The event rate on a target of atomic mass $A$, with cross section per nucleus $\sigma$ is given by: $$dR=\dfrac{N_{A}}{A}\sigma\,mv\,dn$$ where $N_{A}$ is the Avogadro number, $m$ and $v$ are the mass and velocity of the incoming particle and $dn$ is the number density of incoming particles.

This formula can be found in many articles and reviews (for example) concerning dark matter collisions with nuclear matter, but it is non-relativistic.

How can it be generalized in order to contain only Lorentz invariant terms? I mean, since the cross-sections are usually given in terms of the momentum $p_{\mu}$ and contractions $p_{\mu}p^{\mu}$, it is better to integrate $dR$ in momentum space and not writing in terms of the velocities $v.$


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