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Two-Loop Four-Gluon Amplitudes in N=4 Super-Yang-Mills

Z. Bern, J. S. Rozowsky, B. Yan

TL;DR

This paper demonstrates a unitarity-based cutting approach to derive two-loop four-gluon amplitudes in N=4 Super-Yang-Mills, expressing results in terms of scalar integrals and exploiting color decomposition. It provides explicit leading- and subleading-color results in terms of planar and non-planar double-box integrals and argues for the correctness of the method within N=4, supported by SUSY identities and finiteness properties. A central contribution is the proposed all-orders pattern for the leading-color four-gluon amplitudes, constructed from lower-loop graphs via a systematic insertion of new legs, with validations from two- and three-particle cuts up to certain loop orders. The work positions N=4 SYM as a valuable testing ground for higher-loop techniques that may inform QCD calculations, while calling for numerically stable evaluation of the integral functions for practical phenomenology.

Abstract

Using cutting techniques we obtain the two-loop N=4 super-Yang-Mills helicity amplitudes for four-gluon scattering in terms of scalar integral functions. The N=4 amplitudes are considerably simpler than corresponding QCD amplitudes and therefore provide a testing ground for exploring two-loop amplitudes. The amplitudes are constructed directly in terms of gauge invariant quantities and therefore remain relatively compact throughout the calculation. We also present a conjecture for the leading color four-gluon amplitudes to all orders in the perturbative expansion.

Two-Loop Four-Gluon Amplitudes in N=4 Super-Yang-Mills

TL;DR

This paper demonstrates a unitarity-based cutting approach to derive two-loop four-gluon amplitudes in N=4 Super-Yang-Mills, expressing results in terms of scalar integrals and exploiting color decomposition. It provides explicit leading- and subleading-color results in terms of planar and non-planar double-box integrals and argues for the correctness of the method within N=4, supported by SUSY identities and finiteness properties. A central contribution is the proposed all-orders pattern for the leading-color four-gluon amplitudes, constructed from lower-loop graphs via a systematic insertion of new legs, with validations from two- and three-particle cuts up to certain loop orders. The work positions N=4 SYM as a valuable testing ground for higher-loop techniques that may inform QCD calculations, while calling for numerically stable evaluation of the integral functions for practical phenomenology.

Abstract

Using cutting techniques we obtain the two-loop N=4 super-Yang-Mills helicity amplitudes for four-gluon scattering in terms of scalar integral functions. The N=4 amplitudes are considerably simpler than corresponding QCD amplitudes and therefore provide a testing ground for exploring two-loop amplitudes. The amplitudes are constructed directly in terms of gauge invariant quantities and therefore remain relatively compact throughout the calculation. We also present a conjecture for the leading color four-gluon amplitudes to all orders in the perturbative expansion.

Paper Structure

This paper contains 8 sections, 17 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Two-loop color decomposition
  3. Properties of $N=4$ amplitudes
  4. Two-loop cut construction
  5. Computation of amplitudes
  6. Validity of cut construction
  7. Structure of higher loop amplitudes
  8. Conclusions

Figures (8)

  • Figure 1: The two-particle $s$-channel cut has two contributions: one with the four-point one-loop amplitude '1' to the left and the tree amplitude 'T' to the right (a) and the other being the reverse (b).
  • Figure 2: The three-particle $s$-channel cut.
  • Figure 3: The result of evaluating the three-particle $s$-channel cuts in terms of double-box scalar integrals. The dashed lines indicate the cuts.
  • Figure 4: The result for the leading color two-loop amplitude, corresponding to eq. (\ref{['LeadingColorResult']})
  • Figure 5: The non-planar integral function, ${\cal I}_4^{\rm NP}(s_{12}, s_{23})$, appearing in the amplitude.
  • ...and 3 more figures