Conformal coupling of the scalar field with gravity in higher dimensions and invariant powers of the Laplacian

The hierarchy of conformally coupled scalars with the increasing scaling dimensions Δ(k)=k−d/2, k=1,2,3,… connected with the kth Euler density in the corresponding spacetime dimensions d⩾2k is proposed. The corresponding conformal invariant Lagrangian with the kth power of Laplacian for the already known cases k=1,2 is reviewed, and the subsequent case of k=3 is completely constructed and analyzed.