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Towards $B \to V γ$ Decays at NNLO in SCET

Ahmed Ali, Ben D. Pecjak, Christoph Greub

TL;DR

The paper advances NNLO predictions for B→Vγ decays using SCET factorization, delivering complete NNLO vertex corrections for Q7 and Q8 and partial NNLO results for Q1 in the large-β0 limit; it also analyzes hard-spectator contributions and resums large logarithms via renormalization-group evolution. By matching QCD onto SCET_I and then SCET_II, it separates hard and hard-collinear scales, and fixes the SCET soft form factor ζ_{V⊥} through tensor form-factor matching. The resulting predictions for B→K*γ and B_s→φγ show reasonable agreement with data but remain theory-limited primarily by ζ_{V⊥} and incomplete NNLO corrections for Q1 in the spectator sector. Overall, the work validates factorization at NNLO, sharpens theoretical control over radiative B decays, and highlights the nonperturbative inputs needed to further tighten comparisons with experiment.

Abstract

We compute NNLO (${\cal O}(α_s^2)$) corrections to the hard-scattering kernels entering the QCD factorization formula for $B\to Vγ$ decays, where $V$ is a light vector meson. We give complete NNLO results for the dipole operators $Q_7$ and $Q_8$, and partial results for $Q_1$ valid in the large-$β_0$ limit and neglecting the NNLO correction from hard spectator scattering. Large perturbative logarithms in the hard-scattering kernels are identified and resummed using soft-collinear effective theory. We use our results to estimate the branching fractions for $B\to K^*γ$ and $B_s\to φγ$ decays at NNLO and compare them with the current experimental data.

Towards $B \to V γ$ Decays at NNLO in SCET

TL;DR

The paper advances NNLO predictions for B→Vγ decays using SCET factorization, delivering complete NNLO vertex corrections for Q7 and Q8 and partial NNLO results for Q1 in the large-β0 limit; it also analyzes hard-spectator contributions and resums large logarithms via renormalization-group evolution. By matching QCD onto SCET_I and then SCET_II, it separates hard and hard-collinear scales, and fixes the SCET soft form factor ζ_{V⊥} through tensor form-factor matching. The resulting predictions for B→K*γ and B_s→φγ show reasonable agreement with data but remain theory-limited primarily by ζ_{V⊥} and incomplete NNLO corrections for Q1 in the spectator sector. Overall, the work validates factorization at NNLO, sharpens theoretical control over radiative B decays, and highlights the nonperturbative inputs needed to further tighten comparisons with experiment.

Abstract

We compute NNLO () corrections to the hard-scattering kernels entering the QCD factorization formula for decays, where is a light vector meson. We give complete NNLO results for the dipole operators and , and partial results for valid in the large- limit and neglecting the NNLO correction from hard spectator scattering. Large perturbative logarithms in the hard-scattering kernels are identified and resummed using soft-collinear effective theory. We use our results to estimate the branching fractions for and decays at NNLO and compare them with the current experimental data.

Paper Structure

This paper contains 17 sections, 96 equations, 2 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Factorization and the hard-scattering kernels
  3. Vertex corrections
  4. Hard spectator scattering
  5. $Q_8$ at tree level
  6. $Q_8$ at one loop
  7. Numerical analysis
  8. The vertex amplitude
  9. The hard spectator amplitude
  10. The SCET soft function
  11. Branching fractions
  12. Conclusions
  13. Appendix
  14. Matrix elements
  15. The coefficients $\Delta_1 C^{B1(0)}$, $\Delta_7 C^{B1(1)}$ and $j_\perp^{(1)}$
  16. ...and 2 more sections

Figures (2)

  • Figure 1: The lowest-order diagram for spectator scattering with $Q_8$. The double-line represents the incoming $b$ quark and the solid box an insertion of $Q_8$. The photon can be attached to any of the four crosses. Only photon emissions from the light quark emerging from the $Q_8$ insertion contributes at leading power in $1/m_b$.
  • Figure 2: The one-loop corrections to spectator scattering with $Q_8$. The solid box denotes a $Q_8$ insertion and the photon can be attached to any of the crosses.