Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Heavy-to-light B meson form factors at large recoil energy -- spectator-scattering corrections

M. Beneke, D. Yang

TL;DR

The paper computes the complete one-loop spectator-scattering corrections for heavy-to-light $B$ decays at large recoil within SCET by obtaining the jet-functions from the hard-collinear scale and combining them with the previously determined hard-scale coefficients, i.e., $T_i = C^{(B1)}_i \star J$. It finds that hard and hard-collinear corrections are of comparable size (roughly $20$–$40\%$) and that the perturbative expansion remains well-behaved, supporting factorization across scales. Numerically, the combined corrections enhance spectator-scattering by about 50–70% depending on the observable, and the RG-improved coefficients show modest resummation effects; the results also reveal notable differences with QCD sum-rule predictions for form-factor ratios and have implications for radiative, semi-leptonic, and hadronic $B$ decays, including an improved description of color-suppressed amplitudes in $B\to\pi\pi$.

Abstract

We complete the investigation of loop corrections to hard spectator-scattering in exclusive B meson to light meson transitions by computing the short-distance coefficient (jet-function) from the hard-collinear scale. Adding together the two coefficients from matching QCD to SCET_I to SCET_II, we investigate the size of loop effects on the ratios of heavy-to-light meson form factors at large recoil. We find the corrections from the hard and hard-collinear scales to be of approximately the same size, and significant, but the perturbative expansions appear to be well-behaved. Our calculation provides a non-trivial verification of the factorization arguments. We observe considerable differences between the predictions based on factorization in the heavy-quark limit and current QCD sum rule calculations of the form factors. We also include the hard-collinear correction in the B -> pi pi tree amplitudes, and find an enhancement of the colour-suppressed amplitude relative to the colour-allowed amplitude.

Heavy-to-light B meson form factors at large recoil energy -- spectator-scattering corrections

TL;DR

The paper computes the complete one-loop spectator-scattering corrections for heavy-to-light decays at large recoil within SCET by obtaining the jet-functions from the hard-collinear scale and combining them with the previously determined hard-scale coefficients, i.e., . It finds that hard and hard-collinear corrections are of comparable size (roughly –) and that the perturbative expansion remains well-behaved, supporting factorization across scales. Numerically, the combined corrections enhance spectator-scattering by about 50–70% depending on the observable, and the RG-improved coefficients show modest resummation effects; the results also reveal notable differences with QCD sum-rule predictions for form-factor ratios and have implications for radiative, semi-leptonic, and hadronic decays, including an improved description of color-suppressed amplitudes in .

Abstract

We complete the investigation of loop corrections to hard spectator-scattering in exclusive B meson to light meson transitions by computing the short-distance coefficient (jet-function) from the hard-collinear scale. Adding together the two coefficients from matching QCD to SCET_I to SCET_II, we investigate the size of loop effects on the ratios of heavy-to-light meson form factors at large recoil. We find the corrections from the hard and hard-collinear scales to be of approximately the same size, and significant, but the perturbative expansions appear to be well-behaved. Our calculation provides a non-trivial verification of the factorization arguments. We observe considerable differences between the predictions based on factorization in the heavy-quark limit and current QCD sum rule calculations of the form factors. We also include the hard-collinear correction in the B -> pi pi tree amplitudes, and find an enhancement of the colour-suppressed amplitude relative to the colour-allowed amplitude.

Paper Structure

This paper contains 40 sections, 133 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. Heavy-to-light form factors in SCET$_{\rm I}$
  3. Operator basis
  4. Definition of the QCD form factors
  5. Definition of the SCET$_{\rm I}$ form factors
  6. Form factor expressions
  7. Physical form factor scheme
  8. Jet-functions
  9. Set-up of the calculation
  10. Unrenormalized matrix element
  11. Tree
  12. One loop
  13. Ultraviolet counterterms
  14. Matching to SCET${}_{\rm II}$ and extraction of the jet function
  15. SCET${}_{\rm II}$ matrix element
  16. ...and 25 more sections

Figures (6)

  • Figure 1: Tree diagram.
  • Figure 2: 1-loop diagrams. The first diagram summarizes all gluon propagator corrections. Diagrams without numbers are omitted when the calculation is done by expansion of QCD diagrams.
  • Figure 3: $U_a(\tau,\mu_h,\mu)$ for $\mu_h=m_b=4.8$ GeV and $\mu=1.5$ GeV. The upper curves refer to $U_\parallel$, the lower ones to $U_\perp$. Solid lines: exact numerical integration. Dashed lines: approximate solutions.
  • Figure 4: The short-distance coefficients $C^{(B1)}_X(E,\tau,\mu)$ relevant to the form factors $X$ at two representative energy values ($x=1$ (left) and $x=0.6$ (right)) at $\mu=1.5\,$GeV normalized to the tree approximation. Dash-dotted: logarithmic terms at order $\alpha_s$; dashed: full NLO approximation; solid: NLO plus logarithmic summation.
  • Figure 5: $H_X(E)/H_X^{(0)}(E)$ for $X=0+,T+,T_1V$. The two upper curves represent the complete next-to-leading order result with (solid) and without (dashed) the renormalization group summation. The middle (dash-dotted) line shows the jet-function (hard-collinear) correction alone, the lower (dashed) line the hard correction alone.
  • ...and 1 more figures