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Power corrections to B\to X_u l ν(X_sγ) decay spectra in the "shape-function" region

M. Beneke, F. Campanario, T. Mannel, B. D. Pecjak

TL;DR

This work uses soft-collinear effective theory (SCET) to analyze 1/m_b power corrections in inclusive B decays within the shape-function region, showing that factorization persists beyond leading power but requires a substantial set of sub-leading shape-functions and jet-functions. A two-step QCD→SCET→HQET matching framework is developed to separate hard, jet, and soft physics, with a full sub-leading current basis and a comprehensive enumeration of soft operators up to tetra-locality. At tree level, a novel four-quark contribution emerges, and certain simplifications reduce the number of non-perturbative inputs in practice; this four-quark piece can induce observable differences between charged and neutral B decays, potentially at the percent level, while loop corrections will complicate the picture by increasing the number of relevant functions. Overall, the paper lays out the structure of sub-leading factorization in inclusive B decays, provides explicit tree-level results, and highlights where phenomenology must account for power-suppressed effects to achieve precision in extracting CKM parameters like |V_ub|.

Abstract

Using soft-collinear effective theory (SCET), we examine the 1/m_b corrections to the factorization formulas for inclusive semi-leptonic B decays in the endpoint region, where the hadronic final state consists of a single jet. At tree level, we find a new contribution from four-quark operators that was previously assumed absent. Beyond tree level many sub-leading shape-functions are needed to correctly describe the decay process.

Power corrections to B\to X_u l ν(X_sγ) decay spectra in the "shape-function" region

TL;DR

This work uses soft-collinear effective theory (SCET) to analyze 1/m_b power corrections in inclusive B decays within the shape-function region, showing that factorization persists beyond leading power but requires a substantial set of sub-leading shape-functions and jet-functions. A two-step QCD→SCET→HQET matching framework is developed to separate hard, jet, and soft physics, with a full sub-leading current basis and a comprehensive enumeration of soft operators up to tetra-locality. At tree level, a novel four-quark contribution emerges, and certain simplifications reduce the number of non-perturbative inputs in practice; this four-quark piece can induce observable differences between charged and neutral B decays, potentially at the percent level, while loop corrections will complicate the picture by increasing the number of relevant functions. Overall, the paper lays out the structure of sub-leading factorization in inclusive B decays, provides explicit tree-level results, and highlights where phenomenology must account for power-suppressed effects to achieve precision in extracting CKM parameters like |V_ub|.

Abstract

Using soft-collinear effective theory (SCET), we examine the 1/m_b corrections to the factorization formulas for inclusive semi-leptonic B decays in the endpoint region, where the hadronic final state consists of a single jet. At tree level, we find a new contribution from four-quark operators that was previously assumed absent. Beyond tree level many sub-leading shape-functions are needed to correctly describe the decay process.

Paper Structure

This paper contains 19 sections, 76 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Factorization at sub-leading order
  3. 1. Matching to SCET/HQET.
  4. 2. Collinear-soft factorization.
  5. 3. Definition of "shape-functions" and "jet-functions".
  6. 4. Caveats to factorization.
  7. SCET currents
  8. Leading order.
  9. Next-to-leading order.
  10. Next-to-next-to-leading order.
  11. Basis of shape-functions
  12. Example of factorization at order $1/m_b$
  13. Tree approximation
  14. Remarks on factorization beyond tree level
  15. 1. Degree of multi-locality.
  16. ...and 4 more sections

Figures (2)

  • Figure 1: Tree diagrams contributing to the current correlator $T^{\mu\nu}$. Not shown are diagrams that vanish when $n_+ A_c=0$, $n_- A_s=0$, or are symmetric to those shown.
  • Figure 2: Distortion of the $P_+$ spectrum in $B^-\to X_u\ell \nu$ decay by four-quark contributions assuming the model (\ref{['shapemodel']}). The solid central curve is $\hat{S}(P_+)$. The dashed curves correspond to $\lambda=100\,$MeV (long dashes) and $\lambda=500\,$MeV (short dashes). Each pair is for $\varepsilon=\pm 0.1$.