[Paper Review] Forcer: a FORM program for 4-loop massless propagators

Takahiro Ueda, Ben Ruijl+1 more|arXiv (Cornell University)|Jul 25, 2016

Particle physics theoretical and experimental studies38 references13 citations

TL;DR

This paper presents Forcer, a specialized FORM program for efficiently computing four-loop massless propagator-type Feynman integrals using parametric reduction techniques inspired by the three-loop Mincer program. By automating topology classification, reduction flow construction, and symbolic integration-by-parts (IBP) rule derivation—especially for complex topologies—it achieves high performance and correctness, validated by reproducing known results like the four-loop QCD β-function and non-singlet splitting functions.

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.

Motivation & Objective

• To develop a highly efficient, specialized program for four-loop massless propagator-type Feynman integrals, overcoming bottlenecks in general IBP solvers.
• To extend the three-loop Mincer algorithm to the four-loop level, maintaining symbolic parametric reduction while handling increased topological and algebraic complexity.
• To automate the generation of reduction rules and topology classification using graph-theoretic methods and computer-assisted derivations to minimize manual coding errors.
• To validate the program by reproducing established results in quantum field theory, such as the four-loop QCD β-function and splitting functions.
• To enable faster computation of higher-order corrections in quantum chromodynamics, particularly for high-multiplicity and high-moment calculations.

Proposed method

• Representing all four-loop topologies as undirected graphs to systematically classify and detect reducible substructures like one-loop insertions, carpets, triangles, and diamonds via pattern matching.
• Implementing a hierarchical reduction flow that prioritizes standard reductions (e.g., triangle and diamond rules) before applying custom symbolic IBP rules for irreducible topologies.
• Using graph isomorphism detection to eliminate redundant topologies during reduction flow construction, improving code efficiency and maintainability.
• Generating reduction rules symbolically with indices (propagator powers and irreducible numerators) as parameters, enabling reuse across different integral configurations.
• Applying computer-assisted derivation with human-guided optimization to construct efficient reduction schemes for a minimal set of non-trivial topologies.
• Leveraging the background field method and gauge parameter independence to cross-validate results, particularly for the four-loop QCD β-function.

Experimental results

Research questions

• RQ1 How can the four-loop massless propagator integral reduction problem be efficiently solved using a programmatic extension of the three-loop Mincer algorithm?
• RQ2 What automated methods can be used to classify four-loop topologies and construct their reduction flow while minimizing manual intervention?
• RQ3 Which symbolic integration-by-parts (IBP) rules are necessary for irreducible topologies, and how can they be derived efficiently with minimal human error?
• RQ4 Can the program achieve high performance and correctness for high-multiplicity, high-moment calculations such as the four-loop splitting functions?
• RQ5 How can the correctness and robustness of the reduction framework be validated against known physical results?

Key findings

• Forcer successfully computes four-loop massless propagator integrals with high efficiency, reproducing the four-loop QCD β-function in under 10 minutes on a 24-core machine when gauge parameters are dropped.
• The program achieves full correctness by recomputing known results, including the four-loop QCD β-function and lower moments of the non-singlet splitting functions, serving as a robust benchmark.
• The reduction of the three-loop non-planar topology into one master integral and simpler topologies is achieved via a key recurrence relation (equation 10), which systematically reduces complexity by decreasing indices.
• The program’s design ensures that coefficients in reduction rules remain finite (no poles) due to constraints on index values, such as $ n_2 + n_5 + n_8 - 3 \geq 1 $, ensuring numerical stability.
• The use of graph-based topology classification and isomorphism detection significantly reduces code bloat and improves maintainability, especially given the exponential growth in topologies at four loops.
• Forcer demonstrates that a hybrid approach—combining automated topology analysis with human-optimized symbolic IBP rules—yields superior performance compared to general IBP solvers for this specific class of integrals.