Contents

1/N expansion

Perturbative analysis of quantum field theories

How a three gluon vertex would appear in 't Hooft's double index notation. This makes the analogy to a string theory that will appear at large N apparent.
Examples
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Inquantum 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)orSU(N). It consists in deriving an expansion for the properties of the theory in powers of ${\displaystyle 1/N}$, which is treated as a small parameter.

This technique is used in QCD (even though ${\displaystyle N}$ 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[edit]

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. The Lagrangian density is given by

${\displaystyle {\mathcal {L}}={1 \over 2}\partial ^{\mu }\phi _{a}\partial _{\mu }\phi _{a}-{m^{2} \over 2}\phi _{a}\phi _{a}-{\lambda \over 8N}(\phi _{a}\phi _{a})^{2}}$

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;

${\displaystyle {\mathcal {L}}={1 \over 2}\partial ^{\mu }\phi _{a}\partial _{\mu }\phi _{a}-{m^{2} \over 2}\phi _{a}\phi _{a}+{1 \over 2}F^{2}-{{\sqrt {\lambda /N}} \over 2}F\phi _{a}\phi _{a}}$

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.^{[clarification needed]}

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[edit]

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 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[edit]

• G. 't Hooft (1974). "A planar diagram theory for strong interactions". Nuclear Physics B. 72 (3): 461. Bibcode:1974NuPhB..72..461T. doi:10.1016/0550-3213(74)90154-0. Archived from the original on 2006-10-11.