en>ChrisGualtieri: General Fixes using AWB

2013-10-20T02:22:51Z

General Fixes using AWB

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	~~Oscar~~ is ~~what my wife enjoys~~ to ~~contact me~~ and ~~I completely dig~~ that ~~name~~. ~~In her professional life she is~~ a ~~payroll clerk~~ but ~~she~~'s ~~usually wanted her personal business~~. ~~California~~ is ~~where her home~~ is ~~but she needs~~ to ~~transfer simply because~~ of ~~her family members~~. ~~To collect cash~~ is what ~~her family~~ and ~~her appreciate~~.<br><br>~~Also visit my web blog: [http:~~//~~www~~.~~sex~~-~~porn~~-~~tube~~.ch/~~user~~/~~JJeannere~~ http://~~www~~.~~sex~~-~~porn~~-~~tube~~.~~ch/~~]		{{refimprove\|date=November 2012}}
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			In [[theoretical physics]], one often analyzes theories with [[supersymmetry]] which also have
			internal [[gauge symmetry\|gauge symmetries]]. So, it is important to come up with a supersymmetric generalization
			of gauge theories.

			==<math>\mathcal{N}=1</math> SUSY in 4D (with 4 real generators)==
			In four dimensions, the minimal N=1 supersymmetry may be written using a [[superspace]]. This superspace involves four extra fermionic coordinates <math>\theta^1,\theta^2,\bar\theta^1,\bar\theta^2</math>, transforming as a two-component [[spinor]] and its conjugate.

			Every superfield, i.e. a field that depends on all coordinates of the superspace, may be expanded with respect to the new fermionic coordinates. There exists a special kind of superfields, the so-called [[chiral superfield]]s, that only depend on the variables <math>\theta</math> but not their conjugates (more precisely, <math>\overline{D}f=0</math>). However, a [[vector superfield]] depends on all coordinates. It describes a [[gauge field]] and its [[superpartner]], namely a [[Weyl fermion]] that obeys a [[Dirac equation]].

			:<math>
			\begin{matrix}
			V &=& C + i\theta\chi - i \overline{\theta}\overline{\chi} + \frac{i}{2}\theta^2(M+iN)-\frac{i}{2}\overline{\theta^2}(M-iN) - \theta \sigma^\mu \overline{\theta} v_\mu \\
			&&+i\theta^2 \overline{\theta} \left( \overline{\lambda} + \frac{1}{2}\overline{\sigma}^\mu \partial_\mu \chi \right) -i\overline{\theta}^2 \theta \left( \lambda + \frac{i}{2}\sigma^\mu \partial_\mu \overline{\chi} \right) + \frac{1}{2}\theta^2 \overline{\theta}^2 \left( D+ \frac{1}{2}\Box C\right)
			\end{matrix}
			</math>

			V is the vector superfield ('''prepotential''') and is real (<math>\overline{V}=V</math>). The fields on the right hand side are component fields.

			The [[gauge transformation]]s act as
			:<math>
			V \to V + \Lambda + \overline{\Lambda}
			</math>
			where Λ is any chiral superfield.

			It's easy to check that the chiral superfield
			:<math>W_\alpha \equiv -\frac{1}{4}\overline{D}^2 D_\alpha V</math>
			is gauge invariant. So is its complex conjugate <math>\overline{W}_{\dot{\alpha}}</math>.

			A nonSUSY covariant gauge which is often used is the '''[[Wess–Zumino gauge]]'''. Here, C, χ, M and N are all set to zero. The residual gauge symmetries are gauge transformations of the traditional bosonic type.

			A chiral superfield X with a charge of q transforms as
			:<math>X \to e^{q\Lambda}X</math>
			:<math>\overline{X} \to e^{q\overline{\Lambda}}X</math>
			The following term is therefore gauge invariant
			:<math>\overline{X}e^{-qV}X</math>

			<math>e^{-qV}</math> is called a '''bridge''' since it "bridges" a field which transforms under Λ only with a field which transforms under <math>\overline{\Lambda}</math> only.

			More generally, if we have a real gauge group G that we wish to supersymmetrize, we first have to [[complexify]] it to G<sup>c</sup>. e<sup>-qV</sup> then acts a '''compensator''' for the complex gauge transformations in effect absorbing them leaving only the real parts. This is what's being done in the Wess–Zumino gauge.

			===Differential superforms===
			Let's rephrase everything to look more like a conventional [[Yang–Mills]] gauge theory. We have a U(1) gauge symmetry acting upon full superspace with a 1-superform gauge connection A. In the analytic basis for the tangent space, the covariant derivative is given by <math>D_M=d_M+iqA_M</math>. Integrability conditions for chiral superfields with the chiral constraint <math>\overline{D}_{\dot{\alpha}}X=0</math> leave us with <math>\left\{\overline{D}_{\dot{\alpha}}, \overline{D}_{\dot{\beta}} \right\}=F_{\dot{\alpha}\dot{\beta}}=0</math>. A similar constraint for antichiral superfields leaves us with <math>F_{\alpha\beta}=0</math>. This means that we can either gauge fix <math>A_{\dot{\alpha}}=0</math> or <math>A_{\alpha}=0</math> but not both simultaneously. Call the two different gauge fixing schemes I and II respectively. In gauge I, <math>\overline{d}_{\dot{\alpha}}X=0</math> and in gauge II, <math>d_\alpha \overline{X}=0</math>. Now, the trick is to use two different gauges simultaneously; gauge I for chiral superfields and gauge II for antichiral superfields. In order to '''bridge''' between the two different gauges, we need a gauge transformation. Call it e<sup>-V</sup> (by convention). If we were using one gauge for all fields, <math>\overline{X}X</math> would be gauge invariant. However, we need to convert gauge I to gauge II, transforming X to (e<sup>-V</sup>)<sup>q</sup>X. So, the gauge invariant quantity is <math>\overline{X}e^{-qV}X</math>.

			In gauge I, we still have the residual gauge <math>e^\Lambda</math> where <math>\overline{d}_{\dot{\alpha}}\Lambda=0</math> and in gauge II, we have the residual gauge <math>e^{\overline{\Lambda}}</math> satisfying <math>d_\alpha \overline{\Lambda}=0</math>. Under the residual gauges, the bridge transforms as <math>e^{-V}\to e^{-\overline{\Lambda}-V-\Lambda}</math>. Without any additional constraints, the bridge <math>e^{-V}</math> wouldn't give all the information about the gauge field. However, with the additional constraint <math>F_{\dot{\alpha}\beta}</math>, there's only one unique gauge field which is compatible with the bridge modulo gauge transformations. Now, the bridge gives exactly the same information content as the gauge field.

			==Theories with 8 or more SUSY generators (<math>\mathcal{N}>1</math>) ==
			In theories with higher supersymmetry (and perhaps higher dimension), a vector superfield typically describes not only a gauge field and a Weyl fermion but also at least one complex [[scalar field]].

			==References==
			* Stephen P. Martin, ''A Supersymmetry Primer''. [http://arxiv.org/pdf/hep-ph/9709356v6.pdf arxiv.org/pdf/hep-ph/9709356v6.pdf].

			==See also==
			* [[super QCD]]
			* [[superpotential]]
			* [[D-term]]
			* [[F-term]]
			* [[current superfield]]

			{{DEFAULTSORT:Supersymmetric Gauge Theory}}
			[[Category:Supersymmetry]]

en>Omnipaedista at 02:25, 5 March 2012

2012-03-05T02:25:15Z

New page

Oscar is what my wife enjoys to contact me and I completely dig that name. In her professional life she is a payroll clerk but she's usually wanted her personal business. California is where her home is but she needs to transfer simply because of her family members. To collect cash is what her family and her appreciate.<br><br>Also visit my web blog: [http://www.sex-porn-tube.ch/user/JJeannere http://www.sex-porn-tube.ch/]

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en>ChrisGualtieri: General Fixes using AWB

en>Omnipaedista at 02:25, 5 March 2012