  Recent experiments revealed that the plain $s$-wave state without any
sign-reversal emerges in various metals near the 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 ($\chi$-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 so the Migdal
theorem fails. Due to both $\chi$-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".
