1/N expansion

Another application in particle physics is to the study of AdS/CFT correspondence. It is also extensively used in condensed matter physics where it can be used to provide a rigorous basis for mean-field theory.

This expansion technique was first introduced by Gerard 't Hooft in 1974 in an attempt to recast quantum chromodynamics as a string theory.

Example

Starting with a quartic interaction φ⁴, the scalar field φ takes on values in the real vector representation of the orthogonal group 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. The Lagrangian density is given by

${\displaystyle \mathcal{L}=\frac{1}{2}{\mathrm{\partial }}^{\mu }{\varphi }_a{\mathrm{\partial }}_{\mu }{\varphi }_a-\frac{m^2}{2}{\varphi }_a{\varphi }_a-\frac{\lambda }{8N}({\varphi }_a{\varphi }_a{)}^2}$

where ${\displaystyle a}$ runs from 1 to N. Note that N has been absorbed into the coupling strength λ. This is crucial here.

Introducing an auxiliary field F;

${\displaystyle \mathcal{L}=\frac{1}{2}{\mathrm{\partial }}^{\mu }{\varphi }_a{\mathrm{\partial }}_{\mu }{\varphi }_a-\frac{m^2}{2}{\varphi }_a{\varphi }_a+\frac{1}{2}{F}^2-\frac{\sqrt{\lambda /N}}{2}F{\varphi }_a{\varphi }_a}$

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).

Each 4-point vertex contributes λ/N and hence, 1/N. Each flavor cycle contributes N because there are N such flavors to sum over. Note that not all momentum flow cycles are flavor cycles.

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. Performing a 1/N expansion gets more and more accurate in the large N limit. The vacuum energy density is proportional to N, but can be ignored due to non-compliance with general relativity assumptions.

Due to this structure, a different graphical notation to denote the Feynman diagrams can be used. Each flavor cycle can be represented by a vertex. The flavor paths connecting two external vertices are represented by a single vertex. The two external vertices along the same flavor path are naturally paired and can be replaced by a single vertex and an edge (not an F edge) connecting it to the flavor path. The F edges are edges connecting two flavor cycles/paths to each other (or a flavor cycle/path to itself). The interactions along a flavor cycle/path have a definite cyclic order and represent a special kind of graph where the order of the edges incident to a vertex matters, but only up to a cyclic permutation, and since this is a theory of real scalars, also an order reversal (but if we have SU(N) instead of SU(2), order reversals aren't valid). Each F edge is assigned a momentum (the momentum transfer) and there is an internal momentum integral associated with each flavor cycle.