On the consistent disformal couplings to fermions

TL;DR

This work analyzes the ghost-free consistency of disformal couplings to fermions, showing that axial couplings induced by disformal transformations generally introduce higher derivatives and potential Ostrogradsky ghosts. For two scalar fields, consistency demands a degenerate field-space metric, effectively constraining the transformation to a single disformal direction and linking to two-field Horndeski/DHOST theories; the Einstein-frame transformation then yields a two-field DHOST-like structure. Extending to single-field higher derivatives expands the space of consistent disformal models (including generalized U-DHOST) and reveals degeneracy conditions that can preserve second-order dynamics. The study also explores higher-derivative vector couplings, finding that U(1) gauge symmetry imposes strong restrictions (often ghostly), while certain general, invertible cases hint at new vector theories beyond generalized Proca. Overall, the paper provides a systematic framework for constructing ghost-free, multi-field disformal gravity theories and clarifies how higher-derivative couplings interact with degeneracy conditions across scalars and vectors.

Abstract

Disformal couplings to fermions lead to a unique derivative coupling to the axial fermionic current, which contains higher derivatives in general. We derive general conditions on consistent disformal couplings by requiring the absence of higher time derivatives, as they typically lead to ghost degrees of freedom. For a two-scalar field disformal transformation, we show that the consistent disformal coupling must have a degenerate field space metric. This allows us to explore consistent, new two-scalar field modified gravity models. We show that the transformation of the Einstein-Hilbert action leads to two-field Horndeski or two-field DHOST theories. Our formalism also applies to disformal transformations with higher derivatives. We derive the consistent subclasses of disformal transformations that include second derivatives of a scalar field and first derivatives of a vector field that lead to generalized U-DHOST and degenerate beyond generalized Proca theories.