Multi-parton contributions to $\bar B \to X_s \gamma$ at NLO

Many contributions to the decay rate of the inclusive radiative $\bar{B}\rightarrow X_s \gamma$ transition have been calculated to NNLO in QCD during the past decades. However, there are still a few unknown contributions from multi-parton final states which are formally NLO. In the present work, we compute those four-body $b \rightarrow s\, q\, \bar{q}\,\gamma$ contributions at NLO in QCD which need to be supplemented by the five-body $b \rightarrow s\, q\, \bar{q}\, g\,\gamma$ bremsstrahlung. This calculation formally completes the purely perturbative contributions to $\bar{B}\rightarrow X_s \gamma$ at NLO. Our results are obtained by applying modern techniques of integral reduction and evaluation of master integrals. In particular, the analytic integration over the four and five-particle phase space in the presence of a cut on the photon energy turns out to be technically involved. We give our results completely analytically in terms of multiple polylogarithms, including the dependence on the collinear logarithms which arise from the mass-regularisation of collinear divergences. The numerical impact of multi-parton corrections on the $\bar{B}\rightarrow X_s \gamma$ decay rate turns out to be small, owing to a partial cancellation between LO and NLO contributions.