On the Resummed Hadronic Spectra of Inclusive B Decays

TL;DR

This work addresses the challenge of extracting $|V_{ub}|$ from inclusive $B$ decays when experimental cuts push calculations into phase-space boundaries where infrared logs ruin the standard perturbative expansion. It develops a hadronic-variable factorization into hard, jet, and soft functions, enabling threshold resummation at next-to-leading logarithmic accuracy for the cut rates in $B\to X_u e\nu$ and $B\to X_s\gamma$, using Mellin-space techniques and inverse transforms. The authors provide closed-form resummed expressions for the cut rate and extract leading and next-to-leading infrared log contributions to the two-loop differential rate, finding that resummation is crucial for realistic cuts but the controlled region is limited and sensitive to $\bar{\Lambda}$. They also show how the hadronic-mass cut depends on $\bar{\Lambda}$ and discuss nonperturbative Fermi-motion corrections, outlining a path to reduce dependence on nonperturbative input via soft-function information from $B\to X_s\gamma$ end-points. Overall, the paper clarifies the viability and limitations of using the hadronic invariant-mass spectrum to determine $V_{ub}$ and provides a framework for incorporating nonperturbative effects in a controlled way.

Abstract

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_{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 B -> X_s gamma and 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 \barΛ. We discuss the viability of extracting V_{ub} from the hadronic mass spectrum.

Figures (4)

• Figure 1: Reduced diagram for inclusive B decays.
• Figure 2: The rate as a function of the partonic cut. The dotted line is the one loop result, the dashed line is the LL result, while the solid line is the NLL log result. The difference between the one loop and resummed results at large $c_0$ is due to two loop corrections introduced in the resummation. We use $\alpha_s(m_b)=0.21$.
• Figure 3: The rate with a hadronic cut. The dotted line is the one loop result, the dashed line is the LL resummed result, while the solid line is the NLL resummed result. For $\bar{\Lambda}= 0.39\,{\rm GeV}$, we have $\rho_\epsilon=0.0861$. In the region above $\rho_\epsilon$, we run into Landau pole at $c_p = 0.1159$ and interpolate in the small region between $\rho_\epsilon$ and $c_p$.
• Figure 4: The cut rate for several different values of $\bar{\Lambda}$. The dashed line is for $\bar{\Lambda} = 0.28\,{\rm GeV}$, the dot-dashed line is for $\bar{\Lambda} = 0.50\,{\rm GeV}$, while the solid line is for $\bar{\Lambda} = 0.39\,{\rm GeV}$. The dotted line is the one loop result for $\bar{\Lambda} = 0.39\,{\rm GeV}$.