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An antichiral superfield obeys $D_{\alpha }{\Phi }^{†}=0$ and is the Hermitian conjugate of a chiral superfield. It depends naturally on ${\bar{y}}^{\mu }=x^{\mu }-i\theta {\sigma }^{\mu }\bar{\theta }$ and $\bar{\theta }$.

An auxiliary field has no kinetic term and can be eliminated algebraically. The complex field $F$ in a chiral superfield balances bosonic and fermionic degrees of freedom off shell.

Canonical field normalization rescales a field so that its kinetic term has unit coefficient. For a chiral superfield with $K=Z{\Phi }^{†}\Phi$, the change ${\Phi }_c=Z^{1/2}\Phi$ turns holomorphic superpotential parameters into canonical parameters $m_c=m/Z$ and $g_c=g/Z^{3/2}$.

In a four-dimensional theory with unbroken $\mathcal{N}\ge 2$ supersymmetry, a full charged hypermultiplet contains $\mathcal{N}=1$ chiral superfields in $R$ and $\overline{R}$. Their net complex chiral gauge spectrum vanishes. The vector multiplet is in the real adjoint gauge group representation. Half-hypermultiplets in pseudoreal representations do not supply genuinely complex gauge chirality. Thus the chiral matter of the Standard Model can have at most unbroken $\mathcal{N}=1$ supersymmetry in four dimensions.

A four-dimensional N=1 chiral multiplet consists off shell of a complex scalar $\varphi$, a Weyl fermion ${\psi }_{\alpha }$, and a complex auxiliary scalar $F$. A chiral superfield packages their supersymmetry transformations. Eliminating the auxiliary scalar leaves two bosonic and two fermionic physical polarizations; an allowed supersymmetric mass gives equal scalar and fermion masses.

In flat four-dimensional N=1 supersymmetry, let ${\bar{D}}^2={\bar{D}}_{\dot{\alpha }}{\bar{D}}^{\dot{\alpha }}$, contracting the two spinor indices with the antisymmetric epsilon invariant tensor. The supercovariant derivative algebra with left derivatives has $\{{\bar{D}}_{\dot{\alpha }},{\bar{D}}_{\dot{\beta }}\}=0$. There are only two dotted components, so every product of three barred derivatives repeats one component and vanishes. Therefore ${\bar{D}}_{\dot{\alpha }}{\bar{D}}^{2}X=0$ for any superfield $X$: ${\bar{D}}^{2}X$ is a chiral superfield. This map is called chiral projection, although ${\bar{D}}^2$ is not an idempotent projector.

If $\Phi$ is a chiral superfield, holomorphic closure of chiral superfields also makes $W^{\prime }(\Phi )$ chiral. Thus an equation ${\bar{D}}^2{\Phi }^{†}=4W^{\prime }(\Phi )$ is consistent with the chiral constraint: applying any ${\bar{D}}_{\dot{\alpha }}$ gives zero on both sides. Both sides are scalar superfields, and the squared supersymmetric covariant derivative commutes with the supercharges, so the equation is Lorentz invariant and supersymmetry covariant.

A chiral spinor superfield carries a Spinor representation of the Lorentz group index and satisfies ${\bar{D}}_{\dot{\alpha }}{\Psi }_{\beta }=0$. The chiral field-strength superfield is a fermionic example: its lowest component is the Weyl spinor gaugino. Its spinor index distinguishes it from a scalar chiral superfield, even though both obey the same chirality constraint.

A chiral superfield obeys ${\bar{D}}_{\dot{\alpha }}\Phi =0$. In chiral coordinates $y^{\mu }=x^{\mu }+i\bar{\theta }{\bar{\sigma }}^{\mu }\theta$ it expands as

With left Grassmann derivatives, a chiral superfield is a finite expansion in chiral coordinates $y^{\mu }=x^{\mu }+i\theta {\sigma }^{\mu }\bar{\theta }$:Here $\varphi$ is a complex scalar field, $\psi$ a Weyl spinor and $F$ an auxiliary field. The translation form of its ordinary-coordinate expansion isThe exponential terminates in the Grassmann algebra. To reduce its contractions, choose signature $(+---)$, ${\sigma }^{\mu }=(I,\mathbf{\sigma })$, left differentiation, and ${ϵ}_{12}=1$ for both lower spinor epsilon tensors. Define ${\theta }^2={\theta }^{\alpha }{\theta }_{\alpha }$ and ${\bar{\theta }}^2={\bar{\theta }}_{\dot{\alpha }}{\bar{\theta }}^{\dot{\alpha }}$. Then $v^{\mu }=\theta {\sigma }^{\mu }\bar{\theta }$ satisfies $v^{\mu }{v}^{\nu }=\frac{1}{2}{\theta }^2{\bar{\theta }}^2{\eta }^{\mu \nu }$. Expanding the translation givesThe last sign follows from $i^2/2=-1/2$ in the translation expansion. Defining the barred square in the reverse order, ${\bar{\theta }}^{\dot{\alpha }}{\bar{\theta }}_{\dot{\alpha }}$, negates it and writes the same term with a positive quarter coefficient. Spinor contractions and the Grassmann algebra convention must therefore accompany the reduced formula.

With $X^2=0$ and invertible $F_X$, the additional chiral superfield $Y=y+\sqrt{2}\theta \chi +{\theta }^2{F}_Y$ satisfying $XY=0$ has $y=(G\chi )/F_X-(GG)F_Y/(2F_X^2)$. Its Weyl spinor $\chi$ and auxiliary field $F_Y$ remain independent. This eliminates the independent scalar. It does not force $Y^2=0$.

For canonically normalized chiral superfields, the fermion bilinear is $-\frac{1}{2}{W}_{ij}{\psi }_i{\psi }_j+h.c.$ at the chosen vacuum. Its matrix is complex symmetric. The nonnegative physical Weyl spinor masses are its singular values, and their squared sum is $\mathrm{tr}(m^{†}m)$. Using signed eigenvalues of a symmetric mass matrix instead can obscure the physical mass count.

For the MSSM superfield representations augmented by singlet right-handed neutrino chiral superfields $N_i$, the homogeneous cubic gauge-invariant superpotential contains the four Yukawa structures $Q{H}_{u}U$, $Q{H}_{d}D$, $L{H}_{d}E$, $L{H}_{u}N$ and the additional structures $LLE$, $LQD$, $UDD$, $N{H}_d{H}_u$, $NNN$. Weak-doublet pairs contract with the antisymmetric two-index tensor; $UDD$ contracts with the color three-index tensor. Consequently the $LLE$ coefficient is antisymmetric in its two $L$ indices, the $UDD$ coefficient in its two $D$ indices, and the $NNN$ coefficient symmetric. With $L(N)=-1$, the last five structures violate baryon number or lepton number, whereas the first four conserve both. Singlet fields must not be omitted when classifying a general cubic superpotential.

With one canonical ungauged chiral superfield and $\kappa \ne 0$, the auxiliary expectation is nonzero at every scalar value. The potential is flat and positive, so supersymmetry is broken without an isolated minimum. The fermion is a massless Goldstino and the scalar is massless at tree level. Equality of these zero masses does not imply an unbroken supersymmetric vacuum.

An F-term is the highest ${\theta }^2$ component of a chiral superfield, equivalently an integral over chiral half of superspace. Its supersymmetry variation is a spacetime total derivative.

The gauge kinetic function is a symmetric holomorphic function of chiral superfields multiplying the gauge field-strength F-term. Its real part controls gauge kinetic coefficients and its imaginary part controls topological-angle terms. Its Wilsonian perturbative correction is constrained by one-loop exactness of the holomorphic gauge kinetic function; canonical and one-particle-irreducible gauge couplings require separate normalization and infrared analysis.

// Target: supersymmetry.bigb

A Higgs chiral doublet is a weak-doublet chiral superfield containing a Higgs complex scalar field, a higgsino and an auxiliary field. In the MSSM, $H_u$ has hypercharge $+1/2$ and $H_d$ has hypercharge $-1/2$. Their paired charges enable higgsino anomaly cancellation and the holomorphic need for two Higgs chiral doublets.

A higgsino is the Weyl spinor partner of a Higgs scalar field in a chiral superfield. After a supersymmetric Higgs mechanism, its combination along a broken gauge direction mixes with the gaugino in the resulting massive N=1 vector multiplet.

A nonsingular holomorphic function of chiral superfields remains chiral because ${\bar{D}}_{\dot{\alpha }}f(\Phi )={\sum }_i{f}_i(\Phi ){\bar{D}}_{\dot{\alpha }}{\Phi }_i=0$. Dependence on conjugate superfields generally destroys the constraint. This explains the holomorphic form of a superpotential.

A renormalizable MSSM superpotential is a holomorphic function of chiral superfields and cannot use their conjugates. Gauge-invariant up-type Yukawa couplings require $H_u$ of hypercharge $+1/2$, while down-type and charged-lepton Yukawa couplings require $H_d$ of hypercharge $-1/2$. A single doublet and its conjugate cannot play both roles in the superpotential. This is distinct from higgsino anomaly cancellation.

A full four-dimensional $\mathcal{N}=2$ hypermultiplet is a matter supermultiplet equivalent to two $\mathcal{N}=1$ chiral superfields in conjugate gauge group representations. Pseudoreal representations can admit half-hypermultiplets; these do not provide genuinely complex chiral gauge spectra.