Modular invariant models of lepton masses at levels 4 and 5

We explore alternative descriptions of the charged lepton sector in modular invariant models of lepton masses and mixing angles. In addition to the modulus, the symmetry breaking sector of our models includes ordinary flavons. Neutrino mass terms depend only on the modulus and are tailored to minimize the number of free parameters. The charged lepton Yukawa couplings rely upon the flavons alone. We build modular invariant models at levels 4 and 5, where neutrino masses are described both in terms of the Weinberg operator or through a type I seesaw mechanism. At level 4, our models reproduce the hierarchy among electron, muon and tau masses by letting the weights play the role of Froggatt-Nielsen charges. At level 5, our setup allows the treatment of left and right handed charged leptons on the same footing. We have optimized the free parameters of our models in order to match the experimental data, obtaining a good degree of compatibility and predictions for the absolute neutrino masses and the CP violating phases. At a more fundamental level, the whole lepton sector could be correctly described by the simultaneous presence of several moduli. Our examples are meant to make a first step in this direction.This project has received support in part by the MIUR-PRIN project 2015P5SBHT 003 \Search for the Fundamental Laws and Constituents" and by the European Union's Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement Nº 674896 and 690575. The research of F. F. was supported in part by the INFN. The research of J. C. C. was supported by the Spanish MINECO project FPA2016-78220-C3- 1-P, the Junta de Andalucía grant FQM101 and the Spanish MECD grant FPU14