Spontaneous CP violation by modulus τ in A4 model of lepton flavors

Abstract

We discuss the modular A4 invariant model of leptons combining with the generalized CP symmetry. In our model, both CP and modular symmetries are broken spontaneously by the vacuum expectation value of the modulus τ. The source of the CP violation is a non-trivial value of Re[τ] while other parameters of the model are real. The allowed region of τ is in very narrow one close to the fixed point τ = i for both normal hierarchy (NH) and inverted ones (IH) of neutrino masses. The CP violating Dirac phase δCP is predicted clearly in [98°, 110°] and [250°, 262°] for NH at 3 σ confidence level. On the other hand, δCP is in [95°, 100°] and [260°, 265°] for IH at 5 σ confidence level. The predicted ∑mi is in [82, 102] meV for NH and ∑mi = [134, 180] meV for IH. The effective mass 〈mee〉 for the 0νββ decay is predicted in [12.5, 20.5] meV and [54, 67] meV for NH and IH, respectively.

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

“…We show the best fit values of the input parameters and the predictions of the mixing parameters and neutrino masses in table 7 under the assumption of NO neutrino masses. The A 4 modular model with gCP symmetry was recently studied in [30], and a lepton model with 8 parameters was proposed. It corresponds to the D 2 -W 2 model in our work, see table 7 for the numerical results of this model.…”

“…We see that the quark and lepton mass hierarchies require hierarchical values of the coupling constants, this can be naturally achieved by the weighton mechanism in which modular symmetry singlet field with non-vanishing modular weight plays the role of flavon [25]. The mass hierarchies of the charged leptons could possibly be addressed by modular symmetry around certain fixed points of the complex modulus τ , but generally more free parameters are necessary [30,74,75]. The corresponding predictions for masses and mixing parameters of leptons and quarks are given as We find that all of these observables are predicted to lie in the experimentally preferred 3σ ranges.…”

CP symmetry and symplectic modular invariance

“…[39,40]. In particular, CP transformation laws of the modulus τ, of chiral matter multiplets and of modular forms have been determined [48] and several models where CP is spontaneously broken have been constructed [48][49][50][51][52][53][54]. Within the more general case of a multidimensional moduli space, consistent CP definitions have been examined recently in the context of symplectic modular invariant theories, where the relevant flavour group is the Siegel modular group [55,56].…”

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

“…It is worth mentioning that a modular-invariant model with large CP violation around the stabilizer τ = i of the modular A 4 group has been constructed in ref. [54]. Nevertheless, the question why the large CP violation can be generated with a modulus close to τ = i remains to be answered.…”