Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 307 / 2 / a / Solution

A chiral superfield obeysExamples in the Minimal supersymmetric Standard Model include the quark, lepton and Higgs chiral superfields. Such a multiplet contains a complex scalar $\varphi$, a two-component Weyl spinor $\psi$, and a complex auxiliary field $F$; only the scalar and fermion propagate on shell.

The chiral coordinate $y^{\mu }=x^{\mu }+i\theta {\sigma }^{\mu }\bar{\theta }$ is annihilated in the required combination by ${\bar{D}}_{\dot{\alpha }}$, so the solution has the simple formTaylor-expand each component about $x$. Nilpotence truncates the series, and the Grassmann identitygives the chiral-superfield component expansionTherefore