Abstract
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 C P 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.
Highlights
Modular invariance has been invoked as candidate flavour symmetry [10]
We explore alternative descriptions of the charged lepton sector in modular invariant models of lepton masses and mixing angles
This can be intuitively understood by recognizing that the dependence of modular forms on the modulus is nearly exponential and small neutrino mass hierarchies and large mixing angles require a modulus with small imaginary part, which is inadequate to generate the large hierarchies observed among electron, muon and tau masses
Summary
We brefly review the formalism of modular invariant supersymmetric theories [16, 57]. For each level N and for each even nonnegative weight k, there is only a finite number of linearly independent modular forms.. For each level N and for each even nonnegative weight k, there is only a finite number of linearly independent modular forms.2 They span the linear space Mk(Γ(N )). The chiral multiplets φ(I) comprise three generations of lepton singlets Ec and doublets L, the Higgses Hu,d, and gauge invariant flavons φ. We will consider both the case where neutrino masses arise through the Weinberg operator and the case where neutrinos get their masses through the seesaw mechanism. At level 4 our basis does not enjoy this property and a non-canonical solution for X(I) is listed in appendix A
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