k , ω m ) , (9) C ( p , ω n ) = 1 + 43 T ∞ X m = −∞ Z d k (2 π ) (cid:26) ω m ω n σ C ( k , ω m ) D T ( q , Ω nm , (cid:20) − (cid:18) − mn q (cid:19) ω m ω n σ C ( k , ω m ) + 2 qk q Ω nm ω n σ A (cid:21) D L ( q , Ω nm , m g ) (cid:27) , (10)where q = p − k and the propagator functions σ F = σ F ( k , ω m ) are defined by σ F ( k , ω m ) = F ( k , ω m ) k A ( k , ω m ) + ω m C ( k , ω m ) + B ( k , ω m ) (11)for F = A, B and C . The form of the interaction kernel is taken the same as at T = 0,i.e. both the transversal and longitudinal parts consisting of two terms - the infra-red andultraviolet ones. The information on these kernels is even more sparse than in the case of T = 0. While the effective parameters of the kernel in vacuum can be adjusted to some known11xperimental data, e.g. the meson mass spectrum from the BS equation, at finite temperatureone can rely on results of QCD calculations, e.g. by using results of the nonperturbativelattice calculations. There are some indications, cf. [43], that at low temperatures the gluonpropagator is insensitive to the temperature impact, and the interaction can be chosen as at T = 0 with D T = D L [24]. However, in a hot and/or dense medium the gluon is also subject tomedium effects and thereby becomes effectively massive with finite transversal (known also asthe Meissner mass) and longitudinal (Debye or electric) masses. Generally, these masses appearas independent parameters with contributions depending on the considered process [44]. Therole of the Meissner masses in the tDS equation at zero chemical potential is not yet wellestablished and requires separate investigations. This is beyond the goal of the present paperwhere only the Debye mass, m g , is considered. It should be noted that, independently of thevalue of the chemical potential, in most approaches based on the tDS equation within therainbow approximation it is also common practice to ignore the effects of Meissner masses.This is inspired by the results of a tDS equation analysis in the high temperature and densityregion [45] which report that the Meissner mass is of no importance in tDS equation. Atthis level the Debye mass is the only T depending part of the kernel. The Debye mass is welldefined in the weak-coupling regime. In [30, 46–48] it was found that the T -dependence of theDebye mass is in leading order m g = α s π N c + N f ] T , (12)where N c and N f denote the number of active color and flavor degrees of freedom, respectively;the running coupling α s in the one-loop approximation is α s ( E ) ≡ g ( E )4 π = f ( E ) 12 π N c − N f (13)with E being the energy scale. For the temperature range considered in the present paperwe adopt f ( E ) →
2, whichoften employed [30, 46–48] choice for the Debye mass in the tDSequation. It should be noted, however, that such a choice of f ( E ) is not unique. It may varyin some interval, in dependence on the employed method of infra-red regularization [34, 47].Since the Debye mass enters as an additional energy parameter in q = q + Ω mn + m g , whichdetermines the Gaussian form of the IR part of the interaction (3), an increase of m g results ina shift of the tDS solution towards lower temperatures leaving, at the same time, the shape ofthe solution practically unchanged. Accordingly, smaller values of m g shift the solution towardslarger temperatures. Our numerical calculations show that a decrease of m g by a factor of 212esults in a ∼
15% shift of the solution to larger temperatures. The transversal and longitudinalparts of the interaction kernel (5) can be cast in the form D T ( q , Ω mn ,
0) = D IR ( q + Ω mn ) + D UV ( q + Ω mn ) , (14) D L ( q , Ω mn , m g ) = D IR ( q + Ω mn + m g ) + D UV ( q + Ω mn + m g ) . (15)In the present paper we use several sets of parameters for the interaction kernel (3):1) ω = 0 . D = 1 GeV , m u = 5 MeV, m s = 115 MeV; results with these parametersare denoted as AWW (IR term only) [16] and MT1 (IR+UV terms) [2].2) ω = 0 . D = 0 .
93 GeV , m u = 5 MeV, m s = 115 MeV; MT2 [2].3) AWW, MT1 and MT2 with a modified parameter D making it dependent on temperature;at low T it remains constant, equal to the values used in the AWW, MT1 and MT2 sets, whileat large temperatures, where the IR contribution is expected to be screened, the parameter D becomes a decreasing function of T . In this case, since the IR term vanishes, the AWW modelis not applicable. It should be noted that all these models provide values for the vacuum quarkcondensate in a narrow corridor, −h q ¯ q i = (0 . − . , and the correct π and ρ meson masses as quark-antiquark bound states [7]. III. SOLUTIONS OF THE tDS EQUATION IN THE CHIRAL LIMITA. Order parameters
As seen from Eq. (9) in the chiral limit, i.e. at m q = 0, the trivial solution B ( p , ω n ) = 0is possible, known as the Wigner-Weyl mode. It is of separate interest since this is the casewhere the dynamical chiral symmetry breaking is completely disabled. In the present paper,however, we are interested in solutions with finite dynamical quark masses given by the ratio B/A , which enter the BS equation and determine accordingly the hadron bound states in aheat bath. Therefore, we will not consider the Wigner-Weyl mode solution and focus insteadon B = 0 (Nambu-Goldstone mode). It should be also noted that even for B = 0 the sign of B is not defined. Equation (9) is invariant under B → − B . The sign of the solution can be fixedonly by fixing the sign of the initial conditions for B . Here we consider positive values of B .We solve numerically the system of equation (8)-(10) by an iteration procedure. Sincethe UV term in the MT1 and MT2 models is logarithmically divergent, a regularization ofthe integral over the internal momentum and summation over ω n is required. Usually, at13 = 0 one employs an O (4) invariant cutoff Λ. The dependence of the solution on Λ isremoved by choosing a subtraction scheme defined at a renormalization point µ ≤ Λ suchthat A ( p = µ , Λ ) = 1; B ( p = µ , Λ ) = m . In an analogous way one performs therenormalization procedure at finite temperature T [26, 29]. The only difference is that theinternal momentum k is restricted by the condition k + ω n ≤ Λ . At each iteration stepthis requires an interpolation of the previous solution to define the new Gaussian mesh for k max = p Λ − ω n . In our calculations we employ a cubic spline interpolation procedure and amapping k = k x − x (16)for the Gaussian integration with k = 0 .
85 for a mesh of 64 nodes. This provides a ratherlarge cutoff Λ = k max ≃ GeV/c. The summation over ω n is truncated at a large valueof n = N max , where in our calculations N max ∼
320 for low temperatures and N max ∼
250 attemperatures
T > −
100 MeV are utilized. In Figs. 1 and 2 we exhibit the solution of thesystem (8)-(10) for the lowest Matsubara frequency ω = πT in dependence on the momentum | p | at low temperature ( T = 5 MeV, Fig. 1) and higher temperature ( T = 100 MeV, Fig. 2). Acomparison with the vacuum solution [49] shows that qualitatively there is no difference of thesolutions at finite T . To emphasize the dependence on the effective parameters D and ω , inFigs. 1 and 2 we present results of calculations for the two different sets. In Fig. 1, left panel,the solutions A ( p , ω ) are represented by solid and dashed curves, while the solutions C ( p , ω )by dotted and dash-dotted curves for MT1 and MT2 sets, respectively. The solutions B ( p , ω ),left panel, are for MT1 (solid curve) and NT2 (dashed curve). The dependence on the values ofthe effective parameters is seen only at low momenta. In this region, the dependence is mainlydetermined by the slope parameter ω , cf. Eq. (3), which is quite different for the two employedparameter sets. At larger momenta (cf. Fig. 2), the common asymptotics is approached alreadyat | p | >
10 GeV/c. The dependence of the solution on the temperature is of particular interest.It is known that in dense and hot matter there may occur different kind of phase transitions. InSU(3) gauge theory, the deconfinement transition is of first order at T c = O (270) MeV relatedto the center symmetry, while in 2+1 flavor QCD with physical quark masses it is a cross-overat T c = O (150) MeV, see [50–52]. At non-zero baryon density, the liquid-gas phase transition at T c = O (20) MeV matters, and a critical end point of an additional first-order phase transitioncurve is still hypothetical. According to the Columbia plot [53], two-flavor QCD in the chirallimit displays a first-order deconfinement and chiral restoration transition. The corresponding14 A (MT1) A(MT2) C(MT1) C(MT2) A ( p , ) , C ( p , ) P [GeV/c] T=5 MeV
MT1 MT2 B ( p , ) [ G e V ] P [GeV/c] T=5 MeV
FIG. 1: (color online) Solutions of Eqs. (8)-(10) for the lowest Matsubara frequency ω at T = 5MeV in the chiral limit, m q = 0. The solution for ω = 0 . D = 1 GeV is labeled as MT1,while for the parameters ω = 0 . D = 0 . GeV as MT2. Both sets of parameters includethe IR and UV terms. In the left panel the solution for A ( p , ω ) is depicted by solid (MT1) anddashed (MT2) curves, and C ( p , ω ) by dotted and dash-dotted curves, respectively, while the rightpanel exhibits the function B for MT1 (solid) and MT2 (dashed) kernels. MT1 MT2 A ( p , ) P [GeV/c] T=100 MeV C ( p , ) P [GeV/c] T=100 MeV
FIG. 2: (color online) Solutions of Eqs. (8)-(10) in the chiral limit, m q = 0, for the lowest Matsubarafrequency ω = πT at T = 100 MeV for A ( p , ω ) (left panel) and C ( p , ω ) (right panel). Solid (dashed)curves are for MT1 (MT2). Since the quantity B becomes negligibly small, B < − , at T ≥ quantity which characterises such transitions is known as order parameter of the considered15edia. Natural candidates to be considered as order parameters or elements thereof are themass function B ( p , ω n ) and the quark condensate h q ¯ q i , being an integral characteristics of themass B ( p , ω n ) too. Order parameters determine the so-called critical temperature T c or thecross-over region which will serve as an indicator for a possible (phase) transition.At high enough temperatures one expects a chiral restoration. This means that at a certainhigh value of the temperature the mass function B should vanish, indicating a possible phasetransition in the hot matter. The lowest temperature at which B = 0 holds is called the criticaltemperature T c , i.e. for the mass function B ( p , ω n ) the critical temperature can be defined asthat value of T at which the solution B ( p , ω n ) vanishes. Analogously, for the chiral condensate, T c can be determined also as the temperature above which h q ¯ q i vanishes. In principle, thesetwo critical temperatures can be slightly different.The chiral condensate is defined by h q ¯ q i = − N c T ∞ X n = −∞ Z d p (2 π ) T r [ S ( p , ω n )]= − N c T ∞ X n = −∞ Z d p (2 π ) B ( p , ω n ) p A ( p , ω n ) + ω n C ( p , ω n ) + B ( p , ω n ) , (17)where the trace is performed in spinor space. In Fig. 3 we present results of calculations of the T -dependence of the mass solution B (left panel), and the normalized chiral condensate h q ¯ q i (right panel) in the chiral limit m q = 0.One infers from this figure that in a large range of T the solution B (0 , ω ) is a smoothlydecreasing function of T , except for a narrow interval where B (0 , ω ) sharply decreases towardszero, as seen in Fig. 4 where, for the sake of better determination of T for which B (0 , ω ) → B ( B ≤ − ) the convergence of the iteration procedure for Eq. (9) becomes rather poor. Oneneeds to increase essentially the number of iterations in order to achieve the same accuracyin the whole range of considered temperatures. In addition, the actual accurate calculationsin the neighbourhood of T c are restricted by numerical manipulations with quantities close tothe machine zero. These numerical effects hinder a precise determination of T c in the chirallimit, m = 0. In our calculations, we analyse the values of B below 10 − to determine theinterval for T c . An inspection of the numerical results shows that, for the MT1 model, thecritical temperature is about 130 MeV, while for the MT2 model one has about 110 MeV. Ananalogous determination of T c from the chiral condensate provides slightly different values, e.g.16 .03 0.06 0.09 0.12 0.15 T [GeV] MT1 MT2 AWW B ( , ) [ G e V ] T [GeV] |p|=0 MT1 MT2 AWW < qq > / < qq > FIG. 3: (color online) Solutions B ( p , ω ) from (9) in the chiral limit, m q = 0, for the lowest Matsubarafrequency (left panel) and the chiral condensate (17) normalized at low T (right panel) as functionsof T . The dashed and solid curves are for the same effective parameters as in Fig. 1. -25 -15 -5 -60 -50 -40 -30 -20 -10 MT1 MT2 AWW B ( , ) [ G e V ] T [GeV] |p|=0 MT1 MT2 AWW T [GeV] |p|=0 FIG. 4: (color online) Solution B ( p , ω ) from Eq. (9), depicted in a log-scale, in the chiral limit, m q = 0, for the lowest Matsubara frequency. The solid, dashed and dot-dashed curves are for theMT1, MT2 and AWW models, respectively. The left panel exhibits the sharp dropping of B ( p , ω ) inthe vicinity of T ∼ −
105 MeV which continues when extending the scale to even smaller values,as illustrated in the right panel. From the right panel one would infer that T c ≈
110 MeV for the MT2model and T c ≈
130 MeV for the MT1 model. T c = 128 MeV for MT1 and T c = 105 MeV for the MT2 model. The AWW set provides valuesclose to the MT1 model. The obtained values are by 20% smaller than that obtained from17CD calculations [54], which report a cross-over temperature T c in the range T c ∼ [145 . . . ±
9) MeV [55], however, for 2+1 flavor QCD with physicalmasses. It should be also noted that the general feature of the quark condensate, as a functionof the temperature below the chiral transition limit, is established in a model independentway [56, 57] by the low-temperature expansion h q ¯ q i = h q ¯ q i [1 − T / f π − O ( T )] , where f π is the pion decay constant in the chiral limit ( f π ≈
93 MeV). Our results in Fig. 3 are in aqualitative agreement with that. It should be emphasized that the above quoted values for T c stem from the inspection of the numerical results of B and h q ¯ q i at small values. On the linearscale in Fig. 3 however, one infer instead values of about 120 MeV or even less. B. Spectral representation above T c Another important quantity characterizing the hot matter is the spectral representation ofthe retarded quark propagator. The Euclidean propagator can be transferred to Minkowskispace by an analytical continuation of the solution of the gap equation to real energies, S M ( p , ω ) = S ( p , iω n ) | iω n → ω + iη . (18)In Minkowski space, the dispersion relation for the quark propagator determines the spec-tral representation ρ ( p , ω ) which is directly related to the imaginary part of the propagator, ρ ( p , ω ) = − ℑ S M ( p , ω ). It means that in Euclidean space the same spectral density ρ ( p , ω )can be associated to the retarded quark propagator S ( p , iω n ) = 12 π Z ρ ( p , ω ′ ) iω n − ω ′ dω ′ . (19)From this the importance of studying ρ can be inferred. Note that, since the spectral densitycharacterises the propagation of a particle, the dispersion relations in our case are meaningfulonly in the (deconfinement) region T > T c , where quarks can be treated to some extent asquasi-particles. Recall that in Minkowski space the propagator of a free particle can be writtenas S ( p ) = S + + S − = Λ + ω − E p γ + Λ − ω + E p γ , (20)where E p = p p + m is the energy of the particle and Λ ± = E p ± γ [ γp + m ] / E p are theprojection operators on positive and negative energy solutions, respectively, obeying Λ ± Λ ± =18 ± , Λ + + Λ − = 1 , Λ + Λ − = 0. Therefore, for a free quark ρ ± ( ω ) = 12 δ ( ω ∓ E p ) , (21)i.e. the spectral functions ρ ± ( ω ) characterize the propagation of quasi-particles with positive(normal) and negative (abnormal) energies. This can serve as a hint in parametrizing thespectral density ρ ( p , ω ) in Euclidean space. Owing to parity and rotation symmetries, theDirac structure of the quark spectral function at finite temperature is in general decomposedas ρ ( p , ω ) = ρ ( p , ω ) γ + ρ v ( p , ω )( γp ) + ρ s ( p , ω ) . (22)In the present paper we focus to two particular cases.(i) Chiral limit, where the scalar, or ”mass”, part ρ s ( p , ω ) vanishes. In this case, the spectralfunction at negative energies describes the so-called plasmino mode [28]. In the chiral limitthe projection operators Λ ± are of a particularly simple form and the spectral density can bewritten as ρ ( p , ω ) = ρ + ( ω ) (1 + iγ γ ˆ p )2 + ρ − ( ω ) (1 − iγ γ ˆ p ))2 , (23)where ˆ p ≡ p / | p | .(ii) Zero momenta: The projection operators are Λ ± = 1 ± γ ρ ( ω, p = 0) = ρ + ( ω ) 1 + γ ρ − ( ω ) 1 − γ . (24)Note that at zero momenta the energy E of the quark can be associated to a mass m T whichin literature is referred to as the thermal mass, a subject of many investigations within latticeQCD, cf. [28]. Results of such calculations are often considered as ”experimental” data for thecorresponding quantity. This is an important issue, since the model calculations of m T can berelated to ”experimental” data and to serve as a guide in fixing the phenomenological parame-ters and to estimate the applicability of the model which is based directly on parametrisationsand scale settings by vacuum meson physics. It can be shown that the quark propagator canbe written in the same form (23). So, in the chiral limit one can note S ( p , ω n ) = S + ( p , ω n ) (1 + iγ γ ˆ p )2 γ + S − ( p , ω n ) (1 − iγ γ ˆ p ))2 γ , (25)where S ± ( p , ω n ) = − iω n C ( p , ω n ) ± | p | A ( p , ω n )) ω n C ( p , ω n ) + p A ( p , ω n ) . (26)19f one writes the dispersion relations for the model propagators (26) S ± ( p , ω n )) = 12 π Z ρ ( p , ω ) iω n − ω dω, (27)then by inverting (27) one can obtain the (model) spectral density ρ ( p , ω ). Note that in modelcalculations the problem of inverting expressions like (27) is ill posed. Nevertheless, instead ofinverting the equation (27), one can suggest a reliable parametrization for ρ ( p , ω ) which allowsan analytical calculation of the integral over ω and then to minimize the quantity∆ N = 12 N + 1 N X n = − N (cid:12)(cid:12)(cid:12)(cid:12) S ± ( p , ω ) − π Z ρ ± ( p , ω ) iω n − ω dω (cid:12)(cid:12)(cid:12)(cid:12) , (28)where the integral in Eq. (28) must be preliminarily carried out analytically to leave the depen-dence only on ω n and effective parameters. In such a way one can find the effective parametersand estimate the behaviour of ρ ( p , ω ).The simplest parametrization for the spectral function at finite T is suggested by the caseof a free quark propagator (21), i.e. one can expect that ρ ( p , ω ) exhibits two maxima. For thetwo-pole parametrization the spectral function reads ρ + ( p , ω ) = Z ( | p | ) δ ( ω − E ) + Z ( | p | ) δ ( ω + E ) . (29)With such a parametrization the spectral function ρ ( ω, p ) describes the propagation of quasi-particles with the positive energy E and aniti-particles with negative energy E ; the secondterm in (29) is known also as the plasmino mode [28]. The weights Z , ( | p | ) of the normaland plasmino modes play an important role in estimating the phase transitions in hot matter.At zero momenta, one has E = E and Z ( | p | = 0) = Z ( | p | = 0) = 1 /
2. In this case theenergy parameters determine the thermal masses, E , = m T , which are predicted [28] in latticeQCD to be an increasing function of T and at T /T c ≥ m T ∼ . T . This important”experimental” result may be used in choosing the model kernels of the tDS equation. As themomentum | p | increases one expects that the plasmino mode vanishes and Z ( | p | ) → N inEq. (28). Results of calculations are exhibited in Fig. 5. It can be seen that both, MT1 andMT2 models (solid and dashed curves in Fig. 5), provide increasing functions of T . However,the absolute values at T ≥ T c are far from that predicted by QCD [54]. It implies that,while at low temperatures the two models with vacuum parameters, maintain a satisfactorydescription of the quark propagators at T > T c , the interaction kernel requires modifications, cf.20 .0 1.5 2.0 2.5 -6-4-20246 MT1 MT2 MT2+Tdep. E , / T c T/T c |p|=0 FIG. 5: (color online) The scaled thermal masses as function of
T /T c in the chiral limit. The dashedand solid curves are obtained with two different sets of the effective parameters, MT1 and MT2. Thedot-dashed curve is obtained by the modified, T -dependent kernel of Eq. (30). Ref. [24, 25]. Such modifications are inspired by the fact that, at sufficiently large temperatures,thermodynamics should be describable in terms of a weakly interacting quark-gluon gas, and atasymptotically large temperatures all thermodynamic quantities should converge to the idealgas limit (for a discussion of the lattice QCD approaching to the perturbative limit see, e.g.Ref. [50]). It means that, at large temperatures, the IR term in the interaction kernel mustdiminish or even vanish. A first attempt to modify the interaction kernel was done in Ref. [24],where the weight D of the IR term above T c is abruptly (via a step function) replaced by aphenomenological, T -dependent decreasing function. In the present paper we suggest anothermodification of the IR term which smoothly decreases at large T and, at the same time, doesnot affect the IR term at low T . To do so, we introduce a suppression function f ( T ) whichhas a Heavyside step-like behavior at temperatures T ∼ T c , D → D ( T ) = D f ( T ) = D (cid:20) (cid:18) − T − T p β (cid:19)(cid:21) , (30)where the additional adjustable parameters are T p ∼ T c and β as some diffuseness of the IRinteraction. With such a parametrization, at T ≪ T p the weight D of the IR part is as in thevacuum, D ( T ) = D , and at T ≫ T c , D ( T ) →
0. In our calculations we adopt T p = 130 − β = 30 MeV, which assures for both, MT1 and MT2 models, a reasonable behaviourof the thermal masses m T ∼ T at T ≥ T c , see Fig. 5 (cf. also Ref. [54]).Another important characteristic is the behaviour of the plasmino mode as a function of themomentum p . We find, cf. also [24], that, while the energy of the normal mode monotonously21ncreases with | p | (as it should be), the plasmino mode decreases up to a minimum value, thansharply increases approaching the normal mode at large | p | , see Fig. 6-left panel. This doesnot mean at all that the role of plasmino mode is increasing too and becomes of the sameimportance as the normal one. Instead, the weight Z ( | p | ) sharply decreases with increasing | p | , vanishing at large | p | , cf. Fig. 6-right panel. The local minimum of the plasmino mode isrelated to the Van Hove singularity. -2-10123 E , / m T |p| /T Z , |p| /T T/T c =2.5 FIG. 6: (color online) Left panel: Energy of the normal (solid curve) and plasmino mode (dashedcurve) as a function of the scaled momentum computed with the T -dependent interaction (30). Rightpanel: The corresponding weights of the normal (solid curve), and plasmino modes (dashed curve) at T /T c = 2 . IV. SOLUTION OF THE tDS EQUATION AT FINITE BARE MASSES
At finite quark masses the solution of the tDS equation differs from the chiral limit in atleast two aspects. First, the Wigner-Weyl mode is not longer a solution. Second, the integralsover p are logarithmically divergent in the UV region. This means that a regularization andrenormalization procedure is required. To make the results finite one usually uses the Feyn-man method by introducing a cut-off parameter Λ for the integrals, followed by a subsequentreliable subtraction procedure [1, 29, 58]. In approximate models after performing the neces-sary regularizations and renormalizations have been performed, the effective phenomenologicalparameters are fixed in such a way that a bulk of the effects is already included. Numericallyit implies that by choosing a large enough cutoff Λ there is no need for further normaliza-22ions to solve the tDS equation. In our calculations, the integral over | p | is performed up to | p | max ∼ GeV/c which assures an asymptotic behaviour of the solution A ( p , ω n ) → B ( p , ω n ) → m q and C ( p , ω n ) → ∼ . A, B, C one shall bear in mind that additional divergences may becomeapparent for another kind of calculations, and regularization procedures may be still required,as e.g. in calculations of the chiral condensate. At finite quark masses the chiral condensate isin fact quadratically divergent, cf. Eq. (17). This is manifestly seen if one considers the quarkcondensate at T = 0 but m q = 0, h q ¯ q i = − Z d p (2 π ) σ B ( p , p )) = − π Z ˜ p B (˜ p )˜ p A (˜ p ) + B (˜ p ) d ˜ p, (31)where ˜ p = p + p . At large values of the momenta the asymptotic solution of the tDSequation becomes A (˜ p ) → B (˜ p ) → m q and the integral (31) is quadratically divergent.On can regularize it by subtracting at large momenta the asymptotic quark mass m q . Denote B = ˜ B + m q , where at large enough momenta ˜ p max the quantity ˜ B ( p max ) goes to zero whichimplies that at ˜ p > ˜ p max the dynamical chiral symmetry breaking vanishes, i.e. the massfunction receives its asymptotic value B (˜ p max ) ≃ m q . Now, if the cut off parameter is chosenlarge enough, Λ > ˜ p max , then h q ¯ q i (Λ) = − π Z ˜ p d ˜ p " ˜ B (˜ p ) + m q ˜ p A (˜ p ) + B (˜ p ) ≃ h q ¯ q i ren. − π Z ˜ p m q ˜ p + m q d ˜ p = h q ¯ q i ren. − m q π Λ Z x x + ε dx = h q ¯ q i ren. + m q π Λ (cid:2) x − ε ln( x + ε ) (cid:3)(cid:12)(cid:12)(cid:12)(cid:12) ≃ h q ¯ q i ren. − m q π Λ , (32)where ε = m q Λ ∼
0, and the regularized condensate h q ¯ q i ren. does not depend on Λ. In obtaining(32) we put A ∼ B ∼ m q in the second integral. Equation (32) illustrates the quadraticdivergence of the integral and, at the same time, hints to how the subtraction procedure isto be chosen to eliminate this divergence. With this in mind, one can define a renormalized(subtracted) quark condensate as h q ¯ q i l − m l m h h q ¯ q i h = h q ¯ q i lren. − m l m h h q ¯ q i hren. , (33)where m l and m h denote the mass of light, e.g. u , and heavy, e.g. s , quarks, respectively. At m l m h ≪
1, equation (33) determines the required renormalized, cut-off independent light-quark23ondensate. Exactly the same procedure is applied to determine the quark condensate at finite T , see also Ref. [30]. The remaining multiplicative divergences can be removed by normalizingto quark condensate at zero temperature.In Fig. 7 we present the dependence of the mass function B ( , ω ) and the chiral condensate h q ¯ q i on the temperature at finite bare masses, m l = 5 MeV and m h = 115 MeV. From the figureone infers that, since B ( , ω ) and the chiral condensate h q ¯ q i do not exactly vanish at large T , their asymptotic values can not serve for a clear-cut definition of the critical temperature.Instead, one can use the method of the maximum of the chiral susceptibility, i.e. the maximaof the derivatives of B and/or h q ¯ q i with respect to the quark bar mass, as well as the inflectionpoint of the mass function or of the condensate, i.e. the maxima of the corresponding derivativeswith respect to the temperature [30]: χ B ( T ) = d B (0 , ω ) dT ; χ qq ( T ) = d h q ¯ q i dT . (34)The (pseudo-) critical temperature T c is fixed by the condition χ B ( T ) | T = T c = 0 and/or χ qq ( T ) | T = T c = 0. MT1 AWW B ( , ) [ G e V ] T [GeV] |p|=0 < qq > MT1 AWW < qq > / T [GeV] FIG. 7: (color online) The solutions B ( p = 0 , ω ) of the tDS equation for the light-quark mass m l = 5MeV for the lowest Matsubara frequency (left panel) and quark condensate (17) (right panel) as afunction of T . For MT1 (solid) and AWW (dashed) interaction kernels. Figures 7 and 8 clearly demonstrate that the inflection points at finite quark bare massesprovide much smaller (pseudo-) critical temperatures T c ; for all T -independent interactions onehas T c ∼
100 MeV, cf. also Ref. [32]. With the modified interaction, Eq. (30), for which theIR term is screened at large temperatures,
T >
200 MeV, the positions of the inflection points,24 .05 0.10 0.15 -1000-50005001000
T [GeV]
MT1 AWW -1000-50005001000 B [ G e V - ] T [GeV]
MT1 AWW qq [ G e V - ] FIG. 8: (color online) The inflection points (second derivative with respect to temperature) for themass function B ( p = 0 , ω ) (left panel) and for the normalized quark condensate, Eq. (33) (rightpanel), as exhibited in Fig. 7. which occur below T c = O (140) MeV, remain the same. This implies, for a better agreementwith the lattice QCD [29, 54], the interaction kernel, for finite bare quark masses, must acquirean appropriate dependence on the temperature also below T c . Another important issue of ouranalysis of the T -dependence of the IR term is that, starting from a relatively large temperatureof T ∼
100 MeV, the dependence of the IR term is basically governed by the Debye mass whichsuppresses B ( p , ω n ) at large T . This is a hint that in the considered models the Debye masshas to be included only in the perturbative part of the interaction, i.e. in the UV term only. V. IMPACT OF THE IR TERM
Analysing the relative contributions of the IR and UV terms in the interaction we find that,while at T = 0 the UV term can be ignored in considering the hadron ground states [3], atfinite temperatures the ultraviolet behavior can become important.In Fig. 9, left panel, we present the separate contributions of the IR (dashed curve) and UV(dash-dot-dot curve) terms to the tDS solution B ( p = 0 , ω ). It is clearly seen that the absolutecontribution of the UV term becomes sizeably only at large temperatures. However, in the fullkernel (solid curve, with both UV and IR term), the influence of UV part is visible alreadyat low T . This is because of interference effects and effects of higher Matsubara frequenciesin the tDS equation (1). Nevertheless, the overall shape of B ( p = 0 , ω ) and, consequentlythe inflection point, is entirely governed by the IR interaction term. As mentioned above, all25 .00 0.05 0.10 0.15 0.20 MT1 IR UV B ( , ) [ G e V ] T [GeV] |p|=0 MT1 AWW < qq > / < qq > T [GeV] FIG. 9: (color online) Relative contributions of IR and UV terms to the solution B ( p = 0 , ω ) of thetDS equation (1) (left panel) and the quark condensate h q ¯ q i (right panel) with IR term only (dashedcurve), IR+UV terms (solid curve) (right panel). In the left panel, the dashed and dash-dot-dot curvesrepresent the separate contributions of IR and UV terms, respectively. The solid curve is the totalcontribution of the IR+UV terms. Effective parameters of the kernel are for the MT1 model. the considered interactions provide a critical temperature significantly lower than the one inlattice QCD calculations. Obviously, modifications of the kernel in the region above T c similarto Eq. (30) do not affect the behaviour of B ( , ω ) and h q ¯ q i below such temperatures. Thus, amodification of the IR term in the whole range of T is required. A possible modification of thekernel is as follows: (i) in the IR term the Debye mass is omitted, (ii) the parameter D receivesa T -dependence similar to Eq. (30), and (iii) the UV term, being inspired by perturbative QCDcalculations, remains unchanged, i.e. D L ( q , Ω mn , m g ) = D IR ( q + Ω mn ) + D UV ( q + Ω mn + m g ) , (35) D IR ( k ) = 4 π D ( T ) k ω e − k /ω , D UV ( k + m g ) = 8 π γ m F ( k + m g )ln[ τ + (1 + k + m g Λ QCD ) ] . The new effective parameters of the modified kernel should smoothly approach their vacuumvalues as T approaches zero and must provide a suppression of the IR interaction term above the(pseudo-) critical temperature. As in the previous case above, a simple expression simulatingsuch a behaviour may be written by utilizing two suppression functions with a Heavyside stepfunction-like behaviour, one acting below T c and, the second one acting above T c , for example26 ( T ) = D (cid:20) a (cid:26) (cid:18) − T − T p β (cid:19)(cid:27) + b (cid:26) − tanh (cid:18) − T − T p β ′ (cid:19)(cid:27) exp[ − α T ] (cid:21) , (36)where T p , a , b , α , β and β ′ are new adjustable parameters. In Fig. 10 we present an illustrationof the change, according to Eq. (36), of the IR parameter D , computed with a particular choiceof the effective parameters, a = 0 . T p = 0 .
25 GeV, β = 0 .
04 GeV, b = 5, β ′ = 0 .
06 GeV,and α = 10 GeV − . At low temperatures, D is practically equal to its vacuum value, smoothlyincreases with temperature (making the interference with the UV term more pronounced), upto a maximum value ( ∼ D [ G e V ] T [GeV]
MT1 modified MT1
FIG. 10: (color online) A possible dependence of the strength D of the IR term in the MT1 andAWW models on the temperature T (solid curve) in comparison with the case of D = const (dashedline). The solution B ( p = 0 , ω ) and the quark condensate h q ¯ q i with such a modified interaction areexhibited in Fig. 11. The resulting solution B ( p = 0 , ω ), as well as the quark condensate h q ¯ q i for the modifiedinteraction are exhibited in Fig. 11, where the solid and dashed curves are for the T -dependentsolution for the MT1 and AWW models, respectively. It is clearly seen that the inflection pointsare shifted towards larger values of temperature, providing critical temperatures T c ∼
135 GeVfor AWW and T c ∼
140 GeV for MT1, which now are better compatible with the above quotedlattice values. This persuades us that the MT1 and AWW models with proper modifications of27he interaction kernel can provide a reasonable description of the quark propagators and quarkcondensate at finite temperatures. Such a modified interaction can be used then in the BSequation to analyse the behaviour of mesons embedded in a hot environment.
MT1 modified AWW modified AWW B ( , ) [ G e V ] T [GeV] |p|=0 < qq > / < qq > MT1 modified AWW modified AWW T [GeV] FIG. 11: (color online) The tDS solution B ( p = 0 , ω ) (left panel) and the quark condensate h q ¯ q i (right panel) obtained with the modified interaction, Eq. (36). The solid and dashed curves representresults for the MT1 and AWW models, respectively. The dot-dashed curve is for the AWW solutionwith D = const . Obviously, the effective parameters in Eq. (36) can be tuned further to obtain an improvedagreement with lattice calculations. This is not the goal of the present paper. We reiterate thatwe are interested in choosing an effective interaction suitable for solving the BS equation atfinite temperature in a large interval of T , which can allow for performing qualitative analysesof the behaviour of mesons in hot (and dense) matter as well as to infer from this the relevantorder parameters and other conditions for a possible phase transition at large temperature. VI. SUMMARY
We have investigated the impact of various choices of the effective quark-gluon interactionwithin the truncated rainbow approximations on the solution of the truncated Dyson-Schwinger(tDS) equation at finite temperature. The ultimate goal is to establish a reliable interactionkernel adequate in a large range of temperatures which, being used in the Bethe-Salpeterequation, allows for an analysis of the behaviour of hadrons in hot matter, including possible28hase transitions and dissociation effects. For this we investigate to what extent the models,which provide an excellent description of mesons at zero temperatures, can be applied to thetruncated tDS equation at finite temperatures. We find that in the chiral limit at temperaturesbelow a critical value T c both models, with and without the ultraviolet term, describe fairlywell the quark propagators. The critical temperature obtained from the condition of a zero ofthe mass function B and/or of the quark condensate is in agreement with calculations withinthe lattice or unquenched QCD. However, at temperatures above T c the considered models withvacuum parameters fail to describe such important characteristics of the quark propagators asthe quark spectral functions, thermal masses, plasmino mode etc. To achieve agreement ofthe model calculations with QCD lattice results, a modification of the interaction kernels isrequired. A simple change of the interaction is to suppress the contribution of the IR termat large temperatures, thus making the interaction dependent on the temperature. At finitequark masses, the considered models seems to provide too small values for the (pseudo-) criticaltemperature which are by O (50%) smaller than the ones found in lattice QCD. Modificationsof the interaction in the same manner as for the chiral case, i.e. suppressing the IR term above T c , do not affect the values of the (pseudo-) critical temperatures defined by the maximum ofthe susceptibility of the function B or as the inflection point for the quark condensate. Toobtain a larger value of the critical temperature the interaction kernel has to be modified alsofor smaller temperatures, even below T ∼
100 MeV. The Debye mass m g plays a crucial rolein parametrizing the IR term. An inclusion of m g as a Gaussian exponential, cf. Eq. (3),results in an essential suppression of the solution of the tDS equation at T >
100 MeV, makingproblematic the attempts of obtaining larger T c , close to the lattice values. It seems that theDebye mass ought to be included only in the perturbative part of the interaction. The T -dependence of the IR term must be re-parametrized. The results of lattice calculations for the T -dependence of the quark condensate suggest that the T -dependence has to be chosen in sucha way that at small temperatures the IR term approaches its vacuum value remaining constantor smoothly changing up to T ∼ −
150 MeV; then it must be completely screened at largertemperatures.A more detailed parametrization of the IR term requires a separate and meticulous analysisof the tDS equation at finite T and will be done elsewhere.29 cknowledgments This work was supported in part by the Heisenberg - Landau program of the JINR - FRGcollaboration, GSI-FE and BMBF. DSM and LPK appreciate the warm hospitality at theHelmholtz Centre Dresden-Rossendorf.
VII. APPENDIXA. Rainbow truncation
The gap equation can be written as [2] S − ( p ) = Z iγp + Z m bare + Z Z Λ d q (2 π ) g D µν ( p − q ) λ a γ µ S ( q ) λ a ν ( q, p ) , (37)where D µν is the dressed gluon propagator; Γ ν , the quark-gluon vertex; R Λ represents a Poincar´einvariant regularization of the four-dimensional integral, with Λ the regularization mass-scale; m bare (Λ) denotes the current-quark bare massand Z i ( µ , Λ ) stand for the corresponding renor-malisation constants, with µ the renormalisation point, and λ a is a GellMann matrix acting incolor space. The solution of Eq. (37) has the general form S ( p ) − = iγ · pA ( p , µ ) + B ( p , µ )and is renormalized according to S ( p ) − = iγ · p + m ( µ ) at a sufficiently large value of µ , with m ( µ ) the renormalized quark mass at the scale µ . Since the behaviour at high momenta p > is fixed by perturbation theory and the renormalisation flow, in concrete calculations onneeds specify the gap equation at low momenta, i.e. in the infra-red region. As mentionedabove, in studies of the quark DS equation one has to employ reliable model forms of thegluon propagator and quark-gluon vertex, suitable for the whole range of momentum squared p . In rainbow-ladder truncation, which is leading-order in the most widely used scheme, cf.[5, 16, 19], this is achieved by adopting the requirements Z = 1, Z m bare = m , where m is aphenomenological mass parameter, Γ ν ( q, p ) = γ ν and Z g D µν ( k ) = G ( k ) D freeµν ( k ) = (cid:2) D IR ( k ) + 4 π ˜ α QCD ( k ) (cid:3) (cid:18) δ µν − k µ k ν k (cid:19) , (38)where ˜ α QCD ( k ) is a smooth continuation of the perturbative-QCD running coupling to allvalues of spacelike- k fulfilling the constraint of being finite at the origin. The infra-red term D IR ( k ) is constrained by the condition of the consistency with Ward identities for the tDS andtBS equations and to be negligibly small in the perturbative region, i.e. D IR ( k ) ≪ α QCD ( k )30t k ≥ . Otherwise, D IR ( k ) is a pure phenomenological term, the form of whichcan be only qualitatively inferred from lattice calculations or from solution of a (truncated)set of Dyson-Schwinger equations for the quark and gluon propagators within some additionalreasonable approximations. After choosing an explicit form of the interaction, the numericalvalues of the phenomenological parameters are determined from fitting empirical data. [1] P. Maris, C. D. Roberts, Phys. Rev. C (1997) 3369.[2] P. Maris, P. C. Tandy, Phys. Rev. C 60 (1999) 055214.[3] S. M. Dorkin, L. P. Kaptari, B. K¨ampfer, Phys. Rev.
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