$^{12}$C and $\alpha$-clusters, $0^+$ spectrum, and Hoyle-state candidates in $^{24}$Mg
Background: A recent inelastic alpha-scattering experiment [Phys. Rev. Lett. 129, 102701 (2022)] found $0^+$ resonances in $^{24}$Mg on and above the $^{12}$C+$^{12}$C break-up threshold. It has been conjectured that the states have a $^{12}$C+$^{12}$C cluster structure, and play a similar role in accelerating $^{12}$C+$^{12}$C fusion to the manner in which the Hoyle state accelerates production of $^{12}$C in massive stars. Purpose: We wish to build up a quantitative theoretical basis for the considerations of the Hoyle-state paradigm, by calculating the distribution of the $0^+$ states in the shell, as well as in the relevant cluster models. Methods: We determine the spectrum of excited $0^+$ states in $^{24}$Mg nucleus using multiconfigurational dynamical symmetry calculations leading to a unified description of the quartet (or shell), $^{12}$C+$^{12}$C and $^{20}$Ne+$^{4}$He cluster configurations. Results: The density of $0^+$ states in the quartet spectrum is comparable to that found in experiment; however, the density of cluster states is considerably less. Conclusions: The recently observed alpha-scattering resonances do not seem to be simple $^{12}$C+$^{12}$C cluster states, but are more plausibly interpreted as fragmented cluster states due to coupling to quartet excitations, as background states.