Kaluza-Klein mode ($m_{\ell }=2\pi |\ell |/L$)

A Kaluza-Klein mode is a coefficient in a field expansion in eigenfunctions on compact extra dimensions. On a circle of circumference $L$, momentum is $2\pi \ell /L$, and a higher-dimensional massless field has lower-dimensional squared mass $(2\pi \ell /L{)}^2$. Nonzero modes give exponentially suppressed static-potential corrections at distances much larger than $L$.

For a field periodic on a circle of radius $R$, a Fourier series gives modes with quantized internal momentum $n/R$. Each mode is a lower-dimensional field with the displayed squared mass. The entire sequence is the tower; retaining only modes below a chosen energy gives an effective field theory. The tower is distinct from any individual Kaluza-Klein mode.

The compactification scale is the first nonzero internal-mode mass. On a circle it equals $1/R$ for a higher-dimensional massless field. A zero-mode dimensional reduction applies at energies well below it. Warping, nontrivial geometry and boundary conditions can change the relation between the first mass and a geometric radius.

A pointlike pulse can contain frequencies above the compactification gap even when its travel time and distance greatly exceed the circle size. Its retarded circle-compactified massless propagator can resolve winding images and massive-mode oscillations. Suppressing those modes requires low-frequency or long-wavelength sources/measurements, or suitable smearing; mere coordinate separation is not that restriction. The static Green function across a circle compactification gives a precise long-distance zero-mode limit without this ambiguity.

For a periodic scalar on a circle of circumference $2\pi R$, its Fourier series has modes ${\psi }_n(x)e^{inz/R}$. In the fixed-radius metric $d{s}_5^2={\gamma }_{ij}d{x}^{i}d{x}^j+(dz+A_{i}d{x}^i{)}^2$, the five-dimensional Klein-Gordon equation becomes $(D_i{D}^i-M^2-n^2/R^2){\psi }_n=0$ with $D_i={\nabla }_i-in{A}_i/R$. Thus the geometric gauge charge is quantized in units $1/R$ in units $\hslash =1$, or the conserved compact momentum in units $\hslash /R$. A rescaling to a canonically normalized electromagnetic potential changes the corresponding numerical unit of physical electric charge.

On three noncompact spatial dimensions times a circle of circumference $L$, the method of images gives ${\sum }_{j\in \mathbb{Z}}[r^2+j^2{L}^2{]}^{-1}=\pi \coth (\pi r/L)/(Lr)$ at equal compact coordinates. This interpolates from $1/r^2$ at short distances to $\pi /(Lr)$ at long distances. A gravitational circle-radius scalar can change the long-range coefficient if it remains unstabilized.