Hadron physics is notoriously challenging due to non-perturbative QCD. In our work, we use the approximate U-spin symmetry of QCD (unitary rotations between d and s quarks) to get around this problem. Our study reveals a rich mathematical structure underlying the sum rules, which allows us to formulate an algorithm for deriving all U-spin amplitude sum rules to higher orders in the symmetry breaking. This novel approach does not require any explicit calculations and admits a simple diagrammatic interpretation.

Most excitingly, our newest results extend this framework to sum rules between decay rates. Strikingly, much of the underlying structure that we find at the amplitude level persists at the level of rates, even when phase space is taken into account. With the systematic framework in hand, hadronic systems that can in principle support higher-order flavor sum rules become apparent, though further phenomenological analysis is called for.