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

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 A(4) symmetry. The quark mass hierarchies are considered in the vicinity of the fixed point tau = i infinity, tau being the vacuum expectation value of the modulus. We consider first a model in which the up-type and down-type quark mass matrices M-u and M-d involve modular forms of level 3 and weights 6, 4 and 2 and each depends on four constant parameters. Two ratios of these parameters, g(u) and g(d), can be sources of the CP violation. If M-u and M-d depend on the same tau, it is possible to reproduce the up-type and down-type quark mass hierarchies in the considered model for vertical bar g(u)vertical bar similar to 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 tau = i infinity with all constant being in magnitude of the same order and complex g(u) and g(d), if there are two different moduli tau(u) and tau(d) in the up-type and down-type quark sectors. We also consider the case of M-u and M-d depending on the same tau and involving modular forms of weights 8, 4, 2 and 6, 4, 2, respectively, with M-u 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 tau, needed for describing the down-type quark mass hierarchies, is crucial for obtaining the correct up-type quark mass hierarchies.