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Multivariate Residues and Maximal Unitarity

Mads Sogaard, Yang Zhang

TL;DR

This work extends maximal unitarity to cuts that define multidimensional algebraic varieties, enabling the extraction of master-integral coefficients at $3$-loop order for the planar triple box. It develops a comprehensive multivariate-residue toolkit, leverages the Global Residue Theorem to relate infinity and finite poles, and constructs three unique master-contour projectors that isolate the master integrals from a 23-term residue basis. The approach yields explicit expressions for all independent helicity configurations and agrees with prior results in $ N=4$ and related theories, marking a step toward automation of higher-loop amplitude reductions. The framework promises applicability to other multi-loop topologies and to cuts in $D=4-2psilon$, potentially broadening the reach of on-shell methods in perturbative quantum field theory.

Abstract

We extend the maximal unitarity method to amplitude contributions whose cuts define multidimensional algebraic varieties. The technique is valid to all orders and is explicitly demonstrated at three loops in gauge theories with any number of fermions and scalars in the adjoint representation. Deca-cuts realized by replacement of real slice integration contours by higher-dimensional tori encircling the global poles are used to factorize the planar triple box onto a product of trees. We apply computational algebraic geometry and multivariate complex analysis to derive unique projectors for all master integral coefficients and obtain compact analytic formulae in terms of tree-level data.

Multivariate Residues and Maximal Unitarity

TL;DR

This work extends maximal unitarity to cuts that define multidimensional algebraic varieties, enabling the extraction of master-integral coefficients at -loop order for the planar triple box. It develops a comprehensive multivariate-residue toolkit, leverages the Global Residue Theorem to relate infinity and finite poles, and constructs three unique master-contour projectors that isolate the master integrals from a 23-term residue basis. The approach yields explicit expressions for all independent helicity configurations and agrees with prior results in and related theories, marking a step toward automation of higher-loop amplitude reductions. The framework promises applicability to other multi-loop topologies and to cuts in , potentially broadening the reach of on-shell methods in perturbative quantum field theory.

Abstract

We extend the maximal unitarity method to amplitude contributions whose cuts define multidimensional algebraic varieties. The technique is valid to all orders and is explicitly demonstrated at three loops in gauge theories with any number of fermions and scalars in the adjoint representation. Deca-cuts realized by replacement of real slice integration contours by higher-dimensional tori encircling the global poles are used to factorize the planar triple box onto a product of trees. We apply computational algebraic geometry and multivariate complex analysis to derive unique projectors for all master integral coefficients and obtain compact analytic formulae in terms of tree-level data.

Paper Structure

This paper contains 15 sections, 3 theorems, 68 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Multivariate Residues
  3. Calculation of (degenerate) local residues
  4. Global Residue Theorem
  5. Maximal Cuts of Planar Triple Box Integrals
  6. Parametrization of On-Shell Solutions
  7. Composite Leading Singularities
  8. Global Poles and Augmentation of Residues
  9. Integral Reduction Identities
  10. Unique Master Integral Projectors
  11. Examples
  12. Conclusion
  13. Residues at Infinity
  14. Kinematical Configurations of the Planar Triple Box
  15. Planar Triple Box Integration-By-Parts Identities

Key Result

Theorem 1

Let $I=\langle f_1, \dots, f_n \rangle$ be the zero-dimensional ideal generated by $\{f_1,\dots,f_n\}$ and $J=\langle g_1, \dots, g_n \rangle$ be a zero-dimensional ideal such that $J\subset I$. So $g_i = a_{ij} f_j$, where the $a_{ij}$'s are polynomials. Let $A$ be the matrix of $a_{ij}$'s, then fo

Figures (3)

  • Figure 1: Momentum flow for the planar triple box.
  • Figure 2: Infinity diagrams: Inside one diagram, each line (curve) represents a vanishing polynomials in the denominators and each black dot represents a residue at infinity.
  • Figure :

Theorems & Definitions (7)

  • Theorem 1: Transformation law
  • Example 1
  • Example 2
  • Lemma 1
  • proof
  • Theorem 2: Global residue residue, GRT
  • Example 3