Muonic vs electronic dark forces: a complete EFT treatment for atomic spectroscopy

Precision atomic spectroscopy provides a solid model independent bound on the existence of new dark forces among the atomic constituents. We focus on the keV-GeV region investigating the sensitivity to such dark sectors of the recent measurements on muonic atoms at PSI. To this end we develop for the first time, the effective field theory that describes the leading effect of a new (pseudo-)vector or a (pseudo-)scalar particle of any mass at atomic energies. We identify in the Lamb Shift measurement in muonic deuterium ($\mu$D) and the $2s$ Hyperfine Splitting (HFS) in muonic hydrogen ($\mu$H) the most promising measurements to probe respectively spin-independent and spin-dependent new forces. Furthermore, we evaluate the expression of the vector force HFS finding that a future measurement of the $2s$ HFS in regular hydrogen could provide the strongest atomic bound for such a force for masses above 100 MeV.