Into the Amplituhedron

Abstract

We initiate an exploration of the physics and geometry of the amplituhedron, starting with the simplest case of the integrand for four-particle scattering in planar N=4 SYM. We show how the textbook structure of the unitarity double-cut follows from the positive geometry. We also use the geometry to expose the behavior of the multicollinear limit, providing a direct motivation for studying the logarithm of the amplitude. In addition to computing the two and three-loop integrands, we explore various lower-dimensional faces of the amplituhedron, thereby computing non-trivial cuts of the integrand to all loop orders.

34 pages, 23 figures

References in corpus (10)

• Gluon scattering amplitudes at strong coupling
• The Amplituhedron
• 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
• Maximally Supersymmetric Planar Yang-Mills Amplitudes at Five Loops
• Positive Amplitudes In The Amplituhedron
• Loops and trees
• Harmonic R-matrices for Scattering Amplitudes and Spectral Regularization
• Eliminating spurious poles from gauge-theoretic amplitudes

Cited by in corpus (103)

• The Amplituhedron
• Bootstrapping an NMHV amplitude through three loops
• Heptagons from the Steinmann Cluster Bootstrap
• A Periodic Table of Effective Field Theories
• Positive Geometries and Canonical Forms
• 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
• Positive Amplitudes In The Amplituhedron
• Evidence for a Nonplanar Amplituhedron
• Prescriptive Unitarity
• Cosmological Polytopes and the Wavefunction of the Universe
• Local Integrand Representations of All Two-Loop Amplitudes in Planar SYM
• The Double Pentaladder Integral to All Orders
• New relation for AdS amplitudes
• Gravity On-shell Diagrams
• Landau Singularities from the Amplituhedron
• Cutting Deep Into The Amplituhedron
• Multi-Loop Positivity of the Planar $\mathcal{N}=4$ SYM Six-Point Amplitude
• The Correlahedron
• 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
• Nonperturbative Negative Geometries: Amplitudes at Strong Coupling and the Amplituhedron
• On-Shell Diagrams for N = 8 Supergravity Amplitudes
• Dual Conformal Structure Beyond the Planar Limit
• N=4 Scattering Amplitudes and the Deformed Grassmannian
• Integrands of loop amplitudes within loop-tree duality
• Boundaries of Amplituhedra and NMHV Symbol Alphabets at Two Loops
• Accordiohedra as positive geometries for generic scalar field theories
• Integrable Amplitude Deformations for N=4 Super Yang-Mills and ABJM Theory
• The SAGEX Review on Scattering Amplitudes, Chapter 8: Half BPS correlators
• On Positive Geometries of Quartic Interactions II : Stokes polytopes, Lower Forms on Associahedra and Worldsheet Forms
• The Sklyanin Bracket and Cluster Adjacency at All Multiplicity
• All-Helicity Symbol Alphabets from Unwound Amplituhedra
• Positive geometry, local triangulations, and the dual of the Amplituhedron
• On Soft Theorems And Form Factors In N=4 SYM Theory
• The positive geometry for ${\varphi }^p$ interactions
• Properties of scattering forms and their relation to associahedra
• UV cancelations in gravity loop integrands
• The momentum amplituhedron of SYM and ABJM from twistor-string maps
• MultivariateResidues: a Mathematica package for computing multivariate residues
• The orthogonal momentum amplituhedron and ABJM amplitudes
• Scattering Forms, Worldsheet Forms and Amplitudes from Subspaces
• The SAGEX Review on Scattering Amplitudes, Chapter 7: Positive Geometry of Scattering Amplitudes
• All-Loop Four-Point Aharony-Bergman-Jafferis-Maldacena Amplitudes from Dimensional Reduction of the Amplituhedron
• Grassmannians and form factors with $q^2=0$ in N=4 SYM theory
• Amplituhedron meets Jeffrey-Kirwan Residue
• Wilson lines, Grassmannians and Gauge Invariant Off-shell Amplitudes in N=4 SYM
• On Entropy Growth in Perturbative Scattering
• Amplituhedra, and Beyond
• Sign Flip Triangulations of the Amplituhedron
• Supervertices and Non-renormalization Conditions in Maximal Supergravity Theories
• Four dimensional ambitwistor strings and form factors of local and Wilson line operators
• On-shell diagrams and the geometry of planar N < 4 SYM theories
• Integrated negative geometries in ABJM
• Multi-spin soft bootstrap and scalar-vector Galileon
• SYM, Chern-Simons, Wess-Zumino Couplings and their higher derivative corrections in IIA Superstring theory
• On-shell diagrammatics and the perturbative structure of planar gauge theories
• All-loop cuts from the Amplituhedron
• Weights and recursion relations for ${\varphi }^p$ tree amplitudes from the positive geometry
• A Soft Theorem for the Tropical Grassmannian
• One-Loop Integrals from Spherical Projections of Planes and Quadrics
• Emergent unitarity, all-loop cuts and integrations from the ABJM amplituhedron
• Internal boundaries of the loop amplituhedron
• Towards Analytic Structure of Feynman Parameter Integrals with Rational Curves
• Gravity loop integrands from the ultraviolet
• Positivity Sectors and the Amplituhedron
• Multi-Particle Amplitudes from the Four-Point Correlator in Planar N=4 SYM
• The four-loop six-gluon NMHV ratio function
• Non-planar BCFW Grassmannian Geometries
• 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
• Poles At Infinity in On-shell Diagrams
• 4-particle Amplituhedronics for 3-5 Loops
• Triangulation of 2-loop MHV Amplituhedron from Sign Flips
• Emergent unitarity from the amplituhedron
• Grassmannian Integral for General Gauge Invariant Off-shell Amplitudes in N=4 SYM
• On the Positive Geometry of Quartic Interactions III : One Loop Integrands from Polytopes
• A note on generalized hypergeometric functions, KZ solutions, and gluon amplitudes
• Anatomy of the Amplituhedron
• All-loop Mondrian Diagrammatics and 4-particle Amplituhedron
• All-loop Mondrian Reduction of 4-particle Amplituhedron at Positive Infinity
• The twistor Wilson loop and the amplituhedron
• Triangulation-free Trivialization of 2-loop MHV Amplituhedron
• 4-particle Amplituhedron at 3-loop and its Mondrian Diagrammatic Implication
• Expansion of EYM Amplitudes in Gauge Invariant Vector Space
• Illustrations of Integrand-Basis Building at Two Loops
• The Positive orthogonal Grassmannian and loop amplitudes of ABJM
• Logarithmic Singularities and Maximally Supersymmetric Amplitudes
• All Two-Loop, Color-Dressed, Six-Point Amplitude Integrands in sYM
• Connections between Minkowski and Cosmological Correlation Functions
• A note on NMHV form factors from the Graßmannian and the twistor string
• Descent equation for superloop and cyclicity of OPE
• Leading singularities and chambers of Correlahedron
• Scattering amplitudes in gauge theories with and without supersymmetry
• Positivity, Grassmannian Geometry and Simplex-like Structures of Scattering Amplitudes
• Positive Geometry for Stringy Scalar Amplitudes
• Orientations of BCFW Charts on the Grassmannian
• Brane Tilings as On-shell Diagrams
• Logarithms and Volumes of Polytopes
• Extended DBI and its generalizations from graded soft theorems