Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Integrand Reduction for Two-Loop Scattering Amplitudes through Multivariate Polynomial Division

Pierpaolo Mastrolia, Edoardo Mirabella, Giovanni Ossola, Tiziano Peraro

TL;DR

This work introduces a universal integrand-reduction framework for multiloop scattering amplitudes based on multivariate polynomial division modulo a Gröbner basis, yielding complete, process-dependent residues and a recurrence that reconstructs multiparticle pole structures. The authors apply the method to two-loop five-point diagrams in ${ m ot N}=4$ SYM and ${ m ot N}=8$ SUGRA, deriving explicit polynomial residues and identifying the integral bases built from eight-, seven-, and six-denominator topologies. The paper demonstrates both seminumerical and analytic implementations, with the latter using a Laurent expansion to simplify coefficient extraction and subtraction of higher residues. Overall, the approach provides a general, scalable algorithm that can extend integrand reduction to all orders in perturbation theory and to dimensional regularization schemes, offering a robust tool for automating high-order amplitude calculations.

Abstract

We describe the application of a novel approach for the reduction of scattering amplitudes, based on multivariate polynomial division, which we have recently presented. This technique yields the complete integrand decomposition for arbitrary amplitudes, regardless of the number of loops. It allows for the determination of the residue at any multiparticle cut, whose knowledge is a mandatory prerequisite for applying the integrand-reduction procedure. By using the division modulo Groebner basis, we can derive a simple integrand recurrence relation that generates the multiparticle pole decomposition for integrands of arbitrary multiloop amplitudes. We apply the new reduction algorithm to the two-loop planar and nonplanar diagrams contributing to the five-point scattering amplitudes in N=4 super Yang-Mills and N=8 supergravity in four dimensions, whose numerator functions contain up to rank-two terms in the integration momenta. We determine all polynomial residues parametrizing the cuts of the corresponding topologies and subtopologies. We obtain the integral basis for the decomposition of each diagram from the polynomial form of the residues. Our approach is well suited for a seminumerical implementation, and its general mathematical properties provide an effective algorithm for the generalization of the integrand-reduction method to all orders in perturbation theory.

Integrand Reduction for Two-Loop Scattering Amplitudes through Multivariate Polynomial Division

TL;DR

This work introduces a universal integrand-reduction framework for multiloop scattering amplitudes based on multivariate polynomial division modulo a Gröbner basis, yielding complete, process-dependent residues and a recurrence that reconstructs multiparticle pole structures. The authors apply the method to two-loop five-point diagrams in SYM and SUGRA, deriving explicit polynomial residues and identifying the integral bases built from eight-, seven-, and six-denominator topologies. The paper demonstrates both seminumerical and analytic implementations, with the latter using a Laurent expansion to simplify coefficient extraction and subtraction of higher residues. Overall, the approach provides a general, scalable algorithm that can extend integrand reduction to all orders in perturbation theory and to dimensional regularization schemes, offering a robust tool for automating high-order amplitude calculations.

Abstract

We describe the application of a novel approach for the reduction of scattering amplitudes, based on multivariate polynomial division, which we have recently presented. This technique yields the complete integrand decomposition for arbitrary amplitudes, regardless of the number of loops. It allows for the determination of the residue at any multiparticle cut, whose knowledge is a mandatory prerequisite for applying the integrand-reduction procedure. By using the division modulo Groebner basis, we can derive a simple integrand recurrence relation that generates the multiparticle pole decomposition for integrands of arbitrary multiloop amplitudes. We apply the new reduction algorithm to the two-loop planar and nonplanar diagrams contributing to the five-point scattering amplitudes in N=4 super Yang-Mills and N=8 supergravity in four dimensions, whose numerator functions contain up to rank-two terms in the integration momenta. We determine all polynomial residues parametrizing the cuts of the corresponding topologies and subtopologies. We obtain the integral basis for the decomposition of each diagram from the polynomial form of the residues. Our approach is well suited for a seminumerical implementation, and its general mathematical properties provide an effective algorithm for the generalization of the integrand-reduction method to all orders in perturbation theory.

Paper Structure

This paper contains 39 sections, 3 theorems, 85 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Integrand reduction
  3. Integrand recurrence relation
  4. Reducibility criterion
  5. Maximum-cut Theorem
  6. Two-loop integrand reduction
  7. Five-point amplitudes in ${\cal N}=4$ SYM
  8. Residue of the planar pentabox
  9. Residue of the crossed pentabox
  10. Residue of the double pentagon
  11. Seminumerical integrand reduction
  12. Planar pentabox diagram
  13. Eightfold cut.
  14. Sevenfold cut.
  15. Crossed pentabox diagram
  16. ...and 24 more sections

Key Result

Proposition 2.1

The integrand $\mathcal{I}_{i_1\cdots i_n}$ is reducible iff the remainder of the division modulo a Gröbner basis vanishes, i.e. iff ${\cal N}_{i_1\cdots i_n} \in \mathcal{J}_{i_1\cdots i_n}$.

Figures (4)

  • Figure 1: Five-point diagrams entering the amplitudes in $\mathcal{N}=4$ SYM and $\mathcal{N}=8$ SUGRA. They are the pentabox diagram $(a)$, the crossed pentabox diagram $(b)$ and the double pentagon diagram $(c)$. For each diagram, the definition of the denominators is shown as well.
  • Figure 2: Cut diagrams of the sevenfold cuts. Starting from the left, we show the diagram of the cut $(1234568)$, $(1234678)$, $(1234578)$, and $(1235678)$.
  • Figure 3: Cut diagrams of the sevenfold cuts. Starting from the left, we show the diagram of the cut $(1234568)$, $(1234678)$, $(1234578)$, and $(1235678)$.
  • Figure 4: Cut diagrams of the sevenfold cuts. Starting from the left, we show the diagram of the cut $(1234567)$, $(1245678)$, $(1345678)$, and $(2345678)$.

Theorems & Definitions (3)

  • Proposition 2.1
  • Proposition 2.2
  • Theorem 2.1: Maximum cut