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

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