In a model of correlated electrons coupled to phonons, electronic repulsion suppresses the effective electron-phonon coupling by a factor of roughly 2 to 4 at U=2t, while the Fermi-liquid correlation strength is left nearly unchanged.
renormalized Migdal- Eliashberg
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Electron-phonon coupling in correlated materials: insights from the Hubbard-Holstein model
In a model of correlated electrons coupled to phonons, electronic repulsion suppresses the effective electron-phonon coupling by a factor of roughly 2 to 4 at U=2t, while the Fermi-liquid correlation strength is left nearly unchanged.