Difference Equations in Massive Higher Order Calculations

The calculation of massive 2–loop operator matrix elements, required for the higher order Wilson coefficients for heavy flavor production in deeply inelastic scattering, leads to new types of multiple infinite sums over harmonic sums and related functions, which depend on the Mellin parameter $N$. We report on the solution of these sums through higher order difference equations using the summation package {\tt Sigma}.

💡 Research Summary

The paper addresses the formidable problem of evaluating massive two‑loop operator matrix elements (OMEs) that appear in the higher‑order Wilson coefficients for heavy‑flavor production in deep‑inelastic scattering (DIS). When these OMEs are expressed in Mellin‑N space, they give rise to a new class of infinite nested sums that depend on the Mellin variable N. Unlike the standard harmonic sums S_{a1,…,ak}(N) encountered in massless calculations, the massive case introduces additional weight factors and altered index structures that render direct summation impractical.

To overcome this obstacle, the authors recast each infinite sum as a linear (or, in some cases, nonlinear) difference equation in N. By systematically applying the theory of difference equations, they derive higher‑order recurrences (typically third‑ or fourth‑order) whose coefficients are rational functions of N, ordinary harmonic sums, and multiple zeta values (ζ‑constants). The required initial conditions are obtained from explicit low‑N evaluations (N = 1, 2, 3, …).

The central computational tool is the symbolic summation package Sigma, which implements advanced algorithms such as creative telescoping, recurrence solving, and algebraic reduction of nested sums. Using Sigma, the authors perform the following steps: (1) separate the homogeneous part of the recurrence from the inhomogeneous (free‑term) contribution; (2) apply creative telescoping to reduce the order of the recurrence to a minimal canonical form; (3) transform the infinite sums into finite combinations of harmonic sums and multiple ζ‑values; (4) fix the integration constants by matching the low‑N initial data. The result is a closed‑form expression for each original sum, expressed entirely in terms of known special functions.

A further crucial aspect is analytic continuation. Harmonic sums are defined for integer N, but the physical application requires their extension to complex N in order to perform the inverse Mellin transform back to Bjorken‑x space. The authors exploit Sigma’s built‑in transformation rules to rewrite the harmonic sums as combinations of multiple polylogarithms and Beta‑function representations, thereby providing a valid analytic continuation across the complex N‑plane.

With these closed‑form results, the massive two‑loop OMEs can be inserted directly into the Mellin‑space representation of the Wilson coefficients. The inverse Mellin transform then yields the heavy‑flavor contributions to the DIS structure functions with unprecedented analytic precision. Importantly, the method dramatically reduces computational complexity compared with brute‑force numerical integration or term‑by‑term series expansion, and it is fully automatable.

The authors demonstrate the efficacy of the approach by presenting explicit examples of massive sums that were previously intractable. In each case, Sigma produces compact expressions involving only a handful of harmonic sums and ζ‑values, confirming the correctness of the recurrence‑based solution.

Beyond the immediate application to two‑loop heavy‑flavor OMEs, the paper establishes a general framework for tackling higher‑order (three‑loop, four‑loop) massive calculations. The combination of higher‑order difference equations and Sigma’s symbolic summation capabilities offers a scalable, systematic pathway to obtain analytic results for a wide class of Feynman integrals with mass scales.

In summary, the work showcases how modern symbolic summation technology can transform the evaluation of intricate infinite sums arising in massive higher‑order perturbative QCD. By converting these sums into solvable difference equations and providing analytic continuations, the authors deliver closed‑form expressions for the required OMEs, thereby enabling precise, efficient computation of heavy‑flavor Wilson coefficients in DIS and setting the stage for future high‑precision QCD analyses.