Appearance of the Hoyle state and its breathing mode in ${}^{12}\mathrm{C}$ despite strong short-range repulsion of the nucleon-nucleon potential

In the ${}^{12}\mathrm{C}$ nucleus, the Hoyle state $\left(0_2^{+}\right)$ is considered to be an $\alpha$-condensed state, and the $0_3^{+}$ state is considered to be its breathing mode. We investigated whether the $\alpha$ condensation in ${}^{12}\mathrm{C}$ is realized using a microscopic $3\alpha$ cluster model with short-range correlation induced by nucleon-nucleon interactions, where we used the Argonne $v_4^{\prime }$ potential having a short-range repulsion constructed from the realistic Argonne $v_{18}$ potential. Short-range correlation was treated using the unitary correlation operator method, and the Bloch–Brink wave function was adopted as a variational wave function where the $\alpha$ cluster motions were treated by generator coordinates. We obtained four $0^{+}$ states, including short-range correlation, and analyzed them in terms of the Tohsaki–Horiuchi–Schuck–Röpke (THSR) wave function. In addition to the ordinary THSR wave function, we defined a second-order TSHR wave function to describe the $2\hslash \omega$ excitation of the $\alpha$-condensed state. The $0_2^{+}$ state (the Hoyle state) is an $\alpha$-condensed state mainly comprising the ${}^8\mathrm{Be}\left(0^{+}\right)+\alpha (0S$-wave) configuration, and the $0_3^{+}$ state mainly comprises the ${}^8\mathrm{Be}\left(0^{+}\right)+\alpha (2S$-wave) configuration, which is regarded as the breathing mode of the $\alpha$ condensation excited from the Hoyle state.

Figure 1

Original AV4P potential (circles) and the transformed one (squares) obtained via the UCOM in the ${}^{1}E$ (upper-left panel), ${}^{3}E$ (lower-left panel), ${}^{1}O$ (upper-right panel), and ${}^{3}O$ (lower-right panel) channels. The dashed and the solid curves are the fitting curves based on the sum of 15 Gaussians for the original and transformed potentials, respectively.

Figure 2

Energy curves of ${}^4\mathrm{He}$ with the ${\left(0s\right)}^4$ configuration using the AV4' potential as a function of the size parameter $b$. The dashed and solid curves are the energy curves before and after the translation via the UCOM, respectively.

Figure 3

Trajectories of the eigenstates of ${}^8\mathrm{Be}(0^{+}$) in the ${\left(\langle r^2\rangle \right)}^{1/2}$ (root-mean-square radii of eigenstates) and $E^{\prime }$ (eigenvalues measured from the threshold energy of the $2\alpha$ breakup, $E^{\prime }=E-2E_{\alpha }$) plane as the number of generator coordinates increases from 1 fm at 1-fm intervals. Circles, triangles, diamonds, and squares indicate the calculations up to 10, 20, 30, and 40 fm, respectively.

Figure 4

Squared overlaps between the $0S$-THSR wave function and eigenfunctions from the lowest eigenstate to the seventh-lowest eigenstate of ${}^8\mathrm{Be}(0^{+}$) obtained via the GCM calculation up to 40 fm.

Figure 5

Configuration of three $\alpha$ clusters in ${}^{12}\mathrm{C}$ adopted for the GCM calculation, where two $\alpha$ system is projected to the $0^{+}$ state.

Figure 6

Trajectories of the eigenstates of ${}^{12}\mathrm{C}(0^{+}$) in the ${\left(\langle r^2\rangle \right)}^{1/2}\text{−}{E}^{\prime }$ ($=E-3E_{\alpha }$) plane. Green circles, blue triangles, and red diamonds indicate the calculations up to (${\tilde{R}}_1,{\tilde{R}}_2$) = (10 fm, 10 fm), (13 fm, 13 fm), and (16 fm, 16 fm), respectively, from (${\tilde{R}}_1,{\tilde{R}}_2$) = (1 fm, 1 fm) at 1-fm intervals for each generator coordinate.

Figure 7

Contour maps of the squared overlap with the $0S$-THSR wave function for eigenfunctions of the nine lowest eigenstates of ${}^{12}\mathrm{C}$, corresponding to Table 4.

Figure 8

Energy levels for four $0^{+}$ states of ${}^{12}\mathrm{C}$ in the present calculation (BB $+$ UCOM) compared with the experimental data and other theoretical results. The energies were measured from the $3\alpha$ threshold energy. The results labeled “THSR1,” “THSR2,” and “BB (REM)” were taken from Refs. [21], [19, 36], and [37], respectively. The values shown to the right of the energy levels indicate the root-mean-square radii, and the values next to the arrows connecting two levels indicate the values of the matrix elements of a monopole transition between the connected states with units of $e{\mathrm{fm}}^2$. The experimental value of the charge radius for the ground state was converted into the point-proton radius.

Figure 9

Contour maps of the squared overlaps of the $0_3^{+}$ state of ${}^{12}\mathrm{C}$ with the $2S$-THSR wave function with respect to $R_1$ (left panel) and $R_2$ (right panel).

Figure 10

Contour maps of squared overlaps of the $0_4^{+}$ state of ${}^{12}\mathrm{C}$ with the $2S$-THSR wave function with respect to $R_1$ (left panel) and $R_2$ (right panel).

Figure 11

Trajectories of eigenstates of ${}^8\mathrm{Be}\left(2^{+}\right)$ in the ${\left(\langle r^2\rangle \right)}^{1/2}\text{−}{E}^{\prime }$ ($=E-2E_{\alpha }$) plane. Circles, triangles, diamonds, and squares indicate the calculations up to 10, 20, 30, and 40 fm, respectively. Red points indicate the eigenstate with the minimum root-mean-square radius in each GCM calculation.

Figure 12

Squared overlaps between the eigenstate of ${}^8\mathrm{Be}\left(2^{+}\right)$ at $E-E_{2\alpha }=3.08$ MeV and the $2D$-THSR wave function describing an $\alpha$-condensed state with the relative motion excited to $2D_0$ wave. Circles, triangles, diamonds, squares, and inverted triangles indicate the GCM calculations up to 8, 15, 21, 27, and 33 fm, respectively.