Consistency of higher derivative couplings to matter fields in scalar-tensor gravity

TL;DR

This paper examines the coupling of generalized disformal Horndeski theories to matter fields, concluding that only theories related to Horndeski via conventional disformal transformations can be consistently coupled without introducing ghosts.

Contribution

It demonstrates that higher derivative terms in generalized disformal transformations must be excluded to maintain ghost-free matter couplings in scalar-tensor theories.

Findings

Higher derivative terms lead to Ostrogradsky ghosts.

Only conventional disformal transformations preserve ghost-free matter couplings.

Generalized disformal Horndeski theories cannot be minimally coupled to matter without ghosts.

Abstract

Recently, a generalization of invertible disformal transformations containing higher-order derivatives of a scalar field has been proposed in the context of scalar-tensor theories of gravity. By applying this generalized disformal transformation to the Horndeski theory, one can obtain the so-called generalized disformal Horndeski theories which are more general healthy scalar-tensor theories than ever. However, it is unclear whether or not the generalized disformal Horndeski theories can be coupled consistently to matter fields because introducing a matter field could break the degeneracy conditions of higher-order scalar-tensor theories and hence yield the unwanted Ostrogradsky ghost. We investigate this issue and explore the conditions under which a minimal coupling to a matter field is consistent in the generalized disformal Horndeski theories without relying on any particular gauge such as the unitary gauge. We find that all the higher derivative terms in the generalized disformal transformation are prohibited to avoid the appearance of the Ostrogradsky ghost, leading to the conclusion that only the theories that are related to the Horndeski theory through a conventional disformal transformation remain ghost-free in the presence of minimally coupled matter fields.