Threshold Resummed Spectra in B -> Xu l nu Decays in NLO (I)

TL;DR

The paper develops a comprehensive threshold-resummation framework for $B\to X_u l \nu$ decays at NLO, showing that spectra not integrated over hadronic energy can be related to the photon spectrum in $B\to X_s\gamma$ via a universal QCD form factor $\sigma$ and process-dependent short-distance coefficients. It formulates the triple differential distribution, performs resummation in Mellin space with explicit $A_i,B_i,D_i$ coefficients up to NNLO, and provides inverse-transform prescriptions to obtain physical-space results for hadronic-energy, hadron-mass, and hadron-electron spectra. The work distinguishes two classes of spectra: those with unintegrated hadronic energy, which share universal soft-gluon structure with radiative decays, and those with integrated hadronic energy, which exhibit more complex logarithms and require additional matching. It also delivers explicit NLO expressions for coefficient and remainder functions, including an improved scheme for the $w>1$ region to ensure continuity and positivity, and reports quantitative results for average hadron energy and mass-cut effects. Overall, the study clarifies the universality of soft-gluon dynamics in heavy-flavor decays and provides a practical, resum­mation-driven path to precise predictions across related decay channels.

Abstract

We evaluate thresholds resummed spectra in B -> Xu l nu decays in next-to-leading order. We present results for the distribution in E_X and in m_X^2/E_X^2, for the distribution in E_X and E_l and for the distribution in E_X, where E_X and m_X are the energy and the invariant mass of the final hadronic state Xu respectively and E_l is the energy of the charged lepton. We explicitly show that all these spectra (where there is no integration over the hadronic energy) can be directly related to the photon spectrum in B -> Xs gamma via short-distance coefficient functions.