Modular Symmetry with Weighton

Abstract

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 $\tildeϕ$, 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, $q\equiv e^{i2πτ}$. 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 $\tildeϕ$ 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.