Abstract
We discuss fermion mass hierarchies within modular invariant flavour models. We analyse the neighbourhood of the self-dual point τ = i, where modular invariant theories possess a residual Z4 invariance. In this region the breaking of Z4 can be fully described by the spurion ϵ ≈ τ − i, that flips its sign under Z4. Degeneracies or vanishing eigenvalues of fermion mass matrices, forced by the Z4 symmetry at τ = i, are removed by slightly deviating from the self-dual point. Relevant mass ratios are controlled by powers of |ϵ|. We present examples where this mechanism is a key ingredient to successfully implement an hierarchical spectrum in the lepton sector, even in the presence of a non-minimal Kähler potential.
Highlights
JHEP05(2021)242 particle species, has to obey modular invariance and Yukawa couplings become functions of τ
We show that the residual Z4 symmetry can be exploited to reduce the rank of the charged lepton mass matrix at τ = i, with small non-vanishing masses arising from a small departure from the self-dual point
The role of flavour symmetry is played by modular invariance, regarded as a discrete gauge symmetry, circumventing the obstruction concerning fundamental global symmetries
Summary
We shortly review the formalism of supersymmetric modular invariant theories [34, 35] applied to flavour physics [2]. When chiral fermions are considered, Wij and Kimj are both depleted by v/M with respect to the vector-like case, v denoting the gauge symmetry breaking scale. In some cases Ye(τ ) and/or Cν(τ ) are completely determined as a function of τ up to an overall constant, providing a strong potential constraint on the mass spectrum, eq (2.10) Such property does not extend to the Kähler potential K and to the renormalisation factors (zE−c1,/L2). A second constraint on the effect √of the Kähler corrections arises in the vicinity of the fixed points of Γ, τ = i, τ = −1/2 + i 3/2 and τ = i∞, invariant under the action of the elements S, ST and T , respectively.
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