Eq. (3): event rate for spin-dependent photino scattering

Let photinos of mass $m>0$ scatter from nuclei of mass number $A>0$ and mass $M = A\,m_N$ through exchange of a scalar quark of mass $M_{\tilde Q}>0$ whose quark partner has charge $q = Qe$, with $e^2 = 4\pi\alpha$. With spin-averaged squared amplitude $\langle|\mathcal M|^2\rangle = (4q^4/M_{\tilde Q}^4)\,\lambda^2J(J+1)$ and the elastic cross section $\sigma = \frac{m^2M^2}{\pi(m+M)^2}\langle|\mathcal M|^2\rangle$, there is a constant $C$ with $|C/1.1 - 1|\le 0.05$ such that, for all parameters as above and all real $Q$, $\lambda^2J(J+1)$, $\rho$, $v$,

This is the paper's rate formula for dark matter with only spin-dependent interactions.

Formalization Note The factor $\lambda^2J(J+1)$ is a free real parameter; 5% relative tolerance on the printed coefficient.