Lectures on differential equations for Feynman integrals

Abstract

Over the last year significant progress was made in the understanding of the computation of Feynman integrals using differential equations. These lectures give a review of these developments, while not assuming any prior knowledge of the subject. After an introduction to differential equations for Feynman integrals, we point out how they can be simplified using algorithms available in the mathematical literature. We discuss how this is related to a recent conjecture for a canonical form of the equations. We also discuss a complementary approach that allows based on properties of the space-time loop integrands, and explain how the ideas of leading singularities and d-log representations can be used to find an optimal basis for the differential equations. Finally, as an application of the differential equations method we show how single-scale integrals can be bootstrapped using the Drinfeld associator of a differential equation.

40 pages; v3: typos fixed, note at the end of section 4 extended; to appear in J.Phys.A

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• 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
• Differential Equations for Feynman Graph Amplitudes
• Classical Polylogarithms for Amplitudes and Wilson Loops
• Connection between Feynman integrals having different values of the space-time dimension
• Numerical evaluation of multiple polylogarithms
• The All-Loop Integrand For Scattering Amplitudes in Planar N=4 SYM
• FIRE5: a C++ implementation of Feynman Integral REduction
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• Automatic Integral Reduction for Higher Order Perturbative Calculations
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• From polygons and symbols to polylogarithmic functions
• Reduze 2 - Distributed Feynman Integral Reduction
• The elliptic dilogarithm for the sunset graph
• Motivic Amplitudes and Cluster Coordinates
• Numerical Evaluation of Harmonic Polylogarithms
• Presenting LiteRed: a tool for the Loop InTEgrals REDuction
• Analytic treatment of the two loop equal mass sunrise graph
• Two-loop amplitudes and master integrals for the production of a Higgs boson via a massive quark and a scalar-quark loop
• Feynman Diagrams and Differential Equations
• Analytic result for the two-loop six-point NMHV amplitude in N=4 super Yang-Mills theory
• Space-time dimensionality D as complex variable: calculating loop integrals using dimensional recurrence relation and analytical properties with respect to D
• The two-loop sunrise graph in two space-time dimensions with arbitrary masses in terms of elliptic dilogarithms
• Jumpstarting the all-loop S-matrix of planar N=4 super Yang-Mills
• Analytic and Algorithmic Aspects of Generalized Harmonic Sums and Polylogarithms
• Magnus and Dyson Series for Master Integrals
• Massive 3-loop Feynman diagrams reducible to SC* primitives of algebras of the sixth root of unity
• A Two-Loop Five-Gluon Helicity Amplitude in QCD
• On the periods of some Feynman integrals
• Iterative structure of finite loop integrals
• Multiple Elliptic Polylogarithms
• Scattering Amplitudes from Multivariate Polynomial Division
• Schouten identities for Feynman graph amplitudes; the Master Integrals for the two-loop massive sunrise graph
• The three-loop cusp anomalous dimension in QCD
• Evaluating single-scale and/or non-planar diagrams by differential equations
• On the Singularity Structure of Maximally Supersymmetric Scattering Amplitudes
• Sharpening The Leading Singularity
• The Two-Loop Master Integrals for $q\bar{q}\to VV$
• The four-loop cusp anomalous dimension in N=4 super Yang-Mills and analytic integration techniques for Wilson line integrals
• Non-planar master integrals for the production of two off-shell vector bosons in collisions of massless partons
• Analytic results for planar three-loop four-point integrals from a Knizhnik-Zamolodchikov equation
• Simple loop integrals and amplitudes in N=4 SYM
• Application and explicit solution of recurrence relations with respect to space-time dimension
• The one-loop six-dimensional hexagon integral and its relation to MHV amplitudes in N=4 SYM
• Eliminating spurious poles from gauge-theoretic amplitudes
• Master integrals for the two-loop penguin contribution in non-leptonic B-decays
• The massless hexagon integral in D = 6 dimensions
• From the Holomorphic Wilson Loop to `d log' Loop-Integrands of Super-Yang-Mills Amplitudes
• From dlogs to dilogs; the super Yang-Mills MHV amplitude revisited
• Real-Virtual-Virtual contributions to the inclusive Higgs cross section at N3LO
• Massive planar and non-planar double box integrals for light Nf contributions to gg->tt
• Adequate bases of phase space master integrals for $gg\to h$ at NNLO and beyond
• Spanning forest polynomials and the transcendental weight of Feynman graphs
• Cluster Polylogarithms for Scattering Amplitudes
• An Overview of Maximal Unitarity at Two Loops
• Standard and hypergeometric representations for loop diagrams and the photon-photon scattering
• Multiple zeta-values, Galois groups, and geometry of modular varieties
• QCD and High Energy Interactions: Moriond 2016 Theory Summary
• On the Reduction of Singularly-Perturbed Linear Differential Systems
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Cited by in corpus (175)

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• On the maximal cut of Feynman integrals and the solution of their differential equations
• Gravitational Bremsstrahlung from Reverse Unitarity
• Differential equations and dispersion relations for Feynman amplitudes. The two-loop massive sunrise and the kite integral
• Two-loop planar master integrals for Higgs$\to 3$ partons with full heavy-quark mass dependence
• Algorithmic transformation of multi-loop master integrals to a canonical basis with CANONICA
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• Fuchsia: a tool for reducing differential equations for Feynman master integrals to epsilon form
• Collider Physics at the Precision Frontier
• Spin Effects in the Effective Field Theory Approach to Post-Minkowskian Conservative Dynamics
• 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
• Classical Gravity from Loop Amplitudes
• FeynCalc 10: Do multiloop integrals dream of computer codes?
• Decomposition of Feynman Integrals on the Maximal Cut by Intersection Numbers
• Feynman integrals and hyperlogarithms
• Transforming differential equations of multi-loop Feynman integrals into canonical form
• Evidence for a Nonplanar Amplituhedron
• Multiloop corrections for collider processes using auxiliary mass flow
• Analytic results for the planar double box integral relevant to top-pair production with a closed top loop
• The planar double box integral for top pair production with a closed top loop to all orders in the dimensional regularisation parameter
• Four-loop photon quark form factor and cusp anomalous dimension in the large-$N_c$ limit of QCD
• The unequal mass sunrise integral expressed through iterated integrals on ${\overline{\mathcal{M}}}_{1,3}$
• Analytic continuation and numerical evaluation of the kite integral and the equal mass sunrise integral
• Two-Loop Integrals for Planar Five-Point One-Mass Processes
• Simplifying differential equations for multi-scale Feynman integrals beyond multiple polylogarithms
• Determining Feynman integrals with only input from linear algebra
• Differential equations from unitarity cuts: nonplanar hexa-box integrals
• Radiated momentum in the Post-Minkowskian worldline approach via reverse unitarity
• Duals of Feynman Integrals, I: Differential Equations
• Two-Loop Hexa-Box Integrals for Non-Planar Five-Point One-Mass Processes
• Two-Loop Master Integrals for the mixed EW-QCD virtual corrections to Drell-Yan scattering
• Evaluation of Feynman integrals with arbitrary complex masses via series expansions
• Four-loop quark form factor with quartic fundamental colour factor
• Next-to-Leading Order QCD Corrections to the Decay Width $H\to Z\gamma$
• Tropical Monte Carlo quadrature for Feynman integrals
• Baikov representations, intersection theory, and canonical Feynman integrals
• The SAGEX Review on Scattering Amplitudes, Chapter 3: Mathematical structures in Feynman integrals
• Integration-by-parts reductions of Feynman integrals using Singular and GPI-Space
• Macaulay Matrix for Feynman Integrals: Linear Relations and Intersection Numbers
• Direct Solution of Integration-by-Parts Systems
• Two-Loop Master Integrals for the Planar QCD Massive Corrections to Di-photon and Di-jet Hadro-production
• Gravitational Bremsstrahlung with tidal effects in the post-Minkowskian expansion
• Les Houches 2021: Physics at TeV Colliders: Report on the Standard Model Precision Wishlist
• Cosmology meets cohomology
• Scalar QED as a toy model for higher-order effects in classical gravitational scattering
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• Double-real corrections at O(alpha alpha_s) to single gauge boson production
• Mixed QCD-EW corrections for Higgs boson production at $e^{+}{e}^{-}$ colliders
• Classical vs Quantum Eikonal Scattering and its Causal Structure
• NNLO QCD$\times$EW corrections to Z production in the $q\bar{q}$ channel
• Scattering Amplitudes in Quantum Field Theory
• Analytic Computing Methods for Precision Calculations in Quantum Field Theory
• Towards a new approximation for pair-production and associated-production of the Higgs boson
• Semi-analytical calculation of gluon fragmentation into ${}^1{S}_0^{[1,8]}$ quarkonia at next-to-leading order
• Two-loop five-point massless QCD amplitudes within the IBP approach
• The electron self-energy in QED at two loops revisited
• Logarithmic forms and differential equations for Feynman integrals
• Integration by parts identities in integer numbers of dimensions. A criterion for decoupling systems of differential equations
• Leading singularities in Baikov representation and Feynman integrals with uniform transcendental weight
• NLO Corrections to Light-Quark Mixed QCD-EW Contributions to Higgs Production
• Status of Intersection Theory and Feynman Integrals
• Differential equations on unitarity cut surfaces
• IBP reduction coefficients made simple
• Transverse-momentum-dependent wave functions and Soft functions at one-loop in Large Momentum Effective Theory
• A Numerical Routine for the Crossed Vertex Diagram with a Massive-Particle Loop
• Mixed NNLO QCD x electroweak corrections of O(N_f α_s α) to single-W/Z production at the LHC
• Notes on cluster algebras and some all-loop Feynman integrals
• On Epsilon Factorized Differential Equations for Elliptic Feynman Integrals
• Analytic results for planar three-loop integrals for massive form factors
• Truncated cluster algebras and Feynman integrals with algebraic letters
• Next-to-leading-order QCD Corrections to Higgs Production in association with a Jet
• Three-loop mixed QCD-electroweak corrections to Higgs boson gluon fusion
• First look at the evaluation of three-loop non-planar Feynman diagrams for Higgs plus jet production
• Automatic computation of Feynman integrals containing linear propagators via auxiliary mass flow
• Schubert Problems, Positivity and Symbol Letters
• Induced higher-derivative Lorentz-violating Chern-Simons term at finite temperature
• Master integrals for the NNLO virtual corrections to $q\bar{q}\to t\bar{t}$ scattering in QCD: the non-planar graphs
• Two-loop master integrals for the leading QCD corrections to the Higgs coupling to a $W$ pair and to the triple gauge couplings $ZWW$ and ${\gamma }^{\ast }WW$
• A study of Feynman integrals with uniform transcendental weights and the symbology from dual conformal symmetry
• Differential Equations, Associators, and Recurrences for Amplitudes
• What can we learn about QCD and collider physics from N=4 super Yang-Mills?
• Two-loop electroweak corrections to Higgs-gluon couplings to higher orders in the dimensional regularization parameter
• Master Integrals for the two-loop, non-planar QCD corrections to top-quark pair production in the quark-annihilation channel
• Spinning Black Hole Scattering at $\mathcal{O}(G^3{S}^2)$: Casimir Terms, Radial Action and Hidden Symmetry
• Binary Dynamics from Worldline QFT for Scalar-QED
• Analytic integration of soft and collinear radiation in factorised QCD cross sections at NNLO
• The Wilson-loop $d\log$ representation for Feynman integrals
• Classifying post-Minkowskian geometries for gravitational waves via loop-by-loop Baikov
• Two-Loop integrals for CP-even heavy quarkonium production and decays: Elliptic Sectors
• A first look at the function space for planar two-loop six-particle Feynman integrals
• NLO Effects for Doubly Heavy Baryon in QCD Sum Rules
• One-loop pentagon integral in $d$ dimensions from differential equations in $\epsilon$-form
• Pentagon integrals to arbitrary order in the dimensional regulator
• Radiative Classical Gravitational Observables at $\mathcal{O}(G^3)$ from Scattering Amplitudes
• Scattering Amplitudes and N-Body Post-Minkowskian Hamiltonians in General Relativity and Beyond
• One-loop hexagon integral to higher orders in the dimensional regulator
• Analytical calculation for the gluon fragmentation into spin-triplet S-wave quarkonium
• A planar four-loop form factor and cusp anomalous dimension in QCD
• Four Lectures on Euler Integrals
• The property of maximal transcendentality: calculation of master integrals
• Antenna subtraction at NNLO with identified hadrons
• HepLib: A C++ Library for High Energy Physics
• Dissecting polytopes: Landau singularities and asymptotic expansions in $2\to 2$ scattering
• Two-Loop integrals for CP-even heavy quarkonium production and decays
• Gravity loop integrands from the ultraviolet
• Mixed QCD-EW corrections for Higgs leptonic decay via $H{W}^{+}{W}^{-}$ vertex
• Resummation methods for Master Integrals
• Self-duality from twisted cohomology
• Canonical differential equations for the elliptic two-loop five-point integral family relevant to $t\bar{t}+$jet production at leading colour
• The property of maximal transcendentality: calculation of Feynman integrals
• A brief Introduction to Dispersion Relations and Analyticity
• Exact N${}^3$LO results for $q{q}^{\prime }\to H+X$
• Module Intersection and Uniform Formula for Iterative Reduction of One-loop Integrals
• Cyclic Mario Worlds -- Color-Decomposition for One-Loop QCD
• $D$-Module Techniques for Solving Differential Equations in the Context of Feynman Integrals
• Periods and Feynman amplitudes
• Fuchsia and master integrals for splitting functions from differential equations in QCD
• Recoil corrections to the energy levels of hydrogenic atoms
• Bootstrapping spinning two body problem in dynamical Chern-Simons gravity using worldline QFT
• A $QQ\to QQ$ planar doublebox in canonical form
• Maximal transcendental weight contribution of scattering amplitudes
• Conformal Fields from Neural Networks
• The recursive structure of Baikov representations II: the top-down reduction with intersection theory
• A Note on Kinematic Flow and Differential Equations for Two-Site One-Loop Graph in FRW Spacetime
• Classification of Feynman integral geometries for black-hole scattering at 5PM order
• Symmetries of Feynman integrals and the Integration By Parts method
• Master integrals for $\mathcal{O}(\alpha {\alpha }_s)$ corrections to $H\to Z{Z}^{\ast }$
• Fragmentation function of $g\to Q\bar{Q}{(}^3{S}_1^{[8]})$ in soft gluon factorization and threshold resummation
• Learning Feynman integrals from differential equations with neural networks
• An Introduction to Motivic Feynman Integrals
• The dimension-shift conjecture for one-loop amplitudes
• Les Houches 2023 -- Physics at TeV Colliders: Report on the Standard Model Precision Wishlist
• Renormalized $\epsilon$-finite master integrals and their virtues: the three-loop self energy case
• Generalized Polylogarithms in Maple
• The algebraic locus of Feynman integrals
• Bubble diagram through the Symmetries of Feynman Integrals method
• Expansion of hypergeometric functions in terms of polylogarithms with nontrivial variable change
• A walk on sunset boulevard
• Logarithmic Singularities and Maximally Supersymmetric Amplitudes
• Angular integrals with three denominators via IBP, mass reduction, dimensional shift, and differential equations
• A Feynman Integral and its Recurrences and Associators
• Entanglement Entropy with a Time-dependent Hamiltonian
• Calculation of master integrals in terms of elliptic multiple polylogarithms
• IIB horizons
• Fermionic corrections to quark and gluon form factors in four-loop QCD
• NNLO QED contribution to the $\mu e\to \mu e$ elastic scattering
• The HiggsTools Handbook: Concepts and observables for deciphering the Nature of the Higgs Sector
• pfd-parallel, a Singular/GPI-Space package for massively parallel multivariate partial fractioning
• Master Integrals for double real radiation emission in heavy-to-light quark decay
• A new method for finding more symmetry relations of Feynman integrals
• QCD and High Energy Interactions: Moriond 2016 Theory Summary
• One-loop Feynman integrals for $2\to 3$ scattering involving many scales including internal masses
• Two-photon exchange in leptophilic dark matter scenarios
• NNLO mixed EW-QCD corrections to single vector boson production
• Differential equations for Feynman integrals beyond multiple polylogarithms
• From elliptic curves to Feynman integrals
• Two-loop non-planar hexa-box integrals with one massive leg
• QCD Amplitudes: new perspectives on Feynman integral calculus
• Anomalous Ward identities for on-shell amplitudes at the conformal fixed point
• Co-homology of Differential Forms and Feynman diagrams
• LiSK - A C++ Library for Evaluating Classical Polylogarithms and ${\text{Li}}_{22}$
• The Stratification of Rigidity
• Two-loop mixed QCD-EW corrections to neutral current Drell-Yan
• Mixed QCD-EW corrections to $W$-pair production at electron-positron colliders
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• Scalar one-loop 4-point integral with one massless vertex in loop regularization
• Three-loop four-particle QCD amplitudes
• General One-loop Reduction in Generalized Feynman Parametrization Form
• Higher-Order Corrections to Higgs Boson Amplitudes with Full Quark Mass Dependence in Quantum Chromodynamics
• Integral representation for three-loop banana graph