$B$ decays to two pseudoscalars and a generalized $\Delta I = \frac{1}{2}$ rule

We perform an isospin analysis of $B$ decays to two pseudoscalars. The analysis extracts appropriate CKM and short distance loop factors to allow for comparison of non-perturbative QCD effects in the reduced matrix elements of the amplitudes. In decays where penguin diagrams compete with tree-level diagrams we find that the reduced matrix elements of the penguin diagrams, which are singlets or doublets under isospin, are significantly enhanced compared with the triplet and fourplet contributions of the weak Hamiltonian. This similarity to the $\Delta I =\frac{1}{2}$ rule in $K \to \pi \pi$ decays suggests that, more generally, processes mediated by Hamiltonians in lower-dimensional isospin representations see enhancement over higher-dimensional ones in QCD.