At Johannes Gutenberg University Mainz (JGU), physicists have introduced a new way to organize the notorious “Feynman integrals” that underpin high-energy precision predictions. These integrals act as the mathematical backbone for translating particle interactions into numerical results that can be compared with experiments. Until now, a major bottleneck was simply deciding the order in which the integrals should be processed by computer algebra systems.
In the PRISMA++ Cluster of Excellence, Professor Stefan Weinzierl and colleagues report that their approach accelerates computations dramatically—by roughly a factor of 1,000. The impact is practical as well as theoretical: depending on the scattering process, researchers may need to evaluate from thousands up to nearly a million integrals. A speedup at this scale can turn previously infeasible calculations into routine workflows.
The core idea is to sort integrals using intrinsic geometric properties rather than relying on “ad-hoc” labels. Weinzierl likens the method to organizing a library by content: instead of sorting books by superficial metadata, the algorithm “looks inside” each integral by analyzing its geometric structure. This internal geometric viewpoint enables computer algebra programs to simplify the governing equations much more effectively.
Technically, the method is built as a two-step algorithm. First, a new geometric order relation guides the reduction of integrals toward a basis of so-called master integrals. Once expressed in this structured basis, the associated differential equations can be written as a Laurent polynomial in the regularization parameter ε (epsilon).
Second, the team introduces a procedure to “trivialize” the ε-dependence of those differential equations. Together, the steps yield an epsilon-factorized form—an arrangement known to be easier to integrate systematically and reliably. The authors emphasize that the procedure is algorithmic, meaning it can be applied across a wide class of Feynman integrals rather than being tailored to a single problem.
The result is a more scalable computational pipeline for precision calculations in particle physics. Weinzierl notes that the method can enable predictions for far more processes than previously possible, extending the reach of theoretical efforts supporting cutting-edge measurements at facilities such as the Large Hadron Collider.
Subject of Research: Not applicable
Article Title: New algorithms for Feynman integral reduction and epsilon-factorized differential equations
News Publication Date: 15-Jun-2026
Web References: http://dx.doi.org/10.1103/mjpn-61yv
References: 10.1103/mjpn-61yv
Image Credits: Ill./©: JGU
Keywords: Feynman integrals, epsilon-factorized differential equations, master integrals, computer algebra, geometric ordering, particle physics precision, PRISMA++
