Calculations of two-loop virtual corrections tob→sl+l−in the standard model

We present in detail the calculation of the virtual $\mathcal{O}\left({\alpha }_s\right)$ corrections to the inclusive semileptonic rare decay $\overrightarrow{b}{\mathrm{sl}}^{+}{l}^{-}.$ We also include those $\mathcal{O}\left({\alpha }_s\right)$ bremsstrahlung contributions which cancel the infrared and mass singularities showing up in the virtual corrections. In order to avoid large resonant contributions, we restrict the invariant mass squared s of the lepton pair to the range $0.05<~{s/m}_b^2<~\mathrm{0.25.}$ The analytic results are represented as expansions in the small parameters $ŝ{=s/m}_b^2,$ ${z=m}_c^2{/m}_b^2$ and ${s/(4m}_c^2).\mathrm{}$ The new contributions drastically reduce the renormalization scale dependence of the decay spectrum. For the corresponding branching ratio (restricted to the above $ŝ$ range) the renormalization scale uncertainty gets reduced from $\sim \pm 13\%$ to $\sim \pm 6.5\%.$