Effective description of Nelson-Barr models and the theta parameter

A bstract Nelson-Barr theories solve the strong CP problem with CP spontaneously broken at Λ cp 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 Λ cp up to dimension five operators and the relevant matching calculations at the VLQ scale relevant to tracking the contributions to the parameter $$ \overline{\theta} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>θ</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> </mml:math> . In the EFT after the CP breaking scalars have been integrated out, we confirm that the running of $$ \overline{\theta} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>θ</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> </mml:math> 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 $$ \overline{\theta} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>θ</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> </mml:math> at tree-level due to a dimension five operator. Analyzing the experimental constraint from $$ \overline{\theta} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>θ</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> </mml:math> for generic settings, we see that a large separation between Λ cp and the VLQ scale already suppresses the one-loop contribution to $$ \overline{\theta} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>θ</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> </mml:math> sufficiently. We confirm that a simple extension based on a nonconventional CP has vanishing one-loop contribution to $$ \overline{\theta} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>θ</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> </mml:math> . Part of our results can be also applied to generic VLQ extensions of the SM.