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We investigate spectral properties of a fermion coupled with a massive gaugeboson with a mass m at finite temperature (T) in the perturbation theory. Themassive gauge boson is introduced as a U(1) gauge boson in the Stueckelbergformalism with a gauge parameter \alpha. We find that the fermion spectralfunction has a three-peak structure for T \sim m irrespective of the choice ofthe gauge parameter, while it tends to have one faint peak at the origin andtwo peaks corresponding to the normal fermion and anti-plasmino excitationsfamiliar in QED in the hard thermal loop approximation for T \gg m. We showthat our formalism successfully describe the fermion spectral function in thewhole T region with the correct high-T limit except for the faint peak at theorigin, 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 ofthe gauge parameter in the one-loop order, while for T \gg m, the one-loopanalysis is valid only for \alpha \ll 1/g where g is the fermion-boson couplingconstant, implying that the one-loop analysis can not be valid for large gaugeparameters as in the unitary gauge.

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