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

Details On the Scale Uncertainties in the $B \to X_s γ$ Decay On the Scale Uncertainties in the $B \to X_s γ$ Decay

1997-07-26

arxiv.org/abs/hep-ph/9707482v3

Phys.Lett. B414 (1997) 157-165; Erratum-ibid. B434 (1998) 459

Physics

High Energy Physics

High Energy Physics - Phenomenology

11 pages, Latex. The paper is updated by incorporating recently modified results of the literature that were used for the nume

Scientific paper

10.1016/S0370-2693(97)01142-8

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.

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