Feynman graph generation and calculations in the Hopf algebra of Feynman graphs

Abstract

Two programs for the computation of perturbative expansions of quantum field theory amplitudes are provided. feyngen can be used to generate Feynman graphs for Yang–Mills, QED and φk theories. Using dedicated graph theoretic tools feyngen can generate graphs of comparatively high loop orders. feyncop implements the Hopf algebra of those Feynman graphs which incorporates the renormalization procedure necessary to calculate finite results in perturbation theory of the underlying quantum field theory. feyngen is validated by comparison to explicit calculations of zero dimensional quantum field theories and feyncop is validated using a combinatorial identity on the Hopf algebra of graphs. Program summaryProgram title:feyngen, feyncopCatalogue identifier: AEUB_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEUB_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.: 2657No. of bytes in distributed program, including test data, etc.: 22606Distribution format: tar.gzProgramming language: Python.Computer: PC.Operating system: Unix, GNU/Linux.RAM: 64 m bytesClassification: 4.4.External routines: nauty [1], geng, multig (part of the nauty package)Nature of problem:Performing explicit calculations in quantum field theory Feynman graphs are indispensable. Infinities arising in the perturbative calculations make renormalization necessary. On a combinatorial level renormalization can be encoded using a Hopf algebra [2] whose coproduct incorporates the BPHZ procedure. Upcoming techniques initiated an interest in relatively large loop order Feynman diagrams which are not accessible by traditional tools.Solution method:Both programs use the established nauty package to ensure high performance graph generation at high loop orders. feyngen is capable of generating φk-theory, QED and Yang–Mills Feynman graphs and of filtering these graphs for the properties of connectedness, one-particle-irreducibleness, 2-vertex-connectivity and tadpole-freeness. It can handle graphs with fixed external legs as well as those without fixed external legs.feyncop uses basic graph theoretical algorithms to compute the coproduct of graphs encoding their Hopf algebra structure.Running time:All 130516 1PI, φ4, 8-loop diagrams with four external legs can be generated, together with their symmetry factor, by feyngen within eight hours and all 342430 1PI, QED, vertex residue type, 6-loop diagrams can be generated in three days both on a standard end-user PC. feyncop can calculate the coproduct of all 2346 1PI, φ4, 8-loop diagrams with four external legs within ten minutes.