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Two-Loop Helicity Amplitudes for Gluon-Gluon Scattering in QCD and Supersymmetric Yang-Mills Theory

Zvi Bern, Abilio De Freitas, Lance Dixon

TL;DR

This work computes the two-loop helicity amplitudes for $gg \rightarrow gg$ in QCD and presents the full color/helicity dependence, using unitarity cuts and tensor-integral reduction to master integrals within HV/FDH regularization and Catani’s infrared factorization. It provides the finite remainders at two loops, analyzes the $H^{(2)}$ operator (including a nontrivial color-commutator piece) and scheme conversions between HV, CDR, and FDH, and extends the results to pure $N=1$ super-Yang-Mills theory as a useful cross-check and auxiliary input. The paper verifies consistency against prior results, demonstrates the cancellations required at NNLO, and delivers the groundwork for implementing NNLO jet predictions with full color and polarization information. Together, these results bolster precision QCD predictions for jet production and enable polarized-parton analyses through the available helicity amplitudes.

Abstract

We present the two-loop helicity amplitudes for the scattering of two gluons into two gluons in QCD, which are relevant for next-to-next-to-leading order corrections to jet production at hadron colliders. We give the results in the `t Hooft-Veltman and four-dimensional helicity variants of dimensional regularization. Summing our expressions over helicities and colors, and converting to conventional dimensional regularization, gives results in complete agreement with those of Glover, Oleari and Tejeda-Yeomans. We also present the amplitudes for 2 to 2 scattering in pure N=1 supersymmetric Yang-Mills theory.

Two-Loop Helicity Amplitudes for Gluon-Gluon Scattering in QCD and Supersymmetric Yang-Mills Theory

TL;DR

This work computes the two-loop helicity amplitudes for in QCD and presents the full color/helicity dependence, using unitarity cuts and tensor-integral reduction to master integrals within HV/FDH regularization and Catani’s infrared factorization. It provides the finite remainders at two loops, analyzes the operator (including a nontrivial color-commutator piece) and scheme conversions between HV, CDR, and FDH, and extends the results to pure super-Yang-Mills theory as a useful cross-check and auxiliary input. The paper verifies consistency against prior results, demonstrates the cancellations required at NNLO, and delivers the groundwork for implementing NNLO jet predictions with full color and polarization information. Together, these results bolster precision QCD predictions for jet production and enable polarized-parton analyses through the available helicity amplitudes.

Abstract

We present the two-loop helicity amplitudes for the scattering of two gluons into two gluons in QCD, which are relevant for next-to-next-to-leading order corrections to jet production at hadron colliders. We give the results in the `t Hooft-Veltman and four-dimensional helicity variants of dimensional regularization. Summing our expressions over helicities and colors, and converting to conventional dimensional regularization, gives results in complete agreement with those of Glover, Oleari and Tejeda-Yeomans. We also present the amplitudes for 2 to 2 scattering in pure N=1 supersymmetric Yang-Mills theory.

Paper Structure

This paper contains 16 sections, 133 equations.

Table of Contents

  1. Introduction
  2. Review of infrared and color structure
  3. One-loop amplitudes
  4. All orders in $\epsilon$
  5. NNLO cancellations involving ${\cal M}_{gg \to gg}^{(1)}$
  6. Finite remainders
  7. Two-loop amplitudes and finite remainders
  8. Tensor loop integrals
  9. Checks on results
  10. ${{\hbox{\boldmath $H$}}}^{(2)}$ operator
  11. Finite remainders
  12. Scheme conversion at two loops
  13. Two-loop amplitudes in pure $N=1$ super-Yang-Mills theory
  14. Conclusions
  15. Finite remainder functions for QCD
  16. ...and 1 more sections