Counting planar diagrams with various restrictions

TL;DR

This paper derives explicit generating functions for counting planar diagrams with specific vertex restrictions, relevant for renormalization theory, and identifies exact critical lines in the coupling constant plane.

Contribution

It provides exact enumeration formulas for restricted planar diagrams and locates critical lines in the coupling space, advancing combinatorial and renormalization analysis.

Findings

Exact generating functions for restricted planar diagrams

Identification of critical lines in the coupling constants plane

Diagrams with restrictions can be counted exactly

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