A4 modular flavour model of quark mass hierarchies close to the fixed point τ = i∞

Abstract

We study the possibility to generate the quark mass hierarchies as well as the CKM quark mixing and CP violation without fine-tuning in a quark flavour model with modular A4 symmetry. The quark mass hierarchies are considered in the vicinity of the fixed point τ = i∞, τ being the vacuum expectation value of the modulus. We consider first a model in which the up-type and down-type quark mass matrices Mu and Md involve modular forms of level 3 and weights 6, 4 and 2 and each depends on four constant parameters. Two ratios of these parameters, gu and gd, can be sources of the CP violation. If Mu and Md depend on the same τ, it is possible to reproduce the up-type and down-type quark mass hierarchies in the considered model for |gu| ~ mathcal{O} (10) with all other constants being in magnitude of the same order. However, reproducing the CP violation in the quark sector is problematic. A correct description of the quark mass hierarchies, the quark mixing and CP violation is possible close to τ = i∞ with all constant being in magnitude of the same order and complex gu and gd, if there are two different moduli τu and τd in the up-type and down-type quark sectors. We also consider the case of Mu and Md depending on the same τ and involving modular forms of weights 8, 4, 2 and 6, 4, 2, respectively, with Mu receiving a tiny SUSY breaking or higher dimensional operator contribution. Both the mass hierarchies of up-type and down-type quarks as well and the CKM mixing angles and CP violating phase are reproduced successfully with one complex parameter and all parameters being in magnitude of the same order. The relatively large value of Im τ, needed for describing the down-type quark mass hierarchies, is crucial for obtaining the correct up-type quark mass hierarchies.