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Inclusive Decay Rate for $B \to X_d + γ$ in Next-to-Leading Logarithmic Order and CP Asymmetry in the Standard Model

We compute the decay rate for the CKM-suppressed electromagnetic penguin decay $B \to X_d + γ$ (and its charge conjugate) in NLO QCD, including leading power corrections in $1/m_b^2$ and $1/m_c^2$ in the standard model. The average branching ratio of the decay $B \to X_dγ$ and its charge conjugate is estimated to be in the range $6.0 \times 10^{-6} \leq <BR(B \to X_d γ)> \leq 2.6 \times 10^{-5}$, obtained by varying the CKM-Wolfenstein parameters $ρ$ and $η$ in the range $-0.1 \leq ρ\leq 0.4$ and $0.2 \leq η\leq 0.46$ and taking into account other parametric dependence. In the stated range of the CKM parameters, we find the ratio $R(dγ/sγ) = <BR(B \to X_dγ)>/<BR(B \to X_sγ)>$ to lie in the range between 0.017 and 0.074. Theoretical uncertainties in this ratio are found to be small. Hence, this ratio is well suited to provide independent constraints on the CKM parameters. The CP-asymmetry in the $B \to X_d γ$ decay rates is found to be in the range $(7 - 35)%$. Both the decay rates and CP asymmetry are measurable in forthcoming experiments at $B$ factories and possibly at HERA-B.