Revisiting homogeneous electron gas in pursuit of properly normed ab initio Eliashberg theory

We address an issue of how to accurately include the self-energy effect of the screened electron-electron Coulomb interaction in phonon-mediated superconductors from first principles. In the Eliashberg theory for superconductors, self-energy is usually decomposed using the $2\ifmmode\times\else\texttimes\fi{}2$ Pauli matrices in the electron-hole space. We examine how the diagonal (${\ensuremath{\sigma}}₀$ and ${\ensuremath{\sigma}}₃$) components resulting in the quasiparticle correction to the normal state, $Z$ and $\ensuremath{\chi}$ terms, behave in the homogeneous electron gas in order to establish a norm of treating those components in real metallic systems. Within the ${G}₀{W}₀$ approximation, we point out that these components are nonanalytic near the Fermi surface but their directional derivatives and resulting corrections to the quasiparticle velocity are, nevertheless, well defined. Combined calculations using the ${G}₀{W}₀$ approximation and Eliashberg equations show us that the effective mass and pairing strength strikingly depend on both $Z$ and $\ensuremath{\chi}$ in different manners. The calculations without the numerically demanding $\ensuremath{\chi}$ term is thus shown to be incapable of describing the homogeneous electron gas limit. This result poses a challenge to accurate first-principles Eliashberg theory.