Consistent $\alpha$-cluster description of the $^{12}$C$(0^{+}_2)$ “Hoyle” resonance

The near-threshold $^{12}$C$(0^{+}_2)$ resonance provides unique possibility for fast helium burning in stars, as predicted by Hoyle to explain the observed abundance of elements in the Universe. Properties of this resonance are calculated within the framework of the $\alpha$-cluster model whose two-body and three-body effective potentials are tuned to describe the $\alpha-\alpha$ scattering data, the energies of the $0^+_1$ and $0^+_2$ states, and the $0^+_1$-state root-mean-square radius. The extremely small width of the $0^{+}_2$ state, the $0_2^+\to0_1^+$ monopole transition matrix element, and transition radius are found in remarkable agreement with the experimental data. The $0^{+}_2$-state structure is described as a system of three $\alpha$-particles oscillating between the ground-state-like configuration and the elongated chain configuration whose probability exceeds $0.9$.