In the standard model, the renormalization of the QCD vacuum angle $\theta$ is extremely tiny, and small $\theta$ is technically natural. In the general standard model effective field theory (SMEFT), however, $\mathrm{\Delta }\theta$ is quadratically divergent, reflecting the fact that new sources of hadronic $CP$-violation typically produce $\mathcal{O}(1)$ threshold corrections to $\theta$. The observation of such $CP$-violating interactions would therefore be in tension with solutions to the strong $CP$ problem in which $\theta =0$ is an ultraviolet boundary condition, pointing to the Peccei-Quinn mechanism as the explanation for why $\theta$ is small in the infrared. We study the quadratic divergences in $\theta$ arising from dimension-6 SMEFT operators and discuss the discovery prospects for these operators at electric dipole moment experiments, the LHC, and future proton-proton colliders.

• Received 6 November 2018

DOI:https://doi.org/10.1103/PhysRevD.99.015042

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP³.

Published by the American Physical Society