It is also extensively used in condensed matter physics where it can be used to provide a rigorous basis for mean-field theory.
Starting with a simple example — the O(N) φ4 — the scalar field φ takes on values in the real vector representation of O(N).
Using the index notation for the N "flavors" with the Einstein summation convention and because O(N) is orthogonal, no distinction will be made between covariant and contravariant indices.
Introducing an auxiliary field F; In the Feynman diagrams, the graph breaks up into disjoint cycles, each made up of φ edges of the same flavor and the cycles are connected by F edges (which have no propagator line as auxiliary fields do not propagate).
At least perturbatively, the dominant contribution to the 2k-point connected correlation function is of the order (1/N)k-1 and the other terms are higher powers of 1/N.
The vacuum energy density is proportional to N, but can be ignored due to non-compliance with general relativity assumptions.
[clarification needed] Due to this structure, a different graphical notation to denote the Feynman diagrams can be used.