Relative Sizes of Diagrams in B -> pi pi, pi K Decays

Abstract

We show that the neglect of the $(V-A)\times (V+A)$ pieces of the electroweak penguin (EWP) amplitudes in the effective hamiltonian (the Wilson coefficients are very small) allows one to calculate the relative size of some tree and EWP diagrams in $B \to \pi \pi$ and $B\to \pi K$ decays. For both decay classes, tree and EWP amplitudes are related using only isospin. In $B\to \pi \pi$, the ratio $C/T$ is calculated using isospin alone; in $B\to \pi K$ it is found using flavor SU(3) symmetry. These results are obtained by computing explicitly all Wick contractions of all effective operators. Relations among these contractions are found using Fierz identities and final-state symmetry arguments.