Abstract
We perform a comprehensive study of the homogeneous finite modular group $A'_5$ 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 $A'_5$. Then we perform a systematical analysis of the $A'_5$ 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 $A'_5$ 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 $\widetilde{\Gamma}_5\cong A'_5\times Z_5$. A neutrino mass model with $\widetilde{\Gamma}_5$ modular symmetry is presented, and the phenomenological predictions of the model are analyzed numerically.
Highlights
The quark masses and charged lepton masses as well as the neutrino mass squared differences have been precisely measured, yet we still do not know the exact value of the lightest neutrino mass while it is constrained by the cosmology data and the direct neutrino mass measurement experiments through beta decay
It has been established that the modular symmetry group SLð2; ZÞ can be consistently combined with the generalized CP symmetry such that SLð2; ZÞ is enhanced to GLð2; ZÞ [59], and the CP transformation is represented by the matrix
We only show the results for normal ordering (NO) neutrino masses, similar results can be obtained for inverted ordering (IO)
Summary
The quark masses and charged lepton masses as well as the neutrino mass squared differences have been precisely measured, yet we still do not know the exact value of the lightest neutrino mass while it is constrained by the cosmology data and the direct neutrino mass measurement experiments through beta decay. The group theory of A05 and the CG coefficients in our working basis are presented in the Appendix A We give another approach of constructing modular forms of weight 1 and level 5 from. The expressions of linearly independent higher integral weight modular forms with k 1⁄4 4, 5, 6 and higher rational weight modular forms with r 1⁄4 6=5; 7=5; ...; 14=5; 3 are given in Appendix D
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have