Chiral superfield

A (anti-)chiral superfield is a supermultiplet of ${\displaystyle d=4,{\mathcal {N}}=1}$ supersymmetry.

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

In ${\displaystyle d=4,{\mathcal {N}}=1}$ 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 ${\displaystyle \Phi (x,\theta ,{\bar {\theta }})}$ satisfies the covariant constraint ${\displaystyle {\overline {D}}\Phi =0}$, where ${\displaystyle {\bar {D}}}$ is the covariant derivative, given in index notation as

${\displaystyle {\bar {D}}_{\dot {\alpha }}=-{\bar {\partial }}_{\dot {\alpha }}-i\theta ^{\alpha }\sigma _{\alpha {\dot {\alpha }}}^{\mu }\partial _{\mu }.}$

A chiral superfield ${\displaystyle \Phi (x,\theta ,{\bar {\theta }})}$ can then be expanded as

${\displaystyle \Phi (y,\theta )=\phi (y)+{\sqrt {2}}\theta \psi (y)+\theta ^{2}F(y),}$

where ${\displaystyle y^{\mu }=x^{\mu }+i\theta \sigma ^{\mu }{\bar {\theta }}}$. The superfield is independent of the 'conjugate spin coordinates' ${\displaystyle {\bar {\theta }}}$ in the sense that it depends on ${\displaystyle {\bar {\theta }}}$ only through ${\displaystyle y^{\mu }}$. It can be checked that ${\displaystyle {\bar {D}}_{\dot {\alpha }}y^{\mu }=0.}$

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

${\displaystyle \Phi (x,\theta ,{\bar {\theta }})=\phi (x)+{\sqrt {2}}\theta \psi (x)+\theta ^{2}F(x)+i\theta \sigma ^{\mu }{\bar {\theta }}\partial _{\mu }\phi (x)-{\frac {i}{\sqrt {2}}}\theta ^{2}\partial _{\mu }\psi (x)\sigma ^{\mu }{\bar {\theta }}-{\frac {1}{4}}\theta ^{2}{\bar {\theta }}^{2}\square \phi (x).}$

Vector superfield

The vector superfield is a supermultiplet of ${\displaystyle {\mathcal {N}}=1}$ supersymmetry.

A vector superfield (also known as a real superfield) is a function ${\displaystyle V(x,\theta ,{\bar {\theta }})}$ which satisfies the reality condition ${\displaystyle V=V^{\dagger }}$. Such a field admits the expansion

${\displaystyle V=C+i\theta \chi -i{\overline {\theta }}{\overline {\chi }}+{\tfrac {i}{2}}\theta ^{2}(M+iN)-{\tfrac {i}{2}}{\overline {\theta ^{2}}}(M-iN)-\theta \sigma ^{\mu }{\overline {\theta }}A_{\mu }+i\theta ^{2}{\overline {\theta }}\left({\overline {\lambda }}+{\tfrac {i}{2}}{\overline {\sigma }}^{\mu }\partial _{\mu }\chi \right)-i{\overline {\theta }}^{2}\theta \left(\lambda +{\tfrac {i}{2}}\sigma ^{\mu }\partial _{\mu }{\overline {\chi }}\right)+{\tfrac {1}{2}}\theta ^{2}{\overline {\theta }}^{2}\left(D+{\tfrac {1}{2}}\Box C\right).}$

The constituent fields are

• Two real scalar fields ${\displaystyle C}$ and ${\displaystyle D}$
• A complex scalar field ${\displaystyle M+iN}$
• Two Weyl spinor fields ${\displaystyle \chi _{\alpha }}$ and ${\displaystyle \lambda ^{\alpha }}$
• A real vector field (gauge field) ${\displaystyle A_{\mu }}$

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

Using gauge transformations, the fields ${\displaystyle C,\chi }$ and ${\displaystyle M+iN}$ can be set to zero. This is known as Wess-Zumino gauge. In this gauge, the expansion takes on the much simpler form

${\displaystyle V_{\text{WZ}}=\theta \sigma ^{\mu }{\bar {\theta }}A_{\mu }+\theta ^{2}{\bar {\theta }}{\bar {\lambda }}+{\bar {\theta }}^{2}\theta \lambda +{\frac {1}{2}}\theta ^{2}{\bar {\theta }}^{2}D.}$

Then ${\displaystyle \lambda }$ is the superpartner of ${\displaystyle A_{\mu }}$, while ${\displaystyle D}$ is an auxiliary scalar field. It is conventionally called ${\displaystyle D}$, and is known as the D-term.

N = 2

The ${\displaystyle {\mathcal {N}}=2}$ vector or chiral multiplet ${\displaystyle \Psi }$ contains a gauge field ${\displaystyle A_{\mu }}$, two Weyl fermions ${\displaystyle \lambda ,\psi }$, and a scalar ${\displaystyle \phi }$ (which also transform in the adjoint representation of a gauge group). These can also be organised into a pair of ${\displaystyle {\mathcal {N}}=1}$ multiplets, an ${\displaystyle {\mathcal {N}}=1}$ vector multiplet ${\displaystyle W=(A_{\mu },\lambda )}$ and chiral multiplet ${\displaystyle \Phi =(\phi ,\psi )}$. Such a multiplet can be used to define Seiberg–Witten theory concisely.

The ${\displaystyle {\mathcal {N}}=2}$ hypermultiplet or scalar multiplet consists of two Weyl fermions and two complex scalars, or two ${\displaystyle {\mathcal {N}}=1}$ chiral multiplets.

N = 4

The ${\displaystyle {\mathcal {N}}=4}$ vector multiplet contains one gauge field, four Weyl fermions, six scalars, and CPT conjugates. This appears in N = 4 supersymmetric Yang–Mills theory.