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

TL;DR

This paper extends the spurion analysis framework to $ obreak Z_M/ obreak Z_2$ non-invertible selection rules, providing insights into the behavior of coupling zeros at various loop orders and their implications for particle decoupling.

Contribution

It generalizes the spurion analysis to $ obreak Z_M/ obreak Z_2$ NISRs, offering a systematic approach to track couplings and understand zeros at all perturbative orders.

Findings

Systematic labeling scheme for couplings at arbitrary loop orders.

Analysis of low-order and all-order zeros under radiative corrections.

Conjecture on particle decoupling related to non-faithful fusion algebra realization.

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.