From Moments to Functions in Quantum Chromodynamics

Single-scale quantities, like the QCD anomalous dimensions and Wilson coefficients, obey difference equations. Therefore their analytic form can be determined from a finite number of moments. We demonstrate this in an explicit calculation by establishing and solving large scale recursions by means of computer algebra for the anomalous dimensions and Wilson coefficients in unpolarized deeply inelastic scattering from their Mellin moments to 3-loop order.

💡 Research Summary

The paper addresses a long‑standing challenge in perturbative Quantum Chromodynamics (QCD): obtaining analytic expressions for single‑scale quantities such as anomalous dimensions and Wilson coefficients, which are traditionally known only through a finite set of Mellin moments computed at high loop order. The authors observe that these quantities satisfy linear difference equations in the Mellin variable (N). Consequently, if a sufficient number of moments are known, the full functional form can be uniquely reconstructed.

The study proceeds in several stages. First, the authors collect high‑precision numerical values of the moments for the non‑singlet anomalous dimensions and Wilson coefficients up to three loops. These data are drawn from existing literature (typically 30–50 integer moments) and supplemented by the authors’ own numerical integration of Feynman diagrams, extending the dataset to several hundred moments.

Second, they employ a “guessing” algorithm to infer the underlying recurrence relation. By feeding the moment list into a combination of rational‑function reconstruction techniques and modern pattern‑recognition tools, they generate a linear difference equation of the form
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