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Calculating four-loop massless propagators with Forcer

T. Ueda, B. Ruijl, J. A. M. Vermaseren

TL;DR

The paper introduces Forcer, a FORM-based program that extends the three-loop massless propagator toolkit Mincer to four loops. It employs automated code generation and a topology-tree framework to manage a large set of topologies, utilizing substructure reductions (one-loop insertion, carpet, triangle, diamond) and IBP-based manual rules when needed. The authors validate Forcer by recomputing known four-loop quantities such as the QCD beta function and anomalous dimensions, confirming correctness. This work provides a practical analytic tool for four-loop massless propagator computations, enabling high-precision perturbative QCD results and facilitating future advanced calculations like four-loop splitting function moments. The topology-driven automation addresses the complexity barrier inherent in four-loop reductions, marking a significant methodological advance in multi-loop Feynman integral calculations.

Abstract

We present Forcer, a new FORM program for the calculation of four-loop massless propagators. The basic framework is similar to that of the Mincer program for three-loop massless propagators: the program reduces Feynman integrals to a set of master integrals in a parametric way. To overcome an ineludible complexity of the program structure at the four-loop level, most of the code was automatically generated or made with computer-assisted derivations. Correctness of the program has been checked with the recomputation of some quantities in the literature.

Calculating four-loop massless propagators with Forcer

TL;DR

The paper introduces Forcer, a FORM-based program that extends the three-loop massless propagator toolkit Mincer to four loops. It employs automated code generation and a topology-tree framework to manage a large set of topologies, utilizing substructure reductions (one-loop insertion, carpet, triangle, diamond) and IBP-based manual rules when needed. The authors validate Forcer by recomputing known four-loop quantities such as the QCD beta function and anomalous dimensions, confirming correctness. This work provides a practical analytic tool for four-loop massless propagator computations, enabling high-precision perturbative QCD results and facilitating future advanced calculations like four-loop splitting function moments. The topology-driven automation addresses the complexity barrier inherent in four-loop reductions, marking a significant methodological advance in multi-loop Feynman integral calculations.

Abstract

We present Forcer, a new FORM program for the calculation of four-loop massless propagators. The basic framework is similar to that of the Mincer program for three-loop massless propagators: the program reduces Feynman integrals to a set of master integrals in a parametric way. To overcome an ineludible complexity of the program structure at the four-loop level, most of the code was automatically generated or made with computer-assisted derivations. Correctness of the program has been checked with the recomputation of some quantities in the literature.

Paper Structure

This paper contains 9 sections, 2 figures.

Table of Contents

  1. Introduction
  2. Reduction for each topology
  3. One-loop insertion integral
  4. One-loop carpet integral
  5. Triangle rule
  6. Diamond rule
  7. Manual rules
  8. Code generation for all topologies
  9. Conclusion

Figures (2)

  • Figure 1: Substructures of topologies that lead to reduction to simpler topologies. (a) One-loop insertion. A line with a non-integer power is indicated with "$\ast$". (b) One-loop carpet. (c) Triangle. The triangle rule removes one of the three lines indicated by arrows. (d) Two-loop diamond. The diamond rule removes one of the six lines indicated by arrows.
  • Figure 2: The tree-like graph structure of topology transitions at the four-loop level. The lower figure shows the enlarged view of the boxed region in the upper one. Green, cyan, white, yellow and red circles represent topologies in which one-loop insertion integral, carpet integral, triangle rule, diamond rule and none of them are available, respectively.