Spurion Analysis of $\mathbb{Z}_M/\mathbb{Z}_2$ Non-Invertible Selection Rules: Low-Order versus All-Order Zeros

TL;DR

This work extends spurion analysis to non-invertible selection rules arising from $\mathbb{Z}_M/\mathbb{Z}_2$ orbifolds, clarifying how loop corrections induce groupification and how a lifted $\mathbb{Z}_2$-based labeling tracks couplings across all perturbative orders. The authors show that, although the tree-level $\mathbb{Z}_M/\mathbb{Z}_2$ NISRs are exact, radiative effects can generate composite spurions that break these rules in a controlled way, yielding low-order and all-order zeros depending on faithful realization and fusion structure. Using the $\mathbb{Z}_5/\mathbb{Z}_2$ example, they illustrate how certain couplings carry nontrivial raised charges under the lifted group and discuss obstructions when the fusion algebra is not faithfully realized, including implications for effective theories and RG flows. The paper then analyzes Yukawa textures under $\mathbb{Z}_5/\mathbb{Z}_2$ and $\mathbb{Z}_6/\mathbb{Z}_2$, identifying which texture zeros are protected to all orders and which arise at finite loop order, highlighting the predictive power of spurion methods for non-invertible selection rules. Overall, the approach provides a systematic framework to understand hierarchical couplings and radiative zeros in models with NISRs, with potential applications to SM extensions and beyond.

Abstract

Motivated by recent progress in the spurion analysis of non-invertible selection rules (NISRs) arising from near-group fusion algebras, we further generalize the framework to a class of NISRs obtained from $\mathbb{Z}_2$ orbifolding of a $\mathbb{Z}_M$ symmetry, denoted as $\mathbb{Z}_M/\mathbb{Z}_2$. Many structural features are carried over: for instance, our labeling scheme enables systematic tracking of all couplings when constructing composite amplitudes from simpler building blocks at arbitrary loop orders in perturbation theory. Our analysis provides a transparent understanding of both low-order and all-order zeros of couplings under radiative corrections. Furthermore, we examine the fate of low-order zeros when the fusion algebra is not faithfully realized -- a situation not captured by the vanilla argument of ``loop-induced groupification'' -- and formulate a conjecture on the related aspects of particle decoupling and effective theory. Finally, we discuss the low-order versus all-order zeros in Yukawa textures from the perspective of spurion analysis.

Figures (3)

• Figure 1: An intuitive illustration of the basis elements of $\mathbb{Z}_M/\mathbb{Z}_2$ fusion algebra when $M=5$ and $6$, where each class includes a pair of $\mathbb{Z}_M$ elements related by inversion that are identified by the $\mathbb{Z}_2$ orbifolding.
• Figure 2: An example of groupification for the $\mathbb{Z}_5/\mathbb{Z}_2$ fusion algebra. Due to quantum corrections, the particle labeled by the non-invertible element $[g^2]$ appears in the loop and induces the identification (i.e., mixing) between the particles labeled by $[g^1]$ and $[g^0]$ at the one-loop level. By deforming the diagrams, we see that $[g^1]$ and $[g^0]$ become identified by dressing $[g^1]$ produced by the fusion of conjugate pair $([g^2])^2$. This matches Eq. \ref{['eq:groupification']}. The same mechanism applies to other fusion algebras; see e.g. Suzuki:2025oov.
• Figure 3: Radiative generation of the $\phi_0\phi_1$ mixing term at the two-loop order. Here the particles are labeled by the elements in the $\mathbb{Z}_5/\mathbb{Z}_2$ fusion algebra; see Eq. \ref{['eq:exp_particle_labels']}.