On the Scale Uncertainties in the $B \to X_s \gamma$ Decay

Phys.Lett. B414 (1997) 157-165; Erratum-ibid. B434 (1998) 459 We analyze the theoretical uncertainties in $Br(B\to X_s\gamma)$ due to the choice of the high energy matching scale $\mu_W=\ord(\mw)$ and the scale $\mu_t$ at which the running top quark mass is defined: $\mtb(\mu_t)$. 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 $\mu_t\not=\mu_W$. In the leading order the $\mu_W$ and $\mu_t$ uncertainties in $Br(B\to X_s\gamma)$ turn out to be $\pm 13%$ and $\pm 3%$ respectively. We show analytically how these uncertainties are reduced after including next-to-leading QCD corrections. They amount to $\pm 1.1%$ and $\pm 0.4%$ respectively. Reanalyzing the uncertainties due to the scale $\mu_b=\ord(m_b)$ we find that after the inclusion of NLO effects they amount to $\pm 4.3%$ which is a factor 2/3 smaller 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 $Br(B{\to}X_s \gamma) = (3.60 \pm 0.33) \times 10^{-4}$ where the error is dominated by uncertainties in the input parameters. This should be compared with $(3.28 \pm 0.33) \times 10^{-4}$ found by Chetyrkin et al. where the error is shared evenly between the scale and parametric uncertainties.