New Recursions for the Canonical Scalar-Scaffolded Yang-Mills Amplitude

TL;DR

This paper introduces new recursive methods for calculating Yang-Mills amplitudes using a scalar-scaffolding approach, enabling systematic computation of all terms in the Laurent series expansion of the amplitude.

Contribution

It develops novel recursion relations for Yang-Mills amplitudes based on gauge invariance and scalar-scaffolding, extending the computational framework beyond traditional methods.

Findings

Derived a recursion for the full Yang-Mills amplitude.

Established a recursive formula for the Laurent series expansion terms.

Provided Mathematica implementations of the recursions.

Abstract

The recently-developed "scalar-scaffolding" formulation of gluon amplitudes casts the Yang-Mills (YM) amplitude as a well-defined Laurent series expansion in scalar variables, valid for any spacetime dimension and helicity configuration. In this letter, we exploit this new perspective to develop conceptually novel methods of computing YM tree amplitudes. First, using standard gluon factorization to determine all terms with poles, we show how gauge invariance uniquely fixes the piece with no poles (the "contact term") from only terms that have a single pole. This allows us to write a YM recursion not only for the full amplitude but also for the amplitude up to any order in the Laurent series. Next, by imposing gauge invariance for terms with poles, we write down relations which compute numerators recursively in the amplitude's Laurent series expansion. Starting from an initial set of cuts depending only on the $(n-1)$-point amplitude, these formulae allow us to determine the remaining terms in the $n$-point amplitude. Finally, we use this "Laurent series recursion" to derive a recursion solely for the contact term. We speculate on the possibility that this and analogous recursions for any term in the amplitude may be solved. In attached Mathematica notebooks, we give implementations of these three recursions.