Renormalization of Coulomb interactions in s-wave superconductor Na$_x$CoO$_2$

We study the renormalized Coulomb interactions due to retardation effect in Na$_x$CoO$_2$. Although the Morel-Anderson's pseudo potential for $a_{1g}$ orbital $\mu^*_{a1g}$ is relatively large because the direct Coulomb repulsion $U$ is large, that for interband transition between $a_{1g}$ and $e_g'$ orbitals $\mu^*_{a1g,eg'}$ is very small since the renormalization factor for pair hopping $J$ is square of that for $U$. Therefore, the s-wave superconductivity due to valence-band Suhl-Kondo mechanism will survive against strong Coulomb interactions. The interband hopping of Cooper pairs due to shear phonons is essential to understand the superconductivity in Na$_x$CoO$_2$.