Relativistic $<\sigma v_\text{rel}>$ in the calculation of relics
  abundances: a closer look
  In this paper we clarify the relation between the invariant relativistic
relative velocity $V_\text{r}$, the M\o{}ller velocity $\bar{v}$, and the
non-relativistic relative velocity $v_r$. Adopting $V_{\text{r}}$ as the true
physical relative velocity for pair-collisions in a non-degenerate relativistic
gas, we show that in the frame co-moving with the gas (i) the thermally
averaged cross section times relative velocity $<\sigma v_\text{rel}>$ that
appears in the density evolution equation for thermal relics is reformulated
only in terms of $V_\text{r}$ and $\mathcal{P}(V_{\text{r}})$ in a manifestly
Lorentz invariant form; (ii) the frame-dependent issues of the standard
formulation in terms of the M\o{}ller velocity, as well as "superluminal"
relative velocities, are not present in this formulation. Furthermore,
considering the annihilation of dark matter into a particle-antiparticle pair
$f\bar{f}$, in the cases $m_f=0$, $m_f=m$ and $m_f \gg m$, we find that the
coefficients of the low velocity expansion of $<\sigma V_{\text{r}}>$ admit an
exact analytical representation in terms of the Meijer $G$ functions that can
be reduced to combinations of modified Bessel functions of the second kind.
