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

TL;DR

This paper explores a hierarchy of conformally coupled scalar fields in higher dimensions, linking their properties to Euler densities and extending the conformal invariant Lagrangian to higher powers of the Laplacian, including explicit construction for the case k=3.

Contribution

It introduces a new hierarchy of conformally coupled scalars associated with Euler densities and constructs the conformal invariant Lagrangian for higher powers of the Laplacian, especially for k=3.

Findings

Proposed a hierarchy of conformally coupled scalars with increasing scaling dimensions.

Reviewed known cases for k=1,2 and explicitly constructed the case for k=3.

Connected scalar field properties to Euler densities in higher dimensions.

Abstract

The hierarchy of conformally coupled scalars with the increasing scaling dimensions $\Delta_{k}=k-d/2$, $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.