The Extraction of $V_{ub}$ from Inclusive B Decays and the Resummation of End Point Logs

TL;DR

The paper tackles the extraction of |V_ub| from inclusive B decays by addressing endpoint singularities through resummation of leading and next-to-leading endpoint logs and incorporating the leading $\alpha_s$ corrections to the hard amplitude. It develops a factorization framework separating soft, jet, and hard contributions, with a universal soft function that can be constrained via ratios or lattice inputs. Numerical analysis shows that resumming subleading logs can dominate the effect, yielding a 20–50% modification in the endpoint region and highlighting the sensitivity to nonperturbative structure and duality assumptions. The study concludes that, while resummation improves control over theoretical uncertainties, a fully model-independent extraction of |V_ub| remains challenging, potentially achievable at ~30% under optimistic conditions, with significant dependence on higher-order effects and background modeling.

Abstract

In this paper we discuss the theoretical difficulties in extracting $V_{ub}$ using the data from inclusive B decays. Specifically, we address the issue of the end point singularities. We perform the resummation of both the leading and next to leading end point logs and include the leading corrections to the hard scattering amplitude. We find that the resummation is a $20\%-50\%$ effect in the end point region where the resummation is valid. Furthermore, the resummed sub-leading logs dominate the resummed double logs. The consequences of this result for a model independent extraction of the mixing angle $V_{ub}$ are explored.

Figures (4)

• Figure 1: The ratio of $\frac{V_{ub}^2}{V_{bc}^2}$ for which the Nth moments of the leading order spectra are equal.
• Figure 2: The difference between the moments given by the leading $\alpha_s$ correction and the moments of the rate with only the double logs resummed.
• Figure 3: Reduced diagram for B decays.
• Figure 4: The difference between the moments of the one loop result and the resummed result with (dashed) and without (undashed) the resummation of the $\pi^2$, normalized to the one loop result. $N$ varies from $10$ to $30$.