Non-invertible Peccei-Quinn symmetry, natural 2HDM alignment, and the visible axion

A bstract We identify $$ {m}_{12}^2 $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>m</mml:mi> <mml:mn>12</mml:mn> <mml:mn>2</mml:mn> </mml:msubsup> </mml:math> as a spurion of non-invertible Peccei-Quinn symmetry in the type II 2HDM with gauged quark flavor. Thus a UV theory which introduces quark color-flavor monopoles can naturally realize alignment without decoupling and can furthermore revive the Weinberg-Wilczek axion. As an example we consider the SU(9) theory of color-flavor unification, which needs no new fermions. This is the first model-building use of non-invertible symmetry to find a Dirac natural explanation for a small relevant parameter.