Counting planar diagrams with various restrictions Original Research Article

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’ insertions and ‘seagull’ insertions; at a later stage also self-energy insertions are to be removed, and finally also the dressed 3-point insertions 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.