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

Abstract

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 $B\to X_u\ell {\bar{\nu }}_{\ell }$ ($\ell =e,\mu ,\tau$) and $D\to X\ell {\bar{\nu }}_{\ell }$ ($\ell =e,\mu$) decays. The main application of our results is the background subtraction of the $B\to X_u\ell {\bar{\nu }}_{\ell }$ decay in the measurement of the $B\to X_c\ell {\bar{\nu }}_{\ell }$ decay, which is important for the precise extraction of $V_{cb}$ and $R(D^{(\ast )})$. They also play a role in the computation of lifetimes of heavy hadrons.

Comments: 29 pages, 3 figures, 1 ancillary file