assume that the cross section takes a Breit-Wigner form, $$\label{eq:Breit}
\sigma(E) = \frac{g^{4}}{4 \pi} \frac{s}{(s-M^{2})^{2}+M^{2}\Gamma^{2}} \,,$$ where $s = 2 E m_\nu$ and the decay width of the mediator is $\Gamma = g^{2} M / 4\pi$. With this form, we generalize the phenomenological approach of KT87 to include the possibility that a resonance dominates the cross section. In that case, the $t$ channel contribution can be neglected. In the off-resonance case, neglecting the $t$ channel does not change the results much for the cases considered here. For the differential cross section, we take a flat distribution in $E$ between zero and $E^{\prime}$, which corresponds to the case of a scalar mediator. Vector mediators have a different distribution, but we do not consider this case due to strong constraints [@Laha:2013xua; @Karshenboim:2014tka].

This form of the cross section parametrizes all three kinematic regimes of how the mediator mass compares to the interaction energy in the center of momentum frame. For a very light mediator, the cross section is decreasing with neutrino energy, $\sigma \simeq g^{4}/(4\pi s)$, while for a very heavy mediator, the cross section is increasing with neutrino energy, $\sigma \simeq g^{4}s/(4\pi M^{4})$. These two limits correspond to the massless and 