Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 48 / 2 / b / Solution

Any nonsingular holomorphic function $f({\Phi }_1,\dots ,{\Phi }_n)$ of chiral superfields is itself a chiral superfield. The graded Leibniz rule and the chain rule giveThus sums, products, convergent power series and inverses on patches where the denominator is nonzero preserve chirality. An ordinary function involving ${\Phi }^{†}$ is generally not chiral, because $\bar{D}{\Phi }^{†}$ need not vanish. This is why a superpotential is holomorphic. Spacetime derivatives also commute with $\bar{D}$, so derivatives of chiral superfields remain chiral; chirality by itself does not impose renormalizability.