Abstract
We generalize the modular invariance approach to include the half-integral weight modular forms. Accordingly the modular group should be extended to its metaplectic covering group for consistency. We introduce the well-defined half-integral weight modular forms for congruence subgroup $\Gamma(4N)$ and show that they can be decomposed into the irreducible multiplets of finite metaplectic group $\widetilde{\Gamma}_{4N}$. We construct concrete expressions of the half-integral/integral modular forms for $\Gamma(4)$ up to weight 6 and arrange them into the irreducible representations of $\widetilde{\Gamma}_4$. We present three typical models with $\widetilde{\Gamma}_4$ modular symmetry for neutrino masses and mixing, and the phenomenological predictions of each model are analyzed numerically.
Highlights
How to understand the mass hierarchies and flavor mixing patterns of quark and lepton is still one of the greatest challenges in particle physics
We find that the half-integral weight modular forms for Γð4NÞ can be arranged into irreducible multiplets of the finite metaplectic modular group Γ 4N which is the quadruple covering of the inhomogeneous finite modular group Γ4N
In order to discuss the action of the full modular group on the halfintegral modular forms, one should extend the modular group SL2ðZÞ to the metaplectic group Mp2ðZÞ which is the double covering of SL2ðZÞ
Summary
How to understand the mass hierarchies and flavor mixing patterns of quark and lepton is still one of the greatest challenges in particle physics. We focus on the lowest level case of Γð4Þ, and use the corresponding modular forms of half-integral weight to construct lepton mass models. We show that the half-integral weight modular forms of Γð4NÞ arrange themselves into different irreducible multiplets of the finite metaplectic group Γ 4N. It has been shown that the modular forms of integral weight k and level N can be arranged into different irreducible representations of the homogeneous finite modular group Γ0N ≡ Γ=ΓðNÞ up to the factor ðcτ þ dÞk in [52]. The half-integral k=2 weight modular form fðτÞ can be consistently defined for the principal congruence subgroup Γð4NÞ, it is a holomorphic function of τ and satisfies the following condition, fðhτÞ 1⁄4 vkðhÞðcτ þ dÞk=2fðτÞ 1⁄4 vkðhÞJk=2ðh; τÞfðτÞ; ab h1⁄4. This means that Jk=2ðh; τÞ is the correct automorphy factor for Γð4NÞ, this generalized automorphy factor eliminates the ambiguity caused by half-integral weight, and the halfintegral weight modular form defined in Eq (7) really makes sense
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