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

With left Grassmann derivatives, differentiating $\theta {\sigma }^{\mu }\bar{\theta }$ with respect to $\bar{\theta }$ introduces a minus sign. ConsequentlyAt fixed $y$, the chirality condition becomes independence of $\bar{\theta }$. There are only two independent Grassmann variables ${\theta }^{\alpha }$, so the chiral-superfield component expansion terminates:Here $\varphi$ is a complex scalar field, $\psi$ a Weyl spinor, and $F$ a complex auxiliary field. Its four real off-shell bosonic components, two in $\varphi$ and two in $F$, match the four real off-shell fermionic components. Equivalently, in ordinary coordinates the complete expansion is fixed without ambiguous contraction signs byThe exponential terminates because of the Grassmann algebra.