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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 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.

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.

The super-Higgs mechanism is the absorption of the goldstino by the gravitino when local supersymmetry is spontaneously broken. In rigid supersymmetry breaking, the goldstino is a physical massless fermion. In supergravity, its transformation permits a gauge choice removing it; its two degrees of freedom become the longitudinal helicity $\pm \frac{1}{2}$ states of the massive gravitino. The massless gravitino has helicities $\pm \frac{3}{2}$, whereas the massive gravitino has four spin states. This is distinct from the supersymmetric Higgs mechanism, which gives an internal gauge vector a mass and need not break supersymmetry.

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.

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.

Choose canonical vector normalization, $g>0$, and the convention that the Fayet–Iliopoulos term contributes $+\xi D$. In a constant Lorentz-invariant vacuum with vanishing gauge field strength, the gaugino transformation containsA nonzero $⟨D⟩$ therefore shifts a fermion by a constant under supersymmetry: no invariant vacuum can have this expectation. With only D-term breaking, the Goldstino is the gaugino $\lambda$. If F-term expectation values were present too, the Goldstino would instead be the corresponding linear combination of matter fermions and gauginos.

For a genuinely charged single chiral superfield, $q\ne 0$, gauge invariance forbids every nonconstant polynomial superpotential in that field. The renormalizable superspace action is consequentlywhere ${\mathcal{W}}_{\alpha }=-{\bar{D}}^2{D}_{\alpha }V/4$. A constant superpotential is dynamically irrelevant in global supersymmetry. Take the standard vector expansion with highest component ${\theta }^2{\bar{\theta }}^{2}D/2$. In the matter exponential, the term $2gqV$ then contributes $gq|\varphi {|}^{2}D$. The gauge kinetic chiral integral contributes $D^2/2$, and the FI term contributes $\xi D$. ThusThe auxiliary equations are $F=0$ and $D=-(gq|\varphi {|}^2+\xi )$. Completing the square givesThis is the single charged-field D-term breaking model. Its vacuum is found by minimizing over $\rho =|\varphi {|}^2\ge 0$, not by assuming that a nonzero FI constant must break supersymmetry.

For $q\xi >0$, the square cannot be canceled by any $\rho \ge 0$. It is minimized at the origin, whereSupersymmetry breaking occurs while the internal $U(1)$ gauge symmetry remains unbroken. Expanding the scalar potential about the origin givesThere is no superpotential fermion mass and no gaugino-matter mixing at $\varphi =0$. Therefore the chiral multiplet mass splitting isBoth real components of the complex scalar have the same positive squared mass; the gaugino is the massless Goldstino. With this sign convention and $\xi >0$, the breaking condition is $q>0$. Reversing the FI sign convention reverses this quoted charge inequality, while the condition that the scalar cannot cancel the auxiliary-field shift is unchanged.

For $q\xi <0$, the minimum instead satisfiesThe nonzero charged scalar expectation breaks the continuous internal gauge symmetry and preserves supersymmetry. To see the resulting supersymmetric Higgs mechanism, choose a real representative $v>0$ and expand $\varphi =v+(h+ia)/\sqrt{2}$. The scalar kinetic term supplies a vector mass, the D-term potential supplies the radial scalar mass, and the Yukawa interaction $-\sqrt{2}gq{\varphi }^{\ast }\lambda \psi +h.c.$ pairs the two Weyl fermions. They have the common squared massThe phase field $a$ is the absorbed Goldstone mode. There is no Goldstino in this supersymmetric Higgs vacuum; the fields assemble into a massive N=1 vector multiplet. This branch makes explicit that internal gauge breaking and supersymmetry breaking are different phenomena.

If $\xi =0$ and $q\ne 0$, the minimum of $g^2{q}^2|\varphi {|}^4/2$ is $\varphi =0$. Both supersymmetry and the continuous gauge symmetry are then unbroken, and the fields have zero tree-level masses at that vacuum. For the degenerate uncharged case $q=0$ with no superpotential interactions, a nonzero $\xi$ still breaks supersymmetry in the vector sector, but the free chiral multiplet stays unsplit and cannot Higgs the gauge field. An uncharged field would allow additional superpotential terms, outside the genuinely charged model above.

The calculation is classical. A single charged Weyl fermion has nonzero Abelian gauge and mixed gravitational anomaly coefficients proportional to $q^3$ and $q$; a complete quantum gauge theory needs an anomaly-canceling completion. The tree-level branches above refer to the specified field content; extra fields in such a completion can change the vacuum structure.

Every global minimum found for nonzero $\lambda$ has $F_{+}=F_0=F_{-}=D=0$. By energy positivity in global supersymmetry, its zero energy means all supercharges annihilate it. ThereforeThe arbitrary neutral expectation $v$ is a supersymmetric modulus; the nonzero derivatives that would break supersymmetry vanish even though $v$ itself need not vanish. In the exceptional $\lambda =0$ case, every D-flatness minimum likewise has zero F- and D-auxiliaries, so supersymmetry remains unbroken even on the gauge-Higgsed branch. The supersymmetric Higgs mechanism permits internal gauge breaking without supersymmetry breaking. No extra constant or linear superpotential term is needed to find a zero-energy vacuum in this model.

On shell, a massless $\mathcal{N}=1$ vector multiplet contains a gauge boson with two transverse polarizations and a gaugino with two fermionic states. A chiral superfield contains a complex scalar field, hence two real bosonic states, and a Weyl spinor with two fermionic states. Their total is $4$ bosonic and $4$ fermionic states.

Suppose charged scalar fields acquire vacuum expectation values at a supersymmetric vacuum, where F-flatness and D-flatness hold. For each broken generator of the gauge symmetry, the supersymmetric Higgs mechanism combines one massless vector multiplet and one chiral superfield into a massive N=1 vector multiplet. The gauge boson absorbs one real Goldstone boson, acquiring its third polarization. The other real scalar field remains physical. The gaugino and the relevant higgsino mix into the four fermionic states. Thus the spin content isUnbroken supersymmetry gives all these states the same mass through supersymmetric mass degeneracy. Remaining massive chiral superfields instead have spin content $(1/2,0,0)$, with two fermionic and two bosonic states of a common mass. The auxiliary fields ensure matching off shell but are not additional propagating particles in these counts.

The super-Higgs mechanism occurs when local supersymmetry is spontaneously broken in supergravity. Its spin-$3/2$ gauge field, the gravitino, absorbs the spin-$1/2$ goldstino:The extra states have helicities $\pm 1/2$, supplementing $\pm 3/2$. The ordinary supersymmetric Higgs mechanism can preserve supersymmetry while breaking the internal gauge symmetry; the super-Higgs mechanism breaks the local supersymmetry itself. Spontaneously broken global supersymmetry leaves a physical massless goldstino, because there is no dynamical gravitino to absorb it.

When local supersymmetry is spontaneously broken, the gravitino absorbs the goldstino and becomes massive. The two goldstino states provide its longitudinal spin-$1/2$ polarizations. With only global supersymmetry breaking, the goldstino remains a physical massless particle. An ordinary supersymmetric Higgs mechanism instead makes an internal spin-one gauge boson massive and can leave supersymmetry unbroken.