Non-holomorphic modular S4 lepton flavour models

In the formalism of the non-supersymmetric modular invariance approach to the flavour problem the elements of the Yukawa coupling and fermion mass matrices are expressed in terms of polyharmonic Maa ss modular forms of level N in addition to the standard modula forms of the same level and a small number of constant parameters. Non-trivial polyharmonic Maa ss forms exist for zero, negative and positive integer modular weights. Employing the finite modula group S4 as a flavour symmetry group and assuming that the three left-handed lepton doublets furnish a triplet irreducible representation of S4, we construct all possible 7- and 8-parameter lepton flavour models in which the neutrino masses are generated either by the Weinberg effective operator or by the type-I seesaw mechanism. We identify the phenomenologically viable models and obtain predictions for each of these models for the neutrino mass ordering, the absolute neutrino mass scale, the Dirac and Majorana CP-violation phases and, correspondingly, for the sum of neutrino masses and the neutrinoless double beta decay effective Majorana mass. We comment on how these models can be tested and conclude that they are all falsifiable. Detailed analyses are presented in the case of three representative benchmark lepton flavour scenarios.