Axion forces in axion backgrounds

Abstract: Axions can naturally be very light due to the protection of (approximate) shift symmetry. Because of the pseudoscalar nature, the long-range force mediated by the axion at the tree level is spin dependent, which cannot lead to observable effects between two unpolarized macroscopic objects. At the one-loop level, however, the exchange of two axions does mediate a spin-independent force. This force is coherently enhanced when there exists an axion background. In this work, we study the two-axion exchange force in a generic axion background. We find that the breaking of the axion shift symmetry plays a crucial role in determining this force. The background-induced axion force $V_{\rm bkg}$ vanishes in the shift-symmetry restoration limit. The shift symmetry can be broken either explicitly by non-perturbative effects or effectively by the axion background. When the shift symmetry is broken, $V_{\rm bkg}$ scales as $1/r$ and could be further enhanced by a large occupation number of the background axions. We investigate possible probes on this using fifth-force search and atomic spectroscopy experiments.