Thermal corrections to dark matter annihilation with real photon emission/absorption

The dark matter relic density is being increasingly precisely measured. This relic density is theoretically determined by a Boltzmann equation which computes the dark matter distribution according to the (thermally averaged) cross section for annihilation/production of dark matter within a given model. We present here the complete higher order thermal corrections, calculated using real time thermal field theory, to the cross section for the annihilation of dark matter into Standard Model fermions via charged scalars: $\chi \chi \to f \overline{f}$ and $\chi \chi \to f \overline{f} (\gamma)$. The latter process includes real photon emission into, and absorption from, the heat bath at temperature $T$. We use the Grammer and Yennie technique to separate the soft infra-red divergences, which greatly simplifies the calculation. We show explicitly the cancellation of both soft and collinear divergences between real and virtual contributions at next-to-leading order (NLO) in the thermal field theory and remark on the non-trivial nature of the collinear divergences when the thermal contribution from fermions is considered. We present the leading thermal contributions to order ${\cal{O}}(\alpha T^2)$ for the case when the dark matter particle is a Majorana or Dirac type fermion. In the former case, both the leading order (LO) and NLO cross sections are helicity suppressed, while neither is suppressed in the Dirac case. It is interesting that the ratio of NLO to LO cross sections is the same for both Majorana and Dirac type dark matter.