Modular invariant quark and lepton models in double covering of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msub><mml:mi>S</mml:mi><mml:mn>4</mml:mn></mml:msub></mml:math> modular group
Xiang-Gan Liu, Chang-Yuan Yao, Gui-Jun Ding
Abstract
We perform a comprehensive analysis of the homogeneous finite modular group ${\mathrm{\ensuremath{\Gamma}}}_{4}^{\ensuremath{'}}\ensuremath{\equiv}{S}_{4}^{\ensuremath{'}}$, which is the double covering of ${S}_{4}$ group. The weight 1 modular forms of level 4 are constructed in terms of the Dedekind eta function, and they transform as a triplet ${\stackrel{^}{\mathbf{3}}}^{\ensuremath{'}}$ of ${S}_{4}^{\ensuremath{'}}$. The integral weight modular forms until weight 6 are built from the tensor products of weight 1 modular forms. We perform a systematical classification of ${S}_{4}^{\ensuremath{'}}$ modular models for lepton masses and mixing with and without generalized $CP$, where the left-handed leptons are assigned to a triplet of ${S}_{4}^{\ensuremath{'}}$ and right-handed charged leptons transform as singlets under ${S}_{4}^{\ensuremath{'}}$, and we consider both scenarios where the neutrino masses arise from a Weinberg operator or type-I seesaw mechanism. The phenomenological implications of the minimal models for lepton masses, mixing angles, $CP$ violation phases, and neutrinoless double decay are discussed. The ${S}_{4}^{\ensuremath{'}}$ modular symmetry is extended to the quark sector, and we present several predictive models which use nine or ten free parameters including real and imaginary parts of $\ensuremath{\tau}$ to describe quark masses and the Cabibbo-Kobayashi-Maskawa mixing matrix. We give a quark-lepton unified model which can explain the flavor structure of quarks and leptons simultaneously for a common value of $\ensuremath{\tau}$.