$$A_4$$ A 4 modular invariance and the strong CP problem - S. T. Petcov; M. Tanimoto (European Physical Journal C: Particles and Fields)
Abstract We present simple effective theory of quark masses, mixing and CP violation with level $$N=3$$ N = 3 ( $$A_4$$ 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 $${\bar{θ }}$$ 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 $${\bar{θ }}$$ 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 $$A_4$$ 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 .