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Double cover of modular \(S_4\) for flavour model building. (English) Zbl 1508.81967

Summary: We develop the formalism of the finite modular group \(\Gamma_4^\prime \equiv S_4^\prime \), a double cover of the modular permutation group \(\Gamma_4 \simeq S_4\), for theories of flavour. The integer weight \(k > 0\) of the level 4 modular forms indispensable for the formalism can be even or odd. We explicitly construct the lowest-weight (\(k = 1\)) modular forms in terms of two Jacobi theta constants, denoted as \(\varepsilon(\tau)\) and \(\theta(\tau)\), \(\tau\) being the modulus. We show that these forms furnish a 3D representation of \(S_4^\prime\) not present for \(S_4\). Having derived the \(S_4^\prime\) multiplication rules and Clebsch-Gordan coefficients, we construct multiplets of modular forms of weights up to \(k = 10\). These are expressed as polynomials in \(\varepsilon\) and \(\theta\), bypassing the need to search for non-linear constraints. We further show that within \(S_4^\prime\) there are two options to define the (generalised) CP transformation and we discuss the possible residual symmetries in theories based on modular and CP invariance. Finally, we provide two examples of application of our results, constructing phenomenologically viable lepton flavour models.

MSC:

81V15 Weak interaction in quantum theory
81V05 Strong interaction, including quantum chromodynamics
11F41 Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces
20B05 General theory for finite permutation groups
11F11 Holomorphic modular forms of integral weight
70H45 Constrained dynamics, Dirac’s theory of constraints

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