Restoring minimal modular neutrino masses via Kähler $R$-parity violation

Abstract

The possibility of describing neutrino masses through a single multiplet of modular forms in a modular symmetric framework is reanalysed. The minimal model achieving this is Feruglio's model based on modular $A_4$ symmetry, which contains a chiral flavon triplet $ϕ$, with all neutrino masses and mixing determined by a superfield triplet of modular forms $Y(τ)$. We point out that, in local supersymmetry, these superfields by themselves allow an additional $R$-parity-violating contribution to neutrino masses originating from the Kähler potential of the model. The $R$ violation manifests through a bilinear term $ε_i L_i H_u$ in the effective superpotential, with parameters $ε_i$ determined in terms of $Y(τ)$ and its derivatives. The Kähler potential also mixes neutrinos with the fermionic component of $ϕ$, leading to seesaw neutrino masses when these fermions are massive. Two separate cases are analysed, with neutrinos obtaining mass through the Weinberg operator or through the $ϕ$-induced seesaw. Unlike the original model, adding the $R$-violating contribution allows accurate reproduction of neutrino mixing angles and mass ratios for both the normal and inverted orderings in either case, with the Weinberg operator predicting a sum of neutrino masses close to the cosmological limit and the seesaw mechanism predicting ones comfortably below it.