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

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.