1/N expansion
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In
This technique is used in QCD (even though is only 3 there) with the
It is also extensively used in condensed matter physics where it can be used to provide a rigorous basis for mean-field theory.
Example
Starting with a simple example — the
where 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;
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
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.
QCD
QCD is an SU(3)
In the large N limit, we only consider the dominant term. See
References
- doi:10.1016/0550-3213(74)90154-0. Archived from the originalon 2006-10-11.