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We study the properties of the low-lying states at the critical point of thephase transition from U(3) to O(4) symmetry in the U(4) vibron model in detail.By analyzing the general characteristics and comparing the calculated resultsof the energy spectra and the E1, E2 transition rates in E(3) symmetry, in$r^4$ potential model and the finite boson number case in boson space, we findthat the results in the $r^4$ potential demonstrates the characteristic of theclassical limit at the critical point well and the E(3) symmetry over-predictthe energy levels and under-predict the E1 and E2 transition rates of thestates at the critical point. However, the E(3) symmetry may describe part ofthe properties of the system with boson number around 10 to 20. We also confirmthat the 12C+12C system is an empirical evidence of the state at the criticalpoint of the phase transition in the U(4) model when concerning the energies ofthe low-lying resonant states.

Property of the low-lying states at the critical point of the phase transition in U(4) vibron model