On the equivalence of Flavor SU(3) analyses of $B\to PP$ decays

TL;DR

This work addresses the consistency of flavor SU(3) analyses for charmless $B\to PP$ decays by clarifying the complete basis of amplitudes. It introduces ten reduced matrix elements (RMEs) and demonstrates their exact equivalence to the Irreducible Representation Amplitude (IRA) and Topological Diagram Amplitude (TDA) frameworks, including explicit mappings between RMEs and IRA/TDA amplitudes. The analysis highlights that three RMEs originate from electroweak penguins $O_{7,8}$ and, due to small SM Wilson coefficients, can be absorbed via EWP-tree relations, effectively reducing independent amplitudes. It also provides relations for penguin sectors and connects RMEs to QCD factorization-inspired topologies, establishing a solid foundation for consistent SU(3) analyses of heavy-meson decays. The results reconcile prior counts of amplitudes and pave the way for robust, symmetry-based phenomenology in $B\to PP$ decays.

Abstract

We conduct an SU(3) analysis of $B\to PP$ decays based on reduced matrix elements (RMEs), with $P$ being a light pseudoscalar meson excluding $η^{(\prime)}$. We show that a complete basis for the $B\to PP$ decays consists of ten RMEs, where the three RMEs arise from the electroweak penguin operators $O_{7,8}$. In the Standard Model, the relevant Wilson coefficients are small and thus can be neglected. We further demonstrate the equivalence of the RME approach with the irreducible representation amplitude (IRA) and topological diagram amplitude (TDA) methods, and derive relations between the ten RME amplitudes and corresponding IRA/TDA amplitudes. These relations lay a foundation for consistent SU(3) analyses of heavy meson decays.