Conformal couplings of a scalar field to higher curvature terms

TL;DR

This paper introduces a method to construct conformal couplings of scalar fields to higher curvature terms, ensuring second-order equations of motion and covariant transformation under Weyl rescalings, with solutions analogous to Lovelock theories.

Contribution

It provides a new systematic way to couple scalar fields conformally to higher curvature invariants while maintaining second-order field equations.

Findings

Constructed a covariant four-rank tensor involving curvature and derivatives.

Derived second-order equations of motion and energy-momentum tensor.

Obtained solutions for spherically symmetric cases as polynomial equations.

Abstract

We present a simple way of constructing conformal couplings of a scalar field to higher order Euler densities. This is done by constructing a four-rank tensor involving the curvature and derivatives of the field, which transforms covariantly under local Weyl rescalings. The equation of motion for the field, as well as its energy momentum tensor are shown to be of second order. The field equations for the spherically symmetric ansatz are integrated, and for generic non-homogeneous couplings, the solution is given in terms of a polynomial equation, in close analogy with Lovelock theories.