Three-body resonances in theLi6nucleus

Six low-lying states of ${}_{}{}^6{}_{}{}^{}\mathrm{Li}$ have been studied by calculating the trajectories of the first two or three eigenvalues of the kernel of the Faddeev equation as the total energy $E$ increases from $-\infty$ to $\infty +i\epsilon$. In the separable-potential three-body ($\alpha \mathrm{NN}$), model for ${}_{}{}^6{}_{}{}^{}\mathrm{Li}$, the $N-N$ interaction is assumed to be a Yamaguchi-type-central plus tensor force for the triplet $s$-wave part, and the central force alone for the singlet $s$-wave part. On the other hand, the $p$-wave interaction of the $\alpha -N$ system is assumed to be Mitra-type-central plus the spin-orbit force. The integral equations are derived for the spectator functions of the wave function for each state of ${}_{}{}^6{}_{}{}^{}\mathrm{Li}$ in the $\mathrm{LS}$ coupling scheme. Following the mathematical definition of $n$-body resonance, the deformation of the integration contour is performed for the complex eigenvalues and the eigenvalue problems for these equations are solved to determine the binding energy or the resonance energy. The calculated energies of the ground state, the 1st, 2nd, 3rd, 4th, and 5th excited states are -5.558, -1.089, -0.8529, -2.342, 1.431, and -0.2074 MeV; the experimental values are -4.532, -2.347, -0.970, -0.22, 0.84, and 1.17 MeV. The level widths for the 1st, 4th, and 5th excited states are calculated from the trajectories of the eigenvalues obtained above.