Abstract
We consider for the first time level 7 modular invariant flavour models where the lepton mixing originates from the breaking of modular symmetry and couplings responsible for lepton masses are modular forms. The latter are decomposed into irreducible multiplets of the finite modular group Γ7, which is isomorphic to PSL(2, Z7), the projective special linear group of two dimensional matrices over the finite Galois field of seven elements, containing 168 elements, sometimes written as PSL2(7) or Σ(168). At weight 2, there are 26 linearly independent modular forms, organised into a triplet, a septet and two octets of Γ7. A full list of modular forms up to weight 8 are provided. Assuming the absence of flavons, the simplest modular-invariant models based on Γ7 are constructed, in which neutrinos gain masses via either the Weinberg operator or the type-I seesaw mechanism, and their predictions compared to experiment.
Highlights
The latter are decomposed into irreducible multiplets of the finite modular group Γ7, which is isomorphic to PSL(2, Z7), the projective special linear group of two dimensional matrices over the finite Galois field of seven elements, containing 168 elements, sometimes written as PSL2(7) or Σ(168)
If we require that the finite modular group possesses three-dimensional irreducible representations to which the left-handed lepton doublets are assigned, modular symmetry approach with ΓN allows to have N = 3, 4, 5, 7, 8, 16 so in this paper we are considering the smallest example with N > 5, which has interesting properties, as follows
We consider two scenarios: the modular symmetry only acts on the neutrino sector and the charged lepton matrix is diagonal in the first scenario and the modular symmetry acts on both charged lepton and neutrino sector in the second scenario
Summary
We shall construct some typical models for neutrino masses and mixing based on the Γ7 modular symmetry. We will not introduce any flavon field, the flavour symmetry is broken when the complex modulus τ obtains a vacuum expectation value. The Higgs doublets fields Hu,d are assumed to be singlets of Γ7 with vanishing modular weights. The three right-handed (RH) charged leptons E1c,2,3 transform as singlet 1 under Γ7 modular group they are distinguished by the different modular weights k1,2,3. We assign the three generations of left-handed (LH) lepton doublets L and the three right-handed neutrinos N c to two triplets 3 and with the weights kL and kN. We shall employ potentially the lowest weight modular forms as much as possible in order to reduce free parameters
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