Superconductivity in the uniform electron gas: Irrelevance of the Kohn-Luttinger mechanism

We study the Cooper instability in jellium model controlled regime of small to intermediate values Coulomb parameter ${r}_{s}\ensuremath{\le}2$. confirm that superconductivity naturally emerges from purely repulsive interactions described by Kukkonen-Overhauser vertex function. By employing implicit renormalization approach and discrete Lehmann representation we reveal even small-${r}_{s}$ limit, dominant mechanism behind is based on dynamic screening interaction---accurately captured random phase approximation, whereas Kohn-Luttinger contribution negligibly and, thus, not relevant.