Superconductivity in the dilute electron gas

We determine the effective interaction of quasiparticles on the Fermi surface for the electron gas in a rigid positive background. The quasiparticle scattering amplitude is calculated from the two coupled Bethe-Salpeter equations for the two-particle vertex functions in the particle-hole ($p-h$) channels for densities from $r_s=1\mathrm{}$ to $r_s=37\mathrm{}$. The density and spin-density mean fields are fitted to the compressibility and spin susceptibility of Green's-function Monte Carlo calculations and the local-field factors $G\left(q\right)\mathrm{}$ of microscopic models. We find $p$-wave superconductivity for $10<\mathrm{}{r}_s<35\mathrm{}$ and $s$-wave superconductivity for $r_s>35\mathrm{}$.