The kite integral to all orders in terms of elliptic polylogarithms

Abstract

We show that the Laurent series of the two-loop kite integral in $D=4-2\epsilon$ space-time dimensions can be expressed in each order of the series expansion in terms of elliptic generalisations of (multiple) polylogarithms. Using differential equations we present an iterative method to compute any desired order. As an example, we give the first three orders explicitly.

19 pages, v2: version to be published

References in corpus (33)

• Multiloop integrals in dimensional regularization made simple
• Differential Equations for Two-Loop Four-Point Functions
• Differential Equations for Feynman Graph Amplitudes
• Algorithms for the symbolic integration of hyperlogarithms with applications to Feynman integrals
• Reduze - Feynman Integral Reduction in C++
• Lectures on differential equations for Feynman integrals
• 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
• Analytic treatment of the two loop equal mass sunrise graph
• Feynman Diagrams and Differential Equations
• Resolution of singularities for multi-loop integrals
• 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
• Analytical and numerical methods for massive two-loop self-energy diagrams
• Differential equations and dispersion relations for Feynman amplitudes. The two-loop massive sunrise and the kite integral
• - XSummer - Transcendental Functions and Symbolic Summation in Form
• Elliptic integral evaluations of Bessel moments
• A Feynman integral via higher normal functions
• 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
• The two-loop sunrise graph with arbitrary masses
• 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
• Symbolic Expansion of Transcendental Functions
• The iterated structure of the all-order result for the two-loop sunrise integral
• The massless two-loop two-point function
• 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
• Calculation of two-loop self-energies in the electroweak Standard Model
• A numerical test of differential equations for one- and two-loop sunrise diagrams using configuration space techniques
• MPL - a program for computations with iterated integrals on moduli spaces of curves of genus zero

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• Generalised power series expansions for the elliptic planar families of Higgs + jet production at two loops
• An Elliptic Generalization of Multiple Polylogarithms
• A non-planar two-loop three-point function beyond multiple polylogarithms
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• Analytic results for the planar double box integral relevant to top-pair production with a closed top loop
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• Feynman Integrals in Dimensional Regularization and Extensions of Calabi-Yau Motives
• Analytic continuation and numerical evaluation of the kite integral and the equal mass sunrise integral
• Solving differential equations for Feynman integrals by expansions near singular points
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• Embedding Feynman Integral (Calabi-Yau) Geometries in Weighted Projective Space
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• RationalizeRoots: Software Package for the Rationalization of Square Roots
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• Evaluation of Feynman integrals with arbitrary complex masses via series expansions
• Numerical evaluation of iterated integrals related to elliptic Feynman integrals
• Elliptic Polylogarithms and Basic Hypergeometric Functions
• The SAGEX Review on Scattering Amplitudes, Chapter 3: Mathematical structures in Feynman integrals
• Top-bottom interference effects in Higgs plus jet production at the LHC
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• Complete two-loop QCD contributions to the lightest Higgs-boson mass in the MSSM with complex parameters
• Evaluating `elliptic' master integrals at special kinematic values: using differential equations and their solutions via expansions near singular points
• 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 electron self-energy in QED at two loops revisited
• Algorithms for minimal Picard-Fuchs operators of Feynman integrals
• The elliptic double box and symbology beyond polylogarithms
• The complete set of two-loop master integrals for Higgs + jet production in QCD
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• Two-loop diagrams in non-relativistic QCD with elliptics
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• Functional relations for elliptic polylogarithms
• Symbology for elliptic multiple polylogarithms and the symbol prime
• Modular transformations of elliptic Feynman integrals
• Bootstrapping elliptic Feynman integrals using Schubert analysis
• Calabi-Yau Feynman integrals in gravity: $\epsilon$-factorized form for apparent singularities
• Cutting the traintracks: Cauchy, Schubert and Calabi-Yau
• Two-Loop integrals for CP-even heavy quarkonium production and decays: Elliptic Sectors
• Massive kite diagrams with elliptics
• The SAGEX Review on Scattering Amplitudes, Chapter 4: Multi-loop Feynman Integrals
• Iterated integrals over letters induced by quadratic forms
• Rooting Out Letters: Octagonal Symbol Alphabets and Algebraic Number Theory
• Symmetric $\epsilon$- and $(\epsilon +1/2)$-forms and quadratic constraints in "elliptic" sectors
• Two-Loop integrals for CP-even heavy quarkonium production and decays
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• Non-planar elliptic vertex
• Master integrals for $\mathcal{O}(\alpha {\alpha }_s)$ corrections to $H\to Z{Z}^{\ast }$
• The analytic leading color contribution to the Higgs-gluon form factor in QCD at NNLO
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• Electroweak double-box integrals for Moller scattering
• Sunrise integral with two internal masses and pseudo-threshold kinematics in terms of elliptic polylogarithms
• Rationalizing roots: an algorithmic approach
• Addendum: Long-range multi-scalar models at three loops
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• Massive Two-Loop Heavy Particle Diagrams
• Three loop QCD corrections to heavy quark form factors
• Differential equations for Feynman integrals beyond multiple polylogarithms
• The Stratification of Rigidity
• Integral representation for three-loop banana graph
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• Higher-Order Corrections to Higgs Boson Amplitudes with Full Quark Mass Dependence in Quantum Chromodynamics