Isospin-based EWP-tree Relations

TL;DR

This paper demonstrates that SU(2) isospin symmetry alone suffices to derive electroweak-penguin–tree (EWP–tree) relations for six sets of charmless B → PP decays, with ΔS=0 and ΔS=1 cases yielding both similarities and crucial differences relative to the full SU(3)F framework. It shows that, for ΔS=0 decays, SU(2)I EWP–tree relations mimic SU(3)F ones and can be used to incorporate EWP effects into the extraction of the CP phase α from B → ππ, while for ΔS=1 decays, the SU(2)I relations differ markedly from SU(3)F and lead to exact cancellations in certain observables (δ_{πK} = 0). Applying these isospin-based relations to B → πK and related modes, the authors re-fit the data and find that using SU(2)I relations increases the SM tension to about 4–5σ, compared to roughly 2–3σ under SU(3)F relations. The work argues that analyses of isospin-related hadronic B decays should consistently employ SU(2)I EWP–tree relations, significantly impacting interpretations of the ππ α extraction and the πK puzzle and guiding future global fits in flavor physics. Overall, the paper clarifies the proper symmetry-based toolkit for disentangling EWP and tree contributions in hadronic B decays and reveals a larger SM shortfall in πK data when isospin constraints are applied. Key results include exact δ_{πK} = 0 under SU(2)I for B → πK, and the demonstration that ΔS=1 SU(2)I EWP–tree relations yield distinct diagrammatic forms from SU(3)F, with important phenomenological consequences for SM tests in flavor physics.

Abstract

In 1998, it was shown that, if flavor SU(3) symmetry [SU(3)$_F$] is assumed in charmless $B \to PP$ decays ($P$ is a light pseudoscalar meson), some reduced matrix elements involving electroweak penguin (EWP) operators are related to those involving tree operators. Similarly, EWP diagrams are related to tree diagrams. These SU(3)$_F$ EWP-tree relations were recently used in global analyses of $B \to PP$ decays. They have also been used over the years in analyses of the $B \to πK$ puzzle, even though the $B \to πK$ amplitudes are related by isospin symmetry [SU(2)$_I$], and not the full SU(3)$_F$. In this paper, we show that, even if only SU(2)$_I$ is assumed, there are still EWP-tree relations. In $ΔS=0$ decays, these relations are similar to those of SU(3)$_F$, and can be used to take into account the EWP contributions in the extraction of the CP phase $α$ from $B \to ππ$ decays. In $ΔS=1$ decays, the SU(2)$_I$ EWP-tree relations are quite different from those of SU(3)$_F$; when these are used to analyze the $B \to πK$ puzzle, one now finds a 4-5$σ$ discrepancy with the SM, much larger than what was previously found. We argue that, if one analyzes a set of hadronic $B$ decays whose amplitudes are related by isospin, one must use the SU(2)$_I$ EWP-tree relations for that set of decays in the analysis.