Bose–Einstein condensation of dilute alpha clusters above the four-α threshold in 16O in the field theoretical superfluid cluster model

Observed well-developed $\alpha$ cluster states in $^{16}$O located above the four-$\alpha$ threshold are investigated from the viewpoint of Bose–Einstein condensation of $\alpha$ clusters by using a field-theoretical superfluid cluster model in which the order parameter is defined. The experimental energy levels are reproduced well for the first time by calculation. In particular, the observed 16.7 MeV $0_7^+$ and 18.8 MeV $0_8^+$ states with low-excitation energies from the threshold are found to be understood as a manifestation of the states of the Nambu–Goldstone zero-mode operators, associated with the spontaneous symmetry-breaking of the global phase, which is caused by the Bose–Einstein condensation of the vacuum 15.1 MeV $0^+_6$ state with a dilute well-developed $\alpha$ cluster structure just above the threshold. This gives evidence of the existence of the Bose–Einstein condensate of $\alpha$ clusters in $^{16}$O. It is found that the emergence of the energy level structure with a well-developed $\alpha$ cluster structure above the threshold is robust, almost independently of the condensation rate of $\alpha$ clusters under significant condensation rate. The finding of the mechanism that causes the level structure that is similar to $^{12}$C to emerge above the four-$\alpha$ threshold in $^{16}$O reinforces the concept of Bose–Einstein condensation of $\alpha$ clusters in addition to $^{12}$C.