Spectral function of a fermion coupled with a massive vector boson at finite temperature in a gauge invariant formalism

Abstract

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 $\ensuremath{\alpha}$. We find that the fermion spectral function has a three-peak structure for $T\ensuremath{\sim}m$ 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 antiplasmino excitations familiar in QED in the hard thermal loop approximation for $T\ensuremath{\gg}m$. 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 $T\ensuremath{\gg}m$. We clarify that for $T\ensuremath{\sim}m$, the fermion pole is almost independent of the gauge parameter in the one-loop order, while for $T\ensuremath{\gg}m$, the one-loop analysis is valid only for $\ensuremath{\alpha}\ensuremath{\ll}1/g$ where $g$ is the fermion-boson coupling constant, implying that the one-loop analysis cannot be valid for large gauge parameters as in the unitary gauge.