1/N expansion

In quantum field theory and statistical mechanics, the 1/N expansion (also known as the "large-N" expansion) is a particular perturbative analysis of quantum field theories with an internal symmetry group such as SO(N) or SU(N). It consists in deriving an expansion for the properties of the theory in powers of $1/N$, which is treated as a small parameter.

This technique is used in QCD (even though $N$ is only 3 there) with the gauge group SU(3). Another application in particle physics is to the study of AdS/CFT dualities.

It is also extensively used in condensed matter physics where it can be used to provide a rigorous basis for mean field theory.

Example

Let's start with a simple example. Let's look at the O(N) φ⁴. The scalar field φ takes on values in the real vector representation of O(N). Let's use the index notation for the N "flavors" with the Einstein summation convention. Because O(N) is orthogonal, no distinction will be made between covariant and contravariant indices. The Lagrangian density is given by

:$mathcal\{L\}=\{1over\; 2\}partial^mu\; phi\_a\; partial\_mu\; phi\_a-\{m^2over\; 2\}phi\_a\; phi\_a-\{lambdaover\; 8N\}(phi\_a\; phi\_a)^2$

Note that N has been absorbed into the coupling strength λ. This is crucial here.

Let's introduce an auxiliary field F here.

:$mathcal\{L\}=\{1over\; 2\}partial^mu\; phi\_a\; partial\_mu\; phi\_a\; -\{m^2over\; 2\}phi\_a\; phi\_a\; +\{1over\; 2\}F^2-\{sqrt\{lambda\; /N\}over\; 2\}F\; phi\_a\; phi\_a$

Now, it's obvious 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.

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!

It turns out, 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. This means we can do a 1/N expansion, which gets more and more accurate in the large N limit. The vacuum energy density is proportional to N, but since we're not doing general relativity, that can be ignored.

Because of this structure, we can use a different graphical notation to denote the Feynman diagrams. Represent each flavor cycle by a vertex. There are also flavor paths connecting two external vertices. These too are represented by a single vertex. The two external vertices along the same flavor path are naturally paired and we can replace them by a single vertex and draw an edge (not an F edge) connecting it to the flavor path. Now, 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 so, this is 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.

QCD

QCD is an SU(3) gauge theory involving gluons and quarks. The left-handed quarks belong to a triplet representation, the right-handed to an antitriplet representation (after charge-conjugating them) and the gluons to a real adjoint representation. A quark edge is assigned a color (and an orientation!) and a gluon edge is assigned a color pair.

In the large N limit, we only consider the dominant term. See AdS/CFT.

References

[cite journal
author=G. 't Hooft
title=A planar diagram theory for strong interactions
journal=Nuclear Physics B
volume=72
page=461
doi=10.1016/0550-3213(74)90154-0
url=http://igitur-archive.library.uu.nl/phys/2005-0622-152933/UUindex.html
year=1974]