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.