The SAGEX Review on Scattering Amplitudes, Chapter 3: Mathematical structures in Feynman integrals

Abstract

Dimensionally-regulated Feynman integrals are a cornerstone of all perturbative computations in quantum field theory. They are known to exhibit a rich mathematical structure, which has led to the development of powerful new techniques for their computation. We review some of the most recent advances in our understanding of the analytic structure of multiloop Feynman integrals in dimensional regularisation. In particular, we give an overview of modern approaches to computing Feynman integrals using differential equations, and we discuss some of the properties of the functions that appear in the solutions. We then review how dimensional regularisation has a natural mathematical interpretation in terms of the theory of twisted cohomology groups, and how many of the well-known ideas about Feynman integrals arise naturally in this context. This is Chapter 3 of a series of review articles on scattering amplitudes, of which Chapter 0 [arXiv:2203.13011] presents an overview and Chapter 4 [arXiv:2203.13015] contains closely related topics.

arXiv comment62 pages, see also the overview article arXiv:2203.13011. v3: journal version

References in corpus (173)

• The total top quark pair production cross-section at hadron colliders through O(alpha_S^4)
• Harmonic Polylogarithms
• Higgs boson production at hadron colliders in NNLO QCD
• Multiloop integrals in dimensional regularization made simple
• Asymptotic expansion of Feynman integrals near threshold
• Differential Equations for Two-Loop Four-Point Functions
• Classical Polylogarithms for Amplitudes and Wilson Loops
• LiteRed 1.4: a powerful tool for the reduction of the multiloop integrals
• Connection between Feynman integrals having different values of the space-time dimension
• Numerical evaluation of multiple polylogarithms
• Local Integrals for Planar Scattering Amplitudes
• Algorithms for the symbolic integration of hyperlogarithms with applications to Feynman integrals
• FIRE6: Feynman Integral REduction with Modular Arithmetic
• Integral Reduction with Kira 2.0 and Finite Field Methods
• Automatic Integral Reduction for Higher Order Perturbative Calculations
• Lectures on differential equations for Feynman integrals
• Dilepton rapidity distribution in the Drell-Yan process at NNLO in QCD
• A novel approach to integration by parts reduction
• From polygons and symbols to polylogarithmic functions
• Two-Loop Master Integrals for ${\gamma }^{\ast }\to 3$ Jets: The planar topologies
• PolyLogTools - Polylogs for the masses
• Hopf algebras, coproducts and symbols: an application to Higgs boson amplitudes
• Scattering amplitudes over finite fields and multivariate functional reconstruction
• The elliptic dilogarithm for the sunset graph
• FiniteFlow: multivariate functional reconstruction using finite fields and dataflow graphs
• Numerical Evaluation of Harmonic Polylogarithms
• Analytic treatment of the two loop equal mass sunrise graph
• Harmonic Sums and Polylogarithms Generated by Cyclotomic Polynomials
• Higgs Boson Production at Hadron Colliders at N3LO in QCD
• Towards a Basis for Planar Two-Loop Integrals
• Two-loop Integrand Decomposition into Master Integrals and Surface Terms
• Integration-by-parts reductions from unitarity cuts and algebraic geometry
• Elliptic multiple zeta values and one-loop superstring amplitudes
• Space-time dimensionality D as complex variable: calculating loop integrals using dimensional recurrence relation and analytical properties with respect to D
• Soft triple-real radiation for Higgs production at N3LO
• Uniqueness of two-loop master contours
• Motivic Multiple Zeta Values and Superstring Amplitudes
• The two-loop sunrise graph in two space-time dimensions with arbitrary masses in terms of elliptic dilogarithms
• The massless higher-loop two-point function
• Numerical Evaluation of Two-Dimensional Harmonic Polylogarithms
• Feynman Integrals and Intersection Theory
• Analytical and numerical methods for massive two-loop self-energy diagrams
• Analytic and Algorithmic Aspects of Generalized Harmonic Sums and Polylogarithms
• Hexagon functions and the three-loop remainder function
• Positive Geometries and Canonical Forms
• On the maximal cut of Feynman integrals and the solution of their differential equations
• Differential equations and dispersion relations for Feynman amplitudes. The two-loop massive sunrise and the kite integral
• Vector Space of Feynman Integrals and Multivariate Intersection Numbers
• Algorithmic transformation of multi-loop master integrals to a canonical basis with CANONICA
• Scattering Amplitudes from Intersection Theory
• Pentagon functions for massless planar scattering amplitudes
• Fuchsia: a tool for reducing differential equations for Feynman master integrals to epsilon form
• Structural Relations of Harmonic Sums and Mellin Transforms up to Weight w = 5
• A Computer Algebra Toolbox for Harmonic Sums Related to Particle Physics
• Elliptic polylogarithms and iterated integrals on elliptic curves II: an application to the sunrise integral
• Geometric approach to asymptotic expansion of Feynman integrals
• A Feynman integral via higher normal functions
• Critical points and number of master integrals
• Pulling the straps of polygons
• 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
• Maximal cuts and differential equations for Feynman integrals. An application to the three-loop massive banana graph
• Cuts of Feynman Integrals in Baikov representation
• Combinatorics and Topology of Kawai-Lewellen-Tye Relations
• All master integrals for three-jet production at NNLO
• Elliptic Feynman integrals and pure functions
• The kite integral to all orders in terms of elliptic polylogarithms
• The $\epsilon$-form of the differential equations for Feynman integrals in the elliptic case
• A quasi-finite basis for multi-loop Feynman integrals
• Planar Two-Loop Five-Gluon Amplitudes from Numerical Unitarity
• Aspects of Scattering Amplitudes and Moduli Space Localization
• A K3 in phi4
• Traintracks Through Calabi-Yaus: Amplitudes Beyond Elliptic Polylogarithms
• A (Bounded) Bestiary of Feynman Integral Calabi-Yau Geometries
• The two-loop sunrise graph with arbitrary masses
• A New Algorithm For The Generation Of Unitarity-Compatible Integration By Parts Relations
• Schouten identities for Feynman graph amplitudes; the Master Integrals for the two-loop massive sunrise graph
• Simple one-dimensional integral representations for two-loop self-energies: the master diagram
• Pentagon Functions for Scattering of Five Massless Particles
• Interpolation of Dense and Sparse Rational Functions and other Improvements in $\text{FireFly}$
• Iterated Elliptic and Hypergeometric Integrals for Feynman Diagrams
• Analytic Continuation of Mellin Transforms up to two-loop Order
• Evaluating single-scale and/or non-planar diagrams by differential equations
• The Cosmic Galois Group and Extended Steinmann Relations for Planar $\mathcal{N}=4$ SYM Amplitudes
• Constructing d-log integrands and computing master integrals for three-loop four-particle scattering
• Decomposition of Feynman Integrals on the Maximal Cut by Intersection Numbers
• Elliptic symbol calculus: from elliptic polylogarithms to iterated integrals of Eisenstein series
• Two-Loop Four-Gluon Amplitudes with the Numerical Unitarity Method
• Generalised power series expansions for the elliptic planar families of Higgs + jet production at two loops
• An Elliptic Generalization of Multiple Polylogarithms
• Feynman integrals and hyperlogarithms
• Decomposition of Feynman Integrals by Multivariate Intersection Numbers
• The Two-Loop Master Integrals for $q\bar{q}\to VV$
• Transforming differential equations of multi-loop Feynman integrals into canonical form
• Libra: a package for transformation of differential systems for multiloop integrals
• On hyperlogarithms and Feynman integrals with divergences and many scales
• A non-planar two-loop three-point function beyond multiple polylogarithms
• The planar double box integral for top pair production with a closed top loop to all orders in the dimensional regularisation parameter
• Analytic results for the planar double box integral relevant to top-pair production with a closed top loop
• The unequal mass sunrise integral expressed through iterated integrals on ${\overline{\mathcal{M}}}_{1,3}$
• Group structure of the integration-by-part identities and its application to the reduction of multiloop integrals
• Elliptic polylogarithms and Feynman parameter integrals
• Iterated Binomial Sums and their Associated Iterated Integrals
• Analytic continuation and numerical evaluation of the kite integral and the equal mass sunrise integral
• Azurite: An algebraic geometry based package for finding bases of loop integrals
• The iterated structure of the all-order result for the two-loop sunrise integral
• Multiple polylogarithms with algebraic arguments and the two-loop EW-QCD Drell-Yan master integrals
• Two-Loop Integrals for Planar Five-Point One-Mass Processes
• Pentagon Functions for One-Mass Planar Scattering Amplitudes
• Analytic Structure of all Loop Banana Amplitudes
• Diagrammatic Hopf algebra of cut Feynman integrals: the one-loop case
• MultivariateApart: Generalized Partial Fractions
• Automated Solution of First Order Factorizable Systems of Differential Equations in One Variable
• An analytic solution for the equal-mass banana graph
• HandyG -- rapid numerical evaluation of generalised polylogarithms in Fortran
• Two-loop sunset diagrams with three massive lines
• Cuts from residues: the one-loop case
• The algebraic structure of cut Feynman integrals and the diagrammatic coaction
• Calculating Massive 3-loop Graphs for Operator Matrix Elements by the Method of Hyperlogarithms
• Feynman integral relations from parametric annihilators
• Embedding Feynman Integral (Calabi-Yau) Geometries in Weighted Projective Space
• The $l$-loop Banana Amplitude from GKZ Systems and relative Calabi-Yau Periods
• Deriving canonical differential equations for Feynman integrals from a single uniform weight integral
• How to choose master integrals
• Differential equations from unitarity cuts: nonplanar hexa-box integrals
• Duals of Feynman Integrals, I: Differential Equations
• Relations between elliptic multiple zeta values and a special derivation algebra
• The Galois coaction on ${\varphi }^4$ periods
• ep-Finite Basis of Master Integrals for the Integration-By-Parts Method
• The Two Loop Crossed Ladder Vertex Diagram with Two Massive Exchanges
• RationalizeRoots: Software Package for the Rationalization of Square Roots
• Maximal Cuts in Arbitrary Dimension
• Baikov-Lee Representations Of Cut Feynman Integrals
• On the reduction of generalized polylogarithms to ${\text{Li}}_n$ and ${\text{Li}}_{2,2}$ and on the evaluation thereof
• From elliptic multiple zeta values to modular graph functions: open and closed strings at one loop
• Numerical evaluation of iterated integrals related to elliptic Feynman integrals
• On the computation of intersection numbers for twisted cocycles
• The toolbox of modern multi-loop calculations: novel analytic and semi-analytic techniques
• Simplified differential equations approach for Master Integrals
• BOKASUN: a fast and precise numerical program to calculate the Master Integrals of the two-loop sunrise diagrams
• Calculation of Feynman loop integration and phase-space integration via auxiliary mass flow
• Algorithms and tools for iterated Eisenstein integrals
• Star Integrals, Convolutions and Simplices
• Numerical Implementation of Harmonic Polylogarithms to Weight w = 8
• On the status of expansion by regions
• The Dimensional Recurrence and Analyticity Method for Multicomponent Master Integrals: Using Unitarity Cuts to Construct Homogeneous Solutions
• Twisted elliptic multiple zeta values and non-planar one-loop open-string amplitudes
• Superstring amplitudes and the associator
• MPL - a program for computations with iterated integrals on moduli spaces of curves of genus zero
• The elliptic double box and symbology beyond polylogarithms
• Status of Intersection Theory and Feynman Integrals
• From positive geometries to a coaction on hypergeometric functions
• IBP reduction coefficients made simple
• Two-loop leading colour QCD helicity amplitudes for top quark pair production in the gluon fusion channel
• Two-loop diagrams in non-relativistic QCD with elliptics
• Lectures on LHC Physics
• Counting master integrals: integration by parts vs. differential reduction
• Functional relations for elliptic polylogarithms
• Three-loop contributions to the $\rho$ parameter and iterated integrals of modular forms
• Evaluating multi-loop Feynman integrals numerically through differential equations
• Coaction and double-copy properties of configuration-space integrals at genus zero
• Counting master integrals: integration-by-parts procedure with effective mass
• Two-Loop integrals for CP-even heavy quarkonium production and decays: Elliptic Sectors
• An Elliptic Yangian-Invariant, `Leading Singularity'
• Massive kite diagrams with elliptics
• Symmetric $\epsilon$- and $(\epsilon +1/2)$-forms and quadratic constraints in "elliptic" sectors
• $\epsilon$-regular basis for non-polylogarithmic multiloop integrals and total cross section of the process $e^{+}{e}^{-}\to 2(Q\bar{Q})$
• Wrońskian factorizations and Broadhurst-Mellit determinant formulae
• The H-graph with equal masses in terms of multiple polylogarithms
• Total cross sections of $e\gamma \to eX\bar{X}$ processes with $X=\mu ,\gamma ,e$ via multiloop methods
• Master integrals for bipartite cuts of three-loop photon self energy
• Calculation of master integrals in terms of elliptic multiple polylogarithms
• Analytic results for two-loop planar master integrals for Bhabha scattering

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• Complete $N_f^2$ contributions to four-loop pure-singlet splitting functions
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• Numerical evaluation of two-loop QCD helicity amplitudes for $gg\to t\bar{t}g$ at leading colour
• Cutting-Edge Tools for Cutting Edges
• Two-loop Feynman integrals for leading colour $t\bar{t}W$ production at hadron colliders
• Canonical differential equations for the elliptic two-loop five-point integral family relevant to $t\bar{t}+$jet production at leading colour
• Rewording Theoretical Predictions at Colliders with Vacuum Amplitudes
• Three-loop banana integrals with four unequal masses
• On series and integral representations of some NRQCD master integrals
• Conformal Four-Point Integrals: Recursive Structure, Toda Equations and Double Copy
• Les Houches 2023 -- Physics at TeV Colliders: Report on the Standard Model Precision Wishlist
• Differential equations for tree--level cosmological correlators with massive states
• ALPs, the on-shell way
• One-loop corrections in Maxwell-metric-affine bumblebee gravity
• Two-loop master integrals for a planar and a non-planar topology relevant for single top production
• Singularities of Feynman Integrals
• The thermal representation of conformal ladder integrals
• Two-loop form factors for $P$-wave quarkonium production and decay
• Transition Form-Factor for ${\eta }_Q$ at NNLO in the strong coupling ${\alpha }_s$ and with all-order $v^2$ resummation