Vector superfield

A vector superfield is a real superfield, $V=V^{†}$, used to describe a supersymmetric gauge multiplet. In Wess-Zumino gauge its components are a gauge field, a gaugino and a real auxiliary field.

Wess-Zumino gauge uses supergauge freedom to remove the scalar and spinor components of a general vector superfield that do not belong to the physical gauge multiplet.

In four-dimensional $\mathcal{N}=1$ supersymmetry, a gauge-transformation superfield is a dimensionless chiral, Lie-algebra-valued parameter $\Lambda$. Its complex nature allows gauge transformations to preserve chirality.

A supergauge transformation acts on a charged chiral superfield and vector superfield together. In one common convention, ${\Phi }^{\prime }=e^{-2i\Lambda }\Phi$ and $e^{2V^{\prime }}=e^{-2i{\Lambda }^{†}}{e}^{2V}{e}^{2i\Lambda }$.

Eliminating the real auxiliary field $D$ gives a nonnegative D-term scalar potential. For a $U(1)$ theory with charges $q_i$ and a Fayet–Iliopoulos parameter $\xi$, one convention gives $V_D=\frac{1}{2}[g{\sum }_i{q}_i|{\varphi }_i{|}^2+\xi {]}^2$.

A Fayet–Iliopoulos term is a gauge-invariant term linear in the auxiliary $D$ field of an Abelian vector multiplet. Supersymmetry is broken only if no charged-scalar configuration can make the shifted D-term vanish simultaneously with every F-term.

For a non-Abelian vector superfield, the chiral field-strength superfield isIt obeys ${\bar{D}}_{\dot{\alpha }}{W}_{\alpha }=0$ and transforms covariantly under a supergauge transformation.

The four-dimensional supersymmetric Yang-Mills action is the chiral superspace integral of $\mathrm{Tr}(W^{\alpha }{W}_{\alpha })$ plus its Hermitian conjugate. Its component fields are a Yang-Mills gauge field, a gaugino and an auxiliary field.