Strong CP violation: The CP-odd nuclear force and axions

Axions are introduced to reduce the amount of CP violation in strong interactions. In the Standard Model, a small θ¯ term is technically natural but this is no longer true in generic Standard Model extensions. In the framework of the Standard Model Effective Field Theory (SM-EFT), this is reflected by quadratically divergent contributions to θ¯ that signal unsuppressed threshold corrections at the matching scale. The existence then of CP-odd SM-EFT operators necessitates an infrared solution to the strong CP problem, such as a Peccei-Quinn mechanism and the related presence of axions. In the presence of sources of CP violation beyond the QCD θ¯ term, a Peccei-Quinn mechanism leads to CP-violating couplings of axions to Standard Model fields. We investigate the remnant of SM-EFT CP-violating interactions in the form of axion CP-odd couplings and compare the reach of laboratory axion-mediated force experiments and traditional electric dipole moment (EDM) experiments. CP-violating nuclear forces contribute to EDMs in nuclei, diamagnetic atoms, and certain molecules. The naive dimensional analysis predicts these forces to be dominated by long-range one-pion exchange processes with short-range forces entering only at next-to-next-to-leading order in the chiral expansion. Based on renormalization arguments we argue that a consistent picture of CP-violating nuclear forces requires a leading-order short-distance operator contributing to 1S0 - 3P0 transitions due to the attractive and singular nature of the strong tensor force in the 3P0 channel. The short-distance operator leads to O(1) corrections to static and oscillating, relevant for axion searches, electric dipole moments. We discuss strategies on how the finite part of the associated low-energy constant can be determined in the case of CP violation from the QCD θ¯ term, by the connection to the charge-symmetry violation in nuclear systems.