Accidental Peccei-Quinn symmetry from discrete flavour symmetry and Pati-Salam

We show how an accidental U(1) Peccei-Quinn (PQ) symmetry can arise from a discrete A4 family symmetry combined with a discrete flavour symmetry Z3×Z52, 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 A4 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.