Calculation of Massless Feynman Integrals using Harmonic Sums

Abstract

A method for the evaluation of the epsilon expansion of multi-loop massless Feynman integrals is introduced. This method is based on the Gegenbauer polynomial technique and the expansion of the Gamma function in terms of harmonic sums. Algorithms for the evaluation of nested and harmonic sums are used to reduce the expressions to get analytical or numerical results for the expansion coefficients. Methods to increase the precision of numerical results are discussed.

30 pages, 6 figures; Minor typos corrected, references added. Published in Computer Physics Communications

References in corpus (19)

• New features of FORM
• Harmonic sums, Mellin transforms and Integrals
• Harmonic Sums and Mellin Transforms up to two-loop Order
• HypExp, a Mathematica package for expanding hypergeometric functions around integer-valued parameters
• Nested Sums, Expansion of Transcendental Functions and Multi-Scale Multi-Loop Integrals
• QCD Corrections to the $e^{+}{e}^{-}$ Cross Section and the $Z$ Boson Decay Rate
• Numerical evaluation of multi-loop integrals by sector decomposition
• Algebraic Relations Between Harmonic Sums and Associated Quantities
• Analytic two-loop results for selfenergy- and vertex-type diagrams with one non-zero mass
• - XSummer - Transcendental Functions and Symbolic Summation in Form
• Expansion around half-integer values, binomial sums and inverse binomial sums
• Knots and Numbers in ${\varphi }^4$ Theory to 7 Loops and Beyond
• Corrections of order ${\alpha }_s^3$ to $R_{had}$ in pQCD with light gluinos
• The Gegenbauer Polynomial Technique: the evaluation of a class of Feynman diagrams
• The massless two-loop two-point function
• Rhad: A program for the evaluation of the hadronic R-ratio in the perturbative regime of QCD
• Automatic Computation of Feynman Diagrams
• The Cross Section of e^+ e^- Annihilation into Hadrons of Order alpha_s^4 n_f^2 in Perturbative QCD
• Lectures on multiloop calculations

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