Dynamical flavor origin of $\mathbb{Z}_N$ symmetries

Discrete Abelian symmetries ($\mathbb{Z}_N$) are a common "artifact" of beyond the standard model physics models. They provide different avenues for constructing consistent scenarios for lepton and quark mixing patterns, radiative neutrino mass generation as well as dark matter stabilization. We argue that these symmetries can arise from the spontaneous breaking of the Abelian $U(1)$ factors contained in the global flavor symmetry transformations of the gauge invariant kinetic Lagrangian. This will be the case provided the ultra-violet completion responsible for the Yukawa structure involves scalar fields carrying non-trivial $U(1)$ charges. Guided by minimality criteria, we demonstrate the viability of this approach with two examples: first, we derive the "scotogenic" model Lagrangian, and second, we construct a setup where the spontaneous symmetry breaking pattern leads to a $\mathbb{Z}_3$ symmetry which enables dark matter stability as well as neutrino mass generation at the 2-loop order. This generic approach can be used to derive many other models, with residual $\mathbb{Z}_N$ or $\mathbb{Z}_{N_1}\times \cdots \times \mathbb{Z}_{N_k}$ symmetries, establishing an intriguing link between flavor symmetries, neutrino masses and dark matter.

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