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

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 $ϵ\approx \tau -i$, 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 $|ϵ|$. 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 Kähler potential.

Comments: 25 pages, 2 figures, 5 tables; numerical analysis updated using the NuFIT 5.0 results; several comments and references added; typos corrected; matches version published in JHEP