Chemistry - 1/N expansion

In quantum field theory and statistical mechanics, the 1/N expansion is a particular perturbative analysis of quantum field theories with an SO(N) or SU(N) internal symmetry.

This technique is used in QCD (even though N is only 3 there) with the gauge group SU(3).

Another application is to the study of AdS/CFT dualities.

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

$\mathcal{L}={1\over 2}\partial^\mu \phi_a \partial_\mu \phi_a-{m^2\over 2}\phi_a \phi_a-{\lambda\over 8N}(\phi_a \phi_a)^2$

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

Let's introduce an auxilliary field F here.

$\mathcal{L}={1\over 2}\partial^\mu \phi_a \partial_\mu \phi_a -{m^2\over 2}\phi_a \phi_a +{1\over 2}F^2-{\sqrt{\lambda /N}\over 2}F \phi_a \phi_a$

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.

Let's look at some examples:

You get the idea.

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. There are also ghosts with color pairs. Once again, we follow the colors.

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