Scattering Amplitudes from Multivariate Polynomial Division

TL;DR

This paper introduces a polynomial division approach to evaluate scattering amplitudes, providing a recursive algorithm based on algebraic geometry theorems that simplifies multi-particle pole decomposition.

Contribution

It presents a novel recursive method using multivariate polynomial division and algebraic geometry to analyze scattering amplitudes regardless of loop order.

Findings

Recovers known integrand-decomposition formulas for one-loop amplitudes

Provides a polynomial division framework applicable to multi-particle cuts

Shows the residue at maximum-cut is parametrized by solutions of on-shell conditions

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.