Recent experiments revealed that the plain s-wave state without any sign-reversal emerges in various metals near magnetic criticality. To understand this counter-intuitive phenomenon, we study the gap equation for the multiorbital Hubbard–Holstein model, by analyzing the vertex correction (VC) due to the higher-order electron-correlation effects. We find that the phonon-mediated orbital fluctuations are magnified by the VC for the susceptibility (χ-VC). In addition, the charge-channel attractive interaction is enlarged by the VC for the coupling-constant (U-VC), which is significant when the interaction has prominent q-dependences; therefore the Migdal theorem fails. Due to both χ-VC and U-VC, the plain s-wave state is caused by the small electron–phonon interaction near the magnetic criticality against the repulsive Coulomb interaction. We find that the direct Coulomb repulsion for the plain s-wave Cooper pair is strongly reduced by the “multiorbital screening effect”.

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