Counting loop diagrams: computational complexity of higher-order amplitude evaluation

Details Counting loop diagrams: computational complexity of higher-order amplitude evaluation Counting loop diagrams: computational complexity of higher-order amplitude evaluation

2004-06-02

arxiv.org/abs/hep-ph/0406025v3

Eur.Phys.J. C36 (2004) 459-470

Physics

High Energy Physics

High Energy Physics - Phenomenology

27 pages, 17 tables

Scientific paper

10.1140/epjc/s2004-01958-2

We discuss the computational complexity of the perturbative evaluation of scattering amplitudes, both by the Caravaglios-Moretti algorithm and by direct evaluation of the individual diagrams. For a self-interacting scalar theory, we determine the complexity as a function of the number of external legs. We describe a method for obtaining the number of topologically inequivalent Feynman graphs containing closed loops, and apply this to one- and two-loop amplitudes. We also compute the number of graphs weighted by their symmetry factors, thus arriving at exact and asymptotic estimates for the average symmetry factor of diagrams. We present results for the asymptotic number of diagrams up to 10 loops, and prove that the average symmetry factor approaches unity as the number of external legs becomes large.

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