  Using the momentum average (MA) approximation, we find an analytical
expression for the disorder-averaged Green's function of a Holstein polaron in
a three-dimensional simple cubic lattice with random on-site energies. The
on-site disorder is assumed to be weak compared to the kinetic energy of the
polaron, and is treated perturbationally. Within this scheme, the states at the
bottom of the polaron band are found to have an infinite lifetime, signaling a
failure of perturbation theory at these energies. The higher-energy polaron
states have a finite lifetime. We study this lifetime and the disorder-induced
energy shift of these eigenstates for various strengths of disorder and
electron-phonon coupling. We compare our findings to the predictions of Fermi's
golden rule and the average T-matrix method, and find a significant
quantitative discrepancy at strong electron-phonon coupling, where the polaron
lifetime is much shorter than Fermi's golden rule prediction. We attribute this
to the renormalization of the on-site potential by the electron-phonon
coupling.
