Source-linked AI summary
Modern Summation Methods and the Computation of 2- and 3-loop Feynman Diagrams
J. Ablinger, J. Blümlein, S. Klein, C. Schneider
TL;DR
The paper addresses how to simplify definite multi-sums from massive 2- and 3-loop Feynman-diagram calculations. It develops difference-field algorithms that transform them into indefinite nested product-sums and reports success across the concrete calculations studied. The resulting representations extend beyond harmonic sums and S-sums for future computations.
Problem
Method
The method combines Sigma’s ΠΣ∗-difference-field algorithms, telescoping, creative telescoping, recurrence solving, and HarmonicSums transformations.
Source: §1 · p. 1, §2.1 · p. 3Results
Takeaways & Limitations
The resulting product-sum representations substantially extend the class of harmonic sums and S-sums used in subsequent calculations.
Source: §4 · p. 6The approach may fail to represent intermediate sums, derive suitable recurrences, or find enough expressible recurrence solutions, despite succeeding on all encountered examples.
Source: §2.1 · p. 3
Abstract
from arXiv · showhide
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.
1. Introduction
The paper targets simplifying definite multi-sums arising in massive 2- and 3-loop Feynman diagrams into indefinite nested sums and products. Its difference-field machinery combines symbolic summation with later reduction to harmonic sums when possible.
1. Introduction
- The calculations concern massive single-scale 2- and 3-loop Feynman diagrams with local operator insertion, whose expressions depend on Mellin variable N.These diagrams relate to QCD anomalous dimensions and massive operator matrix elements.Source: §1 · p. 1
- Definite nested sums over hypergeometric expressions and harmonic sums are sought in simpler forms that can be processed further in physics.At 2-loop order, completed calculations lead to harmonic sums, whereas this need not hold from 3-loop order onward.Source: §1 · p. 1
- Although intermediate expressions may leave the harmonic-sum solution space, combining the results can return them to harmonic sums or generalized versions.Source: §1 · p. 1
- The developed machinery uses ΠΣ∗-difference-field algorithms, Sigma, and HarmonicSums to transform sums step by step and reduce results to harmonic sums or generalized versions when possible.Eliminating algebraic relations among harmonic sums is part of the combination process.Source: §1 · p. 1
2. Simplification of multisums
The section presents a recursive Sigma-based strategy for converting definite nested multi-sums into indefinite nested product-sums, using creative telescoping, recurrence solving, and depth-optimal difference fields. A worked example applies these steps from an innermost sum through the original triple sum.
2.1. The underlying summation principles
- Creative telescoping searches for parameter-free summand recurrences whose telescoping yields recurrences for the summed function.In Sigma’s generalized setting, both recurrence coefficients and certificates may be indefinite nested product-sum expressions.Source: §2.1 · pp. 2–3
- Depth-optimal ΠΣ∗-difference fields produce algebraically independent representations with minimal nesting depth, simplifying otherwise highly nested recurrence solutions.Without refinement, the worst-case nesting depth can reach d −1.Source: §2.1 · p. 3
- Solutions are combined by matching the first d initial values, after which recurrence uniqueness establishes agreement for all N ≥0 under mild side conditions.Source: §2.1 · p. 3
- Sigma processes summation quantifiers from inner to outer levels, transforming definite multi-sums into indefinite nested product-sums through ΠΣ∗-difference-field algorithms.Creative telescoping derives recurrences, and recurrence solving supplies d’Alembertian solutions that can be simplified by refined telescoping.Source: §2.1 · pp. 2–3
- The strategy can fail when intermediate representations, recurrence searches, or sufficiently many nested product-sum solutions cannot be obtained, although it did not fail in the reported 2- and 3-loop computations.Source: §2.1 · p. 3
2.2. A detailed example
- In the detailed example, the innermost sum is converted to hypergeometric terms, the resulting single and double sums are recursively treated, and the procedure reaches a closed form for the triple sum.The example repeatedly computes and solves recurrences while determining required initial values and constants.Source: §2.2 · p. 4
3. Some recent calculations
The calculations automate the reduction of thousands of multi-sums from three-loop diagrams to indefinite nested sums and algebraically independent harmonic-sum representations. Two example classes show that extensive intermediate sums can collapse to compact final forms.
3. Some recent calculations
- Several thousand multi-sums were simplified mechanically to indefinite nested product-sum expressions using Sigma, HarmonicSums, and the EvaluateMultiSums package.EvaluateMultiSums combines methods from Sigma and HarmonicSums to perform these computations automatically.Source: §3 · p. 4
3.1. All N-Results for 3–Loop Ladder Graphs
- About 1000 sums were produced automatically; Sigma simplified them, transforming the sums into 145 S-sums, whose combination yielded an expression containing 533 S-sums.All 1000 sums can be represented in terms of generalized harmonic sums.Source: §3.1 · p. 5
- Algebraic-independence relations among harmonic and S-sums make all S-sums cancel, leaving a result for the ladder diagram within seconds.HarmonicSums uses precomputed representations up to specified weights to obtain algebraically independent forms.Source: §3.1 · p. 5
3.2. 3-Loop All N-Results for the Nf Contributions
- Combining 768 multi-sums, the final expression for (14) requires 9 MB of memory and is represented by 703 indefinite nested product-sums, including harmonic sums and S-sums.The sums are algebraically independent, so all unexpected sums cancel, yielding a compact form.Source: §3.2 · p. 5
4. Conclusion
The authors show that Sigma transforms multi-sums into indefinite nested product-sums through telescoping, creative telescoping, and recurrence solving in ΠΣ∗-difference fields. The mechanism worked for all reported 2-loop and 3-loop calculations, with EvaluateMultiSums and HarmonicSums supporting automation and representation.
4. Conclusion
- Sigma combines telescoping, creative telescoping, and recurrence solving in ΠΣ∗-difference fields to transform multi-sums into indefinite nested product-sum expressions.These product-sum representations extend beyond harmonic sums and S-sums.Source: §4 · p. 6
- The transformation mechanism worked for all performed 2-loop and 3-loop calculations.EvaluateMultiSums automates the computations, while HarmonicSums helps find harmonic-sum and S-sum representations in optimal form.Source: §4 · p. 6