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 $\tau$. The weighton $\varphi$ is defined as a complete singlet with unit modular weight, leading to fermion mass suppression by powers of $\tilde{\varphi }$, 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, $q\equiv e^{i2\pi \tau }$. 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 $\tilde{\varphi }$ and $q$. We present two examples in detail, based on the modular group $T^{\prime }$, close to the CP boundary of $\tau$, which can address both fermion mass and mixing hierarchies using a weighton field.

Comments: 81 pages, 3 figures