Systematics of U-spin sum rules for systems with direct sums

A rich mathematical structure underlying flavor sum rules has been discovered recently. In this work, we extend these findings to systems with a direct sum of representations. We prove several results for the general case. We derive an algorithm that enables the determination of all U-spin amplitude sum rules at arbitrary order of the symmetry breaking for any system containing a direct sum of the representations 0 ⨁ 1. Potential applications are numerous and include, for example, higher order sum rules for CP-violating charm decays with an arbitrary number of final states.