Superconductivity in electron liquids with and without intermediaries

Solutions to the Eliashberg equation are reported for the one- and two-band electron liquids, with and without phonons, using effective interactions closely constrained by sum rules. For the two-band case, s-wave pairing can arise in the absence of phonons, but not for a single band at typical metallic densities. When phonons and electrons are treated on a completely equal footing, the transition temperatures in several higher-density simple metals are found to be correct to within 20%. For lower-density metals, however, large deviations are found from predictions that make use of the standard approximation μ*≈0.1.