Local integrals for planar scattering amplitudes

Recently, an explicit, recursive formula for the all-loop integrand of planar scattering amplitudes in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$ \mathcal{N} = {4} $\end{document} SYM has been described, generalizing the BCFW formula for tree amplitudes, and making manifest the Yangian symmetry of the theory. This has made it possible to easily study the structure of multi-loop amplitudes in the theory. In this paper we describe a remarkable fact revealed by these investigations: the integrand can be expressed in an amazingly simple and manifestly local form when represented in momentum-twistor space using a set of chiral integrals with unit leading singularities. As examples, we present very-concise expressions for all 2- and 3-loop MHV integrands, as well as all 2-loop NMHV integrands. We also describe a natural set of manifestly IR-finite integrals that can be used to express IR-safe objects such as the ratio function. Along the way we give a pedagogical introduction to the foundations of the subject. The new local forms of the integrand are closely connected to leading singularities — matching only a small subset of all leading singularities remarkably suffices to determine the full integrand. These results strongly suggest the existence of a theory for the integrand directly yielding these local expressions, allowing for a more direct understanding of the emergence of local spacetime physics.