We address an issue of how to accurately include the self energy effect of the screened electron-electron Coulomb interaction in the phonon-mediated superconductors from first principles. In the Eliashberg theory for superconductors, self energy is usually decomposed using the $2times 2$ Pauli matrices in the electron-hole space. We examine how the diagonal ($sigma_{0}$ and $sigma_{3}$) components, which results in the quasiparticle correction to the normal state, 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 non-analytic near the Fermi surface but their directional derivatives and resulting corrections to the quasiparticle velocity are nevertheless well defined. In the low-energy spectrum, we observe large cancellation between effects of these components and, without the numerically more tedious $sigma_{3}$ component, the effective mass is incorrectly increased. Feasible paths to manage this cancellation in the ab initio Eliashberg calculations are discussed.
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