Possible interpretation of the complex expectation values associated with resonances

We propose a possible scheme to interpret the complex expectation values associated with resonances having complex eigenenergies. Using the Green's function for resonances, the expectation value is basically described by the Breit-Wigner distribution as a function of the real excitation energy. In the expression of the complex expectation values for resonances, the real part brings the integral value of the distribution, while the imaginary part produces the deviation from the Breit-Wigner distribution, which explains a shift of the peak in the strength from the resonance energy. We apply the present scheme to several nuclear resonances of ${}^{12}\mathrm{C}$, including the Hoyle state, and neutron/proton-rich nuclei of ${}^6\mathrm{He}, {}^6\mathrm{Be}, {}^8\mathrm{He}$, and ${}^8\mathrm{C}$. In these nuclei, many-body resonances are obtained as the complex-energy eigenstates under the correct boundary condition using the complex scaling method, and their nuclear radii are uniquely evaluated. We discuss the peculiar energy dependence of the strength function of the square radius for the resonances in these nuclei.

Figure 1

Coordinate system of the $\alpha +\alpha +\alpha$ model for ${}^{12}\mathrm{C}$ with three rearrangement channels.

Figure 2

Coordinate systems of $\alpha +N+N$ for ${}^6\mathrm{He}$ and ${}^6\mathrm{Be}$, and $\alpha +N+N+N+N$ for ${}^8\mathrm{He}$ and ${}^8\mathrm{C}$.

Figure 3

Complex energy eigenvalues of the ${}^6\mathrm{Be}$ ($0^{+}$) states obtained using the complex scaling with $\theta ={30}^{\circ }$, measured from the $\alpha +p+p$ threshold energy. Units are in MeV. Double circles indicate the $0^{+}$ resonances. Two kinds of continuum spectra of ${}^5\mathrm{Li}+p$ and $\alpha +p+p$ are obtained.

Figure 4

Strength functions of the square radius of ${}^{12}\mathrm{C}, {}^6\mathrm{He}$, and ${}^6\mathrm{Be}$. (a) and (b): ${}^{12}\mathrm{C}, 0_2^{+}, 0_4^{+}$, and $0_5^{+}$ measured from the $\alpha +\alpha +\alpha$ threshold energy. (c) and (d): ${}^6\mathrm{He}, 0_2^{+}, 1^{+}, 2_1^{+}$, and $2_2^{+}$ measured from the $\alpha +n+n$ threshold energy. (e) and (f): ${}^6\mathrm{Be}{0}_1^{+}, 0_2^{+}, 1^{+}, 2_1^{+}$, and $2_2^{+}$ measured from the $\alpha +p+p$ threshold energy. Units of the vertical axis are ${\mathrm{fm}}^2/\mathrm{MeV}$. Short vertical arrows indicate the resonance energies in Table 1.

Figure 5

Strength functions of the square radius of ${}^8\mathrm{He}$ and ${}^8\mathrm{C}$. (a) and (b): ${}^8\mathrm{He}, 0_2^{+}, 1^{+}, 1^{-}, 2_1^{+}$, and $2_2^{+}$ measured from the $\alpha +n+n+n+n$ threshold energy. (c) and (d): ${}^8\mathrm{C}, 0_1^{+}, 0_2^{+}, 1^{+}, 2_1^{+}$, and $2_2^{+}$ measured from the $\alpha +p+p+p+p$ threshold energy. Units of the vertical axis are in ${\mathrm{fm}}^2/\mathrm{MeV}$. Short vertical arrows indicate the resonance energies in Table 1.