^{3}

We investigate the properties of quark mass functions at finite temperature in quantum chromodynamics calculated by the Schwinger–Dyson equation in the real-time formalism without the instantaneous exchange approximation, in which the loop integration is performed in Minkowski space. In our model, real and imaginary parts of the mass function in the time-like momentum region are directory evaluated. We show that the imaginary part of the mass function has a positive value in some time-like momentum region below the critical temperature for chiral symmetry restoration at finite temperature, which may imply quark confinement.

Chiral phase transition in quantum chromodynamics (QCD) is one of the most interesting phenomena still to be understood. So far, chiral phase transitions have been studied by various methods. One such method is implementation of the Schwinger–Dyson equation (SDE) [

A great deal of work on chiral symmetry breaking has been done with the SDE in the momentum representation, in which a one-loop contribution is integrated over Euclidean space. Some calculations of the mass function for fermions with the SDE have been done in Minkowski space. In Ref. [

At finite temperature and density for equilibrium systems, the imaginary-time formalism (ITF) is implemented, which continues to Euclidean space at the zero temperature limit. On the other hand, the real-time formalism (RTF) for nonequilibrium systems is formulated in Minkowski space. The SDE in the RTF has been studied with the instantaneous exchange approximation (IEA) [

Analytic continuation from Euclidean space to Minkowski space is valid in perturbative calculations if the pole positions in the complex plane of energy are known. However, it is not trivial in the nonperturbative region, particularly when the mass function depends on the energy. So far, the structure of the quark mass function in the strong coupling region in the entire range of energy and momentum space has not been fully studied in Minkowski space at finite temperature and density. In our previous papers [

In Minkowski space, if the imaginary part of the mass function is small, the propagator varies rapidly near resonance peaks, which is one of the difficulties with numerical calculation. In our method, the resonance contributions in momentum integration in Minkowski space are efficiently evaluated. Furthermore, we can directory evaluate real and imaginary parts of the mass function in the time-like momentum region. At zero temperature, we found that the imaginary part of the mass function in some time-like momentum region becomes positive, in which the spectral function has negative value. The positivity violation of the spectral function contradicts the existence of asymptotic fields of the quark [

In this paper, we extend our previous method to calculate the quark mass functions with the SDE in the RTF at finite temperature without the IEA. In

In the RTF, two types of fields specified by 1 and 2 are implemented in the theory, in which the type-1 field is the usual field and the type-2 field corresponds to a ghost field in the heat bath.

We calculate the 1-1 component of a self-energy of quark

The 1-1 component of the quark propagator in the RTF is given as
^{1}

In this paper, we define the quark propagator as

The 1-1 component of the gluon propagator in the RTF is given as

Integrating over the azimuthal angle of the momentum

The real part

In Minkowski space, if the imaginary part of the mass function

In this section, some numerical results are presented. We solve the SDE presented in Eq. (^{2}

First, we evaluate an effective quark mass for the resonance peak of the effective quark propagator at

For each iteration, we calculate the quark mass function normalized as
^{3}

Here,

In

The

As shown in

In order to search for the critical point at which the chiral symmetry is restored, we evaluate

In

The

In

The

The gluon mass in the deep infrared region is nontrivial. Here, we consider two cases with the massive gluon in order to compare with the massless gluon case shown in

The

The

As shown in these figures, though the critical temperatures depend on the gluon masses, the integrated quark mass functions are qualitatively similar behaviors in three cases, in which the imaginary parts of the mass functions

In

The solid and dash-dotted curves denote the

The solid and dash-dotted curves denote the

The solid and dash-dotted curves denote the

In this paper, we have studied quark mass functions solved by the Schwinger–Dyson equation (SDE) at finite temperature in the real-time formalism (RTF) without the instantaneous exchange approximation (IEA). The RTF enables us to evaluate nonequilibrium systems.

In our calculations, we improved the four-momentum integration of the SDE near the resonance peaks, which is one of the difficulties with numerical calculation in Minkowski space.

We defined integrated mass functions as order parameters and examined chiral symmetry restoration at finite temperature in the RTF. In our model, the critical temperature

The gluon mass at finite temperature in the deep infrared region is nontrivial. In this paper, we investigated two cases in order to compare the massless gluon case. We have presented the results for

We found that the imaginary part of the mass function

Furthermore, the real and imaginary parts of the integrated mass functions vanish at the same critical temperature. Therefore, when the temperature

In our calculations, the temperature dependences are not smooth curves. Therefore, we present crude dependences with few data points in order to search for the critical temperature of the chiral symmetry restoration. Within the present accuracy, we cannot determine the order of the phase transition.

We also evaluated the dependences of the squared four-momentum

At finite temperature, we also found that

We may also expect a similar situation for the cases at

Though the model presented in this paper is too simple to evaluate the quark mass function quantitatively, our study suggests that the SDE in the RTF without the IEA seems to be useful to investigate the chiral phase transition in QCD at finite temperature, in which we may directly evaluate the instability of the massive quark state and the signature of quark confinement.

Evaluations of the quark mass function at finite density are difficult in the imaginary-time formalism due to the sign problem, particularly in lattice simulations. In future work, we shall try to extend our method to evaluate the quark mass function at finite density in the RTF.

This work was partially supported by the MEXT-Supported Program for the Strategic Research Foundation at Private Universities, 2014-2017 (S1411024).

Open Access funding: SCOAP

The quark mass function is given by
^{4}

As shown in Eq. (

Here,

Here, the strong coupling constant ^{5}

For

For

The real and imaginary parts of the quark propagator

We implement effective gluon masses for nonperturbative regions obtained by lattice calculation.

In Ref. [

As an example of investigation for the gluon mass effects in SDE, we define the effective gluon masses as

Here, the parameters

These parameters below the critical temperature

^{1} The real and imaginary parts of a propagator

^{2} The initial input parameters are

^{3} We take

^{4} In order to avoid the infrared contributions due to

^{5} We implement the QCD coupling constant