The Weyl-Cartan Gauss-Bonnet gravity

In this paper, we consider the generalized Gauss-Bonnet action in four-dimensional Weyl-Cartan spacetime. In this spacetime, the presence of a torsion tensor and Weyl vector implies that the generalized Gauss-Bonnet action will not be a total derivative in four-dimensional spacetime. It will be shown that the higher than two time derivatives can be removed from the action by choosing a suitable set of parameters. In the special case where only the trace part of the torsion remains, the model reduces to general relativity plus two vector fields, one of which is massless and the other is massive. We will then obtain the healthy region of the five-dimensional parameter space of the theory in some special cases.