Advanced Computer Algebra Algorithms for the Expansion of Feynman Integrals

Abstract

Two-point Feynman parameter integrals, with at most one mass and containing local operator insertions in $4+\ep$-dimensional Minkowski space, can be transformed to multi-integrals or multi-sums over hyperexponential and/or hypergeometric functions depending on a discrete parameter $n$. Given such a specific representation, we utilize an enhanced version of the multivariate Almkvist--Zeilberger algorithm (for multi-integrals) and a common summation framework of the holonomic and difference field approach (for multi-sums) to calculate recurrence relations in $n$. Finally, solving the recurrence we can decide efficiently if the first coefficients of the Laurent series expansion of a given Feynman integral can be expressed in terms of indefinite nested sums and products; if yes, the all $n$ solution is returned in compact representations, i.e., no algebraic relations exist among the occurring sums and products.

14 pages, Proceedings, Loops and Legs in Quantum Field Theory 2012, Wernigerode,D; PoS(2012)

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Cited by in corpus (13)

• Algorithms for the symbolic integration of hyperlogarithms with applications to Feynman integrals
• Analytic and Algorithmic Aspects of Generalized Harmonic Sums and Polylogarithms
• Modern Summation Methods for Loop Integrals in Quantum Field Theory: The Packages Sigma, EvaluateMultiSums and SumProduction
• On the Computation of Form Factors in Massless QCD with Finite Master Integrals
• The O(α_s^3 T_F^2) Contributions to the Gluonic Operator Matrix Element
• The Polarized Transition Matrix Element $A_{gq}(N)$ of the Variable Flavor Number Scheme at $O({\alpha }_s^3)$
• Recent Symbolic Summation Methods to Solve Coupled Systems of Differential and Difference Equations
• Three-Loop Contributions to the Gluonic Massive Operator Matrix Elements at General Values of N
• Higher Order Heavy Quark Corrections to Deep-Inelastic Scattering
• New Results on the 3-Loop Heavy Flavor Corrections in Deep-Inelastic Scattering
• Computer-Assisted Proofs of Some Identities for Bessel Functions of Fractional Order
• 3-Loop Heavy Flavor Corrections in Deep-Inelastic Scattering with Two Heavy Quark Lines
• 3-loop Massive $O(T_F^2)$ Contributions to the DIS Operator Matrix Element $A_{gg}$