Supersymmetric Gauge Theory - SUSY in 4D (with 4 Real Generators)

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, 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 superfields, that only depend on the variables but not their conjugates (more precisely, ). 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.

\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}

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

The gauge transformations act as

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

where Λ is any chiral superfield.

It's easy to check that the chiral superfield

is gauge invariant. So is its complex conjugate .

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

The following term is therefore gauge invariant

is called a bridge since it "bridges" a field which transforms under Λ only with a field which transforms under only.

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

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