Chiral Phase Transitions in QED at Finite Temperature: Dyson-Schwinger Equation Analysis in the Real Time Hard-Thermal-Loop Approximation

Abstract

In order to clarify the essential thermal effects that govern the chiral phase transition at finite temperature, we investigate, in the real-time thermal QED, the consequences of the hard-thermal-loop (HTL) resummed Dyson-Schwinger equation for the physical fermion mass function Σ R . Since Σ R is the mass function of an "unstable" quasi-particle in thermal field theories, it necessarily has non-trivial imaginary parts, together with non-trivial wave function renormalization constants. Analyses so far have completely disregarded this fact, despite it being one of the basic outcomes of thermal field theory. The "approximation", which ignores the imaginary parts of Σ R , gives (non-trivial) constraint equations to be solved simultaneously. For this reason, this approximation cannot be applied consistently except in the trivial case, Σ R = 0. In the present analysis we correctly take this fact into account, and study, in the ladder approximation, the effect of the HTL resummed gauge boson propagator. Our results obtained using numerical analysis, reveal two facts: i) the chiral phase transition is of second order, since the fermion mass is dynamically generated at a critical value of the temperature T c , or at a critical coupling constant α c , without any discontinuity; and ii) the critical temperature T c at a fixed value of α is significantly lower than that obtained previously, which means that the restoration of chiral symmetry occurs at a lower temperature than previously expected. The second fact shows the importance of correctly incorporating the essential thermal effect into the analysis of the chiral phase transition. A procedure for taking account of the gauge invariance in the present approximation is also discussed.

Dynamical mass generation for fermions in quenched quantum electrodynamics at finite temperature

“…It will be interesting to see the impact on results if we include the wave function renormalization F and the function b in which case the analysis should account for the imaginary part of the self-energy [7]. It is also important to repeat the calculation accounting for a more complete fermion-photon vertex, studying the impact of this treatment on the gauge dependence of the critical values for chiral symmetry restoration.…”

“…To pursue it, as a first step we make the approximation where we take only the contribution of the zeroth mode, i.e., n = 0, in the sum over frequencies in Eq. (7) and correspondingly take the same value for ω. We relax this approximation in the following section.…”

“…It is now easy to see that the integral in Eq. (7) can now be expressed in Minkowski space in terms of the functions…”

Renormalization of QED Near Decoupling Temperature

“…It is explicitly checked that, in the existing scheme of calculations, the theory remains renormalizable at ≤ 4 ∼ 2 MeV. However, the perturbative corrections will exceed the original values of QED parameters at higher temperatures and hard thermal loops have to be dealt with, using already developed methods [12][13][14]. However, below the neutrino decoupling temperature, the real part of the propagators is enough to describe the perturbative behavior of the system and doubling of the field is not required.…”

“…At these temperatures, QED coupling starts to play its role in modifying QED parameters for nucleosynthesis. With the help of these effective parameters of QED, the abundance of helium in the early universe can be estimated [12] precisely at a given temperature. The temperature dependent QED corrections to the nucleosynthesis parameters improve the results of the 2 Physics Research International standard big bang model of cosmology [13][14][15][16] and they help to relate the observational data, for example, WMAP (Wilkinson Microwave Anisotropy Probe [15]) with the big bang theory.…”