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

Abstract

We propose an A4 modular invariant flavor model of leptons, in which both CP and modular symmetries are broken spontaneously by the vacuum expectation value of the modulus τ. The value of the modulus τ is restricted by the observed lepton mixing angles and lepton masses for the normal hierarchy of neutrino masses. The predictive Dirac CP phase δCP is in the ranges [0°, 50°], [170°, 175°] and [280°, 360°] for Re [τ] < 0, and [0°, 80°], [185°, 190°] and [310°, 360°] for Re [τ] > 0. The sum of three neutrino masses is predicted in [60, 84] meV, and the effective mass for the 0νββ decay is in [0.003, 3] meV. The modulus τ links the Dirac CP phase to the cosmological baryon asymmetry (BAU) via the leptogenesis. Due to the strong wash-out effect, the predictive baryon asymmetry YB can be at most the same order of the observed value. Then, the lightest right-handed neutrino mass is restricted in the range of M1 = [1.5, 6.5] × 1013 GeV. We find the correlation between the predictive YB and the Dirac CP phase δCP. Only two predictive δCP ranges, [5°, 40°] (Re [τ] > 0) and [320°, 355°] (Re [τ] < 0) are consistent with the BAU.

Electron EDM arising from modulus τ in the supersymmetric modular invariant flavor models

“…Lepton mass matrices of four cases of weights (k e , k µ , k τ ) are completely consistent with observed lepton masses and PMNS mixing angles. Then, the CP violating Dirac phase is predicted clearly at the fixed value of τ [128]. The predicted CP phases of five cases are different as seen in table 3.…”

“…As discussed in subsection 4.1, the CP invariant lepton mass matrices have been given in the A 4 modular symmetry [128]. The tiny neutrino masses are obtained by type-I seesaw.…”

“…The CP invariant lepton mass matrices have been proposed in the A 4 modular symmetry [68,128]. We adopt those ones in order to discuss the soft SUSY breaking terms.…”

“…three generations of the left-handed lepton doublets are assigned to be an A 4 triplet L, and the right-handed charged leptons e c , µ c , and τ c are A 4 singlets 1, 1 and 1 , respectively. The three generations of the right-handed Majorana neutrinos are also assigned to be an A 4 triplet N c [128]. The weight of the superfields of left-handed leptons is fixed to be 1 as a reference value.…”

“…We take the level 3 finite modular groups, Γ 3 for the flavor symmetry since the property of A 4 flavor symmetry has been well known [121][122][123][124][125][126][127]. Indeed, viable CP invariant lepton models have been investigated linking to the leptogenesis JHEP10(2021)183 recently [128]. In this flavor symmetry, we study the electron EDM by fixing τ in the soft SUSY breaking term while the observed lepton masses and PMNS mixing angles are completely reproduced.…”