Effective description of Nelson-Barr models and the theta parameter

Nelson-Barr theories solve the strong CP problem with CP spontaneously broken at Λcp and transmitted to the CP conserving version of the SM through vector-like quarks (VLQs). For an arbitrary number of CP breaking scalars and VLQs, we perform the full one-loop matching calculations at Λcp up to dimension five operators and the relevant matching calculations at the VLQ scale relevant to tracking the contributions to the parameter θ¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \overline{\theta} $$\end{document}. In the EFT after the CP breaking scalars have been integrated out, we confirm that the running of θ¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \overline{\theta} $$\end{document} at one-loop is induced by CP violating dimension five operators and similarly by dimension six operator after matching to the SMEFT. In the latter, an additional contribution enters in the matching to θ¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \overline{\theta} $$\end{document} at tree-level due to a dimension five operator. Analyzing the experimental constraint from θ¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \overline{\theta} $$\end{document} for generic settings, we see that a large separation between Λcp and the VLQ scale already suppresses the one-loop contribution to θ¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \overline{\theta} $$\end{document} sufficiently. We confirm that a simple extension based on a nonconventional CP has vanishing one-loop contribution to θ¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \overline{\theta} $$\end{document}. Part of our results can be also applied to generic VLQ extensions of the SM.