Relativistic and nonrelativistic annihilation of dark matter: a sanity
  check using an effective field theory approach
  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 $v_r$. We show
the correct invariant procedure that gives the nonrelativistic average $\langle
\sigma_{nr} v_r \rangle_{nr}$ coinciding with the large $x$ expansion of
$\langle \sigma v_{\text{rel}}\rangle$ in the comoving frame. We explicitly
formulate flux, cross section, thermal average, collision integral of the
Boltzmann equation in an invariant way using the true relativistic relative
$v_\text{rel}$, showing the uselessness of the M\o{}ller velocity and further
elucidating the conceptual and numerical inconsistencies related with its use.
