The Width of an Electron-Capture Neutrino Wave Packet

We discuss the size of an electron neutrino wave packet emerging from an electron capture decay using the formalism of open quantum systems. This quantitative result is based on methodology that we have previously used to predict the width of an electron antineutrino wave packet from a beta-decaying nucleus, adapted for the different initial states in electron capture and beta decay. These predictions are now becoming relevant to experimental probes of the neutrino width, for example indirectly through precision spectroscopic studies of the nuclear recoil. We provide a translation between a recent Beryllium Electron capture in Superconducting Tunnel junctions Experiment (BeEST) measurement of the daughter nuclear recoil spectrum from $\mathrm{^{7}Be}$ electron capture decay ($e^- + \mathrm{^{7}Be}\rightarrow\mathrm{^{7}Li+\nu_e}$) and a constraint on the outgoing neutrino width. According to our analysis, the direct limit on the neutrino wave packet width from electron capture decay using the recent BeEST result should map to $\sigma_{\nu,x}>6.2\,\mathrm{pm}$. We determine that a hard upper limit on the wave packet width exists at $\sigma_{\nu,x}\sim2.7\,\mathrm{nm}$, based on the relative momentum between the electron and nucleus. However, we find that the localization is most likely driven by the embedding of the decaying atom in the tantalum crystal lattice resulting in a width scale of approximately $\sigma_{\nu,x}\sim10$~pm, motivated by data from X-ray diffraction and M$\ddot{\mathrm{o}}$ssbauer spectroscopy, which lies intriguingly close to the current bound.