New terms for scalar multi-Galileon models and application to $\mathrm{SO}(N)$ and $\mathrm{SU}(N)$ group representations

We investigate a new class of scalar multi-Galileon models, which is not included in the commonly admitted general formulation of generalized multi-Galileons. The Lagrangians of this class of models, some of them having already been introduced in previous works, are specific to multi-Galileon theories, and vanish in the single Galileon case. We examine them in detail, discussing in particular some hidden symmetry properties which can be made explicit by adding total derivatives to these Lagrangians. These properties allow us to describe the possible dynamics for these new Lagrangians in the case of multi-Galileons in the fundamental representation of a $\mathrm{SO}(N)$ and $\mathrm{SU}(N)$ global symmetry group, as well as in the adjoint representation of a $\mathrm{SU}(N)$ global symmetry group. We perform in parallel an exhaustive examination of some of these models, finding a complete agreement with the dynamics obtained using the symmetry properties. Finally, we conclude by discussing what could be the most general multi-Galileon theory, as well as the link between scalar and vector multi-Galileon models.