On the impact of the Migdal effect in reactor CE$\nu$NS experiments

The search for coherent elastic neutrino nucleus scattering (CE$\nu$NS) using reactor antineutrinos represents a formidable experimental challenge, recently boosted by the observation of such a process at the Dresden-II reactor site using a germanium detector. This observation relies on an unexpected enhancement at low energies of the measured quenching factor with respect to the theoretical Lindhard model prediction, which implies an extra observable ionization signal produced after the nuclear recoil. A possible explanation for this additional contribution could be provided by the so-called Migdal effect, which however has never been observed. Here, we study in detail the impact of the Migdal contribution to the standard CE$\nu$NS signal calculated with the Lindhard quenching factor, finding that the former is completely negligible for observed energies below $\sim 0.3\,\mathrm{keV}$ where the signal is detectable, and thus unable to provide any contribution to CE$\nu$NS searches in this energy regime. To this purpose, we compare different formalisms used to describe the Migdal effect that intriguingly show a perfect agreement, making our findings robust.