The last of the simple remainders

Abstract

We compute the n-point two-loop form factors of the half-BPS operators Tr(phi_{AB}^n) in N=4 super Yang-Mills for arbitrary n >2 using generalised unitarity and symbols. These form factors are minimal in the sense that the n^{th} power of the scalar field in the operator requires the presence of at least n on-shell legs. Infrared divergences are shown to exponentiate as for amplitudes, reproducing the known cusp and collinear anomalous dimensions at two loops. We define appropriate infrared-finite remainder functions and compute them analytically for all n. The results obtained by using the known expressions of the integral functions involve complicated combinations of Goncharov multiple polylogarithms, but we show that much simpler expressions can in fact be derived using the symbol of transcendental functions. For n=3 we find a very compact remainder function expressed in terms of classical polylogarithms only. For arbitrary n>3 we are able to write all the remainder functions in terms of a single compact building block, expressed as a sum of classical polylogarithms augmented by two multiple polylogarithms. The decomposition of the symbol into specific components is crucial in order to single out a natural combination of multiple polylogarithms. Finally, we analyse in detail the behaviour of these minimal form factors in collinear and soft limits, which deviates from the usual behaviour of amplitudes and non-minimal form factors.

arXiv comment31 pages, LaTeX

References in corpus (34)

• One-Loop n-Point Gauge Theory Amplitudes, Unitarity and Collinear Limits
• Differential Equations for Two-Loop Four-Point Functions
• Iteration of Planar Amplitudes in Maximally Supersymmetric Yang-Mills Theory at Three Loops and Beyond
• Classical Polylogarithms for Amplitudes and Wilson Loops
• Algorithm FIRE -- Feynman Integral REduction
• Hexagon Wilson loop = six-gluon MHV amplitude
• Planar Amplitudes in Maximally Supersymmetric Yang-Mills Theory
• The Two-Loop Six-Gluon MHV Amplitude in Maximally Supersymmetric Yang-Mills Theory
• From polygons and symbols to polylogarithmic functions
• Two-Loop Master Integrals for ${\gamma }^{\ast }\to 3$ Jets: The planar topologies
• Bootstrapping the three-loop hexagon
• One-Loop N Gluon Amplitudes with Maximal Helicity Violation via Collinear Limits
• The Two-Loop Hexagon Wilson Loop in N = 4 SYM
• Analytic result for the two-loop six-point NMHV amplitude in N=4 super Yang-Mills theory
• All-Order Collinear Behavior in Gauge Theories
• Form factors at strong coupling via a Y-system
• The four-loop remainder function and multi-Regge behavior at NNLLA in planar N=4 super-Yang-Mills theory
• The super-correlator/super-amplitude duality: Part I
• Color-kinematic Duality for Form Factors
• Pulling the straps of polygons
• Analytic two-loop form factors in N=4 SYM
• Form Factors in N=4 Super Yang-Mills and Periodic Wilson Loops
• Harmony of Super Form Factors
• Two-Loop Polygon Wilson Loops in N=4 SYM
• On form factors in N=4 sym
• The three-loop form factor in N=4 super Yang-Mills
• The super-correlator/super-amplitude duality: Part II
• On MHV Form Factors in Superspace for $\mathcal{N}=4$ SYM Theory
• Y-system for form factors at strong coupling in AdS5 and with multi-operator insertions in AdS3
• On super form factors of half-BPS operators in N=4 super Yang-Mills
• Introduction to the GiNaC Framework for Symbolic Computation within the C++ Programming Language
• Wilson Loops @ 3-Loops in Special Kinematics
• All Two-Loop MHV Amplitudes in Multi-Regge Kinematics From Applied Symbology
• Cluster Polylogarithms for Scattering Amplitudes

Cited by in corpus (50)

• Elliptic Feynman integrals and pure functions
• A Three-Point Form Factor Through Five Loops
• Cutting through form factors and cross sections of non-protected operators in N=4 SYM
• The Sudakov form factor at four loops in maximal super Yang-Mills theory
• On-shell Diagrams, Graßmannians and Integrability for Form Factors
• Amplitudes, Form Factors and the Dilatation Operator in $\mathcal{N}=4$ SYM Theory
• On-Shell Methods for the Two-Loop Dilatation Operator and Finite Remainders
• Master integrals for the four-loop Sudakov form factor
• The SU(2|3) dynamic two-loop form factors
• $\text{Tr}(F^3)$ supersymmetric form factors and maximal transcendentality Part I: $\mathcal{N}=4$ super Yang-Mills
• Higgs amplitudes from $\mathcal{N}=4$ super Yang-Mills theory
• Two-Loop SL(2) Form Factors and Maximal Transcendentality
• On Form Factors in N=4 SYM Theory and Polytopes
• An Operator Product Expansion for Form Factors
• Composite Operators in the Twistor Formulation of $\mathcal{N}=4$ SYM Theory
• All Tree-Level MHV Form Factors in $\mathcal{N}=4$ SYM from Twistor Space
• Analytic Two-Loop Higgs Amplitudes in Effective Field Theory and the Maximal Transcendentality Principle
• The SAGEX Review on Scattering Amplitudes, Chapter 8: Half BPS correlators
• Non-planar form factors of generic local operators via on-shell unitarity and color-kinematics duality
• A note on connected formula for form factors
• Full-color three-loop three-point form factors in N=4 SYM
• $\text{Tr}(F^3)$ supersymmetric form factors and maximal transcendentality Part II: $0<\mathcal{N}<4$ super Yang-Mills
• Dual conformal invariance for form factors
• Finite remainders of the Konishi at two loops in $\mathcal{N}=4$ SYM
• Form factor recursion relations at loop level
• On Form Factors and Correlation Functions in Twistor Space
• Bootstrapping a Two-Loop Four-Point Form Factor
• Grassmannians and form factors with $q^2=0$ in N=4 SYM theory
• Wilson lines, Grassmannians and Gauge Invariant Off-shell Amplitudes in N=4 SYM
• Cross-Order Integral Relations from Maximal Cuts
• Two-Loop anomalous dimensions of QCD operators up to dimension-sixteen and Higgs EFT amplitudes
• An Operator Product Expansion for Form Factors III. Finite Coupling and Multi-Particle Contributions
• Hidden Analytic Relations for Two-Loop Higgs Amplitudes in QCD
• Two-Loop QCD Corrections to the Higgs plus three-parton amplitudes with Top Mass Correction
• The Full-Color Two-Loop Four-Gluon Amplitude in $\mathcal{N}=2$ Super-QCD
• Form Factor and Boundary Contribution of Amplitude
• Wilson Loop Duality and OPE for Super Form Factors of Half-BPS Operators
• Complete Form Factors in Yang-Mills from Unitarity and Spinor Helicity in Six Dimensions
• The all-loop conjecture for integrands of reggeon amplitudes in N=4 SYM
• Grassmannian Integral for General Gauge Invariant Off-shell Amplitudes in N=4 SYM
• On Yangian symmetry of scattering amplitudes and the dilatation operator in N=4 super Yang-Mills
• Deciphering the Maximal Transcendentality Principle via Bootstrap
• The Curious Case of Leading Transcendentality: Three Point Form Factors
• Form factors from string amplitudes
• New relations for tree-level form factors and scattering amplitudes
• Analytic two-Loop four-point form factor of the stress-tensor supermultiplet in $\mathcal{N}=4$ SYM
• Analytic Four-Point Lightlike Form Factors and OPE of Null-Wrapped Polygons
• An Operator Product Expansion for Form Factors II. Born level
• Three-loop color-kinematics duality: A 24-dimensional solution space induced by new generalized gauge transformations
• On-shell Methods for Form Factors in N=4 SYM and Their Applications