Infrared finiteness of a thermal theory of scalar electrodynamics to all
  orders
  Models explaining dark matter typically include interactions with charged
scalar and fermion fields. The Infra-Red (IR) finiteness of thermal field
theories of charged fermions (fermionic QED) has been proven to all orders in
perturbation theory. Here we reexamine the IR behaviour of charged scalar
theories at finite temperature. Using the method of Grammer and Yennie, we
identify and factorise the infra-red divergences to all orders in perturbation
theory. The inclusion of IR finite pieces arising from the 4-point interaction
terms of scalars with photon fields is key to the exponentiation. We use this
in a companion paper to prove the IR finiteness of the corresponding thermal
theory which is of relevance in dark matter calculations.
