Title: Hypernuclear constraints on the existence and lifetime of a deeply bound $H$ dibaryon

Abstract: We study to what extent the unique observation of $\Lambda\Lambda$ hypernuclei by their weak decay into known $\Lambda$ hypernuclei, with lifetimes of order 10$^{-10}$ s, rules out the existence of a deeply bound doubly-strange (${\cal S}$=$-$2) $H$ dibaryon. Treating ${_{\Lambda\Lambda}^{~~6}}{\rm He}$ (the Nagara emulsion event) in a realistic $\Lambda-\Lambda-{^4}$He three-body model, we find that the ${_{\Lambda\Lambda}^{~~6}}{\rm He}\to H + {^4{\rm He}}$ strong-interaction lifetime increases beyond 10$^{-10}$ s for $m_H < m_{\Lambda}+m_n$, about 176 MeV below the $\Lambda\Lambda$ threshold, so that such a deeply bound $H$ is not in conflict with hypernuclear data. Constrained by $\Lambda$ hypernuclear $\Delta{\cal S}$=1 nonmesonic weak-interaction decay rates, we follow EFT methods to evaluate the $\Delta{\cal S}$=2 $H\to nn$ weak-decay lifetime of $H$ in the mass range $2m_n \lesssim m_H < m_{\Lambda}+m_n$. The resulting $H$ lifetime is of order 10$^5$ s, many orders of magnitude shorter than required to qualify for a dark-matter candidate.