Generalized Galileons: All scalar models whose curved background extensions maintain second-order field equations and stress tensors

TL;DR

Addressing the problem of creating scalar-tensor theories on curved backgrounds whose equations of motion remain second order, the paper develops a unified covariant Galileon construction. It rewrites flat-space Galileons using the antisymmetric tensor ${\cal A}_{(2n)}$ and then builds a hierarchical family ${\cal L}_{(n+1,p)}$ with curvature insertions ${\cal R}_{(p)}$ and ${\cal S}_{(q)}$, with coefficients ${\cal C}_{(n+1,p)}$ fixed by a recurrence to cancel all higher-derivative terms. The $D=4$ curved-space case is explicitly demonstrated with the nonminimal terms ${\cal L}_{(4,1)}$ and ${\cal L}_{(5,1)}$, and the construction extends to arbitrary $D$ with a closed-form solution ${\cal C}_{(n+1,p)} = (-1/8)^p (n-1)!/[(n-1-2p)!(p!)^2]$. The result is a minimal, Lovelock-like class of covariant Galileons valid in any $D$ and background, offering a transparent framework for second-order scalar-tensor theories with curvature couplings and potential cosmological applications.

Abstract

We extend to curved backgrounds all flat-space scalar field models that obey purely second-order equations, while maintaining their second-order dependence on both field and metric. This extension simultaneously restores to second order the, originally higher derivative, stress tensors as well. The process is transparent and uniform for all dimensions.