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In the local Wilsonian effective action, retain the elementary chiral superfields and a nonzero infrared cutoff. Promote each coefficient in $W={\sum }_A{\lambda }_A{\mathcal{O}}_A(\Phi )$ to a chiral spurion. Holomorphy excludes ${\bar{\lambda }}_A$. Assigning $R({\Phi }_i)=0$ and $R({\lambda }_A)=2$ forces a regular perturbative term of R-charge two to be linear in these spurions. Ordinary flavor charges constrain its field monomial. The perturbative gauge-coupling independence of the Wilsonian superpotential removes gauge-dependent coefficients. With the gauge coupling zero, a single holomorphic vertex cannot close a loop through free $\Phi$-${\Phi }^{†}$ propagators; extra vertices would violate the spurion degree or holomorphy. Its coefficient is therefore the tree coefficient. Gauge anomalies of the formal R-symmetry are tracked by a transforming holomorphic gauge coupling. Keeping the infrared cutoff matters: singular one-particle-irreducible terms and elimination of entire massive fields are different operations.

The background-field holomorphy method is discussed in Seiberg's non-renormalization paper.

A superfield is a field on superspace with a specified Super-Poincaré group transformation law. Being a polynomial in anticommuting coordinates alone does not specify this law or make coefficients a supermultiplet. Conversely, a suitably graded function on full superspace, acted on by coordinate pullback, is an unconstrained scalar superfield. A polynomial with spacetime-independent coefficients can be a special constant-component chiral superfield; dependence on spacetime is not itself the defining criterion. Chirality and reality impose additional conditions.

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.

For left Grassmann derivatives and $D_{\alpha }={\partial }_{\alpha }+i({\sigma }^{\mu }\bar{\theta }{)}_{\alpha }{\partial }_{\mu }$, ${\bar{D}}_{\dot{\alpha }}=-{\bar{\partial }}_{\dot{\alpha }}-i(\theta {\sigma }^{\mu }{)}_{\dot{\alpha }}{\partial }_{\mu }$, the coordinate $y^{\mu }=x^{\mu }+i\theta {\sigma }^{\mu }\bar{\theta }$ gives $D_{\alpha }={\partial }_{\alpha }{|}_y+2i({\sigma }^{\mu }\bar{\theta }{)}_{\alpha }{\partial }_{y^{\mu }}$ and ${\bar{D}}_{\dot{\alpha }}=-{\bar{\partial }}_{\dot{\alpha }}{|}_y$. The odd chain rule gives ${\bar{\partial }}_{\dot{\alpha }}(\theta {\sigma }^{\mu }\bar{\theta })=-(\theta {\sigma }^{\mu }{)}_{\dot{\alpha }}$. A chiral superfield is consequently independent of $\bar{\theta }$ at fixed $y$.

At a vacuum obeying F-flatness and D-flatness, charged scalar field expectation values can break an internal gauge symmetry while preserving supersymmetry. For each broken generator, a massless vector multiplet combines with a chiral superfield into a massive N=1 vector multiplet. The vector absorbs one real Goldstone boson; a real scalar remains, and the gaugino mixes with a higgsino. All four bosonic and four fermionic on-shell states have a common mass.

In an unbroken supersymmetric vacuum, physical boson and fermion partners have equal masses because supercharges commute with four-momentum. For one canonical chiral superfield, fluctuations about $W^{\prime }(v)=0$ have mass $|W^{\prime \prime }(v)|$ for both the complex scalar field and Weyl spinor.

A supersymmetry transformation is the infinitesimal action generated by an odd supercharge, with constant anticommuting spinor parameter in a global theory. In superspace it acts as $\delta \mathcal{S}=(ϵQ+\overline{ϵ}\bar{Q})\mathcal{S}$, with possible conventional overall factors of $i$. On the fermion of a chiral superfield, the transformation includes $\delta \psi =\sqrt{2}ϵF_{\mathrm{aux}}+i\sqrt{2}{\sigma }^{\mu }\overline{ϵ}{\partial }_{\mu }\varphi$. A nonzero vacuum auxiliary field therefore gives an inhomogeneous fermion shift, identifying the Goldstino direction when global supersymmetry is spontaneously broken.