Structure functions for the inclusive semileptonic $b$-quark decay at NNLO: a semi-analytic calculation

TL;DR

This work tackles the NNLO QCD corrections to inclusive $B\to X_u\ell\nu$ decays by extracting the hadronic structure functions $W_i$ through a semi-analytic approach that fuses EFT-derived singular terms from HQET/SCET, known BLM corrections, and a modified parton-level Monte Carlo with phase-space slicing. The authors decompose $W_i$ into a singular EFT part and a regular remainder, modeling the latter with a basis of functions and fitting to differential distributions and moments, while anchoring the result with analytic NNLO total widths and $q^2$ distributions. They validate the method at NLO against exact results and then deliver NNLO fits for $W_i^{(2)}$, achieving agreement with analytic and numerical benchmarks at the per-cent level. The resulting semi-analytic NNLO structure functions enable precise predictions for differential spectra and moments, improving the reliability of $|V_{ub}|$ extractions and inclusive $B$-meson phenomenology.

Abstract

We present a study of the inclusive charmless semileptonic $b$ decay, $b\to X_u\ell\barν$ at next-to-next-to-leading order (NNLO) in perturbative QCD, with the primary aim of extracting the hadronic structure functions $W_i$ at NNLO. The analysis is based on a numerical calculation of the relevant kinematic distributions using a phase-space slicing method to handle infrared-sensitive contributions from real gluon emissions. We use known results from Heavy Quark Effective Theory and Soft-Collinear Effective Theory to extract the singular terms and construct a model for the regular contributions to the structure functions at NNLO, then perform a fit to the numerical results. We use our approximate structure functions to compute various kinematic distributions and moments: the comparison with existing analytic and numerical results shows very good agreement, which is further improved including the available analytic results in the fit.

Figures (6)

• Figure 1: $\mathcal{O}(\alpha_s)$ contributions to the hadronic structure functions $W_i$ as functions of kinematic variables, for three representative values of the lepton invariant mass, $\hat{q}^2=0.1,0.3,0.5$. The bands represent the 68% CL ranges from the fit, which includes the $\hat{q}^2$ differential distribution. The vertical axes show $1-\mathrm{ratio}$, where one takes the ratio of the fitted $W_i^{(1)}$ (numerator) to the analytic result (denominator).
• Figure 2: NLO differential distributions for the $b \to X_u\ell\bar{\nu}$ decay and comparison with the corresponding results from fitted $W^{(1)}_i$, see text for an explanation. The fits employed in this figure are obtained without including information coming from the analytic $\hat{q}^2$ distribution.
• Figure 3: NLO differential distributions for the $b \to X_u\ell\bar{\nu}$ decay and comparison with the corresponding results from fitted $W^{(1)}_i$ (see text). The fits employed in this figure include information coming from the analytic $\hat{q}^2$ distribution.
• Figure 4: NNLO differential decay distributions in $b\to X_u\ell\bar{\nu}$ and comparison with the corresponding results from fitted $W_i^{(2)}$. The fits employed in this Figure are obtained without using information from the analytic results for the $\hat{q}^2$ distribution.
• Figure 5: NNLO differential decay distributions in $b\to X_u\ell\bar{\nu}$ and comparison with the corresponding results from fitted $W_i^{(2)}$. The fits employed in this Figure are include information from the analytic results for the $\hat{q}^2$ distribution.