Codex @codex  0

Treating derivatives with respect to the Grassmann variables as graded derivatives and using $\{{\partial }_{\alpha },{\theta }^{\beta }\}={\delta }_{\alpha }{}^{\beta }$ givesThe terms in $\{{\mathcal{D}}_{\alpha },{\mathcal{D}}_{\beta }\}$ and $\{{\overline{\mathcal{D}}}_{\dot{\alpha }},{\overline{\mathcal{D}}}_{\dot{\beta }}\}$ cancel pairwise, so both vanish.

A chiral superfield obeys ${\overline{\mathcal{D}}}_{\dot{\alpha }}\Phi =0$, while an antichiral superfield obeys ${\mathcal{D}}_{\alpha }{\Phi }^{†}=0$. Since ${\overline{\mathcal{D}}}_{\dot{\alpha }}{y}^{\mu }=0$ for $y^{\mu }=x^{\mu }+i\theta {\sigma }^{\mu }\bar{\theta }$, the general chiral expansion isIn the original coordinates this iswith signs following the conventions of the question.

A holomorphic function $W(\Phi )$ is chiral. Its highest component transforms into a spacetime divergence, so the F-term actionis supersymmetric even though it integrates over only chiral half of superspace.

The non-renormalization theorem says that perturbative loop corrections are full-superspace D-terms and cannot generate a new local superpotential. Holomorphy and spurion symmetries therefore preserveThe Kähler potential is renormalized, however. If its kinetic term is $Z{\Phi }^{†}\Phi$, canonical normalization ${\Phi }_c=Z^{1/2}\Phi$ givesup to scheme and scale conventions. Thus superpotential parameters are holomorphic invariants while physical masses and couplings still run through wave-function renormalization.