Past exam of the mathematics course of the University of Cambridge / 2004 / iii / Paper 50 / 1 / b

TeV-scale softly broken supersymmetry addresses the technical hierarchy problem: equal bosonic and fermionic couplings cancel quadratic ultraviolet contributions to the Higgs mass. Soft mass splittings leave corrections of order $\delta m_H^2\sim g^2{M}_{\mathrm{soft}}^2\log (\Lambda /M_{\mathrm{soft}})/(16{\pi }^2)$, so a weak-scale soft spectrum protects the weak scale against a much higher cutoff. Unbroken four-dimensional extended supersymmetry is incompatible with the observed complex chiral gauge spectrum: charged hypermultiplets contain conjugate representations, while vector multiplets have real adjoint matter. It must therefore be broken to at most $\mathcal{N}=1$ before reproducing low-energy Standard Model chirality.