QCD corrections to the Darwin coefficient in inclusive semileptonic $B\to X_u\ell {\overline{\nu }}_{\ell }$ 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 $B\to X_u\ell {\overline{\nu }}_{\ell }$ ($\ell =e$, $\mu$, $\tau$) and $D\to X\ell {\overline{\nu }}_{\ell }$ ($\ell =e$, $\mu$) decays. The main application of our results is the background subtraction of the $B\to X_u\ell {\overline{\nu }}_{\ell }$ decay in the measurement of the $B\to X_c\ell {\overline{\nu }}_{\ell }$ 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.

Figure 1

$Q(p)q(0)\to Q(p)q(0)$ scattering diagrams contributing to the LO and NLO four-quark operator coefficients ${\mathcal{C}}_i^{hl}$ ($i=1,\dots ,6$) in the HQE of the $H_Q\to X_q\ell {\overline{\nu }}_{\ell }$ decay spectrum, Eq. (23). Circles with crosses stand for insertions of ${\mathcal{L}}_{\mathrm{eff}}$ and the thick line stands for the lepton-antineutrino propagator with mass $q$.

Figure 2

$h_v(0)q(0)\to h_v(0)q(0)$ scattering diagrams contributing to the operator mixing of four-quark operators with themselves at NLO. Circles with crosses stand for insertions of ${\mathcal{O}}_{1,2}^{hl}$.

Figure 3

$Q(p+k_1)\to Q(p+k_2)g(k_1-k_2)$ scattering diagrams contributing to the LO and NLO coefficients ${\overline{\mathcal{C}}}_i$ of power corrections in the HQE of the $H_Q\to X_q\ell {\overline{\nu }}_{\ell }$ decay spectrum, Eq. (23). Black squares stand for $\cancel{v}$ insertions, circles with crosses for insertions of ${\mathcal{L}}_{\mathrm{eff}}$, thick lines for the lepton-antineutrino propagator with mass $q$, and gray dots for possible one gluon insertions with outgoing momentum $k_1-k_2$. After accounting for all one gluon insertions, there are 5 diagrams at LO and 41 diagrams at NLO.