  Simple cubic phosphorus exhibits superconductivity with a maximum $T_c$ of up
to 12 K under pressure. The pressure dependence of $T_c$ cannot be consistently
explained with a simple electron-phonon mechanism, which has stimulated
investigations into the role of electronic correlations and plasmonic
contributions. Here, we solve the gap equation of density functional theory for
superconductors using different electron-electron and electron-phonon
contributions to the kernel. We find that the phonon contribution alone yields
an overestimation of $T_c$, while the addition of the static electronic
contribution results in an underestimation. Taking into account the full
frequency dependence of the screened interaction, the one-shot $GW$
approximation predicts $T_c$ values in good agreement with the experiments in
the pressure range appropriate for the cubic phase. We also explore the use of
quasi-particle bands in the calculation of the electronic and phononic kernels,
and show that this modification significantly improves $T_c$ in the
high-pressure region.
