Determining the closed forms of the $O(a_s^3)$ anomalous dimensions and Wilson coefficients from Mellin moments by means of computer algebra

Single scale quantities, as anomalous dimensions and hard scattering cross sections, in renormalizable Quantum Field Theories are found to obey difference equations of finite order in Mellin space. It is often easier to calculate fixed moments for these quantities compared to a direct attempt to derive them in terms of harmonic sums and their generalizations involving the Mellin parameter $N$. Starting from a sufficiently large number of given moments, we establish linear recurrence relations of lowest possible order with polynomial coefficients of usually high degree. Then these recurrence equations are solved in terms of d’Alembertian solutions where the involved nested sums are represented in optimal nested depth. Given this representation, it is then an easy task to express the result in terms of harmonic sums. In this process we compactify the result such that no algebraic relations occur among the sums involved. We demonstrate the method for the QCD unpolarized anomalous dimensions and massless Wilson coefficients to 3–loop order treating the contributions for individual color coefficients. For the most complicated subproblem 5114 moments were needed in order to produce a recurrence of order 35 whose coefficients have degrees up to 938. About four months of CPU time were needed to establish and solve the recurrences for the anomalous dimensions and Wilson coefficients on a 2 GHz machine requiring less than 10 GB of memory. No algorithm is known yet to provide such a high number of moments for 3–loop quantities. Yet the method presented shows that it is possible to establish and solve recurrences of rather large order and and degree, occurring in physics problems, uniquely, fast and reliably with computer algebra.

💡 Research Summary

The paper presents a systematic computer‑algebra framework for reconstructing closed‑form expressions of three‑loop QCD anomalous dimensions and massless Wilson coefficients from a finite set of Mellin moments. The authors start from the observation that single‑scale quantities in renormalizable quantum field theories satisfy linear difference equations of finite order when expressed in Mellin‑space. While direct analytic calculations of these quantities in terms of harmonic sums and their generalisations are extremely cumbersome at three loops, fixed integer moments are comparatively easy to obtain with existing tools such as MINCER or FORCER.

The methodology proceeds in four stages. First, a sufficiently large number of moments is generated for each colour structure. For the most demanding sub‑problem the authors computed 5 114 moments, which required several weeks of CPU time on a 2 GHz workstation. Second, from these data a linear recurrence relation of minimal order is guessed. The recurrence has the generic form
\