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arXiv · hep-th/9808113

Counting planar diagrams with various restrictions

Gerard 't Hooft
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Abstract

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 $λ$-$g$ plane, where $λ$ and $g$ are effective zero-dimensional coupling constants. These lines can be localized exactly.

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Gerard 't Hooft. 1998-08-19. Counting planar diagrams with various restrictions. https://doi.org/10.1016/s0550-3213(98)00697-x

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