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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.

The massive N=1 vector multiplet contains spins $1,1/2,1/2,0$. Its massless helicity content splits into a CPT-complete vector multiplet with helicities $(1,1/2)$ and $(-1/2,-1)$, and a CPT-complete chiral multiplet with helicities $(1/2,0)$ and $(0,-1/2)$. The longitudinal vector polarization becomes one of the two scalar polarizations; the other comes from the original real scalar. This is the representation-theoretic inverse of the supersymmetric Higgs mechanism.

For a massless supermultiplet, put the momentum along the third axis. The matrix $2\sigma \cdot p$ in the supersymmetry algebra has rank one, so only one fermionic oscillator acts nontrivially. It relates states whose helicities differ by $\frac{1}{2}$. Including the antiparticles required by CPT completion of a supermultiplet gives the following complete spectra.

A chiral multiplet contains a complex scalar $A$ and a Weyl spinor $\psi$. Its scalar particle and antiparticle have helicity $0$; its fermion particle and antiparticle have helicities $-\frac{1}{2}$ and $+\frac{1}{2}$. Therefore $n_B=2=n_F$, with maximum absolute helicity $\frac{1}{2}$.

A supersymmetric vector multiplet contains a massless gauge vector $A_{\mu }$, with helicities $+1,-1$, and a Majorana spinor gaugino $\lambda$, with helicities $+\frac{1}{2},-\frac{1}{2}$. Again $n_B=2=n_F$, now with maximum absolute helicity $1$. The chiral $F$ and vector $D$ auxiliary fields contribute no on-shell states.

The Higgs mechanism is compatible with unbroken supersymmetry. A supersymmetric vacuum may have nonzero scalar vacuum expectation values that break the internal gauge symmetry while obeying both F-flatness and D-flatness. For example, oppositely charged chiral superfields with $W=0$, no Fayet–Iliopoulos term and equal nonzero scalar magnitudes have $F=D=0$ but break the Abelian gauge group. In the supersymmetric Higgs mechanism, a massless vector multiplet combines with the chiral multiplet along the broken gauge direction. One real scalar becomes the longitudinal vector polarization; the other remains a physical real scalar, and the gaugino combines with the Higgsino into a Dirac spinor. The resulting massive vector multiplet has $3+1=4$ bosonic and $4$ fermionic states with a common mass, as in part (i). Any other uneaten chiral multiplets remain separately in the spectrum.

Take canonical kinetic terms, gauge coupling $g>0$, and define the Fayet–Iliopoulos term by $+\xi D$. If $\varphi$ is the scalar component of the charge-$q$ chiral superfield, the relevant auxiliary field Lagrangian isThe algebraic equation eliminates $D$ and gives the D-term scalar potentialSince the superpotential vanishes, $F=0$. In a homogeneous vacuum, the supersymmetry transformation of the gaugino includesThus $D_0\ne 0$ gives an inhomogeneous shift: no nonzero constant supersymmetry parameter leaves this vacuum invariant. The gaugino $\lambda$ is the goldstino, because it is the field shifting under the broken supercharge; there is no $F$ contribution here.

For $q\ne 0$, minimizing over $x=|\varphi {|}^2\ge 0$ separates three cases. If $q\xi <0$, the allowed value $x=-\xi /(gq)$ gives $D_0=V_0=0$: supersymmetry is unbroken while the internal gauge symmetry is Higgsed. If $q\xi >0$, the unconstrained zero of $D$ would have negative $x$, so the minimum is $\varphi =0$ with $D_0=-\xi$ and $V_0={\xi }^2/2$. If $\xi =0$, the minimum $\varphi =0$ is supersymmetric. Consequently, with this explicitly stated sign convention,Reversing the sign used to define the Fayet–Iliopoulos term reverses the printed sign criterion, not the physics. If $q=0$, the scalar is decoupled and flat: any $\xi \ne 0$ still breaks supersymmetry through $D_0=-\xi$, but the charged-field criterion does not apply.

On the broken branch, expand about $\varphi =0$:Writing $\varphi =({\varphi }_1+i{\varphi }_2)/\sqrt{2}$ gives both real scalar modes the squared mass $gq\xi$. There is no superpotential fermion mass and no vacuum expectation value to induce gaugino–matter-fermion mixing. The Abelian gauge symmetry is unbroken, so its vector is also massless. The complete mass spectrum of single charged-field D-term breaking isThe original chiral multiplet therefore has the squared-mass splittingThe original vector multiplet has zero mass splitting, although the nonzero $D_0$ means it is not an unbroken-supersymmetry representation. As a check, the tree-level supertrace mass sum rule gives $\mathrm{Str}{M}^2=2gq\xi$ when the two real scalar degrees of freedom are counted. On the $q=0$ broken branch all the propagating fields are massless.

There is a necessary scope qualification: one nonzero-charge Weyl fermion has an Abelian cubic gauge anomaly proportional to $q^3$ and a mixed gravitational gauge anomaly proportional to $q$. These are the classical vacuum and spectrum calculations requested by the model; a consistent quantum gauge theory needs an anomaly-cancelling completion, whose additional matter must also be included in its vacuum analysis.

In the supersymmetric Higgs mechanism, an internal gauge symmetry is broken by a charged scalar field expectation value while supersymmetry can remain unbroken. A massless vector multiplet absorbs a chiral multiplet to form a massive N=1 vector multiplet: the vector absorbs a scalar Goldstone boson, and the gaugino and matter fermion acquire the matching masses. In the super-Higgs mechanism, local supersymmetry itself is spontaneously broken: the gravitino absorbs the spin-$1/2$ Goldstino and becomes a massive spin-$3/2$ particle. In a flat vacuum this increases its physical states from two to four.

In a massive four-dimensional N=1 supersymmetry representation, the two normalized raising supercharges transform as a spin-half doublet. Start with a Clifford vacuum of superspin $j=1/2$. At occupation levels zero, one and two, the ordinary spins areThe two-creation state has a spin-singlet oscillator factor because the creation operators anticommute. Thus the massive N=1 vector multiplet containsThere are four bosonic polarizations, $3+1$, and four fermionic polarizations, $2+2$, as required by boson-fermion degeneracy in a supermultiplet.

In the massless limit the helicity multiplicities become $+1,-1$, two copies each of $+1/2,-1/2$, and two zero-helicity states. They separate into the four irreducible helicity pairsWith the lower-helicity convention for superhelicity, a pair $(\kappa ,\kappa +1/2)$ has labels $\kappa =1/2,0,-1/2,-1$, respectively. A midpoint convention shifts each label by $1/4$; the explicit helicity pairs remove any naming ambiguity.

Pairing under the CPT theorem, the first and last pairs form a massless vector multiplet, and the middle two form a massless chiral multiplet. This is the massless limit of a massive N=1 vector multiplet. The longitudinal vector polarization supplies one of the chiral multiplet's two real scalar modes; the original spin-zero state supplies the other. Simply discarding the longitudinal polarization would lose a physical degree of freedom and give the wrong decomposition.

In a supersymmetric Higgs mechanism, an internal gauge symmetry breaks while supersymmetry can remain unbroken. A massless vector multiplet absorbs a chiral multiplet and becomes a massive multiplet with spins $(1,\frac{1}{2},\frac{1}{2},0)$: the gauge boson acquires its longitudinal particle polarization from the Goldstone mode. In the super-Higgs mechanism, local supersymmetry itself breaks: the spin-$3/2$ gravitino absorbs the spin-$1/2$ goldstino, gaining two longitudinal polarizations and becoming massive. With only global supersymmetry breaking, the goldstino instead remains a physical massless fermion.