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Renormalization-Group Improved Calculation of the B->Xs+gamma Branching Ratio

Matthias Neubert

TL;DR

This work deploys a renormalization-group–improved multi-scale operator product expansion (MSOPE) to compute the B → Xs γ partial rate with a photon-energy cut, clarifying the role of three scales mb, √(mbΔ), and Δ=mb−2E0. By factorizing into hard, jet, and shape-function components and solving the shape-function evolution in momentum space, the author achieves NNLL resummation and a controlled short-distance expansion of the convolution integral, valid when Δ≫ΛQCD. The analysis shows that the transition from the shape-function region to the MSOPE region is gradual, not abrupt, with significant perturbative uncertainties dominated by the low scale Δ, and finds Br(B → Xs γ) with Eγ≥1.8 GeV around 3.38×10^-4 with substantial theory errors, impacting New Physics constraints. Ratios and observables such as F(E0) and ⟨Eγ⟩ are discussed as probes of low-scale physics, providing a framework to combine MSOPE results with fully inclusive predictions and to assess the imprint of short- and long-distance dynamics on radiative B decays.

Abstract

Using results on soft-collinear factorization for inclusive B-meson decay distributions, a systematic study of the partial $B\to X_sγ$ decay rate with a cut $E_γ> E_0$ on photon energy is performed. For values of $E_0$ below about 1.9 GeV, the rate can be calculated without reference to shape functions using a multi-scale operator product expansion (MSOPE). The transition from the shape-function region to the MSOPE region is studied analytically. The resulting prediction for the $B\to X_sγ$ branching ratio depends on three large scales: $m_b$, $\sqrt{m_bΔ}$, and $Δ=m_b-2E_0$. Logarithms associated with these scales are resummed at next-to-next-to-leading logarithmic order. While power corrections in $Λ_{QCD}/Δ$ turn out to be small, the sensitivity to the scale $Δ\approx 1.1$ GeV (for $E_0\approx 1.8$ GeV) introduces significant perturbative uncertainties, which so far have been ignored. The new theoretical prediction for the $B\to X_sγ$ branching ratio with $E_γ\ge 1.8$ GeV is $Br(B\to X_sγ)=(3.38_{-0.42-0.30}^{+0.31+0.32})\times 10^{-4}$, where the first error is an estimate of perturbative uncertainties and the second one reflects uncertainties in input parameters. With this cut $(89_{-7}^{+6}\pm 1)%$ of all events are contained. The implications of larger theory uncertainties for New Physics searches are briefly explored with the example of the type-II two-Higgs-doublet model, for which the lower bound on the charged-Higgs mass is reduced compared with previous estimates to approximately 200 GeV at 95% confidence level.

Renormalization-Group Improved Calculation of the B->Xs+gamma Branching Ratio

TL;DR

This work deploys a renormalization-group–improved multi-scale operator product expansion (MSOPE) to compute the B → Xs γ partial rate with a photon-energy cut, clarifying the role of three scales mb, √(mbΔ), and Δ=mb−2E0. By factorizing into hard, jet, and shape-function components and solving the shape-function evolution in momentum space, the author achieves NNLL resummation and a controlled short-distance expansion of the convolution integral, valid when Δ≫ΛQCD. The analysis shows that the transition from the shape-function region to the MSOPE region is gradual, not abrupt, with significant perturbative uncertainties dominated by the low scale Δ, and finds Br(B → Xs γ) with Eγ≥1.8 GeV around 3.38×10^-4 with substantial theory errors, impacting New Physics constraints. Ratios and observables such as F(E0) and ⟨Eγ⟩ are discussed as probes of low-scale physics, providing a framework to combine MSOPE results with fully inclusive predictions and to assess the imprint of short- and long-distance dynamics on radiative B decays.

Abstract

Using results on soft-collinear factorization for inclusive B-meson decay distributions, a systematic study of the partial decay rate with a cut on photon energy is performed. For values of below about 1.9 GeV, the rate can be calculated without reference to shape functions using a multi-scale operator product expansion (MSOPE). The transition from the shape-function region to the MSOPE region is studied analytically. The resulting prediction for the branching ratio depends on three large scales: , , and . Logarithms associated with these scales are resummed at next-to-next-to-leading logarithmic order. While power corrections in turn out to be small, the sensitivity to the scale GeV (for GeV) introduces significant perturbative uncertainties, which so far have been ignored. The new theoretical prediction for the branching ratio with GeV is , where the first error is an estimate of perturbative uncertainties and the second one reflects uncertainties in input parameters. With this cut of all events are contained. The implications of larger theory uncertainties for New Physics searches are briefly explored with the example of the type-II two-Higgs-doublet model, for which the lower bound on the charged-Higgs mass is reduced compared with previous estimates to approximately 200 GeV at 95% confidence level.

Paper Structure

This paper contains 14 sections, 72 equations, 3 figures, 6 tables.

Table of Contents

  1. Introduction
  2. QCD factorization theorem
  3. Evolution of the shape function
  4. Short-distance expansion of the convolution integral
  5. Elimination of pole-scheme parameters
  6. Kinematic power corrections
  7. Ratios of decay rates
  8. Ratios insensitive to low-scale physics
  9. Ratios sensitive to low-scale physics
  10. Numerical results
  11. Partial $B\to X_s\gamma$ branching ratio
  12. Event fraction $F(E_0)$
  13. Average photon energy
  14. Conclusions and outlook

Figures (3)

  • Figure 1: Dependence of the three scales $\mu_h=m_b$ (solid), $\mu_i=\sqrt{m_b\Delta}$ (dashed), and $\mu_0=\Delta$ (dash-dotted) on the cutoff $E_0$, assuming $m_b=4.7$ GeV. The gray area at the bottom shows the domain of non-perturbative physics. The light gray band in the center indicates the region where the MSOPE should be applied.
  • Figure 2: Size of the enhanced power correction proportional to $\lambda_1/\Delta^2$ in (\ref{['Sexpand']}) relative to the leading term, as a function of $\Delta=m_b-2E_0$.
  • Figure 3: Sudakov exponents $S(m_b,\mu)$ (black) and $S(1\,\hbox{GeV},\mu)$ (gray) at next-to-next-to-leading order (solid), next-to-leading order (dashed), and leading order (dash-dotted). The solid and dashed curves are nearly indistinguishable.