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Observed well-developed $\alpha$ cluster states in $^{16}$O, located abovethe four $\alpha$ threshold, are investigated from the viewpoint ofBose-Einstein condensation of $\alpha$ clusters by using a field-theoreticalsuperfluid cluster model in which the order parameter is defined. Theexperimental energy levels are reproduced well for the first time bycalculation. 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 beunderstood as a manifestation of the states of the Nambu-Goldstone zero-modeoperators, associated with the spontaneous symmetry breaking of the globalphase, 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 justabove the threshold. This gives evidence of the existence of the Bose-Einsteincondensate of $\alpha$ clusters in $^{16}$O. It is found that the emergence ofthe energy level structure with a well-developed $\alpha$ cluster structureabove the threshold is robust, almost independently of the condensation rate of$\alpha$ clusters under significant condensation rate. The finding of themechanism why the level structure that is similar to $^{12}$C emerges above thefour $\alpha$ threshold in $^{16}$O reinforces the concept of Bose-Einsteincondensation of $\alpha$ clusters in addition to $^{12}$C.

Bose-Einstein condensation of dilute alpha clusters above four $\alpha$ threshold in $^{16}$O in field theoretical superfluid cluster model