Ⅰ. HIGHLIGHTS OF THE YEARRIKEN Accel. Prog. Rep. 56 (2023)
RIKEN
Accel. Prog. Rep. 56 (2023)

α-cluster structure of 12 C from ﬁrst principles†
T. Abe,∗1 T. Otsuka,∗2,∗1,∗3 T. Yoshida,∗4,∗5 Y. Tsunoda,∗4 N. Shimizu,∗4 N. Itagaki,∗6 Y. Utsuno,∗3,∗4J. P. Vary,∗7 P. Maris,∗7 and H. Ueno∗1
The 12 C nucleus is of particular importance in stellar nucleosynthesis. It is produced by triple-α reactions through the second 0+ state (Hoyle state). However, the properties of the Hoyle state are not fully
understood. The Hoyle state is still being studied
actively from both sides of experiments and theories.
On the theoretical side, ab initio approaches for lowenergy nuclear structure calculations have been advanced rapidly in recent years, owing to recent computational and methodological developments. Here, wereport the low-lying structure of 12 C studied by theab initio calculations in the no-core Monte Carlo shellmodel (MCSM).1)
Figure 1 shows the no-core MCSM results comparedto the experiments. The calculations were carried out
in the basis space of Nshell = 7 with the harmonicoscillator energy of h̄ω = 20 MeV. The Daejeon16 N Ninteraction2) was adopted. In Fig. 1, the excitationenergies and transition strengths (B(E2), M (E0)) for++the ground (0+1 ) and two low-lying (21 , 02 ) states are+indicated. For the 21 state, the Q-moment is also depicted in the unit of e fm2 . As seen in Fig. 1, the nocore MCSM calculations (“th” in the ﬁgure) provide
a reasonable agreement with the experimental values(“exp” in the ﬁgure). It is remarkable that the ab initio
calculations for the low-lying states of the twelve-bodysystem can reproduce the experimental data.

Fig. 1. 12 C excitation spectra and the B(E2) (M (E0)) values in the unit of e 2 fm4 (e fm2 ) compared to experiments. For more details, see Ref. 1).
†∗1
∗2
∗3
∗4
∗5
∗6
∗7Condensed from the article in Nat. Commun. 13, 2234(2022)
RIKEN Nishina Center
Department of Physics, University of Tokyo
Advanced Science Research Center, Japan Atomic Energy
AgencyCenter for Nuclear Study, University of Tokyo
Research Organization for Information Science and Technology
Yukawa Institute for Theoretical Physics, Kyoto University
Department of Physics and Astronomy, Iowa State University- S18 -The density in the body-ﬁxed frame (intrinsic density) is closely related to the structure of atomic nuclei.
It is, however, still diﬃcult to observe the intrinsic density experimentally. Thus, the theoretical approaches
are indispensable to investigate the intrinsic densityof atomic nuclei. Figure 2 displays the density distri+butions of the ground (0+1 ) and Hoyle (02 ) states in4comparison with the α-particle ( He) density. For theground state, three α-like clusters are closely lying one
another, being closer to the quantum liquid (i.e., normal nuclear matter). The Hoyle state appears rather
clearly separated three α-like clusters compared to theground state. It is noteworthy that ab initio calculations in the twelve-body system exhibit three-α-likecluster structure of 12 C without any a priori assumptions of α particles.

Fig. 2. (a) Color code of density. (b) Density of the groundstate of the α-particle. (c) Density of the ground stateof 12 C. (d) Density of the Hoyle state of 12 C. See thedetails in Ref. 1).

From the T-plot analysis,3) it is found that theground state is composed of the quantum liquid (normal nuclear matter) with ∼94% probability and theα-like clusters with ∼6% probability, while the Hoyle
state comprises the quantum liquid with ∼33% probability and the α-like clusters with ∼61% probability.
Therefore, the structure of 12 C can be viewed as acrossover of the quantum liquid and clustering in termsof the critical phenomena in quantum many-bod systems. Note that this feature obtained by the T-plot
analysis is veriﬁed by the hierarchical cluster analysis
with dendrogram, that is one of the techniques employed in the machine learning.
Now, we can describe the low-lying structure (staticproperty) of 12 C. From future perspective, we would
like to apply the no-core MCSM wave functions obtained here to the calculations for triple-α reactions(dynamics) so that we can fully understand the process of the 12 C production in stars from ﬁrst principles.
References
1) T. Otsuka et al., Nat. Commun. 13, 2234 (2022).
2) A. M. Shirokov et al., Phys. Lett. B 761, 87 (2016).
3) Y. Tsunoda et al., Phys. Rev. C 89, 031301(R) (2014).