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Register a new userFlavor mixing and CP violation from the interplay of $S_4$ modular group and gCP
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We have performed a systematical analysis of lepton and quark masses models based on $Gamma_4cong S_4$ modular symmetry with gCP symmetry. We have considered both cases that neutrinos are Majorana particles and Dirac particles. All possible nontrivial representation assignments of matter fields are considered, and the most general form of fermion mass matrices are given. The phenomenologically viable models with the lowest number of free parameters together with the results of fit are presented. We find out nine lepton models with seven real free parameters including the real and imaginary parts of modulus for Majorana neutrinos, which can accommodate the lepton masses and neutrino oscillation data. The prediction for leptogenesis is studied in an example lepton model. The observed baryon asymmetry as well as lepton masses and mixing angles can be explained. For Dirac neutrinos, four lepton models with five real free couplings are compatible with experimental data. Ten quark models containing seven couplings are found to be able to accommodate the hierarchical quark masses and mixing angles and CP violation phase. Furthermore, the $S_4$ modular symmetry can provide a unified description of lepton and quark flavor structure, and a benchmark model is presented.
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We study the spontaneous $CP$ violation through the stabilization of the modulus $tau$ in modular invariant flavor models. The $CP$-invaraiant potentential has the minimum only at ${rm Re}[tau] = 0$ or 1/2. From this prediction, we study $CP$ violation in modular invariant flavor models. The physical $CP$ phase is vanishing. The important point for the $CP$ conservation is the $T$ transformation in the modular symmetry. One needs the violation of $T$ symmetry to realize the spontaneous $CP$ violation.
We study the modulus stabilization in an $A_4$ model whose $A_4$ flavor symmetry is originated from the $S_4$ modular symmetry. We can stabilize the modulus so that the $A_4$ invariant superpotential leads to the realistic lepton masses and mixing angles. We also discuss the phenomenological aspect of the present model as a consequence of the modulus stabilization.
We study a flavor model with $A_4$ symmetry which originates from $S_4$ modular group. In $S_4$ symmetry, $Z_2$ subgroup can be anomalous, and then $S_4$ can be violated to $A_4$. Starting with a $S_4$ symmetric Lagrangian at the tree level, the Lagrangian at the quantum level has only $A_4$ symmetry when $Z_2$ in $S_4$ is anomalous. We obtain modular forms of two singlets and a triplet representations of $A_4$ by decomposing $S_4$ modular forms into $A_4$ representations. We propose a new $A_4$ flavor model of leptons by using those $A_4$ modular forms. We succeed in constructing a viable neutrino mass matrix through the Weinberg operator for both normal hierarchy (NH) and inverted hierarchy (IH) of neutrino masses. Our predictions of the CP violating Dirac phase $delta_{CP}$ and the mixing $sin^2theta_{23}$ depend on the sum of neutrino masses for NH.
The prospects of measuring the leptonic angles and CP-odd phases at a neutrino factory are discussed in two scenarios: 1) three active neutrinos as indicated by the present ensemble of atmospheric plus solar data; 2) three active plus one sterile neutrino when the LSND signal is also taken into account. For the latter we develop one and two mass dominance approximations. The appearance of wrong sign muons in long baseline experiments and tau leptons in short baseline ones provides the best tests of CP-violation in scenarios 1) and 2), respectively.
We study the sensitivity of the squark flavor mixing to the CP violating phenomena of $K$, $B^0$ and $B_s$ mesons in the framework of the split-family scenario, where the first and second family squarks are very heavy, ${cal O}(10)$TeV, on the other hand, the third family squark masses are at ${cal O}(1)$TeV. In order to constrain the gluino-sbottom-quark mixing parameters, we input the experimental data of the CP violations of $K$, $B^0$, and $B_s$ mesons, that is $epsilon_K$, $phi_d$, and $phi_s$. The experimental upper bound of the chromo-EDM of the strange quark is also input. In addition, we take account of the observed values $Delta M_{B^0}$, $Delta M_{B_s}$, the CKM mixing $|V_{ub}|$, and the branching ratio of $bto sgamma$. The allowed region of the mixing parameters are obtained as $|delta_{13}^{dL(dR)}|=0sim 0.01$ and $|delta_{23}^{dL(dR)}|=0sim 0.04$. By using these values, the deviations from the SM are estimated in the CP violations of the $B^0$ and $B_s$ decays. The deviation from the SM one is tiny in the CP asymmetries of $B^0to phi K_S$ and $B^0to eta K^0$ due to the chromo-EDM of the strange quark. On the other hand, the CP asymmetries $B_s to phi phi$ and $B_s to phi eta $ could be largely deviated from the SM predictions. We also predict the time dependent CP asymmetry of $B^0to K^0bar K^0$ and the semi-leptonic CP asymmetries of $B^0 to mu ^-X$ and $B_s to mu ^-X$. We expect those precise measurements at Belle II, which will provide us interesting tests for the squark flavor mixing.
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