Effects of $\alpha$-cluster breaking on 3$\alpha$ cluster structures in $^{12}$C

To clarify the effects of $\alpha$-cluster breaking on 3$\alpha$ cluster structures in $^{12}$C, we investigate $^{12}$C using a hybrid model that combines the Brink-Bloch cluster model with the $p_{3/2}$ subshell closure wave function. We have found that $\alpha$-cluster breaking caused by spin-orbit force significantly changes cluster structures of excited $0^{+}$ states through orthogonality to lower states. Spatially developed cluster components of the $0^{+}_{2}$ state are reduced. The $0^{+}_{3}$ state changes from a vibration mode in the bending motion of three $\alpha$ clusters to a chain-like 3$\alpha$ structure having an open triangle configuration. As a result of these structure changes of $0^{+}$ states, the band assignment for the $2^{+}_{2}$ state is changed by the $\alpha$-cluster breaking effect. Namely, in model calculations without the $\alpha$-cluster breaking effect, the $0^{+}_{2}$ state is assigned to be the band-head of the $2^{+}_{2}$ state. However, when we incorporate $\alpha$-cluster breaking caused by the spin-orbit force, the $0^{+}_{3}$ state is regarded as the band-head of the $2^{+}_{2}$ state.