Extra dimensions with light and heavy neutral leptons: An application to CE$\nu$NS

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.