Quark hierarchies and CP violation from the Siegel modular group
We investigate theories of flavour based on genus $g=2$ modular invariance and analyze how fermion mass hierarchies can be generated in this context, in the vicinity of invariant points or regions in moduli space where a residual symmetry is preserved. We apply this mechanism of modular proximity-induced hierarchies to the quark sector, with the vacuum expectation values of the moduli being the only sources of spontaneous breaking of the flavour and CP symmetries. We present a benchmark model where quark mass hierarchies and CP violation are explained, with mass ratios vanishing in the symmetric limit, and quark mixing is reproduced. In this model, the values of the moduli turn out to be close to special values such as $\tau_1 \simeq \omega$ and $\tau_2 \simeq \omega, i$.