Shift-symmetric $SO(N)$ multi-Galileon

A Poincar\`{e} invariant, local scalar field theory in which the Lagrangian and the equation of motion contain only up to second-order derivatives of the fields is called generalized Galileon. The covariant version of it in four dimensions is called Horndeski theory, and has been vigorously studied in applications to inflation and dark energy. In this paper, we study a class of multi-field extensions of the generalized Galileon theory. By imposing shift and $SO(N)$ symmetries on all the currently known multi-Galileon terms in general dimensions, we find that the structure of the Lagrangian is uniquely determined and parameterized by a series of coupling constants. We also study tensor perturbation in the shift-symmetric $SO(3)$ multi-Galileon theory in four dimensions. The tensor perturbations can obtain a mass term stemming from the same symmetry breaking pattern as the solid inflation. We also find that the shift-symmetric $SO(3)$ multi-Galileon theory gives rise to new cubic interactions of the tensor modes, suggesting the existence of a new type of tensor primordial non-Gaussianity.