Fermion Masses and Mixing from Double Cover and Metaplectic Cover of $A_5$ Modular Group

Abstract

We perform a comprehensive study of the homogeneous finite modular group $A_5^{\prime }$ 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 $A_5^{\prime }$. Then we perform a systematical analysis of the $A_5^{\prime }$ 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 $A_5^{\prime }$ 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 $F_1(\tau )$ and $F_2(\tau )$. 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 ${\tilde{\Gamma }}_5\cong A_5^{\prime }\times Z_5$. A neutrino mass model with ${\tilde{\Gamma }}_5$ modular symmetry is presented, and the phenomenological predictions of the model are analyzed numerically.

63 pages, 2 figures

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