Alpha-Core Breakup in the Strong Decay of ${}_{\mathrm{\Lambda }\mathrm{\Lambda }}^6\mathrm{H}\mathrm{e}$ to a Deeply Bound $H$ Dibaryon

We investigate the effect of $\alpha$-core breakup on the strong conversion of ${}_{\mathrm{\Lambda }\mathrm{\Lambda }}^6\mathrm{H}\mathrm{e}$ into a deeply bound $H$ dibaryon. In addition to the coherent $H+{}^4\mathrm{H}\mathrm{e}$ channel, we evaluate the open final states $H+p+{}^3\mathrm{H}$, $H+n+{}^3\mathrm{H}\mathrm{e}$, and $H+d+d$ using a translationally invariant Gaussian cluster description, including spin-isospin recoupling and full nonrelativistic three-body phase-space integrations. The breakup widths are normalized to Gal’s intact-$\alpha$ result. At $B_{\mathrm{\Lambda }\mathrm{\Lambda }}^H=176\mathrm{M}\mathrm{e}\mathrm{V}$, corresponding to $m_H\simeq 2055\mathrm{M}\mathrm{e}\mathrm{V}$, the summed breakup width exceeds the intact-$\alpha$ width by a factor $R_{\mathrm{b}\mathrm{r}}=2.49\times {10}^3$. The resulting inclusive width and lifetime are ${\mathrm{\Gamma }}_{\mathrm{i}\mathrm{n}\mathrm{c}}=3.87\times {10}^{-4}\mathrm{e}\mathrm{V}$ and ${\tau }_{\mathrm{i}\mathrm{n}\mathrm{c}}=1.70\times {10}^{-12}\mathrm{s}$, respectively, compared with ${\tau }_{\alpha }=4.25\times {10}^{-9}\mathrm{s}$ for the intact-$\alpha$ channel alone. The mass-dependent calculation shows that the inclusive lifetime crosses the characteristic hypernuclear weak-decay timescale near $m_H\simeq 2020\mathrm{M}\mathrm{e}\mathrm{V}$ and increases rapidly as the $H$ mass decreases. In the representative dark-matter-motivated interval $1865\le m_H\le 1885\mathrm{M}\mathrm{e}\mathrm{V}$, we obtain $6.59\times {10}^{-4}\lesssim {\tau }_{\mathrm{i}\mathrm{n}\mathrm{c}}\lesssim 6.80\times {10}^{-3}\mathrm{s}$, far exceeding the weak-decay timescale. Thus, although core breakup can dominate the inclusive strong width near $m_H\simeq 2055\mathrm{M}\mathrm{e}\mathrm{V}$, weakly decaying double-$\mathrm{\Lambda }$ hypernuclei remain compatible, within the present framework, with a deeply bound $uuddss$ state in the mass range relevant to sexaquark dark matter.