Exact formula with two dynamically screened electron-phonon couplings for positive phonon-linewidths approximations

In this paper, we present an exact formula for the phonon linewidths involving only dressed electron-phonon couplings and ensuring the positivity property. The formula is designed to account for both nonadiabatic and correlation effects, and it provides an alternative proof that Density Functional Perturbation Theory calculations of phonon linewidths are not affected by a double counting of screening diagrams. Furthermore, we extend the treatment to nonequilibrium scenarios and offer a rigorous justification for employing the phononic Boltzmann equation.