Extended vector-tensor theories
  Recently, several extensions of massive vector theory in curved space-time
have been proposed in many literatures. In this paper, we consider the most
general vector-tensor theories that contain up to two derivatives with respect
to metric and vector field. By imposing a degeneracy condition of the
Lagrangian in the context of ADM decomposition of space-time to eliminate an
unwanted mode, we construct a new class of massive vector theories where five
degrees of freedom can propagate, corresponding to three for massive vector
modes and two for massless tensor modes. We find that the generalized Proca and
the beyond generalized Proca theories up to the quartic Lagrangian, which
should be included in this formulation, are degenerate theories even in curved
space-time. Finally, introducing new metric and vector field transformations,
we investigate the properties of thus obtained theories under such
transformations.
