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Phase transitions in the $sdg$ interacting boson model

A geometric analysis of the $sdg$ interacting boson model is performed. A coherent-state is used in terms of three types of deformation: axial quadrupole ($β_2$), axial hexadecapole ($β_4$) and triaxial ($γ_2$). The phase-transitional structure is established for a schematic $sdg$ hamiltonian which is intermediate between four dynamical symmetries of U(15), namely the spherical ${\rm U}(5)\otimes{\rm U}(9)$, the (prolate and oblate) deformed ${\rm SU}_\pm(3)$ and the $γ_2$-soft SO(15) limits. For realistic choices of the hamiltonian parameters the resulting phase diagram has properties close to what is obtained in the $sd$ version of the model and, in particular, no transition towards a stable triaxial shape is found.