Doublet $1/2^{+}$ resonances of ${}^9\mathrm{B}$ in the complex scaling method

We try to resolve the longstanding controversy over the low-lying $1/2^{+}$ state in ${}^9\mathrm{B}$. Experimentally, the energies for this state show a gap between various measurements done by different groups and some of them reported that the existence of the state is unclear. The situation is similar in the theoretical calculations. We study the existence and structure of the ${}^9\mathrm{B}(1/2^{+}$) state based on our previous method for the $\alpha +\alpha +N$ cluster model with the complex scaling method that well explains the measured photodisintegration cross section for the ${}^9\mathrm{Be}(1/2^{+})$. We find two resonances at the energies of $E_1^{\mathrm{res}}=1.81$ MeV with a decay width $\mathrm{\Gamma }=1.98$ MeV and $E_2^{\mathrm{res}}=2.38$ MeV, $\mathrm{\Gamma }=1.81$ MeV in ${}^9\mathrm{B}$. The charge radii and the three-body channel configurations are calculated to see the properties of two resonances. We also calculate the level density of two resonances, which indicates the difficulty to distinguish them in the energy distribution.

Figure 1

Schematic illustration of three types of Jacobi coordinate sets of the $\alpha +\alpha +p$ system. We call (a)–(c) sets as $Y_a$-, $Y_b$-, and $T$-coordinate systems, respectively.

Figure 2

Distribution of energy eigenvalues of the ${}^9\mathrm{B}(1/2^{+})$ state obtained by using the complex scaling with $\theta ={20}^{\circ }$. Open and filled triangles show the resonance states. The continuum states of the three- and two-body subsystems are given by double, solid, dashed-dotted, and dotted black lines.

Figure 3

Analytical continuation of pole trajectories obtained by decreasing the point charge of a valence proton for the first (filled triangles) and second (open triangles) resonance states in ${}^9\mathrm{B}(1/2^{+}$). The open blue circles are the distribution of the eigenvalues of the ${}^9\mathrm{Be}(1/2^{+}$) state. The open diamonds present the ${}^5\mathrm{Li}+\alpha$ continuum solutions for ${}^9\mathrm{B}$. $\theta ={20}^{\circ }$ is used.

Figure 4

The calculated level density of the two resonances of ${}^9\mathrm{B}(1/2^{+}$) measured from the $\alpha +\alpha +p$ threshold energy. The dotted-dashed line is the contribution of the first broad resonance state, the dashed line is the contribution of the second resonance state, and the solid line is the sum of the two resonance states.