[Paper Review] Evaluation of Multi-Sums for Large Scale Problems

J. Blümlein, A. Hasselhuhn+1 more|arXiv (Cornell University)|Feb 20, 2012

Particle physics theoretical and experimental studies16 citations

TL;DR

This paper presents a fully automated symbolic summation method based on difference field theory to simplify massive multi-sum expressions arising from 3-loop Feynman integrals in quantum field theory. Using the Mathematica packages EvaluateMultiSums and SumProduction, it transforms complex multi-sums into compact nested product-sum expressions in terms of harmonic, S-, and cyclotomic sums, enabling efficient ε-expansion of large-scale amplitudes—demonstrated by a complete 3-loop calculation of fermionic gluonic operator matrix elements.

ABSTRACT

A big class of Feynman integrals, in particular, the coefficients of their Laurent series expansion w.r.t.\ the dimension parameter $\ep$ can be transformed to multi-sums over hypergeometric terms and harmonic sums. In this article, we present a general summation method based on difference fields that simplifies these multi--sums by transforming them from inside to outside to representations in terms of indefinite nested sums and products. In particular, we present techniques that assist in the task to simplify huge expressions of such multi-sums in a completely automatic fashion. The ideas are illustrated on new calculations coming from 3-loop topologies of gluonic massive operator matrix elements containing two fermion lines, which contribute to the transition matrix elements in the variable flavor scheme.

Motivation & Objective

• To develop a fully automated method for simplifying large, complex multi-sum expressions arising from 3-loop Feynman integrals in quantum field theory.
• To enable efficient computation of Laurent series coefficients in ε (dimensional regularization) for massive operator matrix elements.
• To reduce the computational cost and memory footprint of large-scale amplitude calculations by transforming multi-sums into algebraically independent nested product-sums.
• To support mass production of such calculations through scalable, parallelizable routines for multi-sum evaluation and coefficient extraction.
• To provide a systematic framework for representing results in terms of well-known special functions like harmonic, S-, and cyclotomic sums.

Proposed method

• Leverages difference field theory and ΠΣ-fields to symbolically simplify multi-sums from Feynman integrals into indefinite nested product-sum expressions.
• Employs the Sigma package for recurrence-based symbolic summation and creative telescoping to derive and solve linear recurrences for multi-sums.
• Introduces a two-stage pipeline: (1) reduction of 2419 multi-sums to 29 key sums via range synchronization and algebraic basis simplification, and (2) parallel ε-expansion of key sums using EvaluateMultiSums.
• Uses SumProduction to manage large-scale computations by splitting expressions into hypergeometric terms h(n, i1, i2, ε) and rational functions r(n, i1, i2, ε), enabling modular processing.
• Applies harmonic sum and cyclotomic sum reduction via the HarmonicSums package to express final results in terms of S1(n), S2(n), S3(n), etc.
• Employs a fault-tolerant, file-based parallel execution model where each sum is processed independently and results are combined post-hoc.

Experimental results

Research questions

• RQ1 How can multi-sum expressions from 3-loop Feynman integrals be systematically simplified to algebraically independent nested product-sums?
• RQ2 What symbolic summation techniques enable the automatic derivation of Laurent series coefficients in ε for massive operator matrix elements?
• RQ3 Can a scalable, parallelized framework be constructed to handle thousands of multi-sums in large-scale quantum field theory calculations?
• RQ4 What is the performance and memory efficiency of transforming a 2 GB expression of 2419 multi-sums into a compact representation in terms of harmonic and S-sums?
• RQ5 How can pole structures and summation bounds be managed robustly during recursive simplification of multi-sums with complex index dependencies?

Key findings

• The method successfully reduced a 2 GB expression of 2419 multi-sums to a 7.6 MB compact expression containing only 29 sums and 15 rational terms, completing in 6 hours and 53 minutes.
• The ε-expansion of all key sums was computed in parallel in 2 hours and 35 minutes, with each sum's expansion taking time comparable to processing a single typical sum.
• The final result, combining all sub-results, required only 100 KB of memory and was expressed in terms of ζ2, ζ3, (−1)^n, S1(n), S2(n), S3(n), S2,1(n), S3,1(n), and S2,1,1(n).
• The entire calculation, including reduction, expansion, and combination, was completed in approximately 9.5 hours, demonstrating feasibility for large-scale production.
• The method achieved full automation with no failure in recurrence solving, and any failure was traced to suboptimal sum representations, which could be improved to restore success.
• The approach enables the computation of previously intractable 3-loop amplitudes, such as fermionic contributions to gluonic massive operator matrix elements, in a fully symbolic and efficient manner.