Normalization of the Perturbative QCD Corrections for B → X s γ Decay

We study the normalization of perturbative QCD corrections to the inclusive B → Xsγ decay. We propose to set the renormalization scale using the Brodsky-Lepage-Mackenzie (BLM) method. In the proposed method the scale is determined by absorbing the vacuum polarization correction from light fermions to renormalization scale but not including the anomalous dimensions. The BLM scale depends in general on the renormalization scheme and the factorization scale. We find that the BLM scale is insensitive to the factorization scale. In the heavy-quark potential scheme, we find that the BLM scale is μBLM ≈ (0.315 − 0.334)mb when the factorization scale varies from mb/2 to 2mb. 12.38.Bx, 13.40.Hq, 11.10.Gh Typeset using REVTEX E-mail address: [email protected] E-mail address: [email protected] 1 The rare decay process b → sγ has been recently observed at CLEO II [1] at a branching ratio 2.32(±0.57 ± 0.35) × 10 (statistical and systematic errors). The process serves as an excellent probe to the new physics beyond the Standard Model such as charged Higgs [2], supersymmetry [3] or anomalous WWγ coupling [4]. At leading order the process is described by an electromagnetic penguin diagram. It is known that there are large QCD corrections to the penguin diagram. A complete renormalization group-improved perturbative calculation is now available to the leading-logarithm approximation. It is found that the QCD corrections increase the rate by a factor two to three. In the Standard Model the branching ratio to leading-logarithmic accuracy is 2.8(±0.8)×10−4, where the error is dominated by the uncertainty in the QCD renormalization scale varied over mb/2 < μ < 2mb [5,6]. Inclusion of the next-to-leading logarithmic QCD corrections will reduce the theoretical error. However, a complete calculation is still not achieved. With partial next-to-leading corrections known so far, the branching ratio falls to ∼ 1.9× 10. In view of the current status of the theoretical uncertainties as summarized above it is of interest to normalize the leading order QCD corrections and estimate theoretical errors thereof. In this paper we address this issue and determine the renormalization scale according to the Brodsky-Lepage-Mackenzie (BLM) method [7]. So far applications of the BLM method have been mostly restricted to processes involving (partially) conserved currents. In this case QCD correction may be calculated by a perturbative expansion at a fixed order. For example, BLM scale setting has been established for the inclusive semileptonic B decay B → Xqeν [8]. In Ref. [8] the BLM scale in the MS scheme is found μBLM ≈ 0.07mb (0.12mb when running MS b-quark mass [9] is used), indicating a significant QCD correction from higher order terms. The QCD corrections for the radiative B decay is more complicated since the process involves large logarithms arising from QCD effects between the top-quark or W -gauge boson mass and the bottom quark mass scales. The effective Hamiltonian approach takes care of these large logarithms by introducing a factorization scale and separates the longand the short-distance QCD effects. In situations involving large logarithms a physical observable P such as the total decay 2 rate or the branching ratio may be written schematically as