  In order for clarifying what are 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 $\Sigma_R$.
Since $\Sigma_R$ 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. In the present
analysis we correctly respect this fact, and study, in the ladder
approximation, the effect of HTL resummed gauge boson propagator. Our results
with the use of numerical analysis, show the 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 the critical coupling
constant $\alpha_c$, without any discontinuity, and ii) the critical
temperature $T_c$ at fixed value of $\alpha$ is significantly lower than the
previous results, namely the restoration of chiral symmetry occurs at lower
temperature than previously expected. The second fact shows the importance of
correctly taking the essential thermal effect into the analysis of chiral phase
transition, which are, in the previous analyses, neglected due to the
inappropriate approximations. The procedure how to maximally respect the gauge
invariance in the present approximation, is also discussed.
