Two-loop Octagons, Algebraic Letters and $\bar{Q}$ Equations

Abstract

We compute the symbol of the first two-loop amplitudes in planar $\mathcal{N}=4$ SYM with algebraic letters, the eight-point NMHV amplitude (or the dual octagon Wilson loops). We show how to apply $\bar{Q}$ equations for computing the differential of two-loop $n$-point NMHV amplitudes and present the result for n=8 explicitly. The symbol alphabet for octagon consists of 180 independent rational letters and 18 algebraic ones involving Gram-determinant square roots. We comment on all-loop predictions for final entries and aspects of the result valid for all multiplicities.

arXiv comment7 pages, 1 figure, 3 ancillary files in a 7z file

References in corpus (51)

• Review of AdS/CFT Integrability: An Overview
• 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
• Magic identities for conformal four-point integrals
• The All-Loop Integrand For Scattering Amplitudes in Planar N=4 SYM
• Local Integrals for Planar Scattering Amplitudes
• Yangian symmetry of scattering amplitudes in N=4 super Yang-Mills theory
• Planar Amplitudes in Maximally Supersymmetric Yang-Mills Theory
• Dual Superconformal Invariance, Momentum Twistors and Grassmannians
• From polygons and symbols to polylogarithmic functions
• Space-time S-matrix and Flux-tube S-matrix at Finite Coupling
• Bootstrapping the three-loop hexagon
• PolyLogTools - Polylogs for the masses
• Hopf algebras, coproducts and symbols: an application to Higgs boson amplitudes
• Motivic Amplitudes and Cluster Coordinates
• Bootstrapping a Five-Loop Amplitude Using Steinmann Relations
• Analytic result for the two-loop six-point NMHV amplitude in N=4 super Yang-Mills theory
• The Grassmannian Origin Of Dual Superconformal Invariance
• Bootstrapping an NMHV amplitude through three loops
• Superconformal symmetry and two-loop amplitudes in planar N=4 super Yang-Mills
• A Symbol of Uniqueness: The Cluster Bootstrap for the 3-Loop MHV Heptagon
• Heptagons from the Steinmann Cluster Bootstrap
• Jumpstarting the all-loop S-matrix of planar N=4 super Yang-Mills
• 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
• Six-Gluon Amplitudes in Planar $\mathcal{N}=4$ Super-Yang-Mills Theory at Six and Seven Loops
• The Elliptic Double-Box Integral: Massless Amplitudes Beyond Polylogarithms
• Traintracks Through Calabi-Yaus: Amplitudes Beyond Elliptic Polylogarithms
• Cluster adjacency properties of scattering amplitudes
• The Cosmic Galois Group and Extended Steinmann Relations for Planar $\mathcal{N}=4$ SYM Amplitudes
• The Yangian origin of the Grassmannian integral
• Cluster adjacency and the four-loop NMHV heptagon
• Superconformal invariants for scattering amplitudes in N=4 SYM theory
• The Double Pentaladder Integral to All Orders
• Rationalizing Loop Integration
• Cluster adjacency beyond MHV
• Scattering Equations: From Projective Spaces to Tropical Grassmannians
• Bootstrapping two-loop Feynman integrals for planar N=4 sYM
• Boundaries of Amplituhedra and NMHV Symbol Alphabets at Two Loops
• Amplitudes from superconformal Ward identities
• The Sklyanin Bracket and Cluster Adjacency at All Multiplicity
• All-Helicity Symbol Alphabets from Unwound Amplituhedra
• Three-loop octagons and n-gons in maximally supersymmetric Yang-Mills theory
• Cluster Algebras and the Subalgebra Constructibility of the Seven-Particle Remainder Function
• Yangian Invariants and Cluster Adjacency in N=4 Yang-Mills
• Rooting Out Letters: Octagonal Symbol Alphabets and Algebraic Number Theory
• Amplitudes from anomalous superconformal symmetry

Cited by in corpus (32)

• Cluster algebras for Feynman integrals
• How tropical are seven- and eight-particle amplitudes?
• Implications of the Landau Equations for Iterated Integrals
• Loop-by-loop Differential Equations for Dual (Elliptic) Feynman Integrals
• Singularities of eight- and nine-particle amplitudes from cluster algebras and tropical geometry
• The elliptic double box and symbology beyond polylogarithms
• Tropical fans, scattering equations and amplitudes
• The symbol and alphabet of two-loop NMHV amplitudes from $\bar{Q}$ equations
• Symbol Alphabets from Plabic Graphs
• Notes on cluster algebras and some all-loop Feynman integrals
• Truncated cluster algebras and Feynman integrals with algebraic letters
• Symbology for elliptic multiple polylogarithms and the symbol prime
• Symbol Alphabets from Tensor Diagrams
• Schubert Problems, Positivity and Symbol Letters
• Algebraic branch points at all loop orders from positive kinematics and wall crossing
• A study of Feynman integrals with uniform transcendental weights and the symbology from dual conformal symmetry
• A Note on Letters of Yangian Invariants
• Feynman Integrals and Scattering Amplitudes from Wilson Loops
• Bootstrapping a Two-Loop Four-Point Form Factor
• The Wilson-loop $d\log$ representation for Feynman integrals
• The Two-Loop Remainder Function for Eight and Nine Particles
• Symbol Alphabets from Plabic Graphs II: Rational Letters
• Notes on polytopes, amplitudes and boundary configurations for Grassmannian string integrals
• The Three-loop MHV Octagon from $\bar{Q}$ equations
• Comments on all-loop constraints for scattering amplitudes and Feynman integrals
• The two-loop eight-point amplitude in ABJM theory
• The SAGEX Review on Scattering Amplitudes, Chapter 5: Analytic Bootstraps for Scattering Amplitudes and Beyond
• Bootstrapping octagons in reduced kinematics from $A_2$ cluster algebras
• A nice two-loop next-to-next-to-MHV amplitude in $\mathcal{N}=4$ super-Yang-Mills
• Deciphering the Maximal Transcendentality Principle via Bootstrap
• Exploring Reggeon bound states in strongly-coupled $\mathcal{N}=4$ super Yang-Mills
• Analytic Four-Point Lightlike Form Factors and OPE of Null-Wrapped Polygons