Modular symmetry with weighton

A bstract We systematically develop the weighton mechanism for natural quark and charged lepton mass hierarchies in the framework of modular symmetry with a single modulus field τ . The weighton ϕ is defined as a complete singlet with unit modular weight, leading to fermion mass suppression by powers of φ , which is the vacuum expectation value of the field scaled by a flavour cut-off. Further mass and mixing angle suppression comes from powers of the small parameter, q ≡ e i 2 πτ . Assuming some fields transform as triplets under the finite modular symmetry, with general assignments for the other fields, we perform a complete analysis for the levels N = 3 , 4 , 5, expressing fermion masses and mixings in terms of powers of the small parameters φ and q . We present two examples in detail, based on the modular group T ′, close to the CP boundary of τ , which can address both fermion mass and mixing hierarchies using a weighton field.