Overscreening-free electron-phonon interaction in realistic materials

State-of-the-art model Hamiltonians, like Fr\"ohlich's, or density functional theory approaches to electron-phonon interaction suffer from a severe overscreening error. This is due to the incorrect treatment of the screening of the ionic potential. We derive a correct formulation of the electron-phonon interaction and demonstrate its validity by numerically implementing the new scheme in a paradigmatic material: MgB$_2$, a system whose double-gap, low-$T_c$ superconductivity depends on exceptionally high phonon linewidths. We find that the present treatment enhances the linewidths by $57 \%$ with respect to what has been previously reported for the anomalous $E_{2g}$ mode. We further discover that the $A_{2u}$ mode is also anomalous (its strong coupling being completely quenched by the overscreened expression). Our 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.