Momentum Amplituhedron for N=6 Chern-Simons-Matter Theory: Scattering Amplitudes from Configurations of Points in Minkowski Space.

In this Letter, we define the Aharony-Bergman-Jafferis-Maldacena loop momentum amplituhedron, which is a geometry encoding Aharony-Bergman-Jafferis-Maldacena planar tree-level amplitudes and loop integrands in the three-dimensional spinor helicity space. Translating it to the space of dual momenta produces a remarkably simple geometry given by configurations of spacelike separated off-shell momenta living inside a curvy polytope defined by momenta of scattered particles. We conjecture that the canonical differential form on this space gives amplitude integrands, and we provide a new formula for all one-loop n-particle integrands in the positive branch. For higher loop orders, we utilize the causal structure of configurations of points in Minkowski space to explain the singularity structure for known results at two loops.

Momentum amplituhedron meets kinematic associahedron.

In this paper we study a relation between two positive geometries: the momen- tum amplituhedron, relevant for tree-level scattering amplitudes in N = 4 super Yang-Mills theory, and the kinematic associahedron, encoding tree-level amplitudes in bi-adjoint scalar φ 3 theory. We study the implications of restricting the latter to four spacetime dimensions and give a direct link between its canonical form and the canonical form for the momentum amplituhedron. After removing the little group scaling dependence of the gauge theory, we find that we can compare the resulting reduced forms with the pull-back of the associahedron form. In particular, the associahedron form is the sum over all helicity sectors of the reduced momentum amplituhedron forms. This relation highlights the common sin- gularity structure of the respective amplitudes; in particular, the factorization channels, corresponding to vanishing planar Mandelstam variables, are the same. Additionally, we also find a relation between these canonical forms directly on the kinematic space of the scalar theory when reduced to four spacetime dimensions by Gram determinant constraints. As a by-product of our work we provide a detailed analysis of the kinematic spaces relevant for the four-dimensional gauge and scalar theories, and provide direct links between them.

Harmonic R matrices for scattering amplitudes and spectral regularization.

Planar N = 4 supersymmetric Yang-Mills theory appears to be integrable. While this allows one to find this theory's exact spectrum, integrability has hitherto been of no direct use for scattering amplitudes. To remedy this, we deform all scattering amplitudes by a spectral parameter. The deformed tree-level four-point function turns out to be essentially the one-loop R matrix of the integrable N = 4 spin chain satisfying the Yang-Baxter equation. Deformed on-shell three-point functions yield novel three-leg R matrices satisfying bootstrap equations. Finally, we supply initial evidence that the spectral parameter might find its use as a novel symmetry-respecting regulator replacing dimensional regularization. Its physical meaning is a local deformation of particle helicity, a fact which might be useful for a much larger class of nonintegrable four-dimensional field theories.