Robustness of second-order U-spin sum rule bounds near phase-space boundaries

Determine the robustness of bounds on decay rates derived from the second-order U-spin master sum rule—namely, that the sum of Cabibbo-favored and doubly Cabibbo-suppressed CKM-free rates divided by the sum of singly Cabibbo-suppressed CKM-free rates equals one up to second order in U-spin breaking—when applied to systems with kinematically forbidden channels, such as D^0 → P^+ P^+ P^- P^- where D^0 → K^+ K^+ K^- K^- is forbidden, taking into account potential enhancement of symmetry breaking near phase-space boundaries.

Background

The paper derives a universal second-order master sum rule for hadronic weak charm decays: the sum of Cabibbo-favored (CF) and doubly Cabibbo-suppressed (DCS) CKM-free rates divided by the sum of singly Cabibbo-suppressed (SCS) CKM-free rates equals one up to O(ε2) U-spin breaking. This relation is used to test data and to predict unmeasured rates across multiple charm decay systems.

In discussing extensions, the authors propose using the master sum rule to set bounds in systems where some decay channels are kinematically forbidden (e.g., certain four-body D0 decays). They caution, however, that near phase-space boundaries symmetry breaking can be enhanced, potentially affecting the reliability of these bounds. Quantifying this robustness constitutes an explicit unresolved issue.

References

For example, in systems where some decay channels are kinematically forbidden, the second-order master sum rule can be used to set bounds on the allowed rates. This is, for example, the case for D0 \rightarrow P+ P+ P- P- decays, where the D0 \to K+ K+ K- K- mode is kinematically forbidden. It is, however, not clear how robust such bounds are, as near the boundary of phase space one expects enhanced symmetry breaking and correspondingly larger theoretical corrections.

— One Sum To Rule Them All: A Second Order Master Rate Sum Rule for Charm Decays  (2602.22320 - Gavrilova et al., 25 Feb 2026) in Section 6 (Conclusions and outlook), Straightforward extensions paragraph