Investigation of ${}_{12-14}C$ isotopes via Tohsaki-Horiuchi-Schuck-Röpke wave function with cluster breaking

We investigate the ground-state structures of ${}_{12-14}C$ isotopes, which exhibit the coexistence of the cluster and shell-like states, using the cluster breaking Tohsaki-Horiuchi-Schuck-Röpke (CB-THSR) wave function. This approach incorporates cluster breaking nucleon-nucleon pairs, making it particularly suitable for describing the cluster states that feature mixed configurations and dynamic cluster motion. Through variational optimization of THSR bases, we find the coupling between the motions of clusters and valence neutrons in different configurations of ${}_{12-14}C$ isotopes, e.g., dumbbell-like molecular orbits of valence neutrons coupled with an oblate 3-$\alpha$ core, and a prolate 3-$\alpha$ core surrounded by ringlike valence neutron distributions. For describing the coexistence of shell-like structures and cluster formation in the compact cores of ${}_{12-14}C$ isotopes, we introduce CB pairs into the THSR approach to formulate the CB-THSR wave function, which allows a further optimization of the variational wave function and the binding energy. The results suggest oblate and shell-like structures for the ground states of ${}_{12}C$ and ${}_{13-14}C$, respectively. As compared to the original THSR calculation, our new model yields relatively more accurate spectra for ${}_{12-14}C$ isotopes, as well as their matter rms radii. The improved results show that $\alpha$ clusters are broken by the CB pairs, and the CB-THSR wave functions describe a more compact spatial extension of cluster motion, while such strong overlaps between $\alpha$ clusters are not energetically favored in pure cluster models. Finally, we discuss the breaking of $\alpha$ clusters in ${}_{12-14}C$ isotopes via the calculations of momentum distributions and form factors, and it is found that both the particle exchange and the paired excitation should be included for describing the compact cluster states.