The Amplituhedron

Abstract

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.

36 pages, 14 figures

References in corpus (19)

• New Relations for Gauge-Theory Amplitudes
• Gluon scattering amplitudes at strong coupling
• Total positivity, Grassmannians, and networks
• The All-Loop Integrand For Scattering Amplitudes in Planar N=4 SYM
• The Four-Loop Planar Amplitude and Cusp Anomalous Dimension in Maximally Supersymmetric Yang-Mills Theory
• Yangian symmetry of scattering amplitudes in N=4 super Yang-Mills theory
• The Complete Planar S-matrix of N=4 SYM as a Wilson Loop in Twistor Space
• Maximally Supersymmetric Planar Yang-Mills Amplitudes at Five Loops
• Into the Amplituhedron
• Notes on the scattering amplitude / Wilson loop duality
• Gravity from Rational Curves
• Loops and trees
• Harmonic R-matrices for Scattering Amplitudes and Spectral Regularization
• On BCFW shifts of integrands and integrals
• A simple formula for gravitational MHV amplitudes
• Twistor Strings for N=8 Supergravity
• A "Twistor String" Inspired Formula For Tree-Level Scattering Amplitudes in N=8 SUGRA
• The Grassmannian and the Twistor String: Connecting All Trees in N=4 SYM
• Amplitudes at Weak Coupling as Polytopes in AdS_5

Cited by in corpus (309)

• Scattering Forms and the Positive Geometry of Kinematics, Color and the Worldsheet
• The EFT-Hedron
• Loops in AdS from Conformal Field Theory
• On-Shell Recursion Relations for Effective Field Theories
• Into the Amplituhedron
• Heptagons from the Steinmann Cluster Bootstrap
• A Periodic Table of Effective Field Theories
• Analytic form of the two-loop planar five-gluon all-plus-helicity amplitude in QCD
• Positive Geometries and Canonical Forms
• Scattering Amplitudes from Intersection Theory
• Six-Gluon Amplitudes in Planar $\mathcal{N}=4$ Super-Yang-Mills Theory at Six and Seven Loops
• Combinatorics and Topology of Kawai-Lewellen-Tye Relations
• Unwinding the Amplituhedron in Binary
• Aspects of Scattering Amplitudes and Moduli Space Localization
• Positive Amplitudes In The Amplituhedron
• On the Singularity Structure of Maximally Supersymmetric Scattering Amplitudes
• Double copy structure of CFT correlators
• One-Loop Corrections from Higher Dimensional Tree Amplitudes
• Evidence for a Nonplanar Amplituhedron
• Four-gluon scattering at three loops, infrared structure and Regge limit
• Prescriptive Unitarity
• OPE for all Helicity Amplitudes
• The Momentum Amplituhedron
• Cosmological Polytopes and the Wavefunction of the Universe
• Cluster algebras for Feynman integrals
• The two-loop five-point amplitude in $\mathcal{N}=8$ supergravity
• Structure of two-loop SMEFT anomalous dimensions via on-shell methods
• Locality and Unitarity from Singularities and Gauge Invariance
• Local Integrand Representations of All Two-Loop Amplitudes in Planar SYM
• The Double Pentaladder Integral to All Orders
• Cluster adjacency beyond MHV
• Twistor theory at fifty: from contour integrals to twistor strings
• Yangian-invariant fishnet integrals in 2 dimensions as volumes of Calabi-Yau varieties
• Baikov representations, intersection theory, and canonical Feynman integrals
• On-shell Diagrams, Graßmannians and Integrability for Form Factors
• New relation for AdS amplitudes
• Two-loop six gluon all plus helicity amplitude
• Constructing Canonical Feynman Integrals with Intersection Theory
• Gravity On-shell Diagrams
• The Amplituhedron from Momentum Twistor Diagrams
• Amplitudes meet Cosmology: A (Scalar) Primer
• Landau Singularities from the Amplituhedron
• The Correlahedron
• Multi-Loop Positivity of the Planar $\mathcal{N}=4$ SYM Six-Point Amplitude
• Cutting Deep Into The Amplituhedron
• Towards the Amplituhedron Volume
• Stokes Polytopes : The positive geometry for ${\varphi }^4$ interactions
• 1-loop Amplitudes from the Halohedron
• Two-loop Octagons, Algebraic Letters and $\bar{Q}$ Equations
• UV considerations on scattering amplitudes in a web of theories
• All-loop singularities of scattering amplitudes in massless planar theories
• A note on asymptotic symmetries and soft-photon theorem
• Stratifying On-Shell Cluster Varieties: the Geometry of Non-Planar On-Shell Diagrams
• Bootstrapping two-loop Feynman integrals for planar N=4 sYM
• The Amplituhedron and the One-loop Grassmannian Measure
• On Form Factors in N=4 SYM Theory and Polytopes
• On the Positive Geometry of Conformal Field Theory
• On-Shell Diagrams for N = 8 Supergravity Amplitudes
• Nonperturbative Negative Geometries: Amplitudes at Strong Coupling and the Amplituhedron
• Logarithmic forms and differential equations for Feynman integrals
• Cosmological Scattering Equations at Tree-level and One-loop
• $L_{\infty }$-algebras and the perturbiner expansion
• Singularities of eight- and nine-particle amplitudes from cluster algebras and tropical geometry
• The elliptic double box and symbology beyond polylogarithms
• On spinning loop amplitudes in Anti-de Sitter space
• Notes on Scattering Amplitudes as Differential Forms
• N=4 Scattering Amplitudes and the Deformed Grassmannian
• Integrands of loop amplitudes within loop-tree duality
• Dual Conformal Structure Beyond the Planar Limit
• All-Multiplicity Non-Planar MHV Amplitudes in sYM at Two Loops
• $n$-point amplitudes with a single negative-helicity graviton
• Grassmannians for scattering amplitudes in 4d $\mathcal{N}=4$ SYM and 3d ABJM
• Sign variation, the Grassmannian, and total positivity
• Prescriptive Unitarity for Non-Planar Six-Particle Amplitudes at Two Loops
• Boundaries of Amplituhedra and NMHV Symbol Alphabets at Two Loops
• The symbol and alphabet of two-loop NMHV amplitudes from $\bar{Q}$ equations
• Notes on cluster algebras and some all-loop Feynman integrals
• Regularity theorem for totally nonnegative flag varieties
• Integrable Amplitude Deformations for N=4 Super Yang-Mills and ABJM Theory
• Amplitudes at Infinity
• Accordiohedra as positive geometries for generic scalar field theories
• Uniqueness from gauge invariance and the Adler zero
• Full Colour for Loop Amplitudes in Yang-Mills Theory
• Truncated cluster algebras and Feynman integrals with algebraic letters
• Positive configuration space
• Constructing $\mathcal{N}=4$ Coulomb Branch Superamplitudes
• On Positive Geometries of Quartic Interactions II : Stokes polytopes, Lower Forms on Associahedra and Worldsheet Forms
• Scalar-Scaffolded Gluons and the Combinatorial Origins of Yang-Mills Theory
• All-Helicity Symbol Alphabets from Unwound Amplituhedra
• From 4d Ambitwistor Strings to On Shell Diagrams and Back
• On integrands and loop momentum in string and field theory
• Symbology for elliptic multiple polylogarithms and the symbol prime
• The Super BMS Algebra, Scattering and Holography
• Parity duality for the amplituhedron
• Cluster Adjacency for m=2 Yangian Invariants
• On Soft Theorems And Form Factors In N=4 SYM Theory
• Positive geometry, local triangulations, and the dual of the Amplituhedron
• The positive geometry for ${\varphi }^p$ interactions
• Properties of scattering forms and their relation to associahedra
• UV cancelations in gravity loop integrands
• Implications of Superrotations
• Building Bases of Loop Integrands
• The orthogonal momentum amplituhedron and ABJM amplitudes
• Propagators, BCFW Recursion and New Scattering Equations at One Loop
• The momentum amplituhedron of SYM and ABJM from twistor-string maps
• On the Emergence of Lorentz Invariance and Unitarity from the Scattering Facet of Cosmological Polytopes
• From Momentum Amplituhedron Boundaries to Amplitude Singularities and Back
• Ising model and the positive orthogonal Grassmannian
• On-Shell Electroweak Sector and the Higgs Mechanism
• Hidden zeros are equivalent to enhanced ultraviolet scaling and lead to unique amplitudes in Tr(${\varphi }^3$) theory
• Scattering Forms, Worldsheet Forms and Amplitudes from Subspaces
• Algebraic branch points at all loop orders from positive kinematics and wall crossing
• A Note on Letters of Yangian Invariants
• The SAGEX Review on Scattering Amplitudes, Chapter 7: Positive Geometry of Scattering Amplitudes
• What can we learn about QCD and collider physics from N=4 super Yang-Mills?
• Feynman Integrals and Scattering Amplitudes from Wilson Loops
• Amplitude recursions with an extra marked point
• From Six to Four and More: Massless and Massive Maximal Super Yang-Mills Amplitudes in 6d and 4d and their Hidden Symmetries
• All-Loop Four-Point Aharony-Bergman-Jafferis-Maldacena Amplitudes from Dimensional Reduction of the Amplituhedron
• An Etude on Recursion Relations and Triangulations
• Surface Kinematics and "The" Yang-Mills Integrand
• The Wilson-loop $d\log$ representation for Feynman integrals
• Amplituhedron meets Jeffrey-Kirwan Residue
• Grassmannians and form factors with $q^2=0$ in N=4 SYM theory
• Prescriptive Unitarity with Elliptic Leading Singularities
• Two-Loop Five-Point Two-Mass Planar Integrals and Double Lagrangian Insertions in a Wilson Loop
• Wilson lines, Grassmannians and Gauge Invariant Off-shell Amplitudes in N=4 SYM
• On Entropy Growth in Perturbative Scattering
• Positive Geometries and Differential Forms with Non-Logarithmic Singularities I
• Positive Geometries for all Scalar Theories from Twisted Intersection Theory
• Cosmological Polytopes and the Wavefuncton of the Universe for Light States
• Renormalization and non-renormalization of scalar EFTs at higher orders
• Cross-Order Integral Relations from Maximal Cuts
• An Elliptic Yangian-Invariant, `Leading Singularity'
• Recursion relations for scattering amplitudes with massive particles
• Amplituhedra, and Beyond
• Supervertices and Non-renormalization Conditions in Maximal Supergravity Theories
• Sign Flip Triangulations of the Amplituhedron
• From the flat-space S-matrix to the Wavefunction of the Universe
• Towards the Gravituhedron: New Expressions for NMHV Gravity Amplitudes
• Boundaries of the Amplituhedron with amplituhedronBoundaries
• Associahedra for finite type cluster algebras and minimal relations between $\mathbf{g}$-vectors
• Notes on polytopes, amplitudes and boundary configurations for Grassmannian string integrals
• Four dimensional ambitwistor strings and form factors of local and Wilson line operators
• Exposing the threshold structure of loop integrals
• Integrated negative geometries in ABJM
• Hyperbolic Geometry and Amplituhedra in 1+2 dimensions
• Unified Formalism for 6D Superamplitudes Based on a Symplectic Grassmannian
• Uniqueness from locality and BCFW shifts
• On-shell diagrams and the geometry of planar N < 4 SYM theories
• Multi-spin soft bootstrap and scalar-vector Galileon
• The Loop Momentum Amplituhedron
• SYM, Chern-Simons, Wess-Zumino Couplings and their higher derivative corrections in IIA Superstring theory
• A Scattering Amplitude for Massive Particles in AdS
• The ChPT: top-down and bottom-up
• Amplituhedron-like geometries
• On-shell diagrammatics and the perturbative structure of planar gauge theories
• Instantons or Renormalons? A Comment on phi4_4 Theory in the MS Scheme
• Weights and recursion relations for ${\varphi }^p$ tree amplitudes from the positive geometry
• Universal expansions of scattering amplitudes for gravitons, gluons and Goldstone particles
• Bootstrap and Amplitudes: A Hike in the Landscape of Quantum Field Theory
• All-loop cuts from the Amplituhedron
• A Soft Theorem for the Tropical Grassmannian
• Celebrating Loday's Associahedron
• All-multiplicity amplitudes with four massive quarks and identical-helicity gluons
• Gravity loop integrands from the ultraviolet
• One-Loop Integrals from Spherical Projections of Planes and Quadrics
• Momentum Amplituhedron meets Kinematic Associahedron
• The Grassmannian for Celestial Superamplitudes
• Gravitons, Bose-symmetry, and Bonus-scaling
• Wilson Loop Duality and OPE for Super Form Factors of Half-BPS Operators
• Comments on all-loop constraints for scattering amplitudes and Feynman integrals
• Emergent unitarity, all-loop cuts and integrations from the ABJM amplituhedron
• Higher-Point Integrands in N=4 super Yang-Mills Theory
• Towards Analytic Structure of Feynman Parameter Integrals with Rational Curves
• Internal boundaries of the loop amplituhedron
• Color structures and permutations
• Positivity Sectors and the Amplituhedron
• Rewording Theoretical Predictions at Colliders with Vacuum Amplitudes
• Cosmohedra
• Yangian Symmetry for the Tree Amplituhedron
• On Positive Geometry and Scattering Forms for Matter Particles
• Twistors and antipodes in de Sitter space
• Wilson lines in the MHV action
• The two-loop eight-point amplitude in ABJM theory
• Hidden Conformal Invariance of Scalar Effective Field Theories
• Multi-Particle Amplitudes from the Four-Point Correlator in Planar N=4 SYM
• Periods and Feynman amplitudes
• Kleiss-Kuijf Relations from Momentum Amplituhedron Geometry
• On 1D, N = 4 Supersymmetric SYK-Type Models (I)
• Non-planar BCFW Grassmannian Geometries
• Conformally Regulated Direct Integration of the Two-Loop Heptagon Remainder
• ABJM Amplitudes in U-gauge and a Soft Theorem
• The all-loop conjecture for integrands of reggeon amplitudes in N=4 SYM
• Non-Planar On-Shell Diagrams
• On Polytopes and Generalizations of the KLT Relations
• Windows on Quantum Gravity
• Constraining the weights of Stokes Polytopes using BCFW recursions for Phi^4
• Weights, Recursion relations and Projective triangulations for Positive Geometry of scalar theories
• Open associahedra and scattering forms
• Bootstrapping octagons in reduced kinematics from $A_2$ cluster algebras
• $\mathcal{N}=7$ On-Shell Diagrams and Supergravity Amplitudes in Momentum Twistor Space
• Triangulations and Canonical Forms of Amplituhedra: a fiber-based approach beyond polytopes
• $\Delta$-Algebra and Scattering Amplitudes
• Maximal transcendental weight contribution of scattering amplitudes
• The m=2 amplituhedron and the hypersimplex: signs, clusters, triangulations, Eulerian numbers
• Tree-level scattering amplitudes from the amplituhedron
• On the geometry of the orthogonal momentum amplituhedron
• Recursion and worldsheet formulae for 6d superamplitudes
• Covariant color-kinematics duality, Hopf algebras and permutohedra
• Flow-oriented perturbation theory
• Triangulation of 2-loop MHV Amplituhedron from Sign Flips
• Wilson Loop diagrams and Positroids
• Critical varieties in the Grassmannian
• New Worldsheet Formulae for Conformal Supergravity Amplitudes
• Basis shape loci and the positive Grassmannian
• Periods and Superstring Amplitudes
• Grassmannian Integral for General Gauge Invariant Off-shell Amplitudes in N=4 SYM
• 4-particle Amplituhedronics for 3-5 Loops
• Emergent unitarity from the amplituhedron
• Higher-order tails and RG flows due to scattering of gravitational radiation from binary inspirals
• On two notions of total positivity for partial flag varieties
• Poles At Infinity in On-shell Diagrams
• All-loop Mondrian Reduction of 4-particle Amplituhedron at Positive Infinity
• Triangulation-free Trivialization of 2-loop MHV Amplituhedron
• Anatomy of the Amplituhedron
• On the Positive Geometry of Quartic Interactions III : One Loop Integrands from Polytopes
• The twistor Wilson loop and the amplituhedron
• CHY formulae in 4d
• A nice two-loop next-to-next-to-MHV amplitude in $\mathcal{N}=4$ super-Yang-Mills
• All-loop Mondrian Diagrammatics and 4-particle Amplituhedron
• One-loop integrand from generalised scattering equations
• Exploring straight infinite Wilson lines in the Self Dual and the MHV Lagrangians
• A note on generalized hypergeometric functions, KZ solutions, and gluon amplitudes
• Expansion of EYM Amplitudes in Gauge Invariant Vector Space
• Vertex operators for an expanding universe
• Diagrammar in an Extended Theory of Gravity
• 4-particle Amplituhedron at 3-loop and its Mondrian Diagrammatic Implication
• Learning the Simplicity of Scattering Amplitudes
• Pushforwards via Scattering Equations with Applications to Positive Geometries
• Dualities for Ising networks
• The hypersimplex canonical forms and the momentum amplituhedron-like logarithmic forms
• Illustrations of Integrand-Basis Building at Two Loops
• Geometric description of the Schrödinger equation in 3n+1 dimensional configuration space
• The end of spacetime
• UV consistency conditions for CHY integrands
• Logarithmic Singularities and Maximally Supersymmetric Amplitudes
• All Two-Loop, Color-Dressed, Six-Point Amplitude Integrands in sYM
• Nonplanar On-shell Diagrams and Leading Singularities of Scattering Amplitudes
• Triangulations for ABHY Polytopes and Recursions for Tree and Loop Amplitudes
• Two particle entanglement and its geometric duals
• A note on NMHV form factors from the Graßmannian and the twistor string
• Connections between Minkowski and Cosmological Correlation Functions
• Descent equation for superloop and cyclicity of OPE
• $L_{\infty }$ and $A_{\infty }$ structures: then and now
• Large $N$ analytical functional bootstrap I: 1D CFTs and total positivity
• The Amplituhedron BCFW Triangulation
• From hypertoric geometry to bordered Floer homology via the m=1 amplituhedron
• Leading singularities and chambers of Correlahedron
• SU(5) Grand Unified Theory, its Polytopes and 5-fold Symmetric Aperiodic Tiling
• Locally Finite Observables in sYM
• The KLT relations in unimodular gravity
• Shelling the m=1 amplituhedron
• Positivity, Grassmannian Geometry and Simplex-like Structures of Scattering Amplitudes
• Gravitation in terms of observables
• Grassmannian Integrals in Minkowski Signature, Amplitudes, and Integrability
• Loop Amplitudes in the Coulomb Branch of $\mathcal{N}=4$ Super-Yang-Mills Theory
• Combinatorics of Essential Sets for Positroids
• Scattering amplitudes in gauge theories with and without supersymmetry
• Generalised wavefunction coefficients and acyclonesto-cosmohedra
• Positive Geometry for Stringy Scalar Amplitudes
• Quantum deformation of planar amplitudes
• An octahedron of complex null rays, and conformal symmetry breaking
• Gravitation in terms of observables 2: the algebra of fundamental observables
• Triangular tessellations of one-loop scattering amplitudes in ${\varphi }^3$ theory
• Deformed graphical zonotopes
• Cluster algebraic description of entanglement patterns for the BTZ black hole
• Propagator positivity bounds for cosmological correlators
• Orientations of BCFW Charts on the Grassmannian
• The magical "Born Rule" & quantum "measurement": Implications for Physics
• Integrands of Less-Supersymmetric Yang-Mills at One Loop
• Rate of cluster decomposition via Fermat-Steiner point
• Flag manifolds over semifields
• An SU(3) Yang-Mills Structure for Electron-Phonon Interactions Resulting from Strong Electron Correlations in 2D hexagonal lattices
• Boundary measurement and sign variation in real projective space
• Symplectic Grassmannians, dual conformal symmetry and 4-point amplitudes in 6D
• Helicity Amplitudes for production of massive gravitino/goldstino
• On p-adic Frobenius lifts and p-adic periods, from a Deformation Theory viewpoint
• Extended DBI and its generalizations from graded soft theorems
• Plabic Tangles and Cluster Promotion Maps
• Bipartite Graphs, On-Shell Diagrams, and Bipartite Field Theories: a Computational Package in Mathematica
• Non-Abelian basis tensor gauge theory
• A First Step Towards Effectively Nonperturbative Scattering Amplitudes in the Perturbative Regime
• Generalized relative locality and causal sets
• Brane Tilings as On-shell Diagrams
• On the Four-Loop Higgs--Gluon Form Factor in Nonlocal Quantum Field Theory
• Higher-Point Correlators in N=4 SYM: Ten-Dimensional Null Polygons
• The Geometry of BCFW for ABJM Loop Integrands
• Cyclicity of All Anti-NMHV and N${}^2$MHV Tree Amplitudes in N=4 SYM
• Lorentz Invariance of Basis Tensor Gauge Theory
• Defects in Spin Chains via Cluster Categories
• Leading singularities in higher-derivative Yang-Mills theory and quadratic gravity
• Tropicalized quantum field theory and global tropical sampling
• Coherent States and Particle Scattering in Loop Quantum Gravity
• On-shell Correlators and Color-Kinematics Duality in Curved Symmetric Spacetimes
• On the Integrable Structure of Super Yang-Mills Scattering Amplitudes
• Learning scattering amplitudes by heart
• Towards Amplituhedron via One-Dimensional Theory
• Smooth Splitting and Zeros from On-Shell Recursion