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.

In this paper we calculated the n-point hard-thermal-loop (HTL) vertex functions in QCD/QED for n= 2, 3 and 4 in the physical representation in the real time formalism (RTF). The result showed that the n-point HTL vertex functions can be classified into two groups, a) those with odd numbers of external retarded indices, and b) the others with even numbers of external retarded indices. The n-point HTL vertex functions with one retarded index, which obviously belong to the first group a), are nothing but the HTL vertex functions that appear in the imaginary time formalism (ITF), and vise versa. All the HTL vertex functions belonging to the first group a) are of O(g 2 T 2 ) , and satisfy among them the simple QED-type Ward-Takahashi identities, as in the ITF. Those vertex functions belonging to the second group b) never appear in the ITF, namely their existence is characteristic of the RTF, and their HTL's have the high temperature behavior of O(g 2 T 3 ), one-power of T higher than usual. Despite this difference we could verify that those HTL vertex functions belonging to the second group b) also satisfy among themselves the QED-type Ward-Takahashi identities, thus guaranteeing the gauge invariance of the HTL's in the real time thermal QCD/QED.

Chiral phase transition in thermal QCD is studied by using the Dyson-Schwinger (DS) equation in the real time hard thermal loop approximation. Our results on the critical temperature and the critical coupling are significantly different from those in the preceding analyses in the ladder DS equation, showing the importance of properly taking into account the essential thermal effects, namely the Landau damping and the unstable nature of thermal quasiparticles.