Conventions in this section follow the notes by . A general complex superfield \Phi(x, \theta, \bar \theta) in d = 4, \mathcal{N} = 1 supersymmetry can be expanded as :\Phi(x, \theta, \bar\theta) = \phi(x) + \theta\chi(x) + \bar\theta \bar\chi'(x) + \bar \theta \sigma^\mu \theta V_\mu(x) + \theta^2 F(x) + \bar \theta^2 \bar F'(x) + \bar\theta^2 \theta\xi(x) + \theta^2 \bar\theta \bar \xi' (x) + \theta^2 \bar\theta^2 D(x), where \phi, \chi, \bar \chi' , V_\mu, F, \bar F', \xi, \bar \xi', D are different complex fields. This is not an
irreducible supermultiplet, and so different constraints are needed to isolate irreducible representations.
Chiral superfield A (anti-)chiral superfield is a supermultiplet of d=4, \mathcal{N} = 1 supersymmetry. In four dimensions, the minimal \mathcal{N}=1 supersymmetry may be written using the notion of
superspace. Superspace contains the usual space-time coordinates x^{\mu}, \mu=0,\ldots,3, and four extra fermionic coordinates \theta_\alpha,\bar\theta^\dot\alpha with \alpha, \dot\alpha = 1,2, transforming as a two-component (Weyl)
spinor and its conjugate. In d = 4,\mathcal{N} = 1
supersymmetry, a
chiral superfield is a function over
chiral superspace. There exists a projection from the (full) superspace to chiral superspace. So, a function over chiral superspace can be
pulled back to the full superspace. Such a function \Phi(x, \theta, \bar\theta) satisfies the covariant constraint \overline{D}\Phi=0, where \bar D is the covariant derivative, given in index notation as :\bar D_\dot\alpha = -\bar\partial_\dot\alpha - i\theta^\alpha \sigma^\mu_{\alpha\dot\alpha}\partial_\mu. A chiral superfield \Phi(x, \theta, \bar\theta) can then be expanded as : \Phi (y , \theta ) = \phi(y) + \sqrt{2} \theta \psi (y) + \theta^2 F(y), where y^\mu = x^\mu + i \theta \sigma^\mu \bar{\theta} . The superfield is independent of the 'conjugate spin coordinates' \bar\theta in the sense that it depends on \bar\theta only through y^\mu. It can be checked that \bar D_\dot\alpha y^\mu = 0. The expansion has the interpretation that \phi is a complex scalar field, \psi is a Weyl spinor. There is also the auxiliary complex scalar field F, named F by convention: this is the
F-term which plays an important role in some theories. The field can then be expressed in terms of the original coordinates (x,\theta, \bar \theta) by substituting the expression for y: :\Phi(x, \theta, \bar\theta) = \phi(x) + \sqrt{2} \theta \psi (x) + \theta^2 F(x) + i\theta\sigma^\mu\bar\theta\partial_\mu\phi(x) - \frac{i}{\sqrt{2}}\theta^2\partial_\mu\psi(x)\sigma^\mu\bar\theta - \frac{1}{4}\theta^2\bar\theta^2\square\phi(x).
Antichiral superfields Similarly, there is also
antichiral superspace, which is the complex conjugate of chiral superspace, and
antichiral superfields. An antichiral superfield \Phi^\dagger satisfies D \Phi^\dagger = 0, where :D_\alpha = \partial_\alpha + i\sigma^\mu_{\alpha\dot\alpha}\bar\theta^\dot\alpha\partial_\mu. An antichiral superfield can be constructed as the complex conjugate of a chiral superfield.
Actions from chiral superfields For an action which can be defined from a single chiral superfield, see
Wess–Zumino model.
Vector superfield The vector superfield is a supermultiplet of \mathcal{N} = 1 supersymmetry. A vector superfield (also known as a real superfield) is a function V(x,\theta,\bar\theta) which satisfies the reality condition V = V^\dagger. Such a field admits the expansion :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). The constituent fields are • Two real scalar fields C and D • A complex scalar field M + iN • Two Weyl spinor fields \chi_\alpha and \lambda^\alpha • A real vector field (
gauge field) A_\mu Their transformation properties and uses are further discussed in
supersymmetric gauge theory. Using gauge transformations, the fields C, \chi and M + iN can be set to zero. This is known as
Wess–Zumino gauge. In this gauge, the expansion takes on the much simpler form : 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. Then \lambda is the
superpartner of A_\mu, while D is an auxiliary scalar field. It is conventionally called D, and is known as the
D-term. ==Scalars==