Iterated Elliptic and Hypergeometric Integrals for Feynman Diagrams

Abstract

We calculate 3-loop master integrals for heavy quark correlators and the 3-loop QCD corrections to the $\rho$-parameter. They obey non-factorizing differential equations of second order with more than three singularities, which cannot be factorized in Mellin-$N$ space either. The solution of the homogeneous equations is possible in terms of convergent close integer power series as ${}_2{F}_1$ Gauß hypergeometric functions at rational argument. In some cases, integrals of this type can be mapped to complete elliptic integrals at rational argument. This class of functions appears to be the next one arising in the calculation of more complicated Feynman integrals following the harmonic polylogarithms, generalized polylogarithms, cyclotomic harmonic polylogarithms, square-root valued iterated integrals, and combinations thereof, which appear in simpler cases. The inhomogeneous solution of the corresponding differential equations can be given in terms of iterative integrals, where the new innermost letter itself is not an iterative integral. A new class of iterative integrals is introduced containing letters in which (multiple) definite integrals appear as factors. For the elliptic case, we also derive the solution in terms of integrals over modular functions and also modular forms, using $q$-product and series representations implied by Jacobi's ${\vartheta }_i$ functions and Dedekind's $\eta$-function. The corresponding representations can be traced back to polynomials out of Lambert--Eisenstein series, having representations also as elliptic polylogarithms, a $q$-factorial $1/{\eta }^k(\tau )$, logarithms and polylogarithms of $q$ and their $q$-integrals. Due to the specific form of the physical variable $x(q)$ for different processes, different representations do usually appear. Numerical results are also presented.

arXiv comment68 pages LATEX, 10 Figures

References in corpus (85)

• Harmonic Polylogarithms
• Differential Equations for Two-Loop Four-Point Functions
• Differential Equations for Feynman Graph Amplitudes
• Dimensionally Regulated Pentagon Integrals
• Harmonic sums, Mellin transforms and Integrals
• The third-order QCD corrections to deep-inelastic scattering by photon exchange
• Reduze - Feynman Integral Reduction in C++
• Harmonic Sums and Mellin Transforms up to two-loop Order
• The Multiple Zeta Value Data Mine
• Dimensionally Regulated One-Loop Integrals
• Complete Corrections of O(αα_s) to the Decay of the Z Boson into Bottom Quarks
• Nested Sums, Expansion of Transcendental Functions and Multi-Scale Multi-Loop Integrals
• Reduze 2 - Distributed Feynman Integral Reduction
• The elliptic dilogarithm for the sunset graph
• Two-loop two-point functions with masses: asymptotic expansions and Taylor series, in any dimension
• Massive Feynman diagrams and inverse binomial sums
• Analytic treatment of the two loop equal mass sunrise graph
• Harmonic Sums and Polylogarithms Generated by Cyclotomic Polynomials
• MATAD: a program package for the computation of MAssive TADpoles
• High-precision calculation of multi-loop Feynman integrals by difference equations
• Elliptic multiple zeta values and one-loop superstring amplitudes
• Algebraic Relations Between Harmonic Sums and Associated Quantities
• The two-loop sunrise graph in two space-time dimensions with arbitrary masses in terms of elliptic dilogarithms
• Mellin Moments of the {$O({\alpha }_s^3$)} Heavy Flavor Contributions to unpolarized Deep-Inelastic Scattering at $Q^2\gg m^2$ and Anomalous Dimensions
• Analytic and Algorithmic Aspects of Generalized Harmonic Sums and Polylogarithms
• 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
• Two-loop planar master integrals for Higgs$\to 3$ partons with full heavy-quark mass dependence
• A Computer Algebra Toolbox for Harmonic Sums Related to Particle Physics
• Elliptic integral evaluations of Bessel moments
• A Feynman integral via higher normal functions
• Maximal cuts and differential equations for Feynman integrals. An application to the three-loop massive banana graph
• The kite integral to all orders in terms of elliptic polylogarithms
• Expansion around half-integer values, binomial sums and inverse binomial sums
• Calculating Three Loop Ladder and V-Topologies for Massive Operator Matrix Elements by Computer Algebra
• Multiple Elliptic Polylogarithms
• A K3 in phi4
• The two-loop sunrise graph with arbitrary masses
• The 3-Loop Non-Singlet Heavy Flavor Contributions and Anomalous Dimensions for the Structure Function $F_2(x,Q^2)$ and Transversity
• The 3-Loop Pure Singlet Heavy Flavor Contributions to the Structure Function $F_2(x,Q^2)$ and the Anomalous Dimension
• 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
• Computer Algebra Algorithms for Special Functions in Particle Physics
• The O(α_s^3) Massive Operator Matrix Elements of O(n_f) for the Structure Function F_2(x,Q^2) and Transversity
• Further applications of the Container Method
• A non-planar two-loop three-point function beyond multiple polylogarithms
• Iterated Binomial Sums and their Associated Iterated Integrals
• The iterated structure of the all-order result for the two-loop sunrise integral
• The Transition Matrix Element A_{gq}(N) of the Variable Flavor Number Scheme at O(α_s^3)
• Modern Summation Methods for Loop Integrals in Quantum Field Theory: The Packages Sigma, EvaluateMultiSums and SumProduction
• Two-loop sunset diagrams with three massive lines
• Calculating Massive 3-loop Graphs for Operator Matrix Elements by the Method of Hyperlogarithms
• Modern Summation Methods and the Computation of 2- and 3-loop Feynman Diagrams
• A Direct Test of Perturbative QCD at Small x
• Massive 3-loop Ladder Diagrams for Quarkonic Local Operator Matrix Elements
• Picard-Fuchs equations for Feynman integrals
• On the evaluation of a certain class of Feynman diagrams in x-space: Sunrise-type topologies at any loop order
• The Logarithmic Contributions to the O(α_s^3) Asymptotic Massive Wilson Coefficients and Operator Matrix Elements in Deeply Inelastic Scattering
• The Two Loop Crossed Ladder Vertex Diagram with Two Massive Exchanges
• Precise numerical evaluation of the two loop sunrise graph Master Integrals in the equal mass case
• The $O({\alpha }_s^3{n}_f{T}_F^2{C}_{A,F})$} Contributions to the Gluonic Massive Operator Matrix Elements
• The QCD form factor of heavy quarks at NNLO
• The Three-Loop Splitting Functions $P_{qg}^{(2)}$ and $P_{gg}^{(2,N_F)}$
• Towards multiple elliptic polylogarithms
• Elliptic Polylogarithms and Basic Hypergeometric Functions
• BOKASUN: a fast and precise numerical program to calculate the Master Integrals of the two-loop sunrise diagrams
• Calculation of two-loop self-energies in the electroweak Standard Model
• Generalized Harmonic, Cyclotomic, and Binomial Sums, their Polylogarithms and Special Numbers
• The Method of Arbitrarily Large Moments to Calculate Single Scale Processes in Quantum Field Theory
• Ising n-fold integrals as diagonals of rational functions and integrality of series expansions
• A numerical test of differential equations for one- and two-loop sunrise diagrams using configuration space techniques
• Numerical evaluation of the general massive 2-loop sunrise self-mass master integrals from differential equations
• Crossing Probabilities and Modular Forms
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• Transformations of some Gauss hypergeometric functions
• Algebraic transformations of Gauss hypergeometric functions
• Elliptic integral evaluation of a Bessel moment by contour integration of a lattice Green function
• Dihedral Gauss hypergeometric functions
• A brief Introduction to Dispersion Relations and Analyticity
• An Analytic Approach to Sunset Diagrams in Chiral Perturbation Theory: Theory and Practice
• Transformations and invariants for dihedral Gauss hypergeometric functions
• Transformations of algebraic Gauss hypergeometric functions
• Nucleon Structure, Duality and Elliptic Theta Functions
• On computing the generalized Lambert series
• Transformations of hypergeometric elliptic integrals

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• The inverse Mellin transform via analytic continuation
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• Manifestly Causal Loop-Tree Duality
• A Calabi-Yau-to-Curve Correspondence for Feynman Integrals
• The analytic leading color contribution to the Higgs-gluon form factor in QCD at NNLO
• Electroweak double-box integrals for Moller scattering
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• Reclassifying Feynman integrals as special functions
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