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.

33 pages, 1 figure, revised version, typo corrected, accepted in JHEP

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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