Evaluating four-loop conformal Feynman integrals by D-dimensional differential equations

Abstract

We evaluate a four-loop conformal integral, i.e. an integral over four four-dimensional coordinates, by turning to its dimensionally regularized version and applying differential equations for the set of the corresponding 213 master integrals. To solve these linear differential equations we follow the strategy suggested by Henn and switch to a uniformly transcendental basis of master integrals. We find a solution to these equations up to weight eight in terms of multiple polylogarithms. Further, we present an analytical result for the given four-loop conformal integral considered in four-dimensional space-time in terms of single-valued harmonic polylogarithms. As a by-product, we obtain analytical results for all the other 212 master integrals within dimensional regularization, i.e. considered in D dimensions.

10 pages, 2 figures

References in corpus (34)

• Harmonic Polylogarithms
• 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
• Algorithm FIRE -- Feynman Integral REduction
• FIRE5: a C++ implementation of Feynman Integral REduction
• Algorithms for the symbolic integration of hyperlogarithms with applications to Feynman integrals
• Space-time S-matrix and Flux-tube S-matrix at Finite Coupling
• Two-Loop Master Integrals for ${\gamma }^{\ast }\to 3$ Jets: The planar topologies
• FIESTA 4: optimized Feynman integral calculations with GPU support
• Thermodynamic Bubble Ansatz
• Two-Loop Master Integrals for ${\gamma }^{\ast }\to 3$ Jets: The non-planar topologies
• FIRE4, LiteRed and accompanying tools to solve integration by parts relations
• The massless higher-loop two-point function
• Partial non-renormalisation of the stress-tensor four-point function in N=4 SYM and AdS/CFT
• From correlation functions to Wilson loops
• Three-Loop Four-Point Correlator in N=4 SYM
• Anomalous dimensions in N=4 SYM theory at order g^4
• From correlation functions to scattering amplitudes
• Leading singularities and off-shell conformal integrals
• Constructing the correlation function of four stress-tensor multiplets and the four-particle amplitude in N=4 SYM
• Hidden symmetry of four-point correlation functions and amplitudes in N=4 SYM
• Iterative structure of finite loop integrals
• A quasi-finite basis for multi-loop Feynman integrals
• Evaluating single-scale and/or non-planar diagrams by differential equations
• On hyperlogarithms and Feynman integrals with divergences and many scales
• Analytic results for two-loop master integrals for Bhabha scattering I
• All three-loop four-point correlators of half-BPS operators in planar N=4 SYM
• Perturbation Theory at Eight Loops: Novel Structures and the Breakdown of Manifest Conformality
• Non-planar master integrals for the production of two off-shell vector bosons in collisions of massless partons
• Three-point functions in $\mathcal{N}=4$ SYM: the hexagon proposal at three loops
• Analytic results for planar three-loop four-point integrals from a Knizhnik-Zamolodchikov equation
• On the analytic computation of massless propagators in dimensional regularization

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• Transforming differential equations of multi-loop Feynman integrals into canonical form
• Energy-energy correlations at next-to-next-to-leading order
• Ten dimensional symmetry of N=4 SYM correlators
• All five-loop planar four-point functions of half-BPS operators in $\mathcal{N}=4$ SYM
• The SAGEX Review on Scattering Amplitudes, Chapter 8: Half BPS correlators
• Non-planar data of $\mathcal{N}=4$ SYM
• From correlation functions to event shapes in QCD
• Glue-and-Cut at Five Loops
• Notes on conformal integrals: Coulomb branch amplitudes, magic identities and bootstrap