$$A_4$$ modular invariance and the strong CP problem

Abstract We present simple effective theory of quark masses, mixing and CP violation with level $$N=3$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo>=</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> </mml:math> ( $$A_4$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>A</mml:mi> <mml:mn>4</mml:mn> </mml:msub> </mml:math> ) modular symmetry, which provides solution to the strong CP problem without the need for an axion. The vanishing of the strong CP-violating phase $${\bar{\theta }}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mrow> <mml:mi>θ</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>¯</mml:mo> </mml:mrow> </mml:mover> </mml:math> 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 $$\tau $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>τ</mml:mi> </mml:math> . This provides the requisite large value of the CKM CP-violating phase while the strong CP phase $${\bar{\theta }}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mrow> <mml:mi>θ</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>¯</mml:mo> </mml:mrow> </mml:mover> </mml:math> 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 $$A_4$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>A</mml:mi> <mml:mn>4</mml:mn> </mml:msub> </mml:math> singlets $$\textbf{1}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mn>1</mml:mn> </mml:math> , $${\mathbf{1'}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mn>1</mml:mn> <mml:mo>′</mml:mo> </mml:msup> </mml:math> and $$\mathbf{1''}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mn>1</mml:mn> <mml:mrow> <mml:mo>′</mml:mo> <mml:mo>′</mml:mo> </mml:mrow> </mml:msup> </mml:math> with appropriate weights. The VEV of $$\tau $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>τ</mml:mi> </mml:math> is restricted to reproduce the observed CKM parameters. We discuss cases in which the modulus VEV is close to the fixed points i , $$\omega $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ω</mml:mi> </mml:math> and $$i\infty $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>i</mml:mi> <mml:mi>∞</mml:mi> </mml:mrow> </mml:math> . In particular, we focus on the VEV of $$\tau $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>τ</mml:mi> </mml:math> , which gives the absolute minima of the supergravity-motivated modular- and CP-invariant potentials for the modulus $$\tau $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>τ</mml:mi> </mml:math> , so called, modulus stabilisation. We present a successful model, which is consistent with the modulus stabilisation close to $$\tau =\omega $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>τ</mml:mi> <mml:mo>=</mml:mo> <mml:mi>ω</mml:mi> </mml:mrow> </mml:math> .