Eclectic flavour symmetries without flavons

Abstract

We develop a new framework in which eclectic groups are realised as flavour symmetries without flavons. Focusing on the group $Ω(1)$, we show that it can be realised as a finite image of the integral Jacobi group, i.e. the semidirect product of $H({\mathbb Z})$ with $SL(2,{\mathbb Z})$, acting on two moduli $(τ,z)$. The Heisenberg subgroup $H({\mathbb Z})$ leaves the modulus $τ$ unchanged, and its finite image $Δ(27)$ plays the role of the traditional flavour symmetry. We provide the ingredients needed to construct supersymmetric Jacobi-invariant theories, both in globally supersymmetric theories and in supergravity. We develop a realistic model of fermion masses within this framework and formulate a consistent CP symmetry, which can be imposed at the Lagrangian level and broken spontaneously by the moduli. We discuss how the conventional eclectic-flavour description is recovered in an appropriate limit. We also extend this geometric interpretation to the $T^2/\mathbb Z_4$ and $T^2/\mathbb Z_2$ eclectic building blocks, and clarify why the $T^2/\mathbb Z_6$ case is structurally different.