Scattering in Three Dimensions from Rational Maps

Abstract

The complete tree-level S-matrix of four dimensional $\mathcal{N}=4$ super Yang-Mills and $\mathcal{N}=8$ supergravity has compact forms as integrals over the moduli space of certain rational maps. In this note we derive formulas for amplitudes in three dimensions by using the fact that when amplitudes are dressed with proper wave functions dimensional reduction becomes straightforward. This procedure leads to formulas in terms of rational maps for three dimensional maximally supersymmetric Yang-Mills and gravity theories. The integrand of the new formulas contains three basic structures: Parke-Taylor-like factors, Vandermonde determinants and resultants. Integrating out some of the Grassmann directions produces formulas for theories with less than maximal supersymmetry, which exposes yet a fourth kind of structure. Combining all four basic structures we start a search for consistent S-matrices in three dimensions. Very nicely, the most natural ones are those corresponding to ABJM and BLG theories. We also make a connection between the power of a resultant in the integrand, representations of the Poincaré group, infrared behavior and conformality of a theory. Extensions to other theories in three dimensions and to arbitrary dimensions are also discussed.

21 pages

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• N=6 superconformal Chern-Simons-matter theories, M2-branes and their gravity duals
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• Scattering in Mass-Deformed N>=4 Chern-Simons Models
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Cited by in corpus (90)

• Scattering of Massless Particles: Scalars, Gluons and Gravitons
• Scattering Equations and Matrices: From Einstein To Yang-Mills, DBI and NLSM
• Scattering Equations and KLT Orthogonality
• Ambitwistor strings and the scattering equations
• Scattering Forms and the Positive Geometry of Kinematics, Color and the Worldsheet
• Positive Geometries and Canonical Forms
• Color-Kinematics Duality for QCD Amplitudes
• One-loop amplitudes on the Riemann sphere
• Extensions of Theories from Soft Limits
• Black Hole Formation and Classicalization in Ultra-Planckian 2 -> N Scattering
• Aspects of Scattering Amplitudes and Moduli Space Localization
• Scattering equations and BCJ relations for gauge and gravitational amplitudes with massive scalar particles
• One-Loop Corrections from Higher Dimensional Tree Amplitudes
• Exact solutions for the biadjoint scalar field
• The Polynomial Form of the Scattering Equations
• N-Point Tree-Level Scattering Amplitude in the New Berkovits' String
• Expanding Einstein-Yang-Mills by Yang-Mills in CHY frame
• ABJM amplitudes and the positive orthogonal grassmannian
• Extended solutions for the biadjoint scalar field
• Scattering Equations, Twistor-string Formulas and Double-soft Limits in Four Dimensions
• Scattering Equations: From Projective Spaces to Tropical Grassmannians
• 1-loop Amplitudes from the Halohedron
• Massless scattering at special kinematics as Jacobi polynomials
• Scattering equations and virtuous kinematic numerators and dual-trace functions
• Elliptic scattering equations
• New Formulas for Amplitudes from Higher-Dimensional Operators
• On the solutions of the scattering equations
• Labelled tree graphs, Feynman diagrams and disk integrals
• Comments on the evaluation of massless scattering
• The S Matrix of 6D Super Yang-Mills and Maximal Supergravity from Rational Maps
• New structures in scattering amplitudes: a review
• Notes on Scattering Amplitudes as Differential Forms
• Connected formulas for amplitudes in standard model
• Comments on the $S_N$ orbifold CFT in the large $N$-limit
• CHY Loop Integrands from Holomorphic Forms
• Prescriptive Unitarity for Non-Planar Six-Particle Amplitudes at Two Loops
• M5-Brane and D-Brane Scattering Amplitudes
• A note on connected formula for form factors
• Scattering Equations: Real Solutions and Particles on a Line
• CHY-Graphs on a Torus
• From 4d Ambitwistor Strings to On Shell Diagrams and Back
• All Tree Amplitudes of 6D $(2,0)$ Supergravity: Interacting Tensor Multiplets and the $K3$ Moduli Space
• Quadratic Feynman Loop Integrands From Massless Scattering Equations
• Double Soft Theorem for Perturbative Gravity
• Polynomial reduction and evaluation of tree- and loop-level CHY amplitudes
• The momentum amplituhedron of SYM and ABJM from twistor-string maps
• One-loop Parke-Taylor factors for quadratic propagators from massless scattering equations
• Soft Gravity by Squaring Soft QED on the Celestial Sphere
• Non-planar one-loop Parke-Taylor factors in the CHY approach for quadratic propagators
• Kinematic Numerators and a Double-Copy Formula for N = 4 Super-Yang-Mills Residues
• Positive Geometries for all Scalar Theories from Twisted Intersection Theory
• The Scattering Variety
• On the New Factorizations of Yang-Mills Amplitudes
• Manifest colour-kinematics duality and double-copy in the string-based formalism
• On Chiral Splitting and the Ambitwistor String
• Positroid Stratification of Orthogonal Grassmannian and ABJM Amplitudes
• Electromagnetic Duality and D3-Brane Scattering Amplitudes Beyond Leading Order
• Unified Formalism for 6D Superamplitudes Based on a Symplectic Grassmannian
• Direct Evaluation of $n$-point single-trace MHV amplitudes in 4d Einstein-Yang-Mills theory using the CHY Formalism
• The CHY representation of tree-level primitive QCD amplitudes
• One-loop diagrams with quadratic propagators from the worldsheet
• Minitwistors and 3d Yang-Mills-Higgs theory
• A differential operator for integrating one-loop scattering equations
• A String Deformation of the Parke-Taylor Factor
• The m=2 amplituhedron and the hypersimplex: signs, clusters, triangulations, Eulerian numbers
• On Polytopes and Generalizations of the KLT Relations
• ABJM Amplitudes in U-gauge and a Soft Theorem
• Minimal Basis in Four Dimensions and Scalar Blocks
• Bootstrapping solutions of scattering equations
• New Worldsheet Formulae for Conformal Supergravity Amplitudes
• CHY formulae in 4d
• Spontaneously Broken Yang-Mills-Einstein Supergravities as Double Copies
• New Ambitwistor String Theories
• One-loop integrand from generalised scattering equations
• Asymmetrically twisted strings
• Double Soft Theorem for Generalised Biadjoint Scalar Amplitudes
• Infrared behaviour of the one-loop scattering equations and supergravity integrands
• Analytical amplitudes from numerical solutions of the scattering equations
• Geometric Recursion from Polytope Triangulations and Twisted Homology
• Infrared and Holographic Aspects of the $S$-Matrix in Gauge Theory and Gravity
• Pushforwards via Scattering Equations with Applications to Positive Geometries
• Multi-Regge kinematics and the scattering equations
• Permutation in the CHY-Formulation
• Fermions and the scattering equations
• From Twistor-Particle Models to Massive Amplitudes
• Characterizing the solutions to scattering equations that support tree-level ${\text{N}}^k\text{MHV}$ gauge/gravity amplitudes
• Gravitational Scattering in the High-Energy Limit
• A twistor string for the ABJ(M) theory
• Ambitwistor string amplitudes in light-like linear dilaton background
• Color-factor symmetry and BCJ relations for QCD amplitudes