Bloch-Nordsieck Thermometers: One-loop Exponentiation in Finite Temperature QED

We study the scattering of hard external particles in a heat bath in a real-time formalism for finite temperature QED. We investigate the distribution of the 4-momentum difference of initial and final hard particles in a fully covariant manner when the scale of the process, $Q$, is much larger than the temperature, $T$. Our computations are valid for all $T$ subject to this constraint. We exponentiate the leading infra-red term at one-loop order through a resummation of soft (thermal) photon emissions and absorptions. For $T>0$, we find that tensor structures arise which are not present at $T=0$. These carry thermal signatures. As a result, external particles can serve as thermometers introduced into the heat bath. We investigate the phase space origin of $\log(Q/m)$ and $\log(Q/T)$ terms.