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Only the Weyl fermions in the charged chiral multiplets contribute to the Abelian triangle gauge anomaly. Their cubic coefficient isand the mixed gauge-gravitational coefficient is $2+0-2=0$. Diagrammatically, the oppositely charged $X_{2+}$ and $X_{2-}$ fermion triangles have equal magnitudes and opposite signs. The neutral gaugino also contributes nothing, so the theory is anomaly free.

Gauge invariance and renormalizability permit the most general superpotentialAfter setting the linear and quadratic parameters to zero, the supersymmetric non-renormalization theorem ensures that perturbative quantum corrections do not regenerate them in the Wilsonian superpotential.

Write the scalar components as $x_{+},x_0,x_{-}$. For generic nonzero $\lambda ,y$, the F-term scalar potential and D-term scalar potential, including a Fayet–Iliopoulos term, areEvery term is nonnegative. The F-flat equations force $x_0=0$ and $x_{+}{x}_{-}=0$. If $\xi >0$, the vacuumalso makes $D=0$; if $\xi <0$, interchange $x_{+}$ and $x_{-}$ and use $|x_{+}{|}^2=-\xi /(2g)$. Thus for nonzero $\xi$ the charged vacuum expectation value spontaneously breaks the gauged $U(1)$ through the Higgs mechanism, but $F_i=D=0$ means supersymmetry remains unbroken. For $\xi =0$, the origin preserves both the gauge symmetry and supersymmetry.

With canonical normalization, the renormalizable supersymmetric gauge theory hasThe first term is the gauge-covariant Kähler potential and the second is the Supersymmetric Yang-Mills action. A holomorphic superpotential would also be integrated as $\int d^2\theta \mathcal{W}(\Phi )+h.c.$, but one commuting field in the fundamental representation of $SU(3)$ admits no nonconstant renormalizable gauge-invariant superpotential: the apparent cubic ${ϵ}_{ijk}{\Phi }^i{\Phi }^j{\Phi }^k$ vanishes.

On the sector $0\le \varphi \le 1$, choose the superpotentialThe static energy has the Bogomolny bound completionEquality holds for ${\varphi }^{\prime }=W^{\prime }$, and therefore

The eight Gell-Mann matrices generate $G=SU(3)$, and the three superfields form one fundamental triplet. A renormalizable superpotential may have degree at most three. There is no invariant vector or symmetric rank-two tensor in the fundamental. The only apparent cubic singlet isbut chiral superfields are Grassmann-even and commute, so this contraction of a symmetric product with ${ϵ}^{ijk}$ vanishes identically. Consequently the most general invariant superpotential as the question is written is only a constant,which has no effect in global supersymmetry. In particular, $W=\lambda {\Phi }_1{\Phi }_2{\Phi }_3$ is not invariant under a general $SU(3)$ rotation.

Take an ordinary $U(1)$ and an $R$-symmetry, assigning $q(\Phi )=R(\Phi )=1$. The superpotential has chargesbecause $d^2\theta$ is neutral under the first symmetry and has $R$-charge $-2$. Formal invariance of each term then assigns

The holomorphic coupling is $\tau =\vartheta /(2\pi )+4\pi i/g^2$. Perturbative coefficients are independent of the topological angle $\vartheta$. A holomorphic function invariant under continuous shifts of $\mathrm{Re}\tau$ is constant by the Cauchy-Riemann equations, hence a perturbative local superpotential coefficient cannot depend on $\tau$. Terms in $e^{2\pi i\tau }$ can evade this argument nonperturbatively. This does not forbid renormalization of the separate gauge kinetic F-term, or wave-function renormalization from the Kähler potential.

For the Polonyi model with canonical Kähler potential and $W=m^2(z+\beta )$, $\beta >0$, a constant supersymmetric vacuum with $m\ne 0$ requires real $z=(-\beta \pm \sqrt{{\beta }^2-4})/2$ and exists exactly for $\beta \ge 2$. Its supergravity F-term potential is negative because $W\ne 0$. For $0<\beta <2$, the supergravity auxiliary field cannot vanish. If $m=0$, the model is flat with unbroken supersymmetry. This parameter-dependent statement is sharper than asserting that every linear superpotential breaks supersymmetry in supergravity.

In four spacetime dimensions, power-counting renormalizability restricts an ungauged chiral action to a canonical quadratic Kähler potential, up to transformations, and a superpotential of degree at most three. Terms of larger degree belong to an effective field theory.

For one charged chiral superfield, canonical kinetic terms, no nonconstant gauge-invariant superpotential, and $g>0$, the sign of $q\xi$ selects the vacuum. If $q\xi >0$, the minimum has $\varphi =0$, positive vacuum energy, unbroken internal gauge symmetry and a gaugino Goldstino; the scalar squared mass is $gq\xi$ while its chiral fermion is massless. If $q\xi <0$, a scalar condensate cancels the auxiliary field, preserving supersymmetry while Higgsing the gauge group. These are classical statements; a consistent quantum charged spectrum must cancel its gauge anomalies.

A supergraph is a Feynman diagram formulated with superfield propagators and vertices in superspace. Performing its spinor-derivative algebra organizes perturbative corrections into superspace integration terms and provides a proof of the local superpotential non-renormalization theorem.

Let $k^i$ generate a holomorphic gauge transformation, with $k^i{\partial }_{i}W=-{\kappa }^{2}rW$ and moment map $\mathcal{P}=i(k^i{K}_i-r)$. Substitution of the Kähler covariant derivative of a superpotential gives the displayed identity. Consequently, a finite point with $W\ne 0$ and all $D_{i}W=0$ also has $\mathcal{P}=0$. Pure D-term breaking with vanishing chiral auxiliary fields can therefore evade this argument only where $W=0$ or outside its hypotheses. In conventional two-derivative matter-coupled supergravity, a constant Fayet–Iliopoulos term gauges an R-symmetry and constrains the transformation of $W$; a neutral superpotential and an independently assigned nonzero constant shift are not generally consistent. The gauge-covariance restriction is derived in Van Proeyen's discussion of Fayet–Iliopoulos terms and R-symmetry.

For a canonical chiral superfield, an affine translation $\Phi =\Psi +s$ preserves the kinetic action up to a vanishing full-superspace term. It removes the linear superpotential term exactly when $s$ is a critical point. For $W=\alpha +\kappa \Phi +m{\Phi }^2/2+g{\Phi }^3/6$, the shifted quadratic coefficient is $M=m+gs$ with $M^2=m^2-2g\kappa$. A real shift requires a real root; the complex field permits a complex shift as well.

A supersymmetric action is invariant under supersymmetry up to boundary terms. For ordinary four-dimensional chiral superfields, a full superspace integration of a real Kähler potential and a chiral superspace integration of a superpotential provide the two-derivative ungauged action.

In a canonical Wess–Zumino model with cubic superpotential $g{\Phi }^3/3$, eliminating the auxiliary field gives $V\supset |g{|}^2|\phi {|}^4$ and $\mathcal{L}\supset -g\phi \psi \psi +h.c.$. Thus the coefficient of the quartic scalar field interaction is the squared modulus of this convention for the Yukawa coupling.

A two-derivative four-dimensional $\mathcal{N}=1$ gauge action is specified by a real gauge-invariant Kähler potential, a gauge-invariant holomorphic superpotential, and a holomorphic symmetric gauge kinetic function. A constant Fayet–Iliopoulos term is additionally allowed for an Abelian gauge factor. Renormalizability restricts the matter kinetic function to a quadratic form, the superpotential to degree at most three, and the gauge kinetic function to a constant invariant tensor.

For quadratic and cubic interactions, assigning the couplings chiral spurion transformations makes each monomial formally covariant. The ordinary $U(1)$ leaves $\theta$ neutral, while the R-symmetry gives it charge one and the superpotential charge two. The neutral ratio $g\Phi /m$ and the covariant factor $m{\Phi }^2$ give the symmetry-allowed holomorphic form $m{\Phi }^{2}f(g\Phi /m)$. Perturbative regularity and tree matching are additionally needed for superpotential non-renormalization; symmetry alone leaves an arbitrary function. Fixed numerical couplings need not enjoy these formal symmetries as actual field symmetries.