On the $O(α_s^2)$ corrections to $b \to X_u e \bar ν$ inclusive decays

Abstract

We present $O(α_s^2)$ QCD corrections to the fully-differential decay rate of a $b$-quark into inclusive semileptonic charmless final states. Our calculation provides genuine two-loop QCD corrections, beyond the Brodsky-Lepage-Mackenzie (BLM) approximation, to any infra-red safe partonic observable that can be probed in $b \to X_u e \bar ν$ decays. Kinematic cuts that closely match those used in experiments can be fully accounted for. To illustrate these points, we compute the non-BLM corrections to moments of the hadronic invariant mass and the hadronic energy with cuts on the lepton energy and the hadronic invariant mass. Our results remove one of the sources of theoretical uncertainty that affect the extraction of the CKM matrix element $|V_{ub}|$ from charmless inclusive B-decays.

Figures (3)

• Figure 1: The coefficient of the second order correction to the lepton invariant mass distribution. The solid curve is the analytic result of Ref. cz1.
• Figure 2: The cumulative histogram that shows $L^{(2)}_{00}$ as a function of the cut on the charged lepton energy $E_l > E_{\rm cut}$. No cut on the hadronic invariant mass is applied.
• Figure 3: The cumulative histogram that shows $L^{(2)}_{03}$ as a function of the cut on the charged lepton energy $E_l > E_{\rm cut}$. No cut on the hadronic invariant mass is applied.