^{1}

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In view of the properties of mesons in hot strongly interacting matter, the properties of the solutions of the truncated Dyson-Schwinger equation for the quark propagator at finite temperatures within the rainbow-ladder approximation are analysed in some detail. In Euclidean space within the Matsubara imaginary time formalism, the quark propagator is not longer a O(4) symmetric function and possesses a discrete 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 much more involved. The question whether the interaction kernel known from vacuum calculations can be applied at finite temperatures remains still open. We find that, at low temperatures, the model interaction with vacuum parameters provides a reasonable description of the quark propagator, while at 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 calculations with that from LQCD, the kernel is to be modified in such a way as to screen the infra-red part of the interaction at temperatures larger than

*. For this, we analyse the solutions of the truncated Dyson-Schwinger equation with existing interaction kernels in a large temperature range with particular attention on high temperatures in order to find hints to an adequate temperature dependence of the interaction kernel to be further implemented in the Bethe-Salpeter equation for mesons. This will allow investigating the possible in medium modifications of the meson properties as well as the conditions of quark deconfinement in hot matter.*

*T*_{c }The description of mesons as quark-antiquark bound states within the framework of the Bethe-Salpeter (BS) equation with momentum dependent quark mass functions, determined by the Dyson-Schwinger (DS) equation, is able to explain successfully many spectroscopic data, such as meson masses [

However, due to known technical problems, one restricts oneself to calculations within effective models which specify the dressed vertex function

The model is completely specified once a form is chosen for the effective coupling

Another important property of the DS and BS equations is their explicit Poincaré invariance. This frame-independency of the approach provides a useful tool in studying processes when a rest frame for mesons cannot or needs not be defined.

The merit of the approach is that, once the effective parameters are fixed (usually the effective parameters of the kernel are chosen, cf. Ref. [

At low temperatures the properties of hadrons in nuclear matter are expected to change in comparison with the vacuum ones, however the main quantum numbers, such as spin and orbital momenta, space and inner parities etc. are maintained. The hot environment may modify the hadron masses, life time (decay constant) etc. Contrarily, at sufficiently large temperature in hot and dense strongly interacting matter, phase transitions may occur, related to quark deconfinement phenomena, as e.g. dissociation of hadrons in to quark degrees of freedom. Therefore, these temperature regions are of great interest, both from a theoretical and experimental point of view. Hitherto, the truncated DS and BS formalism has been mostly used 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. [

In the present paper we are interested in a detailed investigation of the quark propagator in the whole range of temperatures, from zero temperatures up to above

In quantum field theory, a system embedded in a heat bath can be described within the imaginary-time formalism, known also as the Matsubara approach [

In the present paper we investigate the prerequisites to the interaction kernel of the DS formalism at finite temperatures to be able to investigate, in a subsequent step, different processes with the challenging problem of changes of meson characteristics at finite temperatures. Our goal is to determine with what extend the rainbow truncation of the DS equation is applicable in a large interval of temperature, starting from low values, with the effective parameters, known to accomplish an excellent description of the hadron properties in vacuum, towards temperatures above the critical values predicted by lattice calculations. We try to find a proper modification of the kernel at higher temperatures to be able to describe the properties of the quark propagator in the whole temperature range. A reliable parametrization of the T-dependence will allow to implemented it directly into the BS equation in the same manner as at

Our paper is organized as follows. In Section 2, we recall the truncated BS and DS equations in vacuum and at finite temperatures. The rainbow approximation for the DS equation kernel in vacuum is introduced and the system of equations for the quark propagator, to be solved at finite temperature, is presented. Numerical solution for the chirally symmetric case is discussed in Section 3, where the chiral quark condensate and spectral representation for the quark propagator are introduced. It is found that, to achieve a reasonable behaviour of the spectral functions above the critical temperature, a modification of the interaction kernel is needed. In Section 4, we consider the solution of the truncated DS equation for finite bare masses. The inflection points of the quark condensate and the mass function are considered as a definition of the pseudo-critical temperature at finite quark masses. The procedure of regularization of integrals in calculating the quark condensate from the solution of the DS equation is discussed in some detail. It is shown that, for finite quark masses, the inflection method determines the pseudo critical temperatures by ~50% smaller than the ones obtained by other approaches, e.g. by lattice QCD calculations. The possibility to reconcile the model and lattice QCD results is considered too. The impact of the infrared term in the interaction kernel in the vicinity and above the critical temperature is also briefly discussed. Summary and conclusions are collected in Section 6. A brief explanation of the meaning of the rainbow-ladder approxiamtion is presented in the Appendix.

^{1}Usually, for quarks of masses

To determine the bound-state mass of a quark-antiquark pair one needs to solve the DS and the homogeneous BS equations, which in the rainbow ladder approximation and in Euclidean space read

where ^{1} _{1}-quark and m_{2}-antiquark follow from the solution of BS equation,

Often, the coupled equations of the quark propagator S, the gluon propagator

Note that the nonperturbative behaviour of the kernel

Following examples in the literature [

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 behaviour of the QCD running coupling. In what follows we restrict ourselves to two models. i) The interaction consists of both the IR and UV terms: Such an interaction is known as the Maris-Tandy (MT) model [

The theoretical treatment of systems at non-zero temperatures differs from the case of zero temperatures. In this case, a preferred frame is determined by the local rest system of the thermal bath. This means that the

Accordingly, the interaction kernel is decomposed in to a transversal and longitudinal part

where

The gap equation has the same form as in case of

Then the system of equations for A, B and C to be solved reads (cf. also Ref. [

where

for

where

with E being the energy scale. For the temperature range considered in the present paper we adopt

In the present paper we use several sets of parameters for the interaction kernel (3):

1)

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, while at 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 model is not applicable. It should be noted that all these models provide values for the vacuum quark condensate in a narrow corridor,

As seen from Equation (9) in the chiral limit, i.e. at

We solve numerically the system of Equations (8)-(10) by an iteration procedure. Since the UV term in the MT1 and MT2 models is logarithmically divergent, a regularization of the integral over the internal momentum and summation over

for the Gaussian integration with

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.

In SU(3) gauge theory, the deconfinement transition is of first order at

At high enough temperatures one expects a chiral restoration. This means that at a certain high value of the temperature the mass function B should vanish, indicating a possible phase transition in the hot matter. The lowest temperature at which

The chiral condensate is defined by

where the trace is performed in spinor space. In

One infers from this figure that in a large range of T the solution ^{−}^{60} to determine the interval for

Another important quantity characterizing the hot matter is the spectral representation of the retarded quark propagator. The Euclidean propagator can be transferred to Min- kowski space by an analytical continuation of the solution of the gap equation to real energies,

In Minkowski space, the dispersion relation for the quark propagator determines the spectral representation

From this the importance of studying

where

i.e. the spectral functions

In the present paper we focus to two particular cases.

i) Chiral limit, where the scalar, or “mass”, part

where

ii) Zero momenta: The projection operators are

Note that at zero momenta the energy E of the quark can be associated to a mass

where

If one writes the dispersion relations for the model propagators (26)

then by inverting (27) one can obtain the (model) spectral density

where the integral in Equation (28) must be preliminarily carried out analytically to leave the dependence only on

The simplest parametrization for the spectral function at finite T is suggested by the case of a free quark propagator (21), i.e. one can expect that

With such a parametrization the spectral function

In our calculations we use the Levenberg-Marquardt algorithm for minimization of

It can be seen that both, MT1 and MT2 models (solid and dashed curves in

propagators at

where the additional adjustable parameters are

Another important characteristic is the behaviour of the plasmino mode as a function of the momentum

At finite quark masses the solution of the tDS equation differs from the chiral limit in at least two aspects. First, the Wigner-Weyl mode is not longer a solution. Second, the integrals over

where

where

In obtaining (32) we put

where

pendent light-quark condensate. Exactly the same procedure is applied to determine the quark condensate at finite T, see also Ref. [

In

The (pseudo-)critical temperature

Analysing the relative contributions of the IR and UV terms in the interaction we find that, while at

In

The new effective parameters of the modified kernel should smoothly approach their vacuum values 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 simulating such a behaviour may be written by utilizing two suppression functions with a Heavyside step function-like behaviour, one acting below

where

The resulting solution

the MT1 and AWW models with proper modifications of the interaction kernel can provide a reasonable description of the quark propagators and quark condensate at finite temperatures. Such a modified interaction can be used then in the BS equation to analyse the behaviour of mesons embedded in a hot environment.

Obviously, the effective parameters in Equation (36) can be tuned further to obtain an improved agreement with lattice calculations. This is not the goal of the present paper. We reiterate that we are interested in choosing an effective interaction suitable for solving the BS equation at finite temperature in a large interval of T, which can allow for performing qualitative analyses of the behaviour of mesons in hot (and dense) matter as well as to infer from this the relevant order parameters and other conditions for a possible phase transition at large temperature.

We have investigated the impact of various choices of the effective quark-gluon interaction within the truncated rainbow approximations on the solution of the truncated Dy- son-Schwinger (tDS) equation at finite temperature. The ultimate goal is to establish a reliable interaction kernel adequate in a large range of temperatures which, being used in the Bethe-Salpeter equation, allows for an analysis of the behaviour of hadrons in hot matter, including possible phase 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 the truncated tDS equation at finite temperatures. We find that in the chiral limit at temperatures below a critical value

A more detailed parametrization of the IR term requires a separate and meticulous analysis of the tDS equation at finite T and will be done elsewhere.

This work was supported in part by the Heisenberg-Landau program of the JINR-FRG collaboration, GSI-FE and BMBF. DSM and LPK appreciate the warm hospitality at the Helmholtz Centre Dresden-Rossendorf.

Dorkin, S.M., Viebach, M., Kaptari, L.P. and Kämpfer, B. (2016) Extending the Truncated Dyson-Schwinger Equation to Finite Temperatures. Journal of Modern Physics, 7, 2071-2097. http://dx.doi.org/10.4236/jmp.2016.715182

The gap equation can be written as [

where

where ^{2} fulfilling the constraint of being finite at the origin. The infra-red term