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

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