Towards unification of quark and lepton flavors in $$A_4$$ modular invariance

Abstract

Abstract We study quark and lepton mass matrices in the $$A_4$$ <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 towards the unification of the quark and lepton flavors. We adopt modular forms of weights 2 and 6 for quarks and charged leptons, while we use modular forms of weight 4 for the neutrino mass matrix which is generated by the Weinberg operator. We obtain the successful quark mass matrices, in which the down-type quark mass matrix is constructed by modular forms of weight 2, but the up-type quark mass matrix is constructed by modular forms of weight 6. The viable region of $$\tau $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>τ</mml:mi> </mml:math> is close to $$\tau =i$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>τ</mml:mi> <mml:mo>=</mml:mo> <mml:mi>i</mml:mi> </mml:mrow> </mml:math> . Lepton mass matrices also work well at nearby $$\tau =i$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>τ</mml:mi> <mml:mo>=</mml:mo> <mml:mi>i</mml:mi> </mml:mrow> </mml:math> , which overlaps with the one of the quark sector, for the normal hierarchy of neutrino masses. In the common $$\tau $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>τ</mml:mi> </mml:math> region for quarks and leptons, the predicted sum of neutrino masses is 87–120 meV taking account of its cosmological bound. Since both the Dirac CP phase $$\delta _{CP}^\ell $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>δ</mml:mi> <mml:mrow> <mml:mi>CP</mml:mi> </mml:mrow> <mml:mi>ℓ</mml:mi> </mml:msubsup> </mml:math> and $$\sin ^2\theta _{23}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mo>sin</mml:mo> <mml:mn>2</mml:mn> </mml:msup> <mml:msub> <mml:mi>θ</mml:mi> <mml:mn>23</mml:mn> </mml:msub> </mml:mrow> </mml:math> are correlated with the sum of neutrino masses, improving its cosmological bound provides crucial tests for our scheme as well as the precise measurement of $$\sin ^2\theta _{23}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mo>sin</mml:mo> <mml:mn>2</mml:mn> </mml:msup> <mml:msub> <mml:mi>θ</mml:mi> <mml:mn>23</mml:mn> </mml:msub> </mml:mrow> </mml:math> and $$\delta _{CP}^\ell $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>δ</mml:mi> <mml:mrow> <mml:mi>CP</mml:mi> </mml:mrow> <mml:mi>ℓ</mml:mi> </mml:msubsup> </mml:math> . The effective neutrino mass of the $$0\nu \beta \beta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>0</mml:mn> <mml:mi>ν</mml:mi> <mml:mi>β</mml:mi> <mml:mi>β</mml:mi> </mml:mrow> </mml:math> decay is $$\langle m_{ee}\rangle =15$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>⟨</mml:mo> <mml:msub> <mml:mi>m</mml:mi> <mml:mrow> <mml:mi>ee</mml:mi> </mml:mrow> </mml:msub> <mml:mo>⟩</mml:mo> <mml:mo>=</mml:mo> <mml:mn>15</mml:mn> </mml:mrow> </mml:math> –31 meV. It is remarked that the modulus $$\tau $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>τ</mml:mi> </mml:math> is fixed at nearby $$\tau =i$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>τ</mml:mi> <mml:mo>=</mml:mo> <mml:mi>i</mml:mi> </mml:mrow> </mml:math> in the fundamental domain of SL(2, Z ), which suggests the residual symmetry $$Z_2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>Z</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:math> in the quark and lepton mass matrices. The inverted hierarchy of neutrino masses is excluded by the cosmological bound of the sum of neutrino masses.

Topics & Concepts