Probing Neutron Skins with KDAR Neutrinos: From Coherent to Diffractive Elastic Neutrino--Nucleus Scattering

We investigate coherent elastic neutrino–nucleus scattering (CE$\nu$NS) induced by pion–decay–at–rest ($\pi$DAR) and kaon–decay–at–rest (KDAR) neutrinos, with emphasis on the transition from strict coherence to the diffractive regime. Organizing CE$\nu$NS observables in terms of the dimensionless variable $qR$, we show that $\pi$DAR measurements remain confined to the near–coherent region for all nuclei, whereas KDAR neutrinos ($E_{\nu }=236$~MeV) extend the kinematics into $qR\gtrsim 1$, where recoil spectra develop genuine shape sensitivity to the nuclear weak form factor. Using representative light, medium–mass, and heavy nuclei (${}^{12}$C, ${}^{40}$Ca, ${}^{48}$Ca, and ${}^{208}$Pb), we examine relevant cross sections and quantify the statistical sensitivity to the neutron skin thickness achievable at a JSNS${}^2$–like facility. For a total exposure of 10~ton$\cdot$year and realistic KDAR fluences, projected $1\sigma$ sensitivities reach $\mathrm{\Delta }{R}_{np}^{(1\sigma )}$ $\simeq 0.09$–$0.02$~fm for ${}^{48}$Ca and $\simeq 0.07$–$0.02$~fm for ${}^{208}$Pb as the fluence increases. These sensitivities are competitive with, and complementary to, parity–violating electron–scattering measurements such as CREX and PREX, while relying on an electroweakly clean neutral–current probe with distinct systematic uncertainties. Our results establish KDAR–based CE$\nu$NS as a quantitatively robust and complementary avenue for probing neutron skins and nuclear weak densities beyond the coherent limit.