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The Photon Spectrum in Upsilon Decays

Sean Fleming, Adam K Leibovich

TL;DR

This paper provides a systematic SCET-based treatment of the endpoint region in radiative Upsilon decays, where NRQCD alone fails due to collinear dynamics. It matches NRQCD onto SCET, identifies the relevant color-singlet and color-octet operators, and proves a factorized form of the endpoint rate into hard, jet, and usoft components. Through an operator product expansion and renormalization-group running, it resums Sudakov logarithms in the color-singlet channel and combines these results with fragmentation and color-octet contributions to predict the full photon spectrum. The resulting predictions show improved agreement with CLEO data near the endpoint, though residual uncertainties remain and further higher-order corrections may be needed for full reconciliation.

Abstract

We present a theoretical prediction for the photon spectrum in radiative Upsilon decay. Parts of the spectrum have already been understood, but an understanding of the endpoint region has remained elusive. In this paper we provide the missing piece, and resolve a controversy in the literature. We treat the endpoint region of Upsilon -> X gamma decay within the framework of the soft-collinear effective theory (SCET). Within this approach the Upsilon structure function arises naturally, and kinematic logarithms are summed by running operators using renormalization group equations. In a previous paper we studied the color-octet contribution to the decay. Here we treat the color-singlet contribution. We combine our result with previous results to obtain the Upsilon -> X gamma spectrum. We find that resumming the color-singlet contribution in the endpoint gives a result that is in much better agreement with the data than the leading order prediction.

The Photon Spectrum in Upsilon Decays

TL;DR

This paper provides a systematic SCET-based treatment of the endpoint region in radiative Upsilon decays, where NRQCD alone fails due to collinear dynamics. It matches NRQCD onto SCET, identifies the relevant color-singlet and color-octet operators, and proves a factorized form of the endpoint rate into hard, jet, and usoft components. Through an operator product expansion and renormalization-group running, it resums Sudakov logarithms in the color-singlet channel and combines these results with fragmentation and color-octet contributions to predict the full photon spectrum. The resulting predictions show improved agreement with CLEO data near the endpoint, though residual uncertainties remain and further higher-order corrections may be needed for full reconciliation.

Abstract

We present a theoretical prediction for the photon spectrum in radiative Upsilon decay. Parts of the spectrum have already been understood, but an understanding of the endpoint region has remained elusive. In this paper we provide the missing piece, and resolve a controversy in the literature. We treat the endpoint region of Upsilon -> X gamma decay within the framework of the soft-collinear effective theory (SCET). Within this approach the Upsilon structure function arises naturally, and kinematic logarithms are summed by running operators using renormalization group equations. In a previous paper we studied the color-octet contribution to the decay. Here we treat the color-singlet contribution. We combine our result with previous results to obtain the Upsilon -> X gamma spectrum. We find that resumming the color-singlet contribution in the endpoint gives a result that is in much better agreement with the data than the leading order prediction.

Paper Structure

This paper contains 12 sections, 100 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Tree level spectrum and fragmentation
  3. The direct rate
  4. Fragmentation contribution
  5. Matching onto SCET
  6. Factorization in Inclusive $\Upsilon \to X \gamma$
  7. The Operator Product Expansion
  8. Summing Sudakov Logarithms
  9. Phenomenology and Discussion
  10. conclusion
  11. SCET operators and Feynman rules
  12. Color-octet Factorization

Figures (8)

  • Figure 1: Matching the decay amplitude for $b\bar{b} \to \gamma + g$ in QCD and SCET. Collinear gluons are represented by a spring with a line through it.
  • Figure 2: Matching the decay amplitude for $b\bar{b} \to \gamma + gg$ in QCD and SCET.
  • Figure 3: Feynman diagram for the leading order jet function.
  • Figure 4: Diagrams needed to calculate the counterterm to the color-singlet operator.
  • Figure 5: The color-singlet rate. The dotted curve is the tree-level direct rate. The dashed curve is the interpolated resummed direct rate. The dot-dashed curve is the fragmentation contribution, and the solid curve is the sum of the resummed rate and the fragmentation rates.
  • ...and 3 more figures