Editing 1/N expansion

{{Short description|Perturbative analysis of quantum field theories}}
{{DISPLAYTITLE:1/''N'' expansion}}
{| class=wikitable align=right width=320
|colspan=2|[[File:Three Gluon Vertex in t'Hooft notation.svg|320px]]
{{Break}}How a three gluon vertex would appear in 't&nbsp;Hooft's double index notation.  This makes the analogy to a string theory that will appear at large N apparent.
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!colspan=2|Examples
|- valign=top align=center
|1<BR>[[Image:1 over N1.png|157px]]
|2<BR>[[Image:1 over N2.png|150px]]
|}

In [[quantum field theory]] and [[statistical mechanics]], the '''1/''N'' expansion''' (also known as the "'''large ''N'''''" '''expansion''') is a particular [[perturbation theory|perturbative]] analysis of quantum field theories with an [[internal symmetry]] [[group theory|group]] such as [[special orthogonal group|SO(N)]] or [[special unitary group|SU(N)]].  It consists in deriving an expansion for the properties of the theory in powers of <math>1/N</math>, which is treated as a small parameter.

This technique is used in [[Quantum chromodynamics|QCD]] (even though <math>N</math> 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==
Starting with a simple example — the [[orthogonal group|O(N)]] [[Quartic interaction|&phi;<sup>4</sup>]] — the scalar field φ takes on values in the [[real number|real]] vector representation of O(N). Using the [[index notation]] for the N "[[Flavour (particle physics)|flavor]]s" 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

:<math>\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</math>

where <math>a</math> runs from 1 to N. Note that N has been absorbed into the [[coupling constant|coupling strength]] λ. This is crucial here.

Introducing an [[auxiliary field]] F;

:<math>\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</math>

In the [[Feynman diagram]]s, the graph breaks up into disjoint [[cycle (graph theory)|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)<sup>k-1</sup> 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.{{Clarify|date=December 2020}}

Due to this structure, a different graphical notation to denote the [[Feynman diagram]]s 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==
{{Main|Quantum chromodynamics}}

QCD is an SU(3) [[gauge theory]] involving [[gluon]]s and [[quark]]s.  The [[Weyl fermion|left-handed quark]]s belong to a [[triplet representation]], the right-handed to an [[antitriplet representation]] (after charge-conjugating them) and the gluons to a [[real number|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==
*{{cite journal
 |author      = G. 't Hooft
 |author-link = Gerardus 't Hooft
 |title       = A planar diagram theory for strong interactions
 |journal     = Nuclear Physics B
 |volume      = 72
 |issue       = 3
 |page        = 461
 |doi         = 10.1016/0550-3213(74)90154-0
 |url         = http://igitur-archive.library.uu.nl/phys/2005-0622-152933/UUindex.html
 |date        = 1974
 |bibcode     = 1974NuPhB..72..461T
 |url-status  = dead
 |archiveurl  = https://web.archive.org/web/20061011161657/http://igitur-archive.library.uu.nl/phys/2005-0622-152933/UUindex.html
 |archivedate = 2006-10-11
}}

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{{DEFAULTSORT:1 N Expansion}}
[[Category:Quantum field theory]]
[[Category:Quantum chromodynamics]]
[[Category:String theory]]
[[Category:Statistical mechanics]]