A General Prescription for Spurion Analysis of Non-Invertible Selection Rules

We formulate a general prescription for spurion analysis in particle-physics models whose selection rules are described by commutative non-invertible fusion algebras. The construction applies to fusion algebras containing non-invertible basis elements that need not be self-conjugate, thereby allowing us to systematically track coupling constants in arbitrary particle scattering processes at tree and loop orders, but without assuming faithful realization of the fusion algebra, or no other quantum numbers for dynamical particles. This unifies and streamlines the previous analysis of near-group fusion algebras and of the $\mathbb{Z}_M/\mathbb{Z}_2$ fusion algebras, and supports the broader viewpoint that the non-invertible selection rules often admit auxiliary descriptions using lifted Abelian groups with a structured set of explicit breaking terms.