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 (${\beta }_2$), axial hexadecapole (${\beta }_4$) and triaxial (${\gamma }_2$). The phase-transitional structure is established for a schematic $sdg$ hamiltonian which is intermediate between four dynamical symmetries of U(15), namely the spherical $\mathrm{U}(5)\otimes \mathrm{U}(9)$, the (prolate and oblate) deformed ${\mathrm{S}\mathrm{U}}_{\pm }(3)$ and the ${\gamma }_2$-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.