Local integrals for planar scattering amplitudes

By N. Arkani-Hamed, J. Bourjaily, F. Cachazo, J. Trnka. 2010

352 citations recorded by the source corpus.

A bstractRecently, an explicit, recursive formula for the all-loop integrand of planar scattering amplitudes in $ \mathcal{N} = {4} $ 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.

Subfields: Nuclear and High Energy Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/0cfe919b7a312331f3b0003821ea72cda4f6a311

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Citations in the corpus

Elliptic symbol calculus: from elliptic polylogarithms to iterated integrals of Eisenstein series (2018, background): “ntial equations with a trivial homogeneous part. The corresponding class of functions is closely related, though not identical, to the notion of ‘pure functions’ encountered in the physics literature [82, 83]. We will comment more precisely on this connection at the end of this section. Consider the”
Uniqueness of two-loop master contours (2012, positive): “[58] where they were used to express the integrand of N = 4 super Yang-Mills amplitudes in a strikingly simple form.”
Collider physics at the precision frontier (2020, methodology): “This is because the UT property [482] allows for a straightforward solution of the system of differential equations in terms of Chen iterated integrals [483, 484].”
Lectures on differential equations for Feynman integrals (2014, methodology): “We refer to reference [13] where this philosophy was introduced and many integrals with these properties were J.”
Scattering amplitudes and Wilson loops in twistor space (2011, background): “These ideas have been explored further in [92, 104].”
Scattering Amplitudes and the Positive Grassmannian (2012, background): “) Similarly, it was found in [164] that 2-loop MHV integrand could be written in the following local form: ∑”
The full four-loop cusp anomalous dimension in N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{N} $$\end{do (2019, neutral): “y structure of the integrand, and its residues, allow one to predict key features of the outcome of the integration, such as the transcendental weight properties of the answer. In practice, following [58, 59], we consider loop integrands of the type appearing in eq. (3.1), and search for integrands ”
Heptagons from the Steinmann cluster bootstrap (2016, background): “[88] for a more recent, compact representation of the same amplitude.”
Two-loop planar master integrals for Higgs → 3 partons with full heavy-quark mass dependence (2016, background): “In particular, it is expected that integrals with constant leading singularities [36] satisfy canonical differential equations.”
On the maximal cut of Feynman integrals and the solution of their differential equations (2016, background): “While a complete understanding of the concept of leading singularity and of a canonical basis are available only in the case of integrals that evaluate to multiple-polylogarithms and generalizations thereof1 [19], it has been suggested that the study of the maximal cut should be the starting point t”