NLO QCD corrections to inclusive semitauonic weak decays of heavy hadrons up to $1\mathrm{/}{m}_b^3$

This paper presents ${\alpha }_s$ corrections to the inclusive semitauonic $B\to X_c\tau {\bar{\nu }}_{\tau }$ decay width and leptonic invariant mass spectrum up to order $1\mathrm{/}{m}_b^3$ in the heavy quark expansion. Analytical results with full dependence on the final-state quark and tau lepton masses are obtained. The results up to $1\mathrm{/}{m}_b^2$ can be extrapolated to the $B\to X_u\tau {\bar{\nu }}_{\tau }$ decay by taking the limit of vanishing final-state quark mass.