= Solution
Take canonical kinetic terms, <gauge coupling> $g>0$, and define the <Fayet–Iliopoulos term> by $+\xi D$. If $\phi$ is the scalar component of the charge-$q$ <chiral superfield>, the relevant <auxiliary field> Lagrangian is
$$
\mathcal L_D=\frac12D^2+D\bigl(gq|\phi|^2+\xi\bigr),\qquad D=-\bigl(gq|\phi|^2+\xi\bigr).
$$
The algebraic equation eliminates $D$ and gives the <D-term scalar potential>
$$
\boxed{V_D=\frac12\bigl(\xi+gq|\phi|^2\bigr)^2.}
$$
Since the <superpotential> vanishes, $F=0$. In a homogeneous vacuum, the <supersymmetry transformation> of the <gaugino> includes
$$
\delta\lambda_\alpha=i\epsilon_\alpha D+(\sigma^{\mu\nu}\epsilon)_\alpha F_{\mu\nu}=i\epsilon_\alpha D_0.
$$
Thus $D_0\ne0$ gives an inhomogeneous shift: no nonzero constant <supersymmetry> parameter leaves this vacuum invariant. \b[The gaugino $\lambda$ is the goldstino], because it is the field shifting under the broken <supercharge>; there is no $F$ contribution here.
For $q\ne0$, minimizing over $x=|\phi|^2\geq0$ 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 $\phi=0$ with $D_0=-\xi$ and $V_0=\xi^2/2$. If $\xi=0$, the minimum $\phi=0$ is <supersymmetric>. Consequently, with this explicitly stated sign convention,
$$
\boxed{q\xi>0\quad\Longleftrightarrow\quad\text{supersymmetry breaking}\qquad(q\ne0).}
$$
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 $\xi\ne0$ still breaks <supersymmetry> through $D_0=-\xi$, but the charged-field criterion does not apply.
On the broken branch, expand about $\phi=0$:
$$
V=\frac12\xi^2+gq\xi|\phi|^2+\frac12g^2q^2|\phi|^4.
$$
Writing $\phi=(\phi_1+i\phi_2)/\sqrt2$ 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> is
$$
\boxed{m_{\phi_1}^2=m_{\phi_2}^2=gq\xi,\qquad m_\psi=m_\lambda=m_A=0.}
$$
The original <chiral multiplet> therefore has the squared-mass splitting
$$
\boxed{\Delta m_{\rm chiral}^2=m_\phi^2-m_\psi^2=gq\xi=-gqD_0.}
$$
The 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 $\operatorname{Str}M^2=2gq\xi$ 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.
Back to article page
