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Resummation in Heavy Quark and Jet Cross Sections

Nikolaos Kidonakis, George Sterman

TL;DR

Elastic-limit corrections near the partonic threshold $z=1$ can dominate hadronic cross sections across orders in perturbation theory. The authors develop a color-space threshold resummation formalism that factorizes cross sections into hard and soft components encoded by Wilson lines, and they exponentiate large $1-z$ logarithms to all orders; this framework is then generalized from Drell-Yan to QCD processes such as heavy-quark and jet production, including explicit next-to-leading-logarithm results and color-dependent anomalous-dimension matrices. The work provides concrete expressions for resummed cross sections in moment space and discusses the challenges of inverse transforms and infrared renormalons, illustrating how final-state radiation and color exchange modify the resummation compared to purely electroweak cases. Overall, the paper extends the reach of threshold resummation in QCD, offering improved predictive power for high-energy hadronic collisions and sharper tools for testing QCD and estimating backgrounds for new-physics signals.

Abstract

We discuss corrections from the elastic limit (partonic threshold) in hadronic hard-scattering cross sections. We show why these corrections can be large at all orders in perturbation theory, and describe their resummation to arbitrary logarithmic accuracy. In particular, we discuss the role of color exchange in the hard scattering. This enables us to generalize the resummation of the Drell-Yan cross section to QCD reactions. As an example, we give the explicit resummed hard-scattering cross section for heavy-quark production through light quark annihilation, which takes into account next-to-leading logarithms to all orders.

Resummation in Heavy Quark and Jet Cross Sections

TL;DR

Elastic-limit corrections near the partonic threshold can dominate hadronic cross sections across orders in perturbation theory. The authors develop a color-space threshold resummation formalism that factorizes cross sections into hard and soft components encoded by Wilson lines, and they exponentiate large logarithms to all orders; this framework is then generalized from Drell-Yan to QCD processes such as heavy-quark and jet production, including explicit next-to-leading-logarithm results and color-dependent anomalous-dimension matrices. The work provides concrete expressions for resummed cross sections in moment space and discusses the challenges of inverse transforms and infrared renormalons, illustrating how final-state radiation and color exchange modify the resummation compared to purely electroweak cases. Overall, the paper extends the reach of threshold resummation in QCD, offering improved predictive power for high-energy hadronic collisions and sharper tools for testing QCD and estimating backgrounds for new-physics signals.

Abstract

We discuss corrections from the elastic limit (partonic threshold) in hadronic hard-scattering cross sections. We show why these corrections can be large at all orders in perturbation theory, and describe their resummation to arbitrary logarithmic accuracy. In particular, we discuss the role of color exchange in the hard scattering. This enables us to generalize the resummation of the Drell-Yan cross section to QCD reactions. As an example, we give the explicit resummed hard-scattering cross section for heavy-quark production through light quark annihilation, which takes into account next-to-leading logarithms to all orders.

Paper Structure

This paper contains 11 sections, 21 equations, 3 figures.

Table of Contents

  1. Introduction: Perturbative QCD in Perspective
  2. The Elastic Limit in Hard Inclusive Scattering
  3. Factorized Cross Sections
  4. What Threshold?
  5. Why Large?
  6. Why Positive?
  7. Resolution?
  8. Resummed Drell-Yan
  9. Threshold Resummations for Heavy Quarks and Jets
  10. Resummation with Color Exchange
  11. Conclusion

Figures (3)

  • Figure 1: Hard-scattering cross section in cut (unitarity) diagram notation.
  • Figure 2: Schematic representation of moments of the Drell-Yan partonic hard-scattering function.
  • Figure 3: Representation of the factorization of the hard scattering function $\hat{\sigma}$ near the elastic limit. The second part shows the soft-gluon matrix $S_{IJ}$ as a cut diagram for the scattering of incoming ordered exponentials (double lines - the incoming partons in the eikonal approximation) to give outgoing ordered exponentials (bold lines - the outgoing heavy quarks in the eikonal approximation). For simplicity, only a few of the possible gluon interactions with the ordered exponentials are shown.