QCD corrections to the Darwin coefficient in inclusive semileptonic $B\rightarrow X_u \ell \bar{\nu}_\ell$ decays

In this paper we compute $\alpha_s$ corrections to the matching coefficients of the dimension six operators in the Heavy Quark Expansion of the inclusive semileptonic heavy hadron decay rate and leptonic invariant mass spectrum with a massless quark and both a massive or massless lepton in the final state, analytically. The obtained results can be applied to the inclusive semileptonic $B\rightarrow X_u \ell \bar{\nu}_\ell$ ($\ell = e,\,\mu,\,\tau$) and $D\rightarrow X \ell \bar{\nu}_\ell$ ($\ell = e,\,\mu$) decays. The main application of our results is the background subtraction of the $B\rightarrow X_u \ell \bar{\nu}_\ell$ decay in the measurement of the $B\rightarrow X_c\ell \bar{\nu}_\ell$ decay, which is important for the precise extraction of $V_{cb}$ and $R(D^{(*)})$. They also play a role in the computation of lifetimes of heavy hadrons.