Surface Kinematics and "The" Yang-Mills Integrand

TL;DR

This paper introduces a surface kinematics framework that resolves longstanding ambiguities in defining a canonical loop integrand for non-supersymmetric Yang-Mills amplitudes, ensuring gauge invariance and consistent factorization.

Contribution

It proposes a novel surface kinematics approach that generalizes momenta, enabling the definition of a unique, gauge-invariant integrand for Yang-Mills theory at any loop order.

Findings

Defined a surface kinematics formalism for gluon amplitudes.

Constructed a recursive integrand for one-loop all multiplicities.

Presented the simplest two-loop integrand example.

Abstract

It has been a long-standing challenge to define a canonical loop integrand for non-supersymmetric gluon scattering amplitudes in the planar limit. Naive integrands are inflicted with $1/0$ ambiguities associated with tadpoles and massless external bubbles, which destroy integrand-level gauge invariance as well as consistent on-shell factorization on single loop-cuts. In this letter, we show that this essentially kinematical obstruction to defining "the" integrand for Yang-Mills theory has a structural solution, handed to us by the formulation of gluon amplitudes in terms of curves on surfaces. This defines "surface kinematics" generalizing momenta, making it possible to define "the" integrand satisfying both a (surface generalized) notion of gauge-invariance and consistent loop-cuts. The integrand also vanishes at infinity in appropriate directions, allowing it to be recursively computed for non-supersymmetric Yang-Mills theory in any number of dimensions. We illustrate these ideas through one loop for all multiplicity, and for the simplest two-loop integrand.