Invertible disformal transformations with higher derivatives

Abstract

We consider a higher-derivative generalization of disformal transformations and clarify the conditions under which they form a group with respect to the matrix product and the functional composition. These conditions allow us to systematically construct the inverse transformation in a fully covariant manner. Applying the invertible generalized disformal transformation to known ghost-free scalar-tensor theories, we obtain a novel class of ghost-free scalar-tensor theories, whose action contains the third-or higher-order derivatives of the scalar field as well as nontrivial higherorder derivative couplings to the curvature tensor.*1 Relaxing the degeneracy conditions so that the higher-derivative terms are degenerate only in the unitary gauge, we obtain a broader class of scalar-tensor theories [13][14][15][16][17]. In that case, there is an apparent extra DOF in a generic gauge, but it satisfies an elliptic differential equation and hence is an instantaneous (or "shadowy") mode [14,17]. *2 However, in two spacetime dimensions, the Horndeski class is closed under the disformal transformation of the form (1) [31].

Built-in scordatura in U-DHOST

“…In the context of scalar-tensor theories, the first stealth solution called ghost condensation was constructed on Minkowski and de Sitter backgrounds [6,7] and then extended to a Schwarzschild black hole background [8]. The general construction of stealth solutions was then developed [9,10] and perturbations around them [11][12][13] have been extensively studied, with a particular focus on degenerate higher-order scalar-tensor (DHOST) theories [14][15][16][17][18][19][20][21][22], which form a large class of healthy scalartensor theories without Ostrogradsky ghost [23] thanks to particular degeneracy conditions imposed on higherderivative terms [17,[24][25][26][27][28]. A remarkable fact is that perturbations about a stealth Minkowski (or de Sitter) background in DHOST theories are strongly coupled in general [6,29].…”

“…Specifically, the ghost mass is inversely proportional to (the square root of) c U [see (22)], meaning that the ghost is absent if only c U = 0. The class of theories satisfying…”