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

Positive-energy one-particle Poincaré group representations have labels $(m,s)$ for $m>0$, and $(0,h)$ for finite-helicity massless states; individual massive polarizations add $m_s=-s,\dots ,s$. An $\mathcal{N}=1$ massive supermultiplet is labelled by mass $m$ and superspin $j$, with constituent labelsA massless supermultiplet has constituent labelsA CPT completion of a supermultiplet additionally supplies $(0,-h),(0,-h+\frac{1}{2})$, with conjugated internal charges where applicable. Constituent spins/helicities are the labels here, not counts of polarizations.

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.

In a supersymmetric Higgs mechanism, an internal gauge symmetry breaks while supersymmetry can remain unbroken. A massless vector multiplet absorbs a chiral multiplet and becomes a massive multiplet with spins $(1,\frac{1}{2},\frac{1}{2},0)$: the gauge boson acquires its longitudinal particle polarization from the Goldstone mode. In the super-Higgs mechanism, local supersymmetry itself breaks: the spin-$3/2$ gravitino absorbs the spin-$1/2$ goldstino, gaining two longitudinal polarizations and becoming massive. With only global supersymmetry breaking, the goldstino instead remains a physical massless fermion.

Both Kaluza-Klein theory and the brane-world scenario use a higher-dimensional bulk. Ordinary Kaluza–Klein matter propagates in the compact directions, producing observable massive towers; absence of weak-scale charged modes requires roughly $R^{-1}\gtrsim \mathrm{TeV}$, or $R\lesssim 2\times 10^{-19}\mathrm{m}$, and traditional constructions often take Planck-size radii. Brane-world matter is localized while gravity explores the bulk, so large flat dimensions can reach the submillimetre order of magnitude discussed in the 2004 setting. For $n$ flat dimensions, $M_{\mathrm{Pl}}^2\sim M_D^{n+2}(2\pi R{)}^n$. These are model-dependent scales, not universal maxima: warped gravity-localizing models can have a noncompact extra dimension. The historical large-flat-dimension mechanism is described in arxiv.org/abs/hep-ph/9803315.