Resonant states of the neutron-rich $\mathrm{\Lambda }$ hypernucleus ${}_{\mathrm{\Lambda }}{}^7\mathrm{He}$

The structure of the neutron-rich $\mathrm{\Lambda }$ hypernucleus, ${}_{\mathrm{\Lambda }}{}^7\mathrm{He}$ is studied within the framework of an $\alpha +\mathrm{\Lambda }+n+n$ four-body cluster model. We predict second $3/2^{+}$ and $5/2^{+}$ states, corresponding to a $0s\mathrm{\Lambda }$ coupled to the second $2^{+}$ state of ${}^6\mathrm{He}$, as narrow resonant states with widths of $\mathrm{\Gamma }\sim 1$ MeV to be at 0.03 and 0.07 MeV with respect to the $\alpha +\mathrm{\Lambda }+n+n$ threshold. From a separate estimate of the differential cross section for the ${}^7\mathrm{Li}(\gamma ,K^{+}){}_{\mathrm{\Lambda }}{}^7\mathrm{He}$ reaction, we suggest a possibility to observe these states at the Thomas Jefferson National Accelerator Facility (JLab) in the future. We also calculate the second $2^{+}$ state of ${}^6\mathrm{He}$ as a resonant state within the framework of an $\alpha +n+n$ three-body cluster model. Our result is $2.81$ MeV with $\mathrm{\Gamma }=4.63$ MeV with respect to the $\alpha +n+n$ threshold. This energy position is $\sim 1$ MeV higher, and with a much broader decay width, than the recent SPIRAL data. We suggest that an experiment at JLab to search for the second $3/2^{+}$ and $5/2^{+}$ states of ${}_{\mathrm{\Lambda }}{}^7\mathrm{He}$ would provide an opportunity to confirm the second $2^{+}$ state of the core nucleus ${}^6\mathrm{He}$.

Figure 1

The $2_1^{+}$ and $2_2^{+}$ energy eigenvalue distributions of the complex scaled Hamiltonian of ${}^6\mathrm{He}$ with $\theta ={28}^{\circ }$. The energy is measured with respect to the $\alpha +n+n$ threshold. The two solid lines are the $\alpha +n+n$ and ${}^5\mathrm{He}(3/2^{-})+n$ thresholds.

Figure 2

Calculated energy spectra of ${}^6\mathrm{He}$ together with the experimental data. The energies in MeV are measured with respect to the $\alpha +n+n$ threshold. The values in parentheses are the decay widths $\mathrm{\Gamma }$ in MeV.

Figure 3

The $5/2_2^{+}$ eigenvalue distributions of the complex scaled Hamiltonian with $\theta ={15}^{\circ }$. The energy is measured with respect to the $\alpha +\mathrm{\Lambda }+n+n$ breakup threshold. The three solid lines are the ${}_{\mathrm{\Lambda }}^6\mathrm{He}+n,{}_{\mathrm{\Lambda }}^5\mathrm{He}+n+n$, and ${}^5\mathrm{He}(3/2^{-})+n+\mathrm{\Lambda }$ thresholds.

Figure 4

Calculated energy levels of ${}^6\mathrm{He}$ and ${}_{\mathrm{\Lambda }}{}^7\mathrm{He}$. The level energies in MeV are measured with respect to the $\alpha +n+n$ and $\alpha +\mathrm{\Lambda }+n+n$ breakup thresholds. The values in parentheses are decay widths $\mathrm{\Gamma }$ in MeV.