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

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.