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

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 le...

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Bibliographic Details
Published inThe European physical journal. C, Particles and fields Vol. 36; no. 4; pp. 459 - 470
Main Authors van Eijk, E., Kleiss, R., Lazopoulos, A.
Format Journal Article
LanguageEnglish
Published Heidelberg Springer Nature B.V 01.08.2004
Subjects
Algorithms
Amplitudes
Asymptotic properties
Closed loops
Complexity
Feynman diagrams
Graphs
Symmetry
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Summary: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 1- and 2-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.
ISSN:1434-6044
1434-6052
DOI:10.1140/epjc/s2004-01958-2