Modular invariance approach to flavour, CP violation, and fermion mass hierarchies

In modular-invariant models of favour, hierarchical fermion mass matrices may arise solely (without fine-tuning) due to the proximity of the VEV of the modulus τ to the fixed point of the modular group i) τ=expi2π/3 preserving the $Z_{3}^S$^T residual symmetry, or ii) τ=i∞, preserving the $Z_{N}^T$ symmetry, N=3,4,5. This mechanism does not require flavon fields. The fermion mass hierarchies thus generated depend on the decomposition of field representations under the residual symmetry group. The speaker presents lepton flavour models in which the charged-lepton mass hierarchies are naturally obtained. Another problem considered is modulus stabilisation in the framework of the modular symmetry approach to the flavour problem. By analysing simple UV-motivated CP-invariant potentials for the modulus τ, it is found that a class of these potentials has (non-fine-tuned) CP-breaking minima in the vicinity of the point of the $Z_{3}^S$^T residual symmetry, τ≃expi2π/3. Stabilising the modulus at these novel minima breaks spontaneously the CP symmetry and can naturally explain the mass hierarchies of charged leptons and possibly of quarks.