Fermion masses and mixing from the double cover and metaplectic cover of the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msub><mml:mi>A</mml:mi><mml:mn>5</mml:mn></mml:msub></mml:math> modular group
Chang-Yuan Yao, Xiang-Gan Liu, Gui-Jun Ding
Abstract
We perform a comprehensive study of the homogeneous finite modular group ${A}_{5}^{\ensuremath{'}}$ which is the double covering of ${A}_{5}$. The integral weight and level 5 modular forms have been constructed up to weight 6 and they are decomposed into the irreducible representations of ${A}_{5}^{\ensuremath{'}}$. Then we perform a systematical analysis of the ${A}_{5}^{\ensuremath{'}}$ modular models for lepton masses and mixing. The phenomenologically viable models with minimal number of free parameters and the results of fit are presented. We find out 15 models with 9 real free parameters which can accommodate the experimental data of lepton sector. After including generalized $CP$ symmetry, 9 viable models with 7 free parameters are found out. We apply ${A}_{5}^{\ensuremath{'}}$ modular symmetry to the quark sector, and a quark-lepton unification model is given. The framework of modular invariance is extended to include the rational weight modular forms of level 5. The ring of modular forms at level 5 can be generated by two algebraically independent weight $1/5$ modular forms denoted by ${F}_{1}(\ensuremath{\tau})$ and ${F}_{2}(\ensuremath{\tau})$. We give the expressions of the rational weight modular forms of level 5 up to weight 3 and arrange them into the irreducible multiplets of finite metaplectic group ${\stackrel{\texttildelow{}}{\mathrm{\ensuremath{\Gamma}}}}_{5}\ensuremath{\cong}{A}_{5}^{\ensuremath{'}}\ifmmode\times\else\texttimes\fi{}{Z}_{5}$. A neutrino mass model with ${\stackrel{\texttildelow{}}{\mathrm{\ensuremath{\Gamma}}}}_{5}$ modular symmetry is presented, and the phenomenological predictions of the model are analyzed numerically.