Relativistic and nonrelativistic annihilation of dark matter: a sanity check using an effective field theory approach

Understand

We find an exact formula for the thermally averaged cross section times the relative velocity $\langle \sigma v_{\text{rel}} \rangle$ with relativistic Maxwell-Boltzmann statistics.

• The formula is valid in the effective field theory approach when the masses of the annihilation products can be neglected compared with the dark matter mass and cut-off scale.
• The expansion at $x=m/T\gg 1$ directly gives the nonrelativistic limit of $\langle \sigma v_{\text{rel}}\rangle$ which is usually used to compute the relic abundance for heavy particles that decouple when they are nonrelativistic.
• We compare this expansion with the one obtained by expanding the total cross section $\sigma(s)$ in powers of the nonrelativistic relative velocity $v_r$.