Is flavor discrete?

TL;DR

Is flavor discrete? analyzes whether discrete flavor symmetries can explain the observed fermion flavor structures, comparing traditional non-Abelian discrete symmetries with modular symmetry approaches. It argues that minimal realizations fail to fit current data and that achieving compatibility often requires additional freedom—via extended representations, higher modular weights, or contributions from hidden sectors—creating a large discrete parameter space and risks of missing representations. Through examples including $A_4$-based and binary octahedral modular models as well as an $SO(10)$ embedding, the work shows that data fitting typically relies on many continuous parameters and ad hoc choices, limiting predictivity. The authors propose that low-energy flavor may be shaped by hidden sectors or sterile neutrinos connected through a neutrino portal, implying multi-sector dynamics may be necessary, and they highlight the need for new principles or scanning tools (e.g., AMBer) to identify predictive, comprehensive frameworks.

Abstract

Relevance of the discrete symmetries for explanation of the observed flavor structures in the leptonic sector is considered. Achievements of the "traditional'' discrete symmetry approach and the modular symmetry approach are confronted. Minimal models with small number of parameters do not work. Complication of symmetry prescriptions allow to introduce new free parameters and thus describe the data but simultaneously bring two connected problems: (i) problem of "missing" representations, and (ii) problem of selection of certain point in huge discrete parameter space formed by possible charge assignments. Both problems must be addressed in complete model. Alternatively, one can keep minimal symmetry prescription but extend models introducing new physics unrelated to the original flavor symmetry. The low energy predictions can be "polluted" by additional physics such as effects of coupling to hidden sector, RGE running, decoupling of heavy degrees of freedom, mixing with sterile neutrinos, {\it etc.} The set up with discrete symmetries in the Hidden sector which communicates to the visible one via the neutrino portal and the basis fixing symmetry still looks promising.

Figures (5)

• Figure 1: Residual symmetry scheme. Residual symmetries of the mass matrices of neutrino and charged leptons. $T, U, S$ are generators of symmetry transformations of mass matrices, $S_\nu M_\nu S_\nu^T = M_\nu$, etc.
• Figure 2: The multiplets of symmetry group. Colored fields are representations which filled in by particles of the model.
• Figure 3: $SO(10)$ with modular flavor symmetry.
• Figure 4: Bounds on induced elements of neutrino mass matrix from the oscillation experiments as function of 4th neutrino mass. Blue (red) bands corresponds to the allowed ranges of elements of mass matrix in the case of normal (inverted) mass ordering.
• Figure 5: Set up with hidden sector.