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| {{DISPLAYTITLE:1/''N'' expansion}}
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| |colspan=2|[[File:Three Gluon Vertex in t'Hooft notation.png|320px]]<BR>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.
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| !colspan=2|Examples
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| |1<BR>[[Image:1 over N1.png|157px]]
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| |2<BR>[[Image:1 over N2.png|150px]]
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| 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.
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| 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.
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| It is also extensively used in [[condensed matter physics]] where it can be used to provide a rigorous basis for [[mean field theory]].
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| == Example ==
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| Starting with a simple example - the [[orthogonal group|O(N)]] [[Quartic interaction|φ<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
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| :<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> | |
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| 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.
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| Introducing an [[auxiliary field]] F;
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| :<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>
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| 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.
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| 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.
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| 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.
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| 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.
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| == QCD ==
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| {{Main|Quantum chromodynamics}}
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| QCD is a 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.
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| In the large N limit, we only consider the dominant term. See [[AdS/CFT]].
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| ==References==
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| *{{cite journal
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| |author=[[Gerardus 't Hooft|G. 't Hooft]]
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| |title=A planar diagram theory for strong interactions
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| |journal=Nuclear Physics B
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| |volume=72
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| |issue=3
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| |page=461
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| |doi=10.1016/0550-3213(74)90154-0
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| |url=http://igitur-archive.library.uu.nl/phys/2005-0622-152933/UUindex.html
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| |year=1974|bibcode = 1974NuPhB..72..461T }}
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| {{DEFAULTSORT:1 N Expansion}}
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| [[Category:Quantum chromodynamics]]
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| [[Category:String theory]]
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| [[Category:Statistical mechanics]]
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