On the oscillating electric dipole moment induced by axion-fermion couplings

It has been recently claimed that the axion coupling to fermions is responsible for an oscillating electric dipole moment (EDM) in the background of axion dark matter. In this work, we re-examine the derivation of this effect. Contrary to previous studies, we point out the physical relevance of an axion boundary term, which is crucial in restoring the axion shift symmetry and drastically affects the EDM phenomenology. To describe the latter, we introduce the notion of a time-averaged effective axion EDM, which encodes the boundary term and whose magnitude depends on the oscillation regime. For slow oscillations, the boundary term washes out the standard oscillating EDM, resulting in an exact cancellation in the static limit. Conversely, during fast oscillations, the boundary term amplifies the effective EDM relatively to the standard EDM contribution. This observable is especially interesting in the case of the electron EDM. For an $\mathcal{O}(1)$ axion-electron coupling, the overall size of the effective EDM in the regime of intermediate or fast oscillations is comparable to the present static EDM limit.