The Momentum Amplituhedron

In this paper we define a new object, the momentum amplituhedron, which is the long sought-after positive geometry for tree-level scattering amplitudes in N = 4 super Yang-Mills theory in spinor helicity space. Inspired by the construction of the ordinary amplituhedron, we introduce bosonized spinor helicity variables to represent our external kinematical data, and restrict them to a particular positive region. The momentum amplituhedron M n,k is then the image of the positive Grassmannian via a map determined by such kinematics. The scattering amplitudes are extracted from the canonical form with logarithmic singularities on the boundaries of this geometry.Peer reviewedFinal Published versio

On All-loop Integrands of Scattering Amplitudes in Planar N=4 SYM

We study the relationship between the momentum twistor MHV vertex expansion of planar amplitudes in N=4 super-Yang-Mills and the all-loop generalization of the BCFW recursion relations. We demonstrate explicitly in several examples that the MHV vertex expressions for tree-level amplitudes and loop integrands satisfy the recursion relations. Furthermore, we introduce a rewriting of the MHV expansion in terms of sums over non-crossing partitions and show that this cyclically invariant formula satisfies the recursion relations for all numbers of legs and all loop orders.Comment: 34 pages, 17 figures; v2: Minor improvements to exposition and discussion, updated references, typos fixe

Unification of Residues and Grassmannian Dualities

The conjectured duality relating all-loop leading singularities of n-particle N^(k-2)MHV scattering amplitudes in N=4 SYM to a simple contour integral over the Grassmannian G(k,n) makes all the symmetries of the theory manifest. Every residue is individually Yangian invariant, but does not have a local space-time interpretation--only a special sum over residues gives physical amplitudes. In this paper we show that the sum over residues giving tree amplitudes can be unified into a single algebraic variety, which we explicitly construct for all NMHV and N^2MHV amplitudes. Remarkably, this allows the contour integral to have a "particle interpretation" in the Grassmannian, where higher-point amplitudes can be constructed from lower-point ones by adding one particle at a time, with soft limits manifest. We move on to show that the connected prescription for tree amplitudes in Witten's twistor string theory also admits a Grassmannian particle interpretation, where the integral over the Grassmannian localizes over the Veronese map from G(2,n) to G(k,n). These apparently very different theories are related by a natural deformation with a parameter t that smoothly interpolates between them. For NMHV amplitudes, we use a simple residue theorem to prove t-independence of the result, thus establishing a novel kind of duality between these theories.Comment: 56 pages, 11 figures; v2: typos corrected, minor improvement

New differential equations for on-shell loop integrals

We present a novel type of differential equations for on-shell loop integrals. The equations are second-order and importantly, they reduce the loop level by one, so that they can be solved iteratively in the loop order. We present several infinite series of integrals satisfying such iterative differential equations. The differential operators we use are best written using momentum twistor space. The use of the latter was advocated in recent papers discussing loop integrals in N=4 super Yang-Mills. One of our motivations is to provide a tool for deriving analytical results for scattering amplitudes in this theory. We show that the integrals needed for planar MHV amplitudes up to two loops can be thought of as deriving from a single master topology. The master integral satisfies our differential equations, and so do most of the reduced integrals. A consequence of the differential equations is that the integrals we discuss are not arbitrarily complicated transcendental functions. For two specific two-loop integrals we give the full analytic solution. The simplicity of the integrals appearing in the scattering amplitudes in planar N=4 super Yang-Mills is strongly suggestive of a relation to the conjectured underlying integrability of the theory. We expect these differential equations to be relevant for all planar MHV and non-MHV amplitudes. We also discuss possible extensions of our method to more general classes of integrals.Comment: 39 pages, 8 figures; v2: typos corrected, definition of harmonic polylogarithms adde

Local Spacetime Physics from the Grassmannian

A duality has recently been conjectured between all leading singularities of n-particle N^(k-2)MHV scattering amplitudes in N=4 SYM and the residues of a contour integral with a natural measure over the Grassmannian G(k,n). In this note we show that a simple contour deformation converts the sum of Grassmannian residues associated with the BCFW expansion of NMHV tree amplitudes to the CSW expansion of the same amplitude. We propose that for general k the same deformation yields the (k-2) parameter Risager expansion. We establish this equivalence for all MHV-bar amplitudes and show that the Risager degrees of freedom are non-trivially determined by the GL(k-2) "gauge" degrees of freedom in the Grassmannian. The Risager expansion is known to recursively construct the CSW expansion for all tree amplitudes, and given that the CSW expansion follows directly from the (super) Yang-Mills Lagrangian in light-cone gauge, this contour deformation allows us to directly see the emergence of local space-time physics from the Grassmannian.Comment: 22 pages, 13 figures; v2: minor updates, typos correcte

The Kinematic Algebra From the Self-Dual Sector

We identify a diffeomorphism Lie algebra in the self-dual sector of Yang-Mills theory, and show that it determines the kinematic numerators of tree-level MHV amplitudes in the full theory. These amplitudes can be computed off-shell from Feynman diagrams with only cubic vertices, which are dressed with the structure constants of both the Yang-Mills colour algebra and the diffeomorphism algebra. Therefore, the latter algebra is the dual of the colour algebra, in the sense suggested by the work of Bern, Carrasco and Johansson. We further study perturbative gravity, both in the self-dual and in the MHV sectors, finding that the kinematic numerators of the theory are the BCJ squares of the Yang-Mills numerators.Comment: 29 pages, 5 figures. v2: references added, published versio

Multi-Regge kinematics and the moduli space of Riemann spheres with marked points

We show that scattering amplitudes in planar N = 4 Super Yang-Mills in multi-Regge kinematics can naturally be expressed in terms of single-valued iterated integrals on the moduli space of Riemann spheres with marked points. As a consequence, scattering amplitudes in this limit can be expressed as convolutions that can easily be computed using Stokes' theorem. We apply this framework to MHV amplitudes to leading-logarithmic accuracy (LLA), and we prove that at L loops all MHV amplitudes are determined by amplitudes with up to L + 4 external legs. We also investigate non-MHV amplitudes, and we show that they can be obtained by convoluting the MHV results with a certain helicity flip kernel. We classify all leading singularities that appear at LLA in the Regge limit for arbitrary helicity configurations and any number of external legs. Finally, we use our new framework to obtain explicit analytic results at LLA for all MHV amplitudes up to five loops and all non-MHV amplitudes with up to eight external legs and four loops.Comment: 104 pages, six awesome figures and ancillary files containing the results in Mathematica forma

Wilson Loops @ 3-Loops in Special Kinematics

We obtain a compact expression for the octagon MHV amplitude / Wilson loop at 3 loops in planar N=4 SYM and in special 2d kinematics in terms of 7 unfixed coefficients. We do this by making use of the cyclic and parity symmetry of the amplitude/Wilson loop and its behaviour in the soft/collinear limits as well as in the leading term in the expansion away from this limit. We also make a natural and quite general assumption about the functional form of the result, namely that it should consist of weight 6 polylogarithms whose symbol consists of basic cross-ratios only (and not functions thereof). We also describe the uplift of this result to 10 points.Comment: 26 pages. Typos correcte

Differential equations for multi-loop integrals and two-dimensional kinematics

In this paper we consider multi-loop integrals appearing in MHV scattering amplitudes of planar N=4 SYM. Through particular differential operators which reduce the loop order by one, we present explicit equations for the two-loop eight-point finite diagrams which relate them to massive hexagons. After the reduction to two-dimensional kinematics, we solve them using symbol technology. The terms invisible to the symbols are found through boundary conditions coming from double soft limits. These equations are valid at all-loop order for double pentaladders and allow to solve iteratively loop integrals given lower-loop information. Comments are made about multi-leg and multi-loop integrals which can appear in this special kinematics. The main motivation of this investigation is to get a deeper understanding of these tools in this configuration, as well as for their application in general four-dimensional kinematics and to less supersymmetric theories.Comment: 25 pages, 7 figure

The Grassmannian and the Twistor String: Connecting All Trees in N=4 SYM

We present a new, explicit formula for all tree-level amplitudes in N=4 super Yang-Mills. The formula is written as a certain contour integral of the connected prescription of Witten's twistor string, expressed in link variables. A very simple deformation of the integrand gives directly the Grassmannian integrand proposed by Arkani-Hamed et al. together with the explicit contour of integration. The integral is derived by iteratively adding particles to the Grassmannian integral, one particle at a time, and makes manifest both parity and soft limits. The formula is shown to be related to those given by Dolan and Goddard, and generalizes the results of earlier work for NMHV and N^2MHV to all N^(k-2)MHV tree amplitudes in N=4 super Yang-Mills.Comment: 26 page