Forcer: a FORM program for 4-loop massless propagators

TL;DR

The paper presents Forcer, a specialized FORM program that extends the Mincer approach to four-loop massless propagator-type Feynman integrals by implementing parametric IBP reductions to master integrals. It automates topology classification, reduction-flow construction, and the derivation of special reduction rules, using graph-based techniques and symbolic IBP combinations to generate efficient, topology-specific rules. The method is validated by reproducing known four-loop QCD quantities, including the $eta$-function and splitting-function moments, demonstrating substantial gains in reduction speed and scalability for high-order perturbative calculations. This work enables more precise and tractable four-loop computations, with broader implications for high-precision QCD phenomenology.

Abstract

We present a new FORM program for analytically evaluating four-loop massless propagator-type Feynman integrals in an efficient way. Our program Forcer implements parametric reductions of the aforementioned class of Feynman integrals into a set of master integrals and can be considered as a four-loop extension of the three-loop Mincer program. Since the program structure at the four-loop level is highly complicated and the equations easily become lengthy, most of the code was generated in an automatic way or with computer-assisted derivations. We have checked correctness of the program by recomputing already-known quantities in the literature.

Figures (2)

• Figure 1: Substructures of topologies leading to reduction into simpler ones: (a) one-loop insertion, (b) one-loop carpet, (c) triangle and (d) diamond.
• Figure 2: The three-loop non-planar topology.