Scale uncertainties in the B→Xsγ decay

We analyze the theoretical uncertainties in Br(B --> X-s gamma) due to the choice of the high energy matching scale mu(w) = O(M-w) and the scale mu(t) at which the running top quark mass is defined: (m) over bar(t)( mu(t)). To this end we have repeated the calculation of the initial conditions confirming the final results of Adel and Yao and Greub and Hurth and generalizing them to include the dependences on mu(t) and mu(w) with mu(t) not equal mu(w). In the leading order the mu(w), and mu(t) uncertainties in Br(B --> X-s gamma) turn out to be +/- 13% and +/- 3% respectively. We show analytically how these uncertainties are reduced after including next-to-leading QCD corrections. They amount to +/- 0.4% and +/- 0.3% respectively. Reanalyzing the uncertainties due to the scale mu(b) = O(m(b)) we find that after the inclusion of NLO effects they amount to +/- 3.2% which is smaller by a factor of two than claimed in the literature. Including the uncertainties due to input parameters as well as the non-perturbative 1/m(b)(2) and 1/m(c)(2) corrections we find Br(B --> X-s gamma) = (3.48 +/- 0.31) X 10(-4) where the error is dominated by uncertainties in the input parameters. This should be compared with (3.28 +/- 0.33) X 10(-4) found by Chetyrkin et al. where the error is shared evenly between the scale and parametric uncertainties. (C) 1997 Elsevier Science B.V.