Relationship between X(5)-models and the interacting boson model

The connections between the X(5)-models (the original X(5) using an infinite square well, X(5)-$\beta^8$, X(5)-$\beta^6$, X(5)-$\beta^4$, and X(5)-$\beta^2$), based on particular solutions of the geometrical Bohr Hamiltonian with harmonic potential in the $\gamma$ degree of freedom, and the interacting boson model (IBM) are explored. This work is the natural extension of the work presented in [1] for the E(5)-models. For that purpose, a quite general one- and two-body IBM Hamiltonian is used and a numerical fit to the different X(5)-models energies is performed, later on the obtained wave functions are used to calculate B(E2) transition rates. It is shown that within the IBM one can reproduce well the results for energies and B(E2) transition rates obtained with all these X(5)-models, although the agreement is not so impressive as for the E(5)-models. From the fitted IBM parameters the corresponding energy surface can be extracted and it is obtained that, surprisingly, only the X(5) case corresponds in the moderate large N limit to an energy surface very close to the one expected for a critical point, while the rest of models seat a little farther.