Forcer, a FORM program for the parametric reduction of four-loop massless propagator diagrams
B. Ruijl, T. Ueda, J. A. M. Vermaseren
TL;DR
This paper presents Forcer, a FORM-based program for parametric reduction of four-loop massless propagator diagrams to master integrals. It combines precomputed reduction graphs with IBP-derived parametric identities, handling insertions and non-integer propagator powers and exploiting substructures such as the carpet, triangle, and diamond rules to reduce the majority of topologies. The authors propose a two-stage solving strategy (S0 and S1) with Gaussian elimination to suppress growth of rational coefficients in $\epsilon$, document 21 custom reductions and eight master integrals, and demonstrate applications to four-loop splitting functions, beta functions, and Mellin moments. They outline practical workflow to prepare diagrams for Forcer and discuss potential automation via AI techniques like Monte Carlo Tree Search, arguing the method extends toward five-loop calculations with substantial automation.
Abstract
We explain the construction of Forcer, a FORM program for the reduction of four-loop massless propagator-type integrals to master integrals. The resulting program performs parametric IBP reductions similar to the three-loop Mincer program. We show how one can solve many systems of IBP identities parametrically in a computer-assisted manner. Next, we discuss the structure of the Forcer program, which involves recognizing reduction actions for each topology, applying symmetries, and transitioning between topologies after edges have been removed. This part is entirely precomputed and automatically generated. We give examples of recent applications of Forcer, and study the performance of the program. Finally we demonstrate how to use the Forcer package and sketch how to prepare physical diagrams for evaluation by Forcer.