Evaluation of Multi-Sums for Large Scale Problems

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.

💡 Research Summary

The paper addresses a long‑standing bottleneck in high‑order perturbative quantum field theory: the simplification of massive three‑loop Feynman integrals that, after expansion in the dimensional regulator ε, give rise to extremely large multi‑sums over hypergeometric terms and harmonic sums. Traditional manual or semi‑automatic methods quickly become infeasible when the number of nested summations reaches the order of 10⁵–10⁶, as is typical for gluonic massive operator matrix elements (OMEs) containing two fermion lines.

The authors propose a fully automatic symbolic summation framework built on the theory of difference fields. By representing each summand as an element of a difference field, they can apply systematic difference‑operator transformations that move inner summations outward (“inside‑to‑outside” transformation). This process is realized with the Σ‑package, which implements algorithms such as creative telescoping, recurrence solving, and reduction to normal forms. The key steps are:

1. Pre‑processing – map hypergeometric ratios and harmonic‑sum indices to difference‑field variables, ensuring that the summand satisfies the necessary shift‑invariance properties.
2. Difference‑operator application – use Σ’s rsolve and telescoping routines to rewrite inner sums as differences of outer‑sum expressions, thereby generating boundary terms that are automatically simplified.
3. Harmonic‑sum reduction – exploit known algebraic relations among harmonic sums, expressible in terms of multiple zeta values (MZVs), to replace deep nested sums with compact closed forms whenever possible.
4. Parallelisation and memory management – split the overall summation tree into independent sub‑tasks processed on multiple cores; intermediate results are stored in compressed difference‑field normal forms to avoid exponential memory growth.

The methodology is demonstrated on a concrete physics problem: the three‑loop gluonic massive OME with two fermion lines, which contributes to the transition matrix elements in the variable‑flavour-number scheme. The original expressions, before simplification, span several hundred pages and contain thousands of nested sums. After applying the difference‑field based pipeline, the authors obtain a representation that fits within a few dozen pages, consisting solely of indefinite nested sums, products, and a small set of MZVs. The resulting formulas are directly usable for numerical evaluation and for further analytic manipulations required in phenomenological applications.

Beyond the specific example, the authors discuss scalability. The underlying algorithms impose no intrinsic limit on the depth of nesting or the degree of the hypergeometric terms, suggesting that the same framework can be extended to four‑loop calculations or to problems involving several mass scales. The main practical challenges at higher loops are the increased combinatorial explosion of intermediate terms and the need for more sophisticated parallel‑execution strategies, which the authors outline as future work.

In summary, the paper delivers a robust, fully automated toolchain that transforms unwieldy multi‑sum representations of high‑order Feynman integrals into compact expressions built from indefinite nested sums, products, and known constants. By leveraging difference‑field theory, the authors overcome the manual bottleneck that has limited progress in precision QCD calculations, opening the door to systematic treatment of even more complex diagrams in the near future.