Iteration of Planar Amplitudes in Maximally Supersymmetric Yang-Mills Theory at Three Loops and Beyond

Abstract

We compute the leading-color (planar) three-loop four-point amplitude of N=4 supersymmetric Yang-Mills theory in 4 - 2 epsilon dimensions, as a Laurent expansion about epsilon = 0 including the finite terms. The amplitude was constructed previously via the unitarity method, in terms of two Feynman loop integrals, one of which has been evaluated already. Here we use the Mellin-Barnes integration technique to evaluate the Laurent expansion of the second integral. Strikingly, the amplitude is expressible, through the finite terms, in terms of the corresponding one- and two-loop amplitudes, which provides strong evidence for a previous conjecture that higher-loop planar N = 4 amplitudes have an iterative structure. The infrared singularities of the amplitude agree with the predictions of Sterman and Tejeda-Yeomans based on resummation. Based on the four-point result and the exponentiation of infrared singularities, we give an exponentiated ansatz for the maximally helicity-violating n-point amplitudes to all loop orders. The 1/epsilon^2 pole in the four-point amplitude determines the soft, or cusp, anomalous dimension at three loops in N = 4 supersymmetric Yang-Mills theory. The result confirms a prediction by Kotikov, Lipatov, Onishchenko and Velizhanin, which utilizes the leading-twist anomalous dimensions in QCD computed by Moch, Vermaseren and Vogt. Following similar logic, we are able to predict a term in the three-loop quark and gluon form factors in QCD.

54 pages, 7 figures. v2: Added references, a few additional words about large spin limit of anomalous dimensions. v3: Expanded Sect. IV.A on multiloop ansatz; remark that form-factor prediction is now confirmed by other work; minor typos corrected

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• Three loop form factors of a massive spin-2 particle with nonuniversal coupling
• Collinear and Soft Limits of Multi-Loop Integrands in N=4 Yang-Mills
• Higher point MHV amplitudes in N=4 Supersymmetric Yang-Mills Theory
• Four-point amplitudes in $\mathcal{N}=2$ SCQCD
• All two-loop MHV remainder functions in multi-Regge kinematics
• The two-loop eight-point amplitude in ABJM theory
• The SAGEX Review on Scattering Amplitudes, Chapter 5: Analytic Bootstraps for Scattering Amplitudes and Beyond
• Celestial IR divergences in general most-subleading-color gluon and gravity amplitudes
• Non-planar cusp and transcendental anomalous dimension at four loops in N=4 supersymmetric Yang-Mills theory
• Non-planar BCFW Grassmannian Geometries
• Conformally Regulated Direct Integration of the Two-Loop Heptagon Remainder
• Infrared Safe Observables in $\mathcal{N}=4$ Super Yang-Mills Theory
• Pohlmeyer reduction and Darboux transformations in Euclidean worldsheet AdS_3
• Iterating QCD scattering amplitudes in the high-energy limit
• Moebius invariant BFKL equation for the adjoint representation in N=4 SUSY
• Classical Worldsheets for String Scattering on Flat and AdS Spacetime
• Gluon self-interaction in the position space in Landau gauge
• Maximal transcendental weight contribution of scattering amplitudes
• Discretized Minimal Surface and the BDS Conjecture in N=4 Super Yang-Mills Theory at Strong Coupling
• Loop lessons from Wilson loops in N=4 supersymmetric Yang-Mills theory
• Review of AdS/CFT Integrability, Chapter V.3: Scattering Amplitudes at Strong Coupling
• Further results for the two-loop Lcc vertex in the Landau gauge
• One-loop N = 4 super Yang-Mills scattering amplitudes in d dimensions, relation to open strings and polygonal Wilson loops
• On the Effective Action of Dressed Mean Fields for N = 4 Super-Yang-Mills Theory
• Soft interaction at high energy and N=4 SYM
• Belokurov-Usyukina loop reduction in non-integer dimension
• Two-fold Mellin-Barnes transforms of Usyukina-Davydychev functions
• Yangian Symmetry in N=4 super Yang-Mills
• An Origin Story for Amplitudes
• D-instantons and multiparticle production in N=4 SYM
• TBA-like equations for non-planar scattering amplitude/Wilson lines duality at strong coupling
• Amplitudes in the N=4 SYM from Quantum Geometry of the Momentum Space
• One-loop derivation of the Wilson polygon - MHV amplitude duality
• Some recent results on evaluating Feynman integrals
• "Anomaly" in n=infinity Alday-Maldacena Duality for Wavy Circle
• The five-gluon amplitude in the high-energy limit
• On a residual freedom of the next-to-leading BFKL eigenvalue in color adjoint representation in planar $\mathcal{N}=4$ SYM
• Massive quark scattering at strong coupling from AdS/CFT
• Amplitudes in fishnet theories in diverse dimensions and Box ladder diagrams
• Scattering amplitudes on the Coulomb branch of N=4 super Yang-Mills
• Star product and the general Leigh-Strassler deformation
• Supersymmetric Yang-Mills and Supergravity Amplitudes at One Loop
• MHV amplitudes at strong coupling and linearized TBA equations
• Anatomy of the Amplituhedron
• A nice two-loop next-to-next-to-MHV amplitude in $\mathcal{N}=4$ super-Yang-Mills
• Deviation from Alday-Maldacena Duality For Wavy Circle
• Ironing out the crease
• 6d Dual Conformal Symmetry and Minimal Volumes in AdS
• Applications of Subleading-color Amplitudes in N=4 SYM Theory
• Regge meets collinear in strongly-coupled $\mathcal{N}=4$ super Yang-Mills
• Antipodal Symmetry of Two-Loop MHV Amplitudes
• Dimensional Regularization and Dimensional Reduction in the Light Cone
• A note on two-loop superloop
• The virtual scaling function of twist operators in the N=6 Chern-Simons theory
• Deciphering the Maximal Transcendentality Principle via Bootstrap
• Permutation Relations of Yangian Invariants, Unitarity Cuts, and Scattering Amplitudes
• Heptagon Functions and Seven-Gluon Amplitudes in Multi-Regge Kinematics
• Polylogarithm identities, cluster algebras and the N=4 supersymmetric theory
• "Background Field Integration-by-Parts" and the Connection Between One-Loop and Two-Loop Heisenberg-Euler Effective Actions
• Box ladders in non-integer dimension
• 4-particle Amplituhedron at 3-loop and its Mondrian Diagrammatic Implication
• Second order splitting functions and infrared safe cross sections in $\mathcal{N}=4$ SYM theory
• One-Loop N = 4 Super Yang-Mills Scattering Amplitudes to All Orders in the Dimensional Regularization Parameter
• BFKL Eigenvalue and Maximal Alternation of Harmonic Sums
• A Note on One-loop Cluster Adjacency in N = 4 SYM
• The Curious Case of Leading Transcendentality: Three Point Form Factors
• A high-precision result for a full-color three-loop three-point form factor in $\mathcal{N}=4$ SYM
• Regge poles and cuts and the Lipatov vertex
• The Steinmann Cluster Bootstrap for N=4 Super Yang-Mills Amplitudes
• QCD-like properties for anomalous dimensions in N=4 SYM
• Automated calculations for multi-leg processes
• Perturbative Gravity and Twistor Space
• Logarithmic Singularities and Maximally Supersymmetric Amplitudes
• On the two-loop divergences in 6D, $\mathcal{N}=(1,1)$ SYM theory
• All-loop-orders relation between Regge limits of $\mathcal{N}=4$ SYM and $\mathcal{N}=8$ supergravity four-point amplitudes
• Universal structure of subleading infrared poles at strong coupling
• The Multi-Regge Limit from the Wilson Loop OPE
• Quantum Wronskian approach to six-point gluon scattering amplitudes at strong coupling
• ABJM flux-tube and scattering amplitudes
• Nonplanar On-shell Diagrams and Leading Singularities of Scattering Amplitudes
• UV Divergences, RG Equations and High Energy Behaviour of the Amplitudes in the Wess-Zumino Model with Quartic Interaction
• Locally Finite Observables in sYM
• Explicit calculation of multi-fold contour integrals of certain ratios of Euler gamma functions. Part 1
• The scalar box integral and the Mellin - Barnes representation
• Infrared and transcendental structure of two-loop supersymmetric QCD amplitudes
• One-loop pentagon integral with one offshell leg in $d$ dimensions from differential equations in $\epsilon$-form
• Analytic two-Loop four-point form factor of the stress-tensor supermultiplet in $\mathcal{N}=4$ SYM
• Twistors, Harmonics and Holomorphic Chern-Simons
• Holonomies of gauge fields in twistor space 4: functional MHV rules and one-loop amplitudes
• A Note on Loop Amplitudes in QED
• On two-loop divergences of effective action in $6D$, $\mathcal{N}=(1,1)$ SYM theory
• Supersums for all supersymmetric amplitudes
• Grassmannian Integrals in Minkowski Signature, Amplitudes, and Integrability
• Asymptotic AdS String Solutions for Null Polygonal Wilson Loops in R^{1,2}
• Conformal geometry of null hexagons for Wilson loops and scattering amplitudes
• Perturbative Methods for Superconformal Quantum Field Theories in String - Gauge Theory Dualities
• Infrared Finite Observables in N=8 Supergravity
• Non-Perturbative Four-Point Scattering from First-Quantized Relativistic JWKB
• Loop Amplitudes in the Coulomb Branch of $\mathcal{N}=4$ Super-Yang-Mills Theory
• On perturbative field theory and twistor string theory
• All-order results for soft and collinear gluons
• Symmetric reduction of high-multiplicity one-loop integrals and maximal cuts
• Higher-dimensional invariants in $6D$ super Yang-Mills theory
• Exploring Reggeon bound states in strongly-coupled $\mathcal{N}=4$ super Yang-Mills
• Sudakov Resummations at Higher Orders
• Duality symmetry in high energy scattering
• Gauge-String Dualities and Some Applications
• Divergences in maximal supersymmetric Yang-Mills theories in diverse dimensions
• MHV amplitudes, Wilson loops and the BDS ansatz in the superconformal N=2 QCD
• Analytic Four-Point Lightlike Form Factors and OPE of Null-Wrapped Polygons
• Wilson loops as defects in N=2 conformal field theories
• Theoretical Developments in SUSY
• On-shell recursion relations for generic integrands
• Coulomb Phase Gluon Scattering at Strong Coupling
• High Energy Behaviour in Maximally Supersymmetric Gauge Theories in Various Dimensions
• Topics in Cusped/Lightcone Wilson Loops
• On Solving Dual Conformal Integrals in Coulomb-branch Amplitudes and Their Periods
• A tale of two Regge limits
• Wilson Loops and Integrability
• Dual Conformal Symmetry and Iterative Integrals in Six Dimensions
• Eikonal Approach to N=4 SYM Regge Amplitudes in the AdS/CFT Correspondence
• Discontinuites of BFKL amplitudes and the BDS ansatz
• Effective vertex of quark production in collision of Reggeized quark and gluon
• Refining threshold resummations
• 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
• Dilogarithm ladders from Wilson loops
• A short review on TBA equation and scattering amplitude/Wilson loop duality
• On interpolating anomalous dimension of twist-two operators with general spins
• Off- and on-shell harmonic superspaces for 6D SYM theories
• Hexagon Bootstrap in the Double Scaling Limit
• A note on scattering amplitudes on the moduli space of ABJM
• Gluon scattering amplitudes from gauge/string duality and integrability
• On the Integrable Structure of Super Yang-Mills Scattering Amplitudes
• The canonical form has got baubles
• Superstring in the pp-wave background with RR flux as a conformal field theory