Supersymmetric gauge theory

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In

. Supersymmetric gauge theory generalizes this notion.

Gauge theory

A

of the theory.

Quantum chromodynamics and quantum electrodynamics are famous examples of gauge theories.

Supersymmetry

In

. Thus, supersymmetry is considered a strong candidate for the unification of radiation (boson-mediated forces) and matter.

This unification is given by an

${\displaystyle Q}$ (or typically many operators), known as a supercharge or supersymmetry generator, which acts schematically as

${\displaystyle Q|{\text{boson}}\rangle =|{\text{fermion}}\rangle }$
${\displaystyle Q|{\text{fermion}}\rangle =|{\text{boson}}\rangle }$

For instance, the supersymmetry generator can take a photon as an argument and transform it into a photino and vice versa. This happens through translation in the (parameter) space. This superspace is a ${\displaystyle {\mathbb {Z} _{2}}}$-graded vector space ${\displaystyle {\mathcal {W}}={\mathcal {W}}^{0}\oplus {\mathcal {W}}^{1}}$, where ${\displaystyle {\mathcal {W}}^{0}}$ is the bosonic Hilbert space and ${\displaystyle {\mathcal {W}}^{1}}$ is the fermionic Hilbert space.

SUSY gauge theory

The motivation for a supersymmetric version of gauge theory can be the fact that gauge invariance is consistent with supersymmetry. The first examples were discovered by Bruno Zumino and Sergio Ferrara, and independently by Abdus Salam and James Strathdee in 1974.

Both the half-integer spin fermions and the integer spin bosons can become gauge particles. The gauge vector fields and its spinorial superpartner can be made to both reside in the same representation of the internal symmetry group.

Suppose we have a ${\displaystyle U(1)}$ gauge transformation ${\displaystyle V_{\mu }\rightarrow V_{\mu }+\partial _{\mu }A}$, where ${\displaystyle V_{\mu }}$ is a vector field and ${\displaystyle A}$ is the gauge function. The main difficulty in construction of a SUSY Gauge Theory is to extend the above transformation in a way that is consistent with SUSY transformations.

The

rules and thus obtain the action. Which further leads to the equations of motion and hence can provide a complete analysis of the dynamics of the theory.

N = 1 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 ${\displaystyle \theta ^{1},\theta ^{2},{\bar {\theta }}^{1},{\bar {\theta }}^{2}}$, 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

, that only depend on the variables θ but not their conjugates (more precisely, ${\displaystyle {\overline {D}}f=0}$). However, a .

${\displaystyle V=C+i\theta \chi -i{\overline {\theta }}{\overline {\chi }}+{\tfrac {i}{2}}\theta ^{2}(M+iN)-{\tfrac {i}{2}}{\overline {\theta ^{2}}}(M-iN)-\theta \sigma ^{\mu }{\overline {\theta }}v_{\mu }+i\theta ^{2}{\overline {\theta }}\left({\overline {\lambda }}-{\tfrac {i}{2}}{\overline {\sigma }}^{\mu }\partial _{\mu }\chi \right)-i{\overline {\theta }}^{2}\theta \left(\lambda +{\tfrac {i}{2}}\sigma ^{\mu }\partial _{\mu }{\overline {\chi }}\right)+{\tfrac {1}{2}}\theta ^{2}{\overline {\theta }}^{2}\left(D+{\tfrac {1}{2}}\Box C\right)}$

V is the vector superfield (prepotential) and is real (V = V). The fields on the right hand side are component fields.

The

gauge transformations

act as

${\displaystyle V\to V+\Lambda +{\overline {\Lambda }}}$

where Λ is any chiral superfield.

It's easy to check that the chiral superfield

${\displaystyle W_{\alpha }\equiv -{\tfrac {1}{4}}{\overline {D}}^{2}D_{\alpha }V}$

is gauge invariant. So is its complex conjugate ${\displaystyle {\overline {W}}_{\dot {\alpha }}}$.

A non-supersymmetric 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

${\displaystyle X\to e^{q\Lambda }X,\qquad {\overline {X}}\to e^{q{\overline {\Lambda }}}X}$

Therefore Xe^−qVX is gauge invariant. Here e^−qV is called a bridge since it "bridges" a field which transforms under Λ only with a field which transforms under Λ only.

More generally, if we have a real gauge group G that we wish to supersymmetrize, we first have to

it to G^c ⋅ e^−qV 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

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 ${\displaystyle D_{M}=d_{M}+iqA_{M}}$. Integrability conditions for chiral superfields with the chiral constraint

${\displaystyle {\overline {D}}_{\dot {\alpha }}X=0}$

leave us with

${\displaystyle \left\{{\overline {D}}_{\dot {\alpha }},{\overline {D}}_{\dot {\beta }}\right\}=F_{{\dot {\alpha }}{\dot {\beta }}}=0.}$

A similar constraint for antichiral superfields leaves us with F_αβ = 0. This means that we can either gauge fix ${\displaystyle A_{\dot {\alpha }}=0}$ or A_α = 0 but not both simultaneously. Call the two different gauge fixing schemes I and II respectively. In gauge I, ${\displaystyle {\overline {d}}_{\dot {\alpha }}X=0}$ and in gauge II, d_α X = 0. 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^−V (by convention). If we were using one gauge for all fields, XX would be gauge invariant. However, we need to convert gauge I to gauge II, transforming X to (e^−V)^qX. So, the gauge invariant quantity is Xe^−qVX.

In gauge I, we still have the residual gauge e^Λ where ${\displaystyle {\overline {d}}_{\dot {\alpha }}\Lambda =0}$ and in gauge II, we have the residual gauge e^Λ satisfying d_α Λ = 0. Under the residual gauges, the bridge transforms as

${\displaystyle e^{-V}\to e^{-{\overline {\Lambda }}-V-\Lambda }.}$

Without any additional constraints, the bridge e^−V wouldn't give all the information about the gauge field. However, with the additional constraint ${\displaystyle F_{{\dot {\alpha }}\beta }}$, 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 (N > 1)

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.

Examples

Pure supersymmetric gauge theories

• N = 1 Super Yang–Mills
• N = 2 Super Yang–Mills
• N = 4 Super Yang–Mills

Supersymmetric gauge theories with matter

• Super QCD
• MSSM (Minimal supersymmetric Standard Model)
• NMSSM
  (Next-to-minimal supersymmetric Standard Model)

See also

• superpotential
• D-term
• F-term
• current superfield
• Supersymmetric quantum mechanics

References

• Stephen P. Martin. A Supersymmetry Primer,
  arXiv:hep-ph/9709356
  .
• Prakash, Nirmala. Mathematical Perspective on Theoretical Physics: A Journey from Black Holes to Superstrings, World Scientific (2003).
• Kulshreshtha, D. S.; Mueller-Kirsten, H. J. W. (1991). "Quantization of systems with constraints: The Faddeev-Jackiw method versus Dirac's method applied to superfields". Physical Review D. Phys. Rev. D43, 3376-3383. 43 (10): 3376–3383.
  doi:10.1103/PhysRevD.43.3376
  .