Spurion Analysis for Non-Invertible Selection Rules from Near-Group Fusions

We generalize the framework of spurion analysis to a class of selection rules arising from non-invertible fusion algebras in perturbation theory. As a first step toward systematic applications to particle physics, we analyze the near-group fusion algebras, defined by fusion rules built from a finite Abelian group $G$ extended by a single non-invertible element. Notable examples include the Fibonacci and Ising fusion rules. We introduce a systematic scheme for labeling coupling constants at the level of the non-invertible fusion algebra, enabling consistent tracking of couplings when constructing composite amplitudes from simpler building blocks. Our labeling provides a clear interpretation of why the tree-level exact non-invertible selection rules are violated through radiative corrections, a unique phenomenon essential to ``loop-induced groupification''. We also identify the limit where the near-group fusion algebra is lifted to a $G\times \mathbb{Z}_2$ group, which provides an alternative scheme of spurion analysis consistent with the original one based on the near-group algebra. Meanwhile, we highlight the distinctions between the selection rules imposed by the near-group fusion algebra and those from breaking the $G\times \mathbb{Z}_2$ group.