  In model Hamiltonians, like Fr\"ohlich's, the electron-phonon interaction is
assumed to be screened from the beginning. The same occurs when this
interaction is obtained by using the state-of-the-art density functional
perturbation theory as starting point. In this work I formally demonstrate that
these approaches are affected by a severe over-screening error. By using an
out-of-equilibrium Many-Body technique I discuss how to merge the many-body
approach with density-functional perturbation theory in order to correct the
over-screening error. A symmetric statically screened phonon self-energy is
obtained by down-folding the exact Baym-Kadanoff equations. The statically
screened approximation proposed here is shown to have the same long-range
spatial limit of the exact self-energy and to respect the
fluctuation-dissipation theorem. The doubly screened approximation, commonly
used in the literature, is shown, instead, to be over-screened, to violate
several Many-Body properties and to have a wrong spatial long-range decay. The
accuracy of the proposed approximation is tested against the exact solution of
an extended model Fr\"ohlich Hamiltonian and it is applied to a paradigmatic
material: MgB$_2$. I find that the present treatment enhances the linewidths by
$57 \%$ with respect to what has been previously reported for the anomalous
$E_{2g}$ mode. I further discover that the $A_{2u}$ mode is also anomalous (its
strong coupling being completely quenched by the over-screened expression). The
present results deeply question methods based on state-of-the-art approaches
and impact a wide range of fields such as thermal conductivity, phononic
instabilities and non-equilibrium lattice dynamics.
