The Amplituhedron

Perturbative scattering amplitudes in gauge theories have remarkable simplicity and hidden infinite dimensional symmetries that are completely obscured in the conventional formulation of field theory using Feynman diagrams. This suggests the existence of a new understanding for scattering amplitudes where locality and unitarity do not play a central role but are derived consequences from a different starting point. In this note we provide such an understanding for N=4 SYM scattering amplitudes in the planar limit, which we identify as ``the volume" of a new mathematical object--the Amplituhedron--generalizing the positive Grassmannian. Locality and unitarity emerge hand-in-hand from positive geometry.Comment: 36 pages, 14 figure

Unwinding the Amplituhedron in Binary

We present new, fundamentally combinatorial and topological characterizations of the amplituhedron. Upon projecting external data through the amplituhedron, the resulting configuration of points has a specified (and maximal) generalized 'winding number'. Equivalently, the amplituhedron can be fully described in binary: canonical projections of the geometry down to one dimension have a specified (and maximal) number of 'sign flips' of the projected data. The locality and unitarity of scattering amplitudes are easily derived as elementary consequences of this binary code. Minimal winding defines a natural 'dual' of the amplituhedron. This picture gives us an avatar of the amplituhedron purely in the configuration space of points in vector space (momentum-twistor space in the physics), a new interpretation of the canonical amplituhedron form, and a direct bosonic understanding of the scattering super-amplitude in planar N = 4 SYM as a differential form on the space of physical kinematical data.Comment: 42 pages, 13 figure

Amplitudes at Infinity

We investigate the asymptotically large loop-momentum behavior of multi-loop amplitudes in maximally supersymmetric quantum field theories in four dimensions. We check residue-theorem identities among color-dressed leading singularities in $\mathcal{N}=4$ supersymmetric Yang-Mills theory to demonstrate the absence of poles at infinity of all MHV amplitudes through three loops. Considering the same test for $\mathcal{N}=8$ supergravity leads us to discover that this theory does support non-vanishing residues at infinity starting at two loops, and the degree of these poles grow arbitrarily with multiplicity. This causes a tension between simultaneously manifesting ultraviolet finiteness---which would be automatic in a representation obtained by color-kinematic duality---and gauge invariance---which would follow from unitarity-based methods.Comment: 4+1+1 pages; 15 figures; details provided in ancillary Mathematica file

Simple Recursion Relations for General Field Theories

On-shell methods offer an alternative definition of quantum field theory at tree-level, replacing Feynman diagrams with recursion relations and interaction vertices with a handful of seed scattering amplitudes. In this paper we determine the simplest recursion relations needed to construct a general four-dimensional quantum field theory of massless particles. For this purpose we define a covering space of recursion relations which naturally generalizes all existing constructions, including those of BCFW and Risager. The validity of each recursion relation hinges on the large momentum behavior of an n-point scattering amplitude under an m-line momentum shift, which we determine solely from dimensional analysis, Lorentz invariance, and locality. We show that all amplitudes in a renormalizable theory are 5-line constructible. Amplitudes are 3-line constructible if an external particle carries spin or if the scalars in the theory carry equal charge under a global or gauge symmetry. Remarkably, this implies the 3-line constructibility of all gauge theories with fermions and complex scalars in arbitrary representations, all supersymmetric theories, and the standard model. Moreover, all amplitudes in non-renormalizable theories without derivative interactions are constructible; with derivative interactions, a subset of amplitudes is constructible. We illustrate our results with examples from both renormalizable and non-renormalizable theories. Our study demonstrates both the power and limitations of recursion relations as a self-contained formulation of quantum field theory.Comment: 27 pages and 2 figures; v2: typos corrected to match journal versio

Gravity On-shell Diagrams

We study on-shell diagrams for gravity theories with any number of super-symmetries and find a compact Grassmannian formula in terms of edge variables of the graphs. Unlike in gauge theory where the analogous form involves only d log-factors, in gravity there is a non-trivial numerator as well as higher degree poles in the edge variables. Based on the structure of the Grassmannian formula for =8N=8 supergravity we conjecture that gravity loop amplitudes also possess similar properties. In particular, we find that there are only logarithmic singularities on cuts with finite loop momentum and that poles at infinity are present, in complete agreement with the conjecture presented in [1]

Snowmass TF04 Report: Scattering Amplitudes and their Applications

The field of scattering amplitudes plays a central role in elementary-particle physics. This includes various problems of broader interest for collider physics, gravitational physics, and fundamental principles underlying quantum field theory. We describe various applications and theoretical advances pointing towards novel descriptions of quantum field theories. Comments on future prospects are included.Comment: 18 pages, 4 figure