Multi-Loop Positivity of the Planar $\mathcal{N}=4$ SYM Six-Point Amplitude

Abstract

We study the six-point NMHV ratio function in planar $\mathcal{N}=4$ SYM theory in the context of positive geometry. The Amplituhedron construction of the integrand for the amplitudes provides a kinematical region in which the integrand was observed to be positive. It is natural to conjecture that this property survives integration, i.e. that the final result for the ratio function is also positive in this region. Establishing such a result would imply that preserving positivity is a surprising property of the Minkowski contour of integration and it might indicate some deeper underlying structure. We find that the ratio function is positive everywhere we have tested it, including analytic results for special kinematical regions at one and two loops, as well as robust numerical evidence through five loops. There is also evidence for not just positivity, but monotonicity in a "radial" direction. We also investigate positivity of the MHV six-gluon amplitude. While the remainder function ceases to be positive at four loops, the BDS-like normalized MHV amplitude appears to be positive through five loops.

arXiv comment45 pages, 7 figures, 4 ancillary files. v2: typos corrected; version to appear in JHEP

References in corpus (45)

• Harmonic Polylogarithms
• Iteration of Planar Amplitudes in Maximally Supersymmetric Yang-Mills Theory at Three Loops and Beyond
• Transcendentality and Crossing
• Classical Polylogarithms for Amplitudes and Wilson Loops
• Dual superconformal symmetry of scattering amplitudes in N=4 super-Yang-Mills theory
• The Amplituhedron
• What is the Simplest Quantum Field Theory?
• Numerical evaluation of multiple polylogarithms
• HPL, a Mathematica implementation of the harmonic polylogarithms
• Hexagon Wilson loop = six-gluon MHV amplitude
• The Two-Loop Six-Gluon MHV Amplitude in Maximally Supersymmetric Yang-Mills Theory
• The Multiple Zeta Value Data Mine
• Two-Loop Master Integrals for ${\gamma }^{\ast }\to 3$ Jets: The planar topologies
• Space-time S-matrix and Flux-tube S-matrix at Finite Coupling
• Bootstrapping the three-loop hexagon
• Motivic Amplitudes and Cluster Coordinates
• Bootstrapping a Five-Loop Amplitude Using Steinmann Relations
• Thermodynamic Bubble Ansatz
• Analytic treatment of the two loop equal mass sunrise graph
• An exact formula for the radiation of a moving quark in N=4 super Yang Mills
• Analytic result for the two-loop six-point NMHV amplitude in N=4 super Yang-Mills theory
• An Operator Product Expansion for Polygonal null Wilson Loops
• Bootstrapping an NMHV amplitude through three loops
• Uniqueness of two-loop master contours
• Into the Amplituhedron
• A Symbol of Uniqueness: The Cluster Bootstrap for the 3-Loop MHV Heptagon
• Hexagon functions and the three-loop remainder function
• The four-loop remainder function and multi-Regge behavior at NNLLA in planar N=4 super-Yang-Mills theory
• Pulling the straps of polygons
• Non-MHV Tree Amplitudes in Gauge Theory
• Search for a new scalar resonance decaying to a pair of Z bosons in proton-proton collisions at $\sqrt{s}=$ 13 TeV
• Positive Amplitudes In The Amplituhedron
• Space-time S-matrix and Flux-tube S-matrix IV. Gluons and Fusion
• The four-loop cusp anomalous dimension in N=4 super Yang-Mills and analytic integration techniques for Wilson line integrals
• A Cluster Bootstrap for Two-Loop MHV Amplitudes
• The Amplituhedron from Momentum Twistor Diagrams
• Towards the Amplituhedron Volume
• From dlogs to dilogs; the super Yang-Mills MHV amplitude revisited
• Introduction to the GiNaC Framework for Symbolic Computation within the C++ Programming Language
• From the Holomorphic Wilson Loop to `d log' Loop-Integrands of Super-Yang-Mills Amplitudes
• The Amplituhedron and the One-loop Grassmannian Measure
• Hexagon OPE Resummation and Multi-Regge Kinematics
• Systematics of the cusp anomalous dimension
• A simple construction of Grassmannian polylogarithms
• Positivity Sectors and the Amplituhedron

Cited by in corpus (31)

• PolyLogTools - Polylogs for the masses
• Six-Gluon Amplitudes in Planar $\mathcal{N}=4$ Super-Yang-Mills Theory at Six and Seven Loops
• Unwinding the Amplituhedron in Binary
• The Cosmic Galois Group and Extended Steinmann Relations for Planar $\mathcal{N}=4$ SYM Amplitudes
• Embedding Feynman Integral (Calabi-Yau) Geometries in Weighted Projective Space
• A Three-Point Form Factor Through Five Loops
• Rationalizing Loop Integration
• Sequential Discontinuities of Feynman Integrals and the Monodromy Group
• Bootstrapping a Stress-Tensor Form Factor through Eight Loops
• Cutting Deep Into The Amplituhedron
• Folding Amplitudes into Form Factors: An Antipodal Duality
• Loop-by-loop Differential Equations for Dual (Elliptic) Feynman Integrals
• Nonperturbative Negative Geometries: Amplitudes at Strong Coupling and the Amplituhedron
• All-Mass $n$-gon Integrals in $n$ Dimensions
• The Sklyanin Bracket and Cluster Adjacency at All Multiplicity
• Symbology for elliptic multiple polylogarithms and the symbol prime
• Positive geometry, local triangulations, and the dual of the Amplituhedron
• Yangian Invariants and Cluster Adjacency in N=4 Yang-Mills
• Cluster Algebras and the Subalgebra Constructibility of the Seven-Particle Remainder Function
• Bootstrapping elliptic Feynman integrals using Schubert analysis
• The Two-Loop Remainder Function for Eight and Nine Particles
• Two-Loop Five-Point Two-Mass Planar Integrals and Double Lagrangian Insertions in a Wilson Loop
• Intersection theory rules symbology
• Rooting Out Letters: Octagonal Symbol Alphabets and Algebraic Number Theory
• An Infinite Family of Elliptic Ladder Integrals
• Feynman Integrals from Positivity Constraints
• Conformally Regulated Direct Integration of the Two-Loop Heptagon Remainder
• Poles At Infinity in On-shell Diagrams
• A Two-Loop Four-Point Form Factor at Function Level
• A Novel Algorithm for Nested Summation and Hypergeometric Expansions
• Locally Finite Observables in sYM