From polygons and symbols to polylogarithmic functions

Abstract

We present a review of the symbol map, a mathematical tool that can be useful in simplifying expressions among multiple polylogarithms, and recall its main properties. A recipe is given for how to obtain the symbol of a multiple polylogarithm in terms of the combinatorial properties of an associated rooted decorated polygon. We also outline a systematic approach to constructing a function corresponding to a given symbol, and illustrate it in the particular case of harmonic polylogarithms up to weight four. Furthermore, part of the ambiguity of this process is highlighted by exhibiting a family of non-trivial elements in the kernel of the symbol map for arbitrary weight.

75 pages. Mathematica files with the expression of all HPLs up to weight 4 in terms of the spanning set are included

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• The Three-Loop Splitting Functions in QCD: The Singlet Case
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• Numerical evaluation of multiple polylogarithms
• Two-loop light fermion contribution to Higgs production and decays
• HPL, a Mathematica implementation of the harmonic polylogarithms
• The third-order QCD corrections to deep-inelastic scattering by photon exchange
• Two-Loop Master Integrals for ${\gamma }^{\ast }\to 3$ Jets: The planar topologies
• Bootstrapping the three-loop hexagon
• Numerical Evaluation of Harmonic Polylogarithms
• Massive Feynman diagrams and inverse binomial sums
• Harmonic Sums and Polylogarithms Generated by Cyclotomic Polynomials
• The Two-Loop Hexagon Wilson Loop in N = 4 SYM
• Two-loop amplitudes and master integrals for the production of a Higgs boson via a massive quark and a scalar-quark loop
• An Operator Product Expansion for Polygonal null Wilson Loops
• Analytic Results for Virtual QCD Corrections to Higgs Production and Decay
• Two-Loop Master Integrals for ${\gamma }^{\ast }\to 3$ Jets: The non-planar topologies
• The Longitudinal Structure Function at the Third Order
• New results for the epsilon-expansion of certain one-, two- and three-loop Feynman diagrams
• Superconformal symmetry and two-loop amplitudes in planar N=4 super Yang-Mills
• The massless higher-loop two-point function
• Two-loop QCD Corrections to the Heavy Quark Form Factors: the Vector Contributions
• An Analytic Result for the Two-Loop Hexagon Wilson Loop in N = 4 SYM
• Massive 3-loop Feynman diagrams reducible to SC* primitives of algebras of the sixth root of unity
• On Motives Associated to Graph Polynomials
• Extension of HPL to complex arguments
• Pulling the straps of polygons
• Analytical Result for Dimensionally Regularized Massive On-Shell Planar Double Box
• Two-Loop QCD Corrections to the Heavy Quark Form Factors: Anomaly Contributions
• A geometrical angle on Feynman integrals
• On form factors in N=4 sym
• Master Integrals for the 2-loop QCD virtual corrections to the Forward-Backward Asymmetry
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• Analytical Evaluation of Dimensionally Regularized Massive On-Shell Double Boxes
• Analytic integration of real-virtual counterterms in NNLO jet cross sections I
• More loops and legs in Higgs-regulated N=4 SYM amplitudes
• Master integrals for the two-loop light fermion contributions to $gg\to H$ and $H\to \gamma \gamma$
• The one-loop six-dimensional hexagon integral and its relation to MHV amplitudes in N=4 SYM
• Two-Loop Form Factors in QED
• Two-Loop N_F=1 QED Bhabha Scattering Differential Cross Section
• The massless hexagon integral in D = 6 dimensions
• Two-Loop N_F =1 QED Bhabha Scattering: Soft Emission and Numerical Evaluation of the Differential Cross-section
• The one-loop pentagon to higher orders in epsilon
• Wilson Loops @ 3-Loops in Special Kinematics
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• Higgs Boson Production at Hadron Colliders at N3LO in QCD
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• Three-loop corrections to the soft anomalous dimension in multi-leg scattering
• A Symbol of Uniqueness: The Cluster Bootstrap for the 3-Loop MHV Heptagon
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• Hexagon functions and the three-loop remainder function
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• Two-loop planar master integrals for Higgs$\to 3$ partons with full heavy-quark mass dependence
• Color-Kinematics Duality for QCD Amplitudes
• Pentagon functions for massless planar scattering amplitudes
• Elliptic polylogarithms and iterated integrals on elliptic curves II: an application to the sunrise integral
• Collider Physics at the Precision Frontier
• The two-loop five-point amplitude in $\mathcal{N}=4$ super-Yang-Mills theory
• Analytic Form of the Planar Two-Loop Five-Parton Scattering Amplitudes in QCD
• Six-Gluon Amplitudes in Planar $\mathcal{N}=4$ Super-Yang-Mills Theory at Six and Seven Loops
• All master integrals for three-jet production at NNLO
• Elliptic Feynman integrals and pure functions
• The two-loop helicity amplitudes for $q{\bar{q}}^{\prime }\to V_1{V}_2\to 4\mathrm{leptons}$
• Analytic Form of the Planar Two-Loop Five-Gluon Scattering Amplitudes in QCD
• Higgs boson production at the LHC using the $q_T$ subtraction formalism at N${}^3$LO QCD
• Pentagon Functions for Scattering of Five Massless Particles
• Non-abelian $Z$-theory: Berends-Giele recursion for the ${\alpha }^{\prime }$-expansion of disk integrals
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• The Cosmic Galois Group and Extended Steinmann Relations for Planar $\mathcal{N}=4$ SYM Amplitudes
• Three-mass triangle integrals and single-valued polylogarithms
• Elliptic symbol calculus: from elliptic polylogarithms to iterated integrals of Eisenstein series
• Generalised power series expansions for the elliptic planar families of Higgs + jet production at two loops
• The Two-Loop Master Integrals for $q\bar{q}\to VV$
• A non-planar two-loop three-point function beyond multiple polylogarithms
• On hyperlogarithms and Feynman integrals with divergences and many scales
• Multi-Regge kinematics and the moduli space of Riemann spheres with marked points
• Elliptic polylogarithms and Feynman parameter integrals
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• Two-Loop Integrals for Planar Five-Point One-Mass Processes
• Pentagon Functions for One-Mass Planar Scattering Amplitudes
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• Prescriptive Unitarity
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• Analytic result for the nonplanar hexa-box integrals
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• The two-loop five-point amplitude in $\mathcal{N}=8$ supergravity
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• Analytic results for two-loop master integrals for Bhabha scattering I
• Emergence of Calabi-Yau manifolds in high-precision black hole scattering
• The last of the simple remainders
• Two-loop helicity amplitudes for diphoton plus jet production in full color
• From multiple unitarity cuts to the coproduct of Feynman integrals
• The Double Pentaladder Integral to All Orders
• Non-planar master integrals for the production of two off-shell vector bosons in collisions of massless partons
• Two-Loop Hexa-Box Integrals for Non-Planar Five-Point One-Mass Processes
• Baikov-Lee Representations Of Cut Feynman Integrals
• NNLO phase space master integrals for two-to-one inclusive cross sections in dimensional regularization
• On the reduction of generalized polylogarithms to ${\text{Li}}_n$ and ${\text{Li}}_{2,2}$ and on the evaluation thereof
• All-loop infrared-divergent behavior of most-subleading-color gauge-theory amplitudes
• Next-to-Leading Order QCD Corrections to the Decay Width $H\to Z\gamma$
• Sequential Discontinuities of Feynman Integrals and the Monodromy Group
• Bootstrapping the half-BPS line defect CFT in $\mathcal{N}=4$ SYM at strong coupling
• Light-quark two-loop corrections to heavy-quark pair production in the gluon fusion channel
• The SAGEX Review on Scattering Amplitudes, Chapter 3: Mathematical structures in Feynman integrals
• The two-loop five-particle amplitude in $\mathcal{N}=8$ supergravity
• Simplified differential equations approach for Master Integrals
• Scale dependence and collinear subtraction terms for Higgs production in gluon fusion at N3LO
• Di-photon amplitudes in three-loop Quantum Chromodynamics
• Amplitudes meet Cosmology: A (Scalar) Primer
• Bootstrapping a Stress-Tensor Form Factor through Eight Loops
• Folding Amplitudes into Form Factors: An Antipodal Duality
• An analytic result for the two-loop seven-point MHV amplitude in N=4 SYM
• Algorithms and tools for iterated Eisenstein integrals
• Lifting Heptagon Symbols to Functions
• Star Integrals, Convolutions and Simplices
• Two-loop Octagons, Algebraic Letters and $\bar{Q}$ Equations
• The ice cone family and iterated integrals for Calabi-Yau varieties
• Leading-Color Two-Loop QCD Corrections for Three-Photon Production at Hadron Colliders
• Four-graviton scattering to three loops in $\mathcal{N}=8$ supergravity
• Analytic Computing Methods for Precision Calculations in Quantum Field Theory
• Evaluating two-loop non-planar master integrals for Higgs + jet production with full heavy-quark mass dependence
• The double-soft integral for an arbitrary angle between hard radiators
• Towards a new approximation for pair-production and associated-production of the Higgs boson
• Bootstrapping two-loop Feynman integrals for planar N=4 sYM
• From Webs to Polylogarithms
• Two-loop QCD helicity amplitudes for $gg\to Zg$ and $gg\to Z\gamma$
• Logarithmic forms and differential equations for Feynman integrals
• Three-loop helicity amplitudes for diphoton production in gluon fusion
• N-jettiness beam functions at N3LO
• MPL - a program for computations with iterated integrals on moduli spaces of curves of genus zero
• Massive planar and non-planar double box integrals for light Nf contributions to gg->tt
• Two-loop Master Integrals with the Simplified Differential Equations approach
• On spinning loop amplitudes in Anti-de Sitter space
• The complete set of two-loop master integrals for Higgs + jet production in QCD
• Two-Loop Master Integrals for $q\bar{q}\to VV$: the Planar Topologies
• IBP reduction coefficients made simple
• Antipodal Self-Duality for a Four-Particle Form Factor
• Notes on cluster algebras and some all-loop Feynman integrals
• Locally finite two-loop amplitudes for off-shell multi-photon production in electron-positron annihilation
• The symbol and alphabet of two-loop NMHV amplitudes from $\bar{Q}$ equations
• Two-loop master integrals for a planar topology contributing to $pp\to t\bar{t}j$
• The BFKL equation, Mueller-Navelet jets and single-valued harmonic polylogarithms
• Truncated cluster algebras and Feynman integrals with algebraic letters
• The Complete Two-Loop Integrated Jet Thrust Distribution In Soft-Collinear Effective Theory
• Multi-Regge Limit of the Two-Loop Five-Point Amplitudes in $\mathcal{N}=4$ Super Yang-Mills and $\mathcal{N}=8$ Supergravity
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• Numerical Multi-Loop Calculations via Finite Integrals and One-Mass EW-QCD Drell-Yan Master Integrals
• Amplitudes from superconformal Ward identities
• The six-point remainder function to all loop orders in the multi-Regge limit
• Planar three-loop master integrals for $2\to 2$ processes with one external massive particle
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• Two-loop mixed QCD-EW corrections to $gg\to Hg$
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• The SAGEX Review on Scattering Amplitudes, Chapter 4: Multi-loop Feynman Integrals
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• An Infinite Family of Elliptic Ladder Integrals
• The recursive structure of Baikov representations I: generics and application to symbology
• Towards Analytic Structure of Feynman Parameter Integrals with Rational Curves
• Comments on all-loop constraints for scattering amplitudes and Feynman integrals
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