Abstract
We show how quark and lepton mass hierarchies can be reproduced in the framework of modular symmetry. The mechanism is analogous to the Froggatt-Nielsen (FN) mechanism, but without requiring any Abelian symmetry to be introduced, nor any Standard Model (SM) singlet flavon to break it. The modular weights of fermion fields play the role of FN charges, and SM singlet fields with non-zero modular weight called weightons play the role of flavons. We illustrate the mechanism by analysing A4 (modular level 3) models of quark and lepton (including neutrino) masses and mixing, with a single modulus field. We discuss two examples in some detail, both numerically and analytically, showing how both fermion mass and mixing hierarchies emerge from different aspects of the modular symmetry.
Highlights
Quark mixing angles are small, the lepton sector has two large mixing angles θ12, θ23 and one small mixing angle θ13 which is of the same order of magnitude as the quark Cabibbo mixing angle [1]
We show how quark and lepton mass hierarchies can be reproduced in the framework of modular symmetry
The modular weights of fermion fields play the role of FN charges, and Standard Model (SM) singlet fields with non-zero modular weight called weightons play the role of flavons
Summary
A crucial element of the modular invariance approach is the modular form f (τ ) of weight k and level N. The modular form f (τ ) is a holomorphic function of the complex modulus τ and it is required to transform under the action of Γ(N ) as follows, f aτ + b = (cτ + d)kf (τ ) for ∀ a b ∈ Γ(N ). The modular forms of weight k and level N span a linear space of finite dimension. The finite modular group Γ3 is isomorphic to A4 which is the symmetry group of the tetrahedron It contains twelve elements and it is the smallest non-abelian finite group which admits an irreducible three-dimensional representation. The linear space of the modular forms of integral weight k and level N = 3 has dimension k + 1 [16].
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