Effect of cluster-shell competition of $^{12}$C on $E0$ transitions in $^{16}$O

In $^{16}$O, we investigate the relation between the $E0$ monopole transition matrix elements and cluster-shell competition using antisymmetrized quasi cluster model (AQCM), where the dissolution of $alpha$ clusters into quasi clusters due to the effect of the spin-orbit force is introduced. We focus on the structure change when the strength of the spin-orbit force is varied. The ground state dominantly has a compact four $alpha$ structure (doubly magic structure of the $p$ shell), which is rather independent of the strength. However the first excited state ($0_2^+$) has $^{12}$C$+alpha$ cluster structure and this state is largely affected by the increase of the strength; the subclosure configuration of the $p_{3/2}$ shell for the $^{12}$C cluster part becomes more important than three $alpha$ configuration. The $E0$ transition from the ground state with compact four $alpha$ configuration to the $0^+_2$ state with $^{12}$C$+alpha$ cluster configuration is suppressed when increasing the spin-orbit strength. Although the $E0$ operator itself does not have the spin dependence, the matrix element is sensitive to the difference of intrinsic spin structures of the two states. The $E0$ transition characterizes the persistence of four $alpha$ structure in the $0^+_2$ state.