Generalized Proca and its Constraint Algebra

We reconsider the construction of general derivative self-interactions for a massive Proca field. The constructed Lagrangian is such that the vector field propagates at most three degrees of freedom, thus avoiding the ghostly nature of a fourth polarisation. The construction makes use of the well-known condition for constrained systems of having a degenerate Hessian. We briefly discuss the casuistry according to the nature of the existing constraints algebra. We also explore various classes of interesting new interactions that have been recently raised in the literature. For the sixth order Lagrangian that satisfies the constraints by itself we prove its topological character, making such a term irrelevant. There is however a window of opportunity for exploring other classes of fully-nonlinear interactions that satisfy the constraint algebra by mixing terms of various order.