Virtual and bremsstrahlung corrections tob→dl+l−in the standard model

We present the calculation of the virtual and bremsstrahlung corrections of $O\left({\alpha }_s\right)$ to the matrix elements $〈{\mathrm{dl}}^{+}{l}^{-}|O_i|b〉.$ This is the missing piece in the full next-to-next-to-leading logarithmic (NNLL) results for various observables associated with the process $\overrightarrow{B}{X}_d{l}^{+}{l}^{-},$ such as the branching ratio, the CP-rate asymmetry, and the forward-backward asymmetry. This paper is an extension of analogous calculations done by some of us for the process $\overrightarrow{B}{X}_s{l}^{+}{l}^{-}.$ As the contributions of the diagrams induced by the four-quark operators $O_1^u$ and $O_2^u$ with a u quark running in the quark loop are strongly Cabibbo-Kobayashi-Maskawa suppressed, they were omitted in the analysis of $\overrightarrow{B}{X}_s{l}^{+}{l}^{-}.$ This is no longer possible for $\overrightarrow{B}{X}_d{l}^{+}{l}^{-},$ as the corresponding contributions are not suppressed. The main new work therefore consists of calculating the $O\left({\alpha }_s\right)$ corrections to $〈{\mathrm{dl}}^{+}{l}^{-}|O_{1,2}^u|b〉.$ In this paper we restrict ourselves to the range $0.05<~{s/m}_b^2<~0.25$ (s is the invariant mass of the lepton pair), which lies above the ρ and ω resonances and below the $J/\psi$ resonance. We present the analytic results for the mentioned observables related to the process $\overrightarrow{B}{X}_d{l}^{+}{l}^{-}$ as expansions in the small parameters $ŝ{=s/m}_b^2,$ ${z=m}_c^2{/m}_b^2,$ and ${s/(4m}_c^2).\mathrm{}$ In the phenomenological analysis at the end of the paper we discuss the impact of the NNLL corrections on the observables mentioned above.