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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.

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.

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.

A four-dimensional massive $\mathcal{N}=1$ vector multiplet contains one spin-one particle, two spin-one-half particles and one real spin-zero particle. Its on-shell state counts are $3+1=4$ bosonic and $2+2=4$ fermionic. These masses agree in an unbroken supersymmetric vacuum. The supersymmetric Higgs mechanism assembles it from a massless vector multiplet and one chiral superfield per broken gauge generator.

Matter parity assigns odd parity to the quark and lepton chiral superfields and even parity to the Higgs chiral doublets and gauge superfields. It is related to component R-parity by $R_p=P_M(-1{)}^{2s}$. The Lorentz-invariant action contains an even number of fermions, so conserved matter parity is equivalent to conserved R-parity. Products of four matter superfields can be even while violating baryon number; matter parity alone does not exclude higher-dimensional proton-decay operators.

Matter parity assigns $-1$ to every matter chiral superfield, including a singlet $N$ describing a conjugate right-handed neutrino, and $+1$ to Higgs chiral superfields. Thus the quadratic superpotential term $\frac{1}{2}{M}_{ij}{N}_i{N}_j$ is even, although it changes lepton number by two. The higher-degree operators $QQQL$ and $(L{H}_u{)}^2$ are also even and violate baryon number or lepton number. Exact R-parity therefore forbids the renormalizable trilinear violations $LLE$, $LQD$, $UDD$, $N{H}_d{H}_u$, $NNN$, but does not imply conservation of these continuous charges in every allowed interaction.