Resummed hadronic spectra of inclusiveBdecays

In this paper we investigate the hadronic mass spectra of inclusive $B$ decays. Specifically, we study how an upper cut on the invariant mass spectrum, which is necessary to extract $V_{\mathrm{ub}},$ results in the breakdown of the standard perturbative expansion due to the existence of large infrared logs. We first show how the decay rate factorizes at the level of the double differential distribution. Then, we present closed form expressions for the resummed cut rate for the inclusive decays $\overrightarrow{B}{X}_s\gamma$ and $\overrightarrow{B}{X}_{u}e\nu$ at next-to-leading order in the infrared logs. Using these results, we determine the range of cuts for which resummation is necessary, as well as the range for which the resummed expansion itself breaks down. We also use our results to extract the leading and next to leading infrared log contribution to the two loop differential rate. We find that for the phenomenologically interesting cut values, there is only a small region where the calculation is under control. Furthermore, the size of this region is sensitive to the parameter $\overline{\Lambda }.$ We discuss the viability of extracting $V_{\mathrm{ub}}$ from the hadronic mass spectrum.