Spectral Function of Fermion Coupled with Massive Vector Boson at Finite Temperature in Gauge Invariant Formalism

We investigate spectral properties of a fermion coupled with a massive gauge boson with a mass m at finite temperature (T) in the perturbation theory. The massive gauge boson is introduced as a U(1) gauge boson in the Stueckelberg formalism with a gauge parameter \alpha. We find that the fermion spectral function has a three-peak structure for T \sim m irrespective of the choice of the gauge parameter, while it tends to have one faint peak at the origin and two peaks corresponding to the normal fermion and anti-plasmino excitations familiar in QED in the hard thermal loop approximation for T \gg m. We show that our formalism successfully describe the fermion spectral function in the whole T region with the correct high-T limit except for the faint peak at the origin, although some care is needed for choice of the gauge parameter for T \gg m. We clarify that for T \sim m, the fermion pole is almost independent of the gauge parameter in the one-loop order, while for T \gg m, the one-loop analysis is valid only for \alpha \ll 1/g where g is the fermion-boson coupling constant, implying that the one-loop analysis can not be valid for large gauge parameters as in the unitary gauge.