Abstract

A globally converging numerical method to solve coupled sets of non-linear integral equations is presented. Such systems occur, e.g., in the study of Dyson–Schwinger equations of Yang–Mills theory and QCD. The method is based on the knowledge of the qualitative properties of the solution functions in the far infrared and ultraviolet. Using this input, the full solutions are constructed using a globally convergent modified Newton iteration. Two different systems will be treated as examples: The Dyson–Schwinger equations of 3-dimensional Yang–Mills–Higgs theory provide a system of finite integrals, while those of 4-dimensional Yang–Mills theory at high temperatures are only finite after renormalization.

Keywords: Dyson–Schwinger equations; Non-linear integral equations; Globally convergent solution methods; Numerical solution methods; Coupled sets of integral equations

PACS classification codes: 02.30.Rz; 02.60.Cb; 11.15.Tk

Article Outline

1. Introduction

2. Dyson–Schwinger equations

2.1. 3-Dimensional Yang–Mills theory coupled to an adjoint, massive Higgs

2.2. Finite temperature Yang–Mills theory

3. Numerical method

3.1. Analytical foundation

3.2. Expansion and integration

3.3. Micro-, macro-, and super-cycles

3.4. Notes on implementation and optimization

3.5. Example calculation

4. 4-Dimensional Yang–Mills theory

5. Conclusions

Acknowledgements

Appendix A. Integral kernels and tadpoles of the 3-dimensional system

References