Cluster states and their potential contribution to carbon burning in massive stars

Background: A recent inelastic α-scattering experiment [P. Adsley et al., Phys. Rev. Lett. 129, 102701 (2022)] found 0⁺ resonances in ²⁴Mg on and above the ¹²C+¹²C break-up threshold. It has been conjectured that the states have a ¹²C+¹²C cluster structure, and play a similar role in accelerating ¹²C+¹²C fusion to the manner in which the Hoyle state accelerates production of ¹²C in massive stars.Purpose: We aim 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 model, as well as in the relevant cluster models.Methods: We determine the spectrum of excited 0⁺ states in ²⁴Mg nucleus using multiconfigurational dynamical symmetry calculations leading to a unified description of the quartet (or shell), ¹²C+¹²C and ²⁰Ne+⁴He 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 α-scattering resonances do not seem to be simple ¹²C+¹²C cluster states, but are more plausibly interpreted as fragmented cluster states due to coupling to quartet excitations, as background states.