The Mathematical Structure of $U$-Spin Amplitude Sum Rules

We perform a systematic study of $SU(2)$ flavor amplitude sum rules with particular emphasis on $U$-spin. This study reveals a rich mathematical structure underlying the sum rules that allows us to formulate an algorithm for deriving all $U$-spin amplitude sum rules to any order of the symmetry breaking. This novel approach to deriving the sum rules does not require one to explicitly compute the Clebsch-Gordan tables, and allows for simple diagrammatic interpretation. Several examples that demonstrate the application of our novel method to systems that can be probed experimentally are provided.