New description to soft modes in N-alpha cluster states as Bose-Einstein condensation based on quantum field theory with zero mode excitation

In this paper, we consider the Nα cluster state as Bose-Einstein condensation and calculate the energy level for Nα nuclei from ¹²C-⁵²Fe. We take the phenomenological model, and use the quantum field theoretical approach for finite system with spontaneous symmetry breaking. In our approach, the order parameter that is the density amplitude of Bose-Einstein condensation is introduced as the vacuum expectation value of the field operator. We non-perturbatively treat the zero mode operators, which accompany the spontaneous symmetry breaking and the order parameter, because their quantum fluctuations are large for finite system. Then the Hamiltonian for the zero mode operators gives a new type of energy levels with 0⁺ for Nα nuclei, which turn out to be robust for condensation rate and number of α clusters.