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Manuscript Title: Feynman graph generation and calculations in the Hopf algebra of Feynman graphs
Authors: Michael Borinsky
Program title: feyngen, feyncop
Catalogue identifier: AEUB_v1_0
Distribution format: tar.gz
Journal reference: Comput. Phys. Commun. 185(2014)3317
Programming language: Python.
Computer: PC.
Operating system: Unix, GNU/Linux.
RAM: 64m bytes
Keywords: Quantum Field Theory, Feynman graphs, Feynman diagrams, Hopf algebra, Renormalization, BPHZ.
PACS: 11.10.Gh, 11.15.Bt.
Classification: 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.

References:
[1] McKay, B.D., Practical Graph Isomorphism, Congressus Numerantium, 30 (1981) 45-87
[2] Connes and D. Kreimer, Renormalization in quantum eld theory and the RiemannHilbert problem I. The Hopf algebra structure of graphs and the main theorem, Commun. Math. Phys. 210 (2000) 249273. Letters in Mathematical Physics 103 (9) (2013) 9331007