Derivation of matter potential from QFT

I’m trying to derive the matter potential as experienced by neutrinos from QFT. The paper I am looking at is unfortunately not publicly available online (https://doi.org/10.1103/PhysRevD.40.259) and so I cut and paste the relevant part of the derivation I’m struggling with.

But before that, some context. I want to calculate the dispersion relation of the neutrino as it propagates in matter. Its mass is modified by the medium in which it travels.

So the loop correction is from the medium to the neutrino is:

here $f_{F}$ is the Fermi-Dirac distribution (of the electrons in the medium)
and $L$ is $P_L$, the projection operator.

I can go to the rest frame of the medium $u=(1,0,0,0)$ and use the relation
$$delta(k^2-m_e^2)=frac{1}{omega_k}[delta(k^0-omega_k)+ delta(k^0+omega_k)]$$
and using

$$n_e=int frac{dk^3}{(2pi)^3}frac{1}{e^{beta(omega_kpmmu)}+1}$$

one finds

$$S_1=frac{g^2}{4M^2_W}(n_e-n_{overline{e}})gamma^0L.$$

I’m stuck on how to proceed from the first expression. Any help appreciated