NNLO corrections to inclusive semileptonic B decays in the shape-function region

TL;DR

To improve precision in |V_ub| from inclusive B -> X_u l nu decays, the authors compute two-loop QCD corrections to the hard coefficient functions within the SCET-based factorization framework for the shape-function region. They perform a full NNLO analysis: two-loop QCD form-factor calculations, UV renormalization, and a matching to extract H_ij, including infrared subtractions and charm-mass effects. The results, which agree with prior independent calculations and demonstrate IR cancellation, complete the NNLO perturbative input for this decay channel and indicate moderate NNLO corrections with manageable charm-mass contributions. Together with known NNLO jet-function results, this work enables a more precise theoretical description of inclusive B decays in the shape-function region and paves the way for reduced uncertainties in |V_ub|.

Abstract

We compute 2-loop QCD corrections to the hard coefficient functions which arise in the factorization formula for B -> X_u l nu decays in the shape-function region. Our calculation provides the last missing piece required for a NNLO analysis of inclusive semileptonic B decays, which may significantly reduce the theoretical uncertainty in the extraction of the CKM matrix element |V_ub|. Among the technical aspects, we find that the 2-loop hard coefficient functions are free of infrared singularities as predicted by the factorization framework. We perform a brief numerical analysis of the NNLO corrections and include a discussion on charm mass effects.

Figures (6)

• Figure 1: 2-loop diagrams. Diagrams that result from mirroring at the vertical axis are not shown. The bubble in the last diagram represents the 1-loop gluon self-energy.
• Figure 2: Scalar 2-loop Master Integrals. We use dashed (double) lines for massless (massive) propagators. Dashed/solid/double external lines correspond to virtualities $0/u m_b^2/m_b^2$, respectively. Dotted propagators are taken to be squared.
• Figure 3: 2-loop diagram with massive fermion loop.
• Figure 4: Additional Master Integrals with propagators of mass $m_c$ (wavy line). Notation as in Figure \ref{['fig:MIs']}.
• Figure 5: Wilson coefficients $C_i$ at the scale $\mu=m_b$ as a function of $u$ (the momentum transfer is given by $q^2=(1-u)m_b^2$). The dashed lines refer to the NLO results and the solid lines to the NNLO results with $z=m_c/m_b=0$ (orange/light gray) and $z=0.3$ (blue/dark gray).