  We explore the possibility of relating extra dimensions with light and heavy
Dirac-type neutral leptons and develop a framework for testing them in various
laboratory experiments. The Kaluza-Klein modes in the large extra dimension
models of the light neutral leptons could mix with the standard model neutrinos
and produce observable effects in the oscillation experiments. We show that the
chirality flipping up-scattering processes occurring through either neutrino
magnetic dipole moment or the weakly coupled scalar interactions can also
produce heavy Kaluza-Klein modes of the corresponding right-handed neutral
leptons propagating in one or more extra dimensions. However, to conserve the
four-dimensional energy-momentum, their masses must be below the maximum energy
of the neutrinos in the initial state. The appreciable size of extra dimensions
connected with these heavy neutral leptons can thus affect the cross-sections
of these processes. This framework applies to any up-scattering process. Our
work here focuses only on its application to the coherent elastic
neutrino-nucleus scattering process. We derive constraints on the size of extra
dimensions using the COHERENT data in oscillation and up-scattering processes.
For model with one large extra dimension for the light neutral leptons, we
obtain the limits, $R \sim 3 \ \mu$m (NH) and $R \sim 2.5 \ \mu$m (IH), on the
size of extra dimension corresponding to the absolute mass limit, $m_{0} \leq 3
\times 10^{-3}$ eV at 90$\%$ C.L. from the short-baseline oscillations. Using
the up-scattering process for heavy neutral leptons, we obtain new parameter
spaces between the size of extra dimensions and parameters of the dipole or
scalar interactions.
