Positive Amplitudes In The Amplituhedron

Abstract

The all-loop integrand for scattering amplitudes in planar N = 4 SYM is determined by an "amplitude form" with logarithmic singularities on the boundary of the amplituhedron. In this note we provide strong evidence for a new striking property of the superamplitude, which we conjecture to be true to all loop orders: the amplitude form is positive when evaluated inside the amplituhedron. The statement is sensibly formulated thanks to the natural "bosonization" of the superamplitude associated with the amplituhedron geometry. However this positivity is not manifest in any of the current approaches to scattering amplitudes, and in particular not in the cellulations of the amplituhedron related to on-shell diagrams and the positive grassmannian. The surprising positivity of the form suggests the existence of a "dual amplituhedron" formulation where this feature would be made obvious. We also suggest that the positivity is associated with an extended picture of amplituhedron geometry, with the amplituhedron sitting inside a co-dimension one surface separating "legal" and "illegal" local singularities of the amplitude. We illustrate this in several simple examples, obtaining new expressions for amplitudes not associated with any triangulations, but following in a more invariant manner from a global view of the positive geometry.

27 pages, 10 figures

References in corpus (12)

• 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
• Bootstrapping an NMHV amplitude through three loops
• MHV Diagrams in Momentum Twistor Space
• Eliminating spurious poles from gauge-theoretic amplitudes
• The Amplituhedron from Momentum Twistor Diagrams
• Positroids, Plabic Graphs, and Scattering Amplitudes in Mathematica
• Efficient Tree-Amplitudes in N=4: Automatic BCFW Recursion in Mathematica
• Dual-Conformal Regularization of Infrared Loop Divergences and the Chiral Box Expansion
• Anatomy of the Amplituhedron
• Amplituhedron cells and Stanley symmetric functions

Cited by in corpus (62)

• Analytic form of the two-loop planar five-gluon all-plus-helicity amplitude in QCD
• Positive Geometries and Canonical Forms
• Unwinding the Amplituhedron in Binary
• The Cosmic Galois Group and Extended Steinmann Relations for Planar $\mathcal{N}=4$ SYM Amplitudes
• Evidence for a Nonplanar Amplituhedron
• The Momentum Amplituhedron
• Gauge and Gravity Amplitude Relations
• Gravity On-shell Diagrams
• Amplitudes meet Cosmology: A (Scalar) Primer
• Landau Singularities from the Amplituhedron
• Towards the Amplituhedron Volume
• The Correlahedron
• Cutting Deep Into The Amplituhedron
• Multi-Loop Positivity of the Planar $\mathcal{N}=4$ SYM Six-Point Amplitude
• The Hunt for Primordial Interactions in the Large Scale Structures of the Universe
• Steinmann Relations and the Wavefunction of the Universe
• Stratifying On-Shell Cluster Varieties: the Geometry of Non-Planar On-Shell Diagrams
• 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
• Accordiohedra as positive geometries for generic scalar field theories
• 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
• MultivariateResidues: a Mathematica package for computing multivariate residues
• The SAGEX Review on Scattering Amplitudes, Chapter 7: Positive Geometry of Scattering Amplitudes
• Amplituhedron meets Jeffrey-Kirwan Residue
• Positive Geometries for all Scalar Theories from Twisted Intersection Theory
• Two-Loop Five-Point Two-Mass Planar Integrals and Double Lagrangian Insertions in a Wilson Loop
• Positive Geometries and Differential Forms with Non-Logarithmic Singularities I
• The Two-Loop Remainder Function for Eight and Nine Particles
• The Geometry of Cosmological Correlators
• Sign Flip Triangulations of the Amplituhedron
• Amplituhedra, and Beyond
• Generalized Cuts of Feynman Integrals in Parameter Space
• Towards the Gravituhedron: New Expressions for NMHV Gravity Amplitudes
• Exposing the threshold structure of loop integrals
• Multi-spin soft bootstrap and scalar-vector Galileon
• On-shell diagrams and the geometry of planar N < 4 SYM theories
• On-shell diagrammatics and the perturbative structure of planar gauge theories
• All-loop cuts from the Amplituhedron
• Gravity loop integrands from the ultraviolet
• Positivity Sectors and the Amplituhedron
• Non-planar BCFW Grassmannian Geometries
• Non-Planar On-Shell Diagrams
• $\Delta$-Algebra and Scattering Amplitudes
• Tree-level scattering amplitudes from the amplituhedron
• Open associahedra and scattering forms
• Flow-oriented perturbation theory
• Introduction to the number of $e$-folds in slow-roll inflation
• On Polytopes and Generalizations of the KLT Relations
• Scalar induced gravitational waves review
• Triangulation of 2-loop MHV Amplituhedron from Sign Flips
• The twistor Wilson loop and the amplituhedron
• A Novel Algorithm for Nested Summation and Hypergeometric Expansions
• $L_{\infty }$ and $A_{\infty }$ structures: then and now
• Adjoints and Canonical Forms of Tree Amplituhedra
• Positive Geometry for Stringy Scalar Amplitudes
• Towards a unitary formulation of quantum field theory in curved spacetime: the case of de Sitter spacetime
• Positivity, Grassmannian Geometry and Simplex-like Structures of Scattering Amplitudes
• Logarithms and Volumes of Polytopes
• Extended DBI and its generalizations from graded soft theorems