Renormalization and dimensional regularization for a scalar field with Gauss-Bonnet-type coupling to curvature

We consider a scalar field with a Gauss-Bonnet-type coupling to the curvature in a curved space-time. For such a quadratic coupling to the curvature, the metric energy-momentum tensor does not contain derivatives of the metric of orders greater than two. We obtain the metric energy momentum tensor and find the geometric structure of the first three counterterms to the vacuum averages of the energy-momentum tensors for an arbitrary background metric of an N-dimensional space-time. In a homogeneous isotropic space, we obtain the first three counterterms of the n-wave procedure, which allow calculating the renormalized values of the vacuum averages of the energy momentum tensors in the dimensions N = 4, 5. Using dimensional regularization, we establish that the geometric structures of the counterterms in the n-wave procedure coincide with those in the effective action method.