Modular invariant dynamics and fermion mass hierarchies around τ = i

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 approximate to 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 vertical bar epsilon vertical bar. 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 Kahler potential.

Automated summary

In modular invariant flavour models, the breaking of residual Z(4) invariance near the self-dual point tau = i can be described by correcting epsilon, which changes sign under Z(4). Although the Z(4) symmetry at tau = i may lead to degeneracies or vanishing eigenvalues of fermion mass matrices, this can be eliminated by deviating slightly from the self-dual point. The relevant mass ratios are controlled by powers of the absolute value of epsilon.