In this paper, we propose a general approach for describing hadronic B decays. Using this method, all amplitudes for such decays can be expressed in terms of contractions, though the matrix elements are not evaluated. Many years ago, Buras and Silvestrini proposed a similar approach. However, our technique goes beyond theirs in several ways. First, we include recent theoretical and experimental developments which indicate which contractions are negligible, and which are expected to be smaller than others. Second, we show that all B-decay diagrams can be simply expressed in terms of contractions. This constitutes a formal proof that the diagrammatic method is rigorous. Third, we show that one reproduces the relations between tree and electroweak-penguin diagrams described by Neubert and Rosner, and by Gronau, Pirjol and Yan. Fourth, although the previous results hold to all orders in αs, we show that it is also possible to work order-by-order in this approach. In this way it is possible to make a connection with the matrix-element evaluation methods of QCD factorization (QCDfac) and perturbative QCD (pQCD). Finally, using the contractions approach, we reevaluate the question of whether there is a "B → πK puzzle." At [Formula: see text], we find that the diagram ratio |C′/T′| is about 0.17, a factor of 10 too small to explain all the B → πK data. Both QCDfac and pQCD find that, at [Formula: see text], the value of |C′/T′| may be raised to only about 2–3 times its lowest-order value. We therefore conclude that, assuming the effect is not a statistical fluctuation, it is likely that the value of |C′/T′| is similar to its [Formula: see text] result, and that there really is a B → πK puzzle.
S
-wave
Kπ
contributions to the hadronic charmonium
B
decays in the perturbative QCD approach