Modern Summation Methods and the Computation of 2- and 3-loop Feynman Diagrams

TL;DR

The paper develops a general symbolic summation framework based on $\Pi\Sigma^*$-difference fields to transform definite multi-sums arising from 2- and 3-loop massive single-scale Feynman diagrams with local operator insertions into indefinite nested sums and products. It combines creative telescoping, recurrence solving, and algebraic simplification (via HarmonicSums) to express results in harmonic sums and their generalized $S$-sums, including all-N results for challenging 3-loop ladder graphs and $N_f$ contributions. The approach is implemented in the Sigma and EvaluateMultiSums toolchain, enabling automated, compact representations and revealing cancellations that yield minimal, independent bases of sums. These methods extend analytic capabilities in QCD-related computations and are poised to handle even higher-loop calculations beyond harmonic sums. The work demonstrates robust, scalable reduction of thousands of sums to manageable, physically meaningful expressions.

Abstract

By symbolic summation methods based on difference fields we present a general strategy that transforms definite multi-sums, e.g., in terms of hypergeometric terms and harmonic sums, to indefinite nested sums and products. We succeeded in this task with all our concrete calculations of 2--loop and 3--loop massive single scale Feynman diagrams with local operator insertion.