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Hepta-Cuts of Two-Loop Scattering Amplitudes

Simon Badger, Hjalte Frellesvig, Yang Zhang

TL;DR

This work develops a practical framework to compute seven-propagator hepta-cuts at two loops for 2→2 scattering in non-supersymmetric gauge theories. It builds a unitarity-based integrand parameterisation using Gram-matrix constraints, fits coefficients from products of tree-level amplitudes evaluated at on-shell solutions, and then reduces the resulting integrand to master integrals via standard IBP methods. The method is applied to planar double boxes, non-planar crossed boxes, and penta-box topologies, with explicit results for gluon-gluon scattering in Yang–Mills theory with adjoint fermions and scalars, consistently connecting to known SUSY results. The approach provides a concrete, extensible route toward full two-loop amplitude decompositions and potential extensions to higher loops or D-dimensional cuts, offering a practical tool for NNLO predictions.

Abstract

We present a method for the computation of hepta-cuts of two loop scattering amplitudes. Four dimensional unitarity cuts are used to factorise the integrand onto the product of six tree-level amplitudes evaluated at complex momentum values. Using Gram matrix constraints we derive a general parameterisation of the integrand which can be computed using polynomial fitting techniques. The resulting expression is further reduced to master integrals using conventional integration by parts methods. We consider both planar and non-planar topologies for 2 to 2 scattering processes and apply the method to compute hepta-cut contributions to gluon-gluon scattering in Yang-Mills theory with adjoint fermions and scalars.

Hepta-Cuts of Two-Loop Scattering Amplitudes

TL;DR

This work develops a practical framework to compute seven-propagator hepta-cuts at two loops for 2→2 scattering in non-supersymmetric gauge theories. It builds a unitarity-based integrand parameterisation using Gram-matrix constraints, fits coefficients from products of tree-level amplitudes evaluated at on-shell solutions, and then reduces the resulting integrand to master integrals via standard IBP methods. The method is applied to planar double boxes, non-planar crossed boxes, and penta-box topologies, with explicit results for gluon-gluon scattering in Yang–Mills theory with adjoint fermions and scalars, consistently connecting to known SUSY results. The approach provides a concrete, extensible route toward full two-loop amplitude decompositions and potential extensions to higher loops or D-dimensional cuts, offering a practical tool for NNLO predictions.

Abstract

We present a method for the computation of hepta-cuts of two loop scattering amplitudes. Four dimensional unitarity cuts are used to factorise the integrand onto the product of six tree-level amplitudes evaluated at complex momentum values. Using Gram matrix constraints we derive a general parameterisation of the integrand which can be computed using polynomial fitting techniques. The resulting expression is further reduced to master integrals using conventional integration by parts methods. We consider both planar and non-planar topologies for 2 to 2 scattering processes and apply the method to compute hepta-cut contributions to gluon-gluon scattering in Yang-Mills theory with adjoint fermions and scalars.

Paper Structure

This paper contains 33 sections, 120 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Review of Generalised Unitarity at One-Loop
  3. Quadruple Cuts
  4. Triple Cuts
  5. Integrand Representations of Two Loop Amplitudes
  6. Hepta-cuts of Two-Loop Amplitudes
  7. Integrand Parameterisations from Gram Matrix Constraints
  8. The Double Box
  9. Solutions to the on-shell constraints
  10. Integration by parts identities
  11. The Crossed Box
  12. Solutions to the on-shell constraints
  13. Integration by parts identities
  14. The Penta-Box
  15. Solutions to the on-shell constraints
  16. ...and 18 more sections

Figures (6)

  • Figure 1: Conventions for the momentum flow in the one-loop box.
  • Figure 2: Conventions for the momentum flow in the planar double box.
  • Figure 3: Conventions for the momentum flow in the non-planar crossed box.
  • Figure 4: Conventions for the momentum flow in the penta-box.
  • Figure 5: Contributions to the double box in Yang-Mills theories of gluons, $n_f$ adjoint fermions and $n_s$ scalars. Gluons are represented as solid lines, fermions as solid lines with arrows and scalars as dashed lines.
  • ...and 1 more figures