 Solution

In conventional Kaluza-Klein theory, bulk fields have a Kaluza-Klein tower with masses of order $1/R$. If ordinary matter propagates in the extra dimension, the absence of accessible excited modes requires a small $R$: a TeV compactification scale corresponds to roughly $10^{-19}$ m; Planck-scale compactification would be much smaller, of order the Planck length. This is a scale estimate, not a universal radius limit independent of the fields and spectrum.

An unwarped brane-world scenario localizes ordinary fields on a brane while gravity explores a large compact bulk. The volume relationmakes gravity weak without a fundamental Planck-scale cutoff. In the original large-extra-dimension model, $M_{\ast }\sim \mathrm{TeV}$ permits approximately millimetre/submillimetre dimensions for $n=2$; more dimensions are smaller. Ordinary-field KK bounds no longer limit these gravity-only radii, although gravitational and other constraints do. This replaces the large fundamental scale hierarchy by a large compact volume needing stabilization.

A warped brane-world scenario instead uses a position-dependent metric factor. In the two-brane Randall–Sundrum model, $m_{\mathrm{IR}}=e^{-kL}{m}_0$, so $kL\sim 35$ generates a weak scale from a Planck-sized input without a millimetre proper separation; for $k$ of Planck order, $L$ is tens of inverse-Planck lengths. The single-brane version even allows infinite proper size through gravity localization in a noncompact warped dimension; it is not by itself the two-brane redshift hierarchy construction. Thus warped proper size can be unbounded, while finite hierarchy models need specified warping and radius stabilization.

Supersymmetry addresses the hierarchy problem differently: boson/fermion cancellations protect scalar masses against quadratic sensitivity to high scales, with residual corrections controlled by soft supersymmetry breaking. It stabilizes a chosen hierarchy rather than explaining its initial size. Volume dilution and the warped redshift mechanism explain scale separation geometrically; neither automatically determines its moduli or removes all tuning.