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Factorization for Power Corrections to B->Xs gamma and B-> Xu e nu

Keith S. M. Lee, Iain W. Stewart

TL;DR

This work uses Soft-Collinear Effective Theory to derive a complete factorization framework for Lambda_QCD/m_b power corrections in the B-decay endpoint region, covering B->Xs gamma and B->Xu l nu up to ${\cal O}(\lambda^2)$. It identifies new, previously neglected shape functions, including four-quark operators, whose contributions can be numerically large and substantially affect |V_{ub}| extractions from endpoint spectra. The results organize corrections into hard, jet, and soft components with explicit convolution structures, enabling systematic perturbative resummation and improved uncertainty estimates. The findings emphasize the need to account for these subleading shape-function effects in phenomenology and point to future work on perturbative corrections, resummation, and experimental constraints to bound these power corrections.

Abstract

We derive factorization theorems for Lambda_QCD/mb power corrections to inclusive B-decays in the endpoint region, where mX^2 ~mb Lambda_QCD. In B-> Xu e nu our results are for the full triply differential rate. A complete enumeration of Lambda_QCD/mb corrections is given. We point out the presence of new Lambda_QCD/mb-suppressed shape functions, which arise at tree level with a $4 pi$-enhanced coefficient, and show that these previously neglected terms induce an additional large uncertainty for current inclusive methods of measuring |Vub| that depend on the endpoint region of phase space.

Factorization for Power Corrections to B->Xs gamma and B-> Xu e nu

TL;DR

This work uses Soft-Collinear Effective Theory to derive a complete factorization framework for Lambda_QCD/m_b power corrections in the B-decay endpoint region, covering B->Xs gamma and B->Xu l nu up to . It identifies new, previously neglected shape functions, including four-quark operators, whose contributions can be numerically large and substantially affect |V_{ub}| extractions from endpoint spectra. The results organize corrections into hard, jet, and soft components with explicit convolution structures, enabling systematic perturbative resummation and improved uncertainty estimates. The findings emphasize the need to account for these subleading shape-function effects in phenomenology and point to future work on perturbative corrections, resummation, and experimental constraints to bound these power corrections.

Abstract

We derive factorization theorems for Lambda_QCD/mb power corrections to inclusive B-decays in the endpoint region, where mX^2 ~mb Lambda_QCD. In B-> Xu e nu our results are for the full triply differential rate. A complete enumeration of Lambda_QCD/mb corrections is given. We point out the presence of new Lambda_QCD/mb-suppressed shape functions, which arise at tree level with a -enhanced coefficient, and show that these previously neglected terms induce an additional large uncertainty for current inclusive methods of measuring |Vub| that depend on the endpoint region of phase space.

Paper Structure

This paper contains 30 sections, 266 equations, 4 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Basic Ingredients
  3. Weak Effective Hamiltonians
  4. Hadronic Tensors and Decay Rates
  5. Light-cone Hadronic Variables and Endpoint Kinematics
  6. OPE and Partonic Variables
  7. SCET ingredients
  8. Leading-Order Factorization
  9. Vanishing Time-Ordered Products at ${\cal O}(\lambda)$
  10. Factorization at Next-to-Leading Order
  11. Switching to hadronic variables at order $\lambda^2$
  12. Time-Ordered Products at order $\lambda^2$
  13. Tree-Level Matching Calculations
  14. NLO Soft Operators and Shape Functions
  15. Factorization Calculations at ${\cal O}(\lambda^2)$
  16. ...and 15 more sections

Figures (4)

  • Figure 1: Comparison of the ratio of annihilation contributions to the lowest-order result. In the total decay rate, b) is $\sim 16\pi^2(\Lambda^3/m_b^3) \Delta B\simeq 0.02$, while c) is $\sim 4\pi\alpha_s(m_b)(\Lambda^3/m_b^3)\simeq 0.003$ when compared to a). In the endpoint region, b) is $\sim 16\pi^2 (\Lambda^2/m_b^2)\Delta B\simeq 0.16$, a large correction, while c) becomes $\sim 4\pi\alpha_s(\mu_J)(\Lambda/m_b)\epsilon' \simeq 0.6 \epsilon'$, a potentially large correction.
  • Figure 2: Allowed phase space for $B\to X_u \ell\bar{\nu}$, where $m_\pi < m_X < m_B$. The second figure shows the same regions using the variables defined in Eq. (\ref{['ybaru']}). We indicate the region where charm contamination enters, $m_X > m_D$, and the region of phase space where annihilation contributions enter. Also shown is the region where the SCET expansion converges, which is taken to be $u_H/\overline y_H\le 0.2$ and corresponds to $m_X^2/(4E_X^2) \lesssim 0.14$.
  • Figure 3: Momentum routing for leading-order insertion of currents.
  • Figure 4: Momentum routing for a) $T^{(2L)}$ and b) $T^{(2q)}$. In a), the operator with one gluon comes from $[i{\cal D}\!\!\!\!\slash_{us}^\perp i{\cal D}\!\!\!\!\slash_{us}^\perp]$.