Inclusive Decay Rate for B → X d + γ in Next - to - Leading Logarithmic Order and CP Asymmetry in the Standard Model

We compute the decay rate for the Cabibbo-Kobayashi-Maskawa (CKM)-suppressed electromagnetic penguin decay B → Xd + γ (and its charge conjugate) in the next-to-leading order in QCD, including leading power corrections in 1/mb and 1/m 2 c in the standard model. The average branching ratio 〈B(B → Xd + γ)〉 of the decay B → Xd + γ and its charge conjugate B → Xd + γ is estimated to be in the range 6.0 × 10 ≤ 〈B(B → Xd + γ)〉 ≤ 2.6 × 10, obtained by varying the CKM-Wolfenstein parameters ρ and η in the range −0.1 ≤ ρ ≤ 0.4 and 0.2 ≤ η ≤ 0.46 and taking into account other parametric dependence. In the NLL approximation and in the stated range of the CKM parameters, we find the ratio R(dγ/sγ) ≡ 〈B(B → Xdγ)〉/〈B(B → Xsγ) to lie in the range 0.017 ≤ R(dγ/sγ) ≤ 0.074. Theoretical uncertainties in this ratio are estimated and found to be small. Hence, this ratio is well suited to provide independent constraints on the CKM parameters. The CP-asymmetry in the decay rates, defined as aCP (B → Xdγ) ≡ (Γ(B → Xdγ) − Γ(B → Xdγ))/(Γ(B → Xdγ) + Γ(B → Xdγ)), 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. Work partially supported by Schweizerischer Nationalfonds.