Quantum Induced Broadening- A Challenge For Cosmic Neutrino Background Discovery

A recent preprint by Cheipesh {\it et al.} pointed out that the zero-point motion of Tritium atoms bound to Graphene may blur the measured energies of $\beta$ electrons. Smearing due to zero point motion is well known. Such an effect features in studies of the $\beta$ spectrum expected in experiments like KATRIN using diatomic Tritium. The recent preprint may, however, challenge new planned experiments seeking to discover the Cosmic Neutrino Background (CNB) neutrinos (and/or other neutrinos of masses smaller than $0.1$ eV) which plan to use Tritium adsorbed onto Graphene or other materials. Our paper clarifies these issues and examines the more general problem of smearing induced by quantum uncertainty. We find that the effect of Cheipesh {\it et al.} is reduced considerably. The importance of the chemical evolution of the $^{3}$H atom hosting the Tritium nucleus into a tightly bound neutral $^{3}$He atom is emphasized. We estimate the excess blurring caused by the dense spectrum near the lowest state of the Graphene or other hosts of the Tritium atom, generated by the electronic response to the "sudden" escape of the $\beta$ electron. Our analysis suggests yet larger effects and difficulties facing many experiments searching for small mass neutrinos. We speculate on a possible experimental setup which could minimize quantum broadening.