The Weyl-Cartan Gauss-Bonnet gravity

In this paper, we consider the generalized Gauss-Bonnet action in $4$-dimensional Weyl-Cartan space-time. In this space-time, the presence of torsion tensor and Weyl vector implies that the generalized Gauss-Bonnet action will not be a total derivative in four dimension space-time. It will be shown that the higher than two time derivatives can be removed from the action by choosing suitable set of parameters. In the special case where only the trace part of the torsion remains, the model reduces to GR plus two vector fields. One of which is massless and the other is massive. We will then obtain the healthy region of the 5-dimensional parameter space of the theory in some special cases.