Black hole solutions in shift-symmetric degenerate higher-order scalar-tensor theories

We show that most classes of shift-symmetric degenerate higher-order scalar-tensor (DHOST) theories which satisfy certain degeneracy conditions are not compatible with the conditions for the existence of exact black hole solutions with a linearly time-dependent scalar field whose canonical kinetic term takes a constant value. Combined with constraints from the propagation speed of gravitational waves, our results imply that cubic DHOST theories are strongly disfavoured and that pure quadratic theories are likely to be the most viable class of DHOST theories. We find exact static and spherically symmetric (Schwarzschild and Schwarzschild-(anti-)de Sitter) black hole solutions in all shift-symmetric higher-derivative scalar-tensor theories which contain up to cubic order terms of the second-order derivatives of the scalar field, especially the full class of Horndeski and Gleyzes-Langlois-Piazza-Vernizzi theories. After deriving the conditions for the coupling functions in the DHOST Lagrangian that allow the exact solutions, we clarify their compatibility with the degeneracy conditions.