Kohn-Luttinger Correction to Tc in a Phonon Superconductor

Weak coupling theory predicts the critical temperature of a phonon superconductor to be $T_c = 1.13 e^{-3/2} \omega_D e^{-{1/\lambda}}$, where $\omega_D$ is the Debye frequency, $\lambda$ is the dimensionless electron-phonon coupling constant, and the factor $e^{-3/2}$ comes from fermionic self-energy and frequency dependence of the interaction. Other corrections are small either in $\omega_D/E_F$, by Migdal's theorem, or in $\lambda$. However, this formula assumes that $\omega_D \ll E_F$, where $E_F$ is the Fermi energy. We obtain $T_c$ in the dilute regime, when the Fermi energy is smaller than $\omega_D$. We argue that in this situation Migdal's theorem is no longer valid, and Kohn-Luttinger-type corrections to the pairing interaction must be included to obtain the correct prefactor for $T_c$.