A₄ modular invariance and the strong CP problem

We present simple effective theory of quark masses, mixing and CP violation with level N=3 (A₄) modular symmetry, which provides solution to the strong CP problem without the need for an axion. The vanishing of the strong CP-violating phase is ensured by assuming CP to be a fundamental symmetry of the Lagrangian of the theory. The CP symmetry is broken spontaneously by the vacuum expectation value (VEV) of the modulus. This provides the requisite large value of the CKM CP-violating phase while the strong CP phase remains zero or is tiny. Within the considered framework we discuss phenomenologically viable quark mass matrices with three types of texture zeros, which are realized by assigning both the left-handed and right-handed quark fields to A₄ singlets 1, 1' and 1'' with appropriate weights. The VEV of is restricted to reproduce the observed CKM parameters. We discuss, in particular, cases in which the modulus VEV is close to the fixed points i, and i, which are broadly consistent with the results on modulus stabilisation.