Modular Invariant Dynamics and Fermion Mass Hierarchies around $\tau = i$

TL;DR

This paper explores how fermion mass hierarchies can naturally emerge in modular invariant models near the self-dual point , using residual symmetries and small deviations to generate realistic mass spectra.

Contribution

It introduces a mechanism where breaking residual  symmetry near  leads to hierarchical fermion masses, applicable even with non-minimal Khler potentials.

Findings

Mass ratios controlled by powers of 

Residual  symmetry constrains fermion mass degeneracies

Deviations from  generate realistic hierarchies

Abstract

We discuss fermion mass hierarchies within modular invariant flavour models. We analyse the neighbourhood of the self-dual point $\tau=i$, where modular invariant theories possess a residual $Z_4$ invariance. In this region the breaking of $Z_4$ can be fully described by the spurion $\epsilon \approx \tau - i$, that flips its sign under $Z_4$. Degeneracies or vanishing eigenvalues of fermion mass matrices, forced by the $Z_4$ symmetry at $\tau=i$, are removed by slightly deviating from the self-dual point. Relevant mass ratios are controlled by powers of $|\epsilon|$. We present examples where this mechanism is a key ingredient to successfully implement an hierarchical spectrum in the lepton sector, even in the presence of a non-minimal K\"ahler potential.