Towards the extended SU(2) Proca theory

In this work, we explore the construction of the most general vector-tensor theory with a SU(2) global symmetry in the vector sector as a proposal for a modified theory of gravity. We start with a general Lagrangian containing terms involving symmetric and/or antisymmetric combinations of the covariant derivative of the vector field plus an arbitrary function of the vector field times the Ricci scalar. Then, we study the degeneracy of the full theory to determine whether it can be healthy or not. We find relations among some of the free functions in the theory that are necessary for the healthiness of the theory in correspondence with the several ways in which the kinetic matrix can be made degenerate. Finally, we take the scalar limit and present additional conditions on the free functions that are necessary for the healthiness of the longitudinal modes.