CrasyDSE: A framework for solving Dyson–Schwinger equations

Abstract

Dyson–Schwinger equations are important tools for non-perturbative analyses of quantum field theories. For example, they are very useful for investigations in quantum chromodynamics and related theories. However, sometimes progress is impeded by the complexity of the equations. Thus automating parts of the calculations will certainly be helpful in future investigations. In this article we present a framework for such an automation based on a C++ code that can deal with a large number of Green functions. Since also the creation of the expressions for the integrals of the Dyson–Schwinger equations needs to be automated, we defer this task to a Mathematica notebook. We illustrate the complete workflow with an example from Yang–Mills theory coupled to a fundamental scalar field that has been investigated recently. As a second example we calculate the propagators of pure Yang–Mills theory. Our code can serve as a basis for many further investigations where the equations are too complicated to tackle by hand. It also can easily be combined with DoFun, a program for the derivation of Dyson–Schwinger equations.1Program summaryProgram title: CrasyDSECatalogue identifier: AEMY _v1_0Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AEMY_v1_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 49030No. of bytes in distributed program, including test data, etc.: 303958Distribution format: tar.gzProgramming language: Mathematica 8 and higher, C++.Computer: All on which Mathematica and C++ are available.Operating system: All on which Mathematica and C++ are available (Windows, Unix, Mac OS).Classification: 11.1, 11.4, 11.5, 11.6.Nature of problem: Solve (large) systems of Dyson–Schwinger equations numerically.Solution method: Create C++ functions in Mathematica to be used for the numeric code in C++. This code uses structures to handle large numbers of Green functions.Unusual features: Provides a tool to convert Mathematica expressions into C++ expressions including conversion of function names.Running time: Depending on the complexity of the investigated system solving the equations numerically can take seconds on a desktop PC to hours on a cluster.

Vertex functions of Coulomb gauge Yang-Mills theory

“…The final result is obtained by a fixed point iteration. All calculations w ere perform ed with the crasydse fram ew ork [20], Further num erical details can also be found in Ref. [20] and references therein.…”

“…Hence there is only one relevant dressing function for the full ghost-gluon vertex, which can be chosen as antighost and gluon momenta and the angle between them will be used: D£cA(|p|, |k |,a). To obtain a scalar integral equation for the dressing function, the CRDSE (17) gpjtjiWpi' (20) This results in the following integral equation:…”

Gauge bosons at zero and finite temperature

“…Last, but not least, most of these steps are rather similar, and can thus be automatized [240], though still great care is required when dealing with both the physical and numerical particulars of a given system, as discussed here.…”

An exploratory study of Yang-Mills three-point functions at non-zero temperature

“…Computation were performed with DoFun [46,47] and CrasyDSE [48] using HPC Clusters at the University of Graz. Feynman diagrams were created with Jaxodraw [49].…”