Counting planar diagrams with various restrictions

Details Counting planar diagrams with various restrictions Counting planar diagrams with various restrictions

1998-08-19

arxiv.org/abs/hep-th/9808113v1

Nucl.Phys. B538 (1999) 389-410

Physics

High Energy Physics

High Energy Physics - Theory

24 pages TeX, 16 figures Postscript

Scientific paper

10.1016/S0550-3213(98)00697-X

Explicit expressions are considered for the generating functions concerning the number of planar diagrams with given numbers of 3- and 4-point vertices. It is observed that planar renormalization theory requires diagrams with restrictions, in the sense that one wishes to omit `tadpole' inserions and `seagull' insertions; at a later stage also self-energy insertions are to be removed, and finally also the dressed 3-point inserions and the dressed 4-point insertions. Diagrams with such restrictions can all be counted exactly. This results in various critical lines in the $\lambda$-$g$ plane, where $\lambda$ and $g$ are effective zero-dimensional coupling constants. These lines can be localized exactly.

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