Some simple theories of gravity with propagating torsion

We consider antisymmetric metric-affine theories of gravity with a Lagrangian containing the most general terms up to dimension four and search for theories that are ghost- and tachyon-free when expanded around flat space. We find new examples that propagate only the graviton and one other massive degree of freedom of spin zero, one, or two. These models require terms of the form (∇T)2 in the Lagrangian, that have been largely ignored in the literature.