A₄ modular invariance and the strong CP problem

We present simple effective theory of quark masses, mixing and CP violation with level $$N=3$$ N = 3 ( A₄ A 4 ) modular symmetry, which provides solution to the strong CP problem without the need for an axion. The vanishing of the strong CP-violating phase θ ̄ θ ¯ 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 τ τ . This provides the requisite large value of the CKM CP-violating phase while the strong CP phase θ ̄ θ ¯ 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 A 4 singlets 1 1 , 1' 1 ′ and 1'' 1 ′ ′ with appropriate weights. The VEV of τ τ is restricted to reproduce the observed CKM parameters. We discuss cases in which the modulus VEV is close to the fixed points i , ω ω and i∞ i ∞ . In particular, we focus on the VEV of τ τ , which gives the absolute minima of the supergravity-motivated modular- and CP-invariant potentials for the modulus τ τ , so called, modulus stabilisation. We present a successful model, which is consistent with the modulus stabilisation close to τ =ω τ = ω .