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Relativistic $<σv_\text{rel}>$ 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 Møller 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 $<σv_\text{rel}>$ 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 Møller 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 $<σV_{\text{r}}>$ 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.