The symbol and alphabet of two-loop NMHV amplitudes from $\bar{Q}$ equations

Abstract

We study the symbol and the alphabet for two-loop NMHV amplitudes in planar $\mathcal{N}=4$ super-Yang-Mills from the $\bar{Q}$ equations, which provide a first-principle method for computing multi-loop amplitudes. Starting from one-loop N${}^2$MHV ratio functions, we explain in detail how to use $\bar{Q}$ equations to obtain the total differential of two-loop $n$-point NMHV amplitudes, whose symbol contains letters that are algebraic functions of kinematics for $n\ge 8$. We present explicit formula with nice patterns for the part of the symbol involving algebraic letters for all multiplicities, and we find $17-2m$ multiplicative-independent letters for a given square root of Gram determinant, with $0\le m\le 4$ depending on the number of particles involved in the square root. We also observe that these algebraic letters can be found as poles of one-loop four-mass leading singularities with MHV or NMHV trees. As a byproduct of our algebraic results, we find a large class of components of two-loop NMHV, which can be written as differences of two double-pentagon integrals, particularly simple and absent of square roots. As an example, we present the complete symbol for $n=9$ whose alphabet contains $59\times 9$ rational letters, in addition to the $11\times 9$ independent algebraic ones. We also give all-loop NMHV last-entry conditions for all multiplicities.

arXiv comment33 pages, 1 figure, revised version, typo corrected, accepted in JHEP

References in corpus (56)

• Review of AdS/CFT Integrability: An Overview
• Iteration of Planar Amplitudes in Maximally Supersymmetric Yang-Mills Theory at Three Loops and Beyond
• Classical Polylogarithms for Amplitudes and Wilson Loops
• Dimensionally Regulated Pentagon Integrals
• 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
• Algorithms for the symbolic integration of hyperlogarithms with applications to Feynman integrals
• 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
• Thermodynamic Bubble Ansatz
• Scattering into the fifth dimension of N=4 super Yang-Mills
• 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
• Notes on the scattering amplitude / Wilson loop duality
• Jumpstarting the all-loop S-matrix of planar N=4 super Yang-Mills
• 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
• Unwinding the Amplituhedron in Binary
• 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
• Yangians, Grassmannians and T-duality
• 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
• RationalizeRoots: Software Package for the Rationalization of Square Roots
• Cluster adjacency beyond MHV
• Rationalizing Loop Integration
• How tropical are seven- and eight-particle amplitudes?
• Landau Singularities and Symbology: One- and Two-loop MHV Amplitudes in SYM Theory
• Lifting Heptagon Symbols to Functions
• Two-loop Octagons, Algebraic Letters and $\bar{Q}$ Equations
• All-loop singularities of scattering amplitudes in massless planar theories
• Bootstrapping two-loop Feynman integrals for planar N=4 sYM
• Logarithmic forms and differential equations for Feynman integrals
• Notes on Scattering Amplitudes as Differential Forms
• Boundaries of Amplituhedra and NMHV Symbol Alphabets at Two Loops
• Symbol Alphabets from Plabic Graphs
• Three-loop octagons and n-gons in maximally supersymmetric Yang-Mills theory
• Yangian Invariants and Cluster Adjacency in N=4 Yang-Mills
• A Note on Letters of Yangian Invariants
• Rooting Out Letters: Octagonal Symbol Alphabets and Algebraic Number Theory

Cited by in corpus (26)

• Implications of the Landau Equations for Iterated Integrals
• Singularities of eight- and nine-particle amplitudes from cluster algebras and tropical geometry
• The elliptic double box and symbology beyond polylogarithms
• Notes on cluster algebras and some all-loop Feynman integrals
• Truncated cluster algebras and Feynman integrals with algebraic letters
• Symbol Alphabets from Tensor Diagrams
• Symbology for elliptic multiple polylogarithms and the symbol prime
• Constraints on Sequential Discontinuities from the Geometry of On-shell Spaces
• Schubert Problems, Positivity and Symbol Letters
• Algebraic branch points at all loop orders from positive kinematics and wall crossing
• Feynman Integrals and Scattering Amplitudes from Wilson Loops
• Bootstrapping elliptic Feynman integrals using Schubert analysis
• The Wilson-loop $d\log$ representation for Feynman integrals
• Bootstrapping a Two-Loop Four-Point Form Factor
• The Two-Loop Remainder Function for Eight and Nine Particles
• Symbol Alphabets from Plabic Graphs II: Rational Letters
• The Three-loop MHV Octagon from $\bar{Q}$ equations
• The recursive structure of Baikov representations I: generics and application to symbology
• Comments on all-loop constraints for scattering amplitudes and Feynman integrals
• The SAGEX Review on Scattering Amplitudes, Chapter 5: Analytic Bootstraps for Scattering Amplitudes and Beyond
• The two-loop eight-point amplitude in ABJM theory
• 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