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In [[theoretical physics]] a '''nonrenormalization theorem''' is a limitation on how a certain quantity in the classical description of a [[quantum field theory]] may be modified by [[renormalization]] in the full quantum theory.  Renormalization theorems are common in theories with a sufficient amount of [[supersymmetry]], usually at least 4 [[supercharges]].

Perhaps the first nonrenormalization theorem was introduced by [[Marcus T. Grisaru]], [[Martin Rocek]] and [[Warren Siegel]] in their 1979 paper [http://www.slac.stanford.edu/spires/find/hep/www?j=NUPHA,B159,429 Improved methods for supergraphs].

==Nonrenormalization in supersymmetric theories and holomorphy==

Nonrenormalization theorems in supersymmetric theories are often consequences of the fact that certain objects  must have a holomorphic dependence on the [[quantum field]]s and [[coupling constant]]s.  In this case the nonrenormalization theory is said to be a consequence of [[holomorphy]].

The more supersymmetry a theory has, the more renormalization theorems apply.  Therefore a renormalization theorem that is valid for a theory with <math>\mathcal{N}</math> supersymmetries will also apply to any theory with more than <math>\mathcal{N}</math> supersymmetries.

===Examples in 4-dimensional theories===

In 4 dimensions the number <math>\mathcal{N}</math> counts the number of 4-component [[Majorana spinor|Majorana]] spinors of supercharges.  Some examples of nonrenormalization theorems in 4-dimensional supersymmetric theories are:

In an <math>\mathcal{N}=1</math> 4D SUSY theory involving only chiral superfields, the [[superpotential]] is immune from renormalization.  With an arbitrary field content it is immune from renormalization in perturbation theory but may be renormalized by nonperturbative effects such as [[instantons]].

In an <math>\mathcal{N}=2</math> 4D SUSY theory the [[moduli space]] of the [[supermultiplet|hypermultiplets]], called the [[Higgs branch]], has a [[hyper-Kähler manifold|hyper-Kähler metric]] and is not renormalized.   In the article [http://www.slac.stanford.edu/spires/find/hep/www?j=NUPHA,B255,569  Lagrangians of N=2 Supergravity - Matter Systems] it was further shown that this metric is independent of the [[scalar (physics)|scalars]] in the [[vector superfield|vector multiplets]].  They also proved that the metric of the [[Coulomb branch]], which is a rigid special [[Kähler manifold]] parametrized by the scalars in <math>\mathcal{N}=2</math> vector multiplets, is independent of the scalars in the hypermultiplets.  Therefore the vacuum manifold is locally a product of a Coulomb and Higgs branch.  The derivations of these statements appear in [http://arxiv.org/abs/hep-th/9603042 The Moduli Space of N=2 SUSY QCD and Duality in N=1 SUSY QCD].

In an <math>\mathcal{N}=2</math> 4D SUSY theory the superpotential is entirely determined by the matter content of the theory.  Also there are no perturbative corrections to the &beta;-function beyond one-loop, as was shown in 1983 in the article [http://www.slac.stanford.edu/spires/find/hep/www?eprint=hep-th/0108200  Superspace Or One Thousand and One Lessons in Supersymmetry] by [[Sylvester James Gates]], Marcus Grisaru, Martin Rocek and Warren Siegel.

In [[N=4 super Yang-Mills|<math>\mathcal{N}=4</math> super Yang-Mills]] the &beta;-function is zero for all couplings, meaning that the theory is [[conformal field theory|conformal]].  This was demonstrated perturbatively by [[Martin Sohnius]] and [[Peter West]] in the 1981 article [http://www.slac.stanford.edu/spires/find/hep/www?j=PHLTA,B100,245  Conformal Invariance in N=4 Supersymmetric Yang-Mills Theory] under certain symmetry assumptions on the theory, and then with no assumptions by [[Stanley Mandelstam]] in the 1983 article [http://www.slac.stanford.edu/spires/find/hep/www?j=NUPHA,B213,149  Light Cone Superspace and the Ultraviolet Finiteness of the N=4 Model].  The full nonperturbative proof by [[Nathan Seiberg]] appeared in the 1988 article [http://www.slac.stanford.edu/spires/find/hep/www?j=PHLTA,B206,75  Supersymmetry and Nonperturbative beta Functions].

===Examples in 3-dimensional theories===

In 3 dimensions the number <math>\mathcal{N}</math> counts the number of 2-component [[Majorana spinor|Majorana]] spinors of supercharges.

When <math>\mathcal{N}=1</math> there is no holomorphicity and few exact results are known.

When <math>\mathcal{N}=2</math> the superpotential cannot depend on the [[linear multiplet]]s and in particular is independent of the [[Fayet-Iliopoulos term]]s (FI) and [[Majorana spinor|Majorana mass]] terms.   On the other hand the [[central charge]] is independent of the chiral multiplets, and so is a linear combination of the FI and Majorana mass terms.  These two theorems were stated and proven in [http://arxiv.org/abs/hep-th/9703110 Aspects of N=2 Supersymmetric Gauge Theories in Three Dimensions].

When <math>\mathcal{N}=3</math>, unlike <math>\mathcal{N}=2</math>, the [[R-symmetry]] is the [[nonabelian group]] SU(2) and so the [[representation theory|representation]] of each [[quantum field|field]] is not renormalized.  In a [[super conformal field theory]] the [[conformal dimension]] of a [[chiral superfield|chiral multiplet]] is entirely determined by its R-charge, and so these conformal dimensions are not renormalized.  Therefore matter fields have no [[wave function renormalization]] in <math>\mathcal{N}=3</math> superconformal field theories, as was shown in [http://arxiv.org/abs/hep-th/9902033 On Mirror Symmetry in Three Dimensional Abelian Gauge Theories].  These theories consist of vector multiplets and [[hypermultiplet]]s.  The hypermultiplet metric is [[hyperkähler]] and may not be lifted by quantum corrections, but its metric may be modified.  No [[renormalization|renormalizable]] interaction between hyper and abelian vector multiplets is possible except for [[Chern-Simons theory|Chern-Simons terms]].

When <math>\mathcal{N}=4</math>, unlike <math>\mathcal{N}=3</math> the hypermultiplet metric may no longer be modified by quantum corrections.

===Examples in 2-dimensional theories===
In <math>\mathcal{N}=(2,2)</math> [[linear sigma model]]s, which are superrenormalizable [[abelian group|abelian]] [[gauge theories]] with matter in [[chiral superfield|chiral supermultiplet]]s, [[Edward Witten]] has argued in [http://www.slac.stanford.edu/spires/find/hep/www?eprint=hep-th/9301042  Phases of N=2 theories in two-dimensions] that the only divergent quantum correction is the [[logarithm]]ic one-loop correction to the FI term.

==Nonrenormalization from a quantization condition==

In supersymmetric and nonsupersymmetric theories, the nonrenormalization of a quantity subject to the [[Dirac quantization condition]] is often a consequence of the fact that possible renormalizations would be inconsistent with the quantization condition, for example the quantization of the level of a [[Chern-Simons theory]] implies that it may only be renormalized at one-loop.  In the 1994 article [http://arxiv.org/abs/hep-th/9401053  Nonrenormalization Theorem for Gauge Coupling in 2+1D] the authors find the renormalization of the level can only be a finite shift, independent of the energy scale, and extended this result to topologically massive theories in which one includes a [[kinetic energy|kinetic term]] for the [[gluon]]s.  In [http://arxiv.org/abs/0704.3740 Notes on Superconformal Chern-Simons-Matter Theories] the authors then showed that this shift needs to occur at one loop, because any renormalization at higher loops would introduce inverse powers of the level, which are nonintegral and so would be in conflict with the quantization condition.

==References==
*[http://arxiv.org/abs/hep-ph/9309335 N. Seiberg (1993) "Naturalness Versus Supersymmetric Non-renormalization Theorems"]

==External links==
*[http://thp.uni-bonn.de/Research/susy.html Non-Renormalization Theorems in Supersymmetry]

[[Category:Supersymmetry]]