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

TL;DR

This paper proposes a hierarchy of conformally invariant scalar–gravity couplings in spacetime dimensions $d\geq 2k$, where the scalar has scaling dimension $\Delta_{(k)}=k-d/2$ and each level $k$ is tied to the $k$-th Euler density $E_{(k)}$. The authors articulate a general correspondence between the conformal scalars $\varphi_{(k)}$ and $E_{(k)}$, and they explicitly construct the $k=3$ case by a careful Noether/Weyl variation procedure, incorporating Weyl invariants built from the Weyl tensor and its contractions. They show that the resulting invariant actions contain a leading $k$-th order Laplacian term $\varphi_{(k)}\Box^{k}\varphi_{(k)}$ and a curvature-coupled piece proportional to $E_{(k)}\varphi_{(k)}^{2}$, with coefficients determined by $\Delta_{(k)}$, and they identify auxiliary invariants (including holographic-like contributions) needed for exact conformal invariance. The results yield, for $k=1,2,3$, explicit actions and support a general conjecture for the form of $S_{(k)}^{E_{(k)}}$ at arbitrary $k$, offering a universal framework with potential applications in higher-dimensional conformal field theory, trace anomalies, and AdS/CFT holography.

Abstract

The hierarchy of conformally coupled scalars with the increasing scaling dimensions $Δ_{k}=k-d/2$, $k=1,2,3,... $ connected with the $k$-th Euler density in the corresponding space-time dimensions $d\geq 2k$ is proposed. The corresponding conformal invariant Lagrangian with the $k$-th 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.