Modular flavour symmetries and modulus stabilisation

A bstract We study the problem of modulus stabilisation in framework modular symmetry approach to flavour problem. By analysing simple UV-motivated CP-invariant potentials for τ we find that a class these has (non-fine-tuned) CP-breaking minima vicinity point $$ {\mathrm{\mathbb{Z}}}_3^{ST} ℤ 3 ST residual symmetry, ≃ e 2 πi /3 . Stabilising at novel breaks spontaneously CP and can naturally explain mass hierarchies charged leptons possibly quarks.