A Microscopic Description of the Elusive Hoyle State

Using the symplectic Sp(3,\textbf{R}) symmetry inherent to nuclear dynamics together with a novel many-nucleon interaction, we are able to reproduce low-lying spectral features of $^{12}$C, including the Hoyle state energy, and to gain a further understanding of the underlying physics. We employ a no-core symplectic model for symmetry-preserving interactions--with Sp(3,\textbf{R}) the underpinning symmetry --that offers a microscopic description of nuclei in terms of mixed shape deformations and allows for the inclusion of higher-lying configurations currently inaccessible to ab initio shell models. Our interaction is effectively realized by an exponential dependence on the quadrupole-quadrupole two-body interaction. We were able to reproduce the energies of the ground state rotational band, the Hoyle state, and the next excited $0^+$ state, along with the $B(E2: 2_1^+ \rightarrow 0^+_{\rm g.st.})$ transition strength for $^{12}$C. The success of this work indicates the importance of alpha-cluster structures in the $^{12}$C nucleus and the inclusion of hierarchical many-body interactions.