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 $L$.

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.