Accidental Peccei-Quinn Symmetry from Discrete Flavour Symmetry and Pati-Salam

Abstract

We show how an accidental $U(1)$ Peccei-Quinn (PQ) symmetry can arise from a discrete $A_4$ family symmetry combined with a discrete flavour symmetry ${\mathbb{Z}}_3\times {\mathbb{Z}}_5^2$, in a realistic Pati-Salam unified theory of flavour. Imposing only these discrete flavour symmetries, the axion solution to the strong $CP$ problem is protected from PQ-breaking operators to the required degree. A QCD axion arises from a linear combination of $A_4$ triplet flavons, which are also responsible for fermion flavour structures due to their vacuum alignments. We find that the requirement of an accidental PQ symmetry arising from a discrete flavour symmetry constrains the form of the Yukawa matrices, providing a link between flavour and the strong $CP$ problem. Our model predicts specific flavour-violating couplings of the flavourful axion and thus puts a strong limit on the axion scale from kaon decays.

Comments: 14 pages, 2 figures