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Modern Summation Methods for Loop Integrals in Quantum Field Theory: The Packages Sigma, EvaluateMultiSums and SumProduction

A large class of Feynman integrals, like e.g., two-point parameter integrals with at most one mass and containing local operator insertions, can be transformed to multi-sums over hypergeometric expressions. In this survey article we present a difference field approach for symbolic summation that enables one to simplify such definite nested sums to indefinite nested sums. In particular, the simplification is given -if possible- in terms of harmonic sums, generalized harmonic sums, cyclotomic harmonic sums or binomial sums. Special emphasis is put on the developed packages Sigma, EvaluateMultiSums and SumProduction that assist in the task to perform these simplifications completely automatically for huge input expressions.