Isospin-based EWP-tree relations

In 1998, it was shown that, if flavor SU(3) symmetry SU(3)F is assumed in charmless B → 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 → PP decays. They have also been used over the years in analyses of the B → πK puzzle, even though the B → 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 ∆S = 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 → ππ decays. In ∆S = 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 → πK puzzle, one now finds a 4–5σ discrepancy with the Standard Model, 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.