Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Scattering Amplitudes from Multivariate Polynomial Division

Pierpaolo Mastrolia, Edoardo Mirabella, Giovanni Ossola, Tiziano Peraro

TL;DR

Introduces a novel algebraic approach to scattering amplitudes by formulating integrands as multivariate polynomials in loop momenta. The authors derive a loop-order independent recurrence using multivariate polynomial division modulo Gröbner bases and the Weak Nullstellensatz to obtain the complete multi-pole decomposition. They apply the method to dimensionally regulated one-loop amplitudes, reproducing the known integrand-decomposition formula, and prove a Maximum-cut Theorem that links the residue to the number of cut solutions. The framework intertwines unitarity ideas with computational algebraic geometry to provide a general, constructive reconstruction of amplitudes.

Abstract

We show that the evaluation of scattering amplitudes can be formulated as a problem of multivariate polynomial division, with the components of the integration-momenta as indeterminates. We present a recurrence relation which, independently of the number of loops, leads to the multi-particle pole decomposition of the integrands of the scattering amplitudes. The recursive algorithm is based on the Weak Nullstellensatz Theorem and on the division modulo the Groebner basis associated to all possible multi-particle cuts. We apply it to dimensionally regulated one-loop amplitudes, recovering the well-known integrand-decomposition formula. Finally, we focus on the maximum-cut, defined as a system of on-shell conditions constraining the components of all the integration-momenta. By means of the Finiteness Theorem and of the Shape Lemma, we prove that the residue at the maximum-cut is parametrised by a number of coefficients equal to the number of solutions of the cut itself.

Scattering Amplitudes from Multivariate Polynomial Division

TL;DR

Introduces a novel algebraic approach to scattering amplitudes by formulating integrands as multivariate polynomials in loop momenta. The authors derive a loop-order independent recurrence using multivariate polynomial division modulo Gröbner bases and the Weak Nullstellensatz to obtain the complete multi-pole decomposition. They apply the method to dimensionally regulated one-loop amplitudes, reproducing the known integrand-decomposition formula, and prove a Maximum-cut Theorem that links the residue to the number of cut solutions. The framework intertwines unitarity ideas with computational algebraic geometry to provide a general, constructive reconstruction of amplitudes.

Abstract

We show that the evaluation of scattering amplitudes can be formulated as a problem of multivariate polynomial division, with the components of the integration-momenta as indeterminates. We present a recurrence relation which, independently of the number of loops, leads to the multi-particle pole decomposition of the integrands of the scattering amplitudes. The recursive algorithm is based on the Weak Nullstellensatz Theorem and on the division modulo the Groebner basis associated to all possible multi-particle cuts. We apply it to dimensionally regulated one-loop amplitudes, recovering the well-known integrand-decomposition formula. Finally, we focus on the maximum-cut, defined as a system of on-shell conditions constraining the components of all the integration-momenta. By means of the Finiteness Theorem and of the Shape Lemma, we prove that the residue at the maximum-cut is parametrised by a number of coefficients equal to the number of solutions of the cut itself.

Paper Structure

This paper contains 7 sections, 3 theorems, 29 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Multivariate polynomial division
  3. Reducibility criterion.
  4. Integrand Recursion Formula
  5. One-loop integrand decomposition
  6. The Maximum-cut Theorem
  7. Conclusions

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 (1)

  • Figure 1: The on-shell diagrams in the picture are examples of maximum-cuts. The first diagram in the left column represents the 5-ple cut of the 5-point one-loop dimensionally regulated amplitude. All the other on-shell diagrams are considered in four dimensions. For each of them, the general structure of the residue $\Delta$ (according to the Shape Lemma) and the corresponding value of $n_s$ are provided.

Theorems & Definitions (5)

  • Proposition 2.1
  • Proposition 2.2
  • proof
  • Theorem 4.1: Maximum cut
  • proof