Green function theory of dirty two-band superconductivity

We study the effects of random nonmagnetic impurities on the superconducting transition temperature $T_c$ in a two-band superconductor, where we assume the equal-time spin-singlet s-wave pair potential in each conduction band and the hybridization between the two bands as well as the band asymmetry. In the clean limit, the phase of hybridization determines the stability of two states: called $s_{++}$ and $s_{+-}$. The interband impurity scatterings decrease $T_c$ of the two states exactly in the same manner when the Hamiltonian preserves time-reversal symmetry. We find that a superconductor with larger hybridization shows more moderate suppression of $T_c$. This effect can be explained by the presence of odd-frequency Cooper pairs which are generated by the band hybridization in the clean limit and are broken by impurities.