Hadronic Two-body Charmless B Decays

In the absence of first-principles calculations for hadronic matrix elements, it is customary to evaluate the matrix elements under the factorization hypothesis so that 〈O(μ)〉 is factorized into the product of two matrix elements of single currents, governed by decay constants and form factors. However, the naive factorized amplitude is not renormalization scaleand γ5 schemeindependent as the scale and scheme dependence of Wilson coefficients are not compensated by that of the factorized hadronic matrix elements. In principle, the scale and scheme problems with naive factorization will not occur in the full amplitude since 〈O(μ)〉 involves vertex-type and penguin-type corrections to the hadronic matrix elements of the 4-quark operator renormalized at the scale μ. Formally, one can write