Systematics of U-Spin Sum Rules for Systems with Direct Sums

TL;DR

This paper extends the mathematical framework of flavor sum rules to systems with direct sums of representations, providing an algorithm to determine all U-spin amplitude sum rules at any order of symmetry breaking, with applications in CP-violating charm decays.

Contribution

It introduces a general algorithm for deriving U-spin sum rules in systems with direct sum representations, advancing the theoretical understanding of flavor symmetries.

Findings

Derived an algorithm for all-order U-spin sum rules

Extended sum rule analysis to systems with direct sum representations

Potential applications in CP violation studies in charm decays

Abstract

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 $0 \oplus 1$. Potential applications are numerous and include, for example, higher order sum rules for CP-violating charm decays with an arbitrary number of final states.