Editing Supermultiplet (section)

=== Vector superfield ===
The vector superfield is a supermultiplet of <math>\mathcal{N} = 1</math> supersymmetry.

A vector superfield (also known as a real superfield) is a function <math>V(x,\theta,\bar\theta)</math> which satisfies the reality condition <math>V = V^\dagger</math>. Such a field admits the expansion

:<math>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} A_\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).</math>

The constituent fields are
* Two real scalar fields <math>C</math> and <math>D</math>
* A complex scalar field <math>M + iN</math>
* Two Weyl spinor fields <math>\chi_\alpha</math> and <math>\lambda^\alpha</math>
* A real vector field ([[gauge field]]) <math>A_\mu</math>

Their transformation properties and uses are further discussed in [[supersymmetric gauge theory]].

Using gauge transformations, the fields <math>C, \chi</math> and <math>M + iN</math> can be set to zero. This is known as [[Wess–Zumino gauge]]. In this gauge, the expansion takes on the much simpler form
:<math> V_{\text{WZ}} = \theta\sigma^\mu\bar\theta A_\mu + \theta^2 \bar\theta \bar\lambda + \bar\theta^2 \theta \lambda + \frac{1}{2}\theta^2\bar\theta^2 D. </math>
Then <math>\lambda</math> is the [[superpartner]] of <math>A_\mu</math>, while <math>D</math> is an auxiliary scalar field. It is conventionally called <math>D</math>, and is known as the [[D-term]].