A Simple way to generate high order vacuum graphs

Kajantie K,, Laine M, Schroeder Y (2002)
Phys.Rev. D 65(4): 045008.

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We describe an efficient practical procedure for enumerating and regrouping vacuum Feynman graphs of a given order in perturbation theory. The method is based on a combination of Schwinger-Dyson equations and the two-particle-irreducible (skeleton) expansion. The regrouping leads to skeletons containing only free propagators, together with ring diagrams containing all the self-energy insertions. As a consequence, relatively few diagrams need to be drawn and integrations carried out at any single stage of the computation and, in low dimensions, overlapping ultraviolet/infrared subdivergences can be cleanly isolated. As an illustration we enumerate the graphs contributing to the 4-loop free energy in QCD, explicitly in a continuum and more compactly in a lattice regularization.
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Kajantie K,, Laine M, Schroeder Y. A Simple way to generate high order vacuum graphs. Phys.Rev. D. 2002;65(4):045008.
Kajantie, K. ,, Laine, M., & Schroeder, Y. (2002). A Simple way to generate high order vacuum graphs. Phys.Rev. D, 65(4), 045008. doi:10.1103/PhysRevD.65.045008
Kajantie, K. ,, Laine, M., and Schroeder, Y. (2002). A Simple way to generate high order vacuum graphs. Phys.Rev. D 65, 045008.
Kajantie, K.,, Laine, M., & Schroeder, Y., 2002. A Simple way to generate high order vacuum graphs. Phys.Rev. D, 65(4), p 045008.
K., Kajantie, M. Laine, and Y. Schroeder, “A Simple way to generate high order vacuum graphs”, Phys.Rev. D, vol. 65, 2002, pp. 045008.
Kajantie, K.,, Laine, M., Schroeder, Y.: A Simple way to generate high order vacuum graphs. Phys.Rev. D. 65, 045008 (2002).
Kajantie, K.,, Laine, Mikko, and Schroeder, York. “A Simple way to generate high order vacuum graphs”. Phys.Rev. D 65.4 (2002): 045008.
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