Editing Supermultiplet (section)

=== Chiral superfield ===
A (anti-)chiral superfield is a supermultiplet of <math>d=4, \mathcal{N} = 1</math> supersymmetry.

In four dimensions, the minimal <math>\mathcal{N}=1</math> supersymmetry may be written using the notion of [[superspace]]. Superspace contains the usual space-time coordinates <math>x^{\mu}</math>, <math>\mu=0,\ldots,3</math>, and four extra fermionic coordinates <math>\theta_\alpha,\bar\theta^\dot\alpha</math> with <math>\alpha, \dot\alpha = 1,2</math>, transforming as a two-component (Weyl) [[spinor]] and its conjugate.

In <math>d = 4,\mathcal{N} = 1</math> [[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 [[Differential geometry|pulled back]] to the full superspace. Such a function <math>\Phi(x, \theta, \bar\theta)</math> satisfies the covariant constraint <math>\overline{D}\Phi=0</math>, where <math>\bar D</math> is the covariant derivative, given in index notation as
:<math>\bar D_\dot\alpha = -\bar\partial_\dot\alpha - i\theta^\alpha \sigma^\mu_{\alpha\dot\alpha}\partial_\mu.</math>

A chiral superfield <math>\Phi(x, \theta, \bar\theta)</math> can then be expanded as

:<math> \Phi (y , \theta ) = \phi(y) + \sqrt{2} \theta \psi (y) + \theta^2 F(y),</math>
where <math> y^\mu = x^\mu + i \theta \sigma^\mu \bar{\theta} </math>. The superfield is independent of the 'conjugate spin coordinates' <math>\bar\theta</math> in the sense that it depends on <math>\bar\theta</math> only through <math>y^\mu</math>. It can be checked that <math>\bar D_\dot\alpha y^\mu = 0.</math>

The expansion has the interpretation that <math>\phi</math> is a complex scalar field, <math>\psi</math> is a Weyl spinor. There is also the auxiliary complex scalar field <math>F</math>, named <math>F</math> 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 <math>(x,\theta, \bar \theta)</math> by substituting the expression for <math>y</math>:
:<math>\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).</math>

==== Antichiral superfields ====

Similarly, there is also '''antichiral superspace''', which is the complex conjugate of chiral superspace, and '''antichiral superfields'''.

An antichiral superfield <math>\Phi^\dagger</math> satisfies <math>D \Phi^\dagger = 0,</math> where
:<math>D_\alpha = \partial_\alpha + i\sigma^\mu_{\alpha\dot\alpha}\bar\theta^\dot\alpha\partial_\mu.</math>

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]].