Double cover of modular S4 for flavour model building

Abstract

We develop the formalism of the finite modular group Γ 4 ≡ S 4 , a double cover of the modular permutation group Γ 4 S 4 , for theories of flavour. The integer weight k > 0 of the level 4 modular forms indispensable for the formalism can be even or odd. We explicitly construct the lowest-weight (k = 1) modular forms in terms of two Jacobi theta constants, denoted as ε(τ ) and θ(τ ), τ being the modulus. We show that these forms furnish a 3D representation of S 4 not present for S 4 . Having derived the S 4 multiplication rules and Clebsch-Gordan coefficients, we construct multiplets of modular forms of weights up to k = 10. These are expressed as polynomials in ε and θ, bypassing the need to search for non-linear constraints. We further show that within S 4 there are two options to define the (generalised) CP transformation and we discuss the possible residual symmetries in theories based on modular and CP invariance. Finally, we provide two examples of application of our results, constructing phenomenologically viable lepton flavour models.

Leptonic sum rules from flavour models with modular symmetries

“…Due to the interplay of the sum rules we find a clearly preferred region for the mass scale in the case of NO in particular for SR 1 and 4. This also explains why in [44] the authors find a preference for a mass scale. Nevertheless, in all four cases the cosmological upper bound on the mass scale can be saturated in the 3σ range.…”

“…It is also interesting to note that in many cases the best fit value for the expectation value of the modulus, τ , is near a symmetry point, see, for instance, [44,45], where formalisms have been developed which allows to expand around symmetry points.…”

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

“…A clear prediction of the neutrino mixing angles and the CP violating phase was given in the simple lepton mass matrices with the A 4 modular symmetry [28]. On the other hand, the Double Covering groups T [34,35] and S 4 [36,37] were realized in the modular symmetry. Furthermore, modular forms for ∆ (96) and ∆(384) were constructed [38], and the extension of the traditional flavor group was discussed with modular symmetries [39].…”

Adjoint SU(5) GUT model with modular S4 symmetry

“…Besides the topics on the fermion flavor structures, the related phenomenological issues has been discussed JHEP03(2021)002 in those works, such as the dark matter models [27], baryon number violation [26] and leptogenesis [22,33,34]. On the other side, the double covering of finite modular group which is homogeneous, has been used as flavor symmetry as well [46][47][48][49][50][51].…”