Modular invariant A4 models for quarks and leptons with generalized CP symmetry

Abstract

We perform a systematical analysis of the A4 modular models with generalized CP for the masses and flavor mixing of quarks and leptons, and the most general form of the quark and lepton mass matrices is given. The CP invariance requires all couplings real in the chosen basis and thus the vacuum expectation value of the modulus τ uniquely breaks both the modular symmetry and CP symmetry. The phenomenologically viable models with minimal number of free parameters and the results of fit are presented. We find 20 models with 7 real free parameters that can accommodate the experimental data of lepton sector. We then apply A4 modular symmetry to the quark sector to explain quark masses and CKM mixing matrix, the minimal viable quark model is found to contain 10 free real parameters. Finally, we give two predictive quark-lepton unification models which use only 16 real free parameters to explain the flavor patterns of both quarks and leptons.

Modular origin of mass hierarchy: Froggatt-Nielsen like mechanism

“…1 Indeed, various modular invariant models which successfully reproduce the SM have been constructed in these years, e.g. for Γ 2 [14][15][16][17], Γ 3 [1,2,14,[17][18][19][20][21][22][23][24], Γ 4 [17,20,[25][26][27][28], Γ 5 [17,28,29], Γ 7 [30], and the double covering groups of the modular groups [17,[31][32][33][34][35][36][37]. A combined symmetry of the modular symmetry with the conventional flavor symmetries or CP-symmetry is also considered in [38][39][40][41][42][43][44][45].…”

Modulus τ linking leptonic CP violation to baryon asymmetry in A4 modular invariant flavor model

“…A viable lepton model was proposed in the modular A 4 symmetry [80], in which the CP violation is realized by fixing τ , that is, the breaking of the modular symmetry. Afterward, the systematic search of the viable A 4 model was done [81].…”

Explicit perturbations to the stabilizer τ = i of modular $$ {A}_5^{\prime } $$ symmetry and leptonic CP violation

“…Once the modulus τ acquires its vev, the Yukawa couplings are determined and thus leptonic flavor mixing pattern is obtained. Finite modular groups, such as Γ 2 S 3 [13][14][15], Γ 3 A 4 [12,[16][17][18][19][20][21][22][23][24][25][26][27], Γ 4 S 4 [28][29][30][31][32][33], Γ 5 A 5 [34][35][36], Γ 7…”