Existence and disappearance of conical singularities in Gleyzes-Langlois-Piazza-Vernizzi theories

In a class of Gleyzes-Langlois-Piazza-Vernizzi (GLPV) theories, we derive both vacuum and interior Schwarzschild solutions under the condition that the derivatives of a scalar field $\phi$ with respect to the radius $r$ vanish. If the parameter $\alpha_{\rm H}$ characterizing the deviation from Horndeski theories approaches a non-zero constant at the center of a spherically symmetric body, we find that the conical singularity arises at $r=0$ with the Ricci scalar given by $R=-2\alpha_{\rm H}/r^2$. This originates from violation of the geometrical structure of four-dimensional curvature quantities. The conical singularity can disappear for the models in which the parameter $\alpha_{\rm H}$ vanishes in the limit that $r \to 0$. We propose explicit models without the conical singularity by properly designing the classical Lagrangian in such a way that the main contribution to $\alpha_{\rm H}$ comes from the field derivative $\phi'(r)$ around $r=0$. We show that the extension of covariant Galileons with a diatonic coupling allows for the recovery of general relativistic behavior inside a so-called Vainshtein radius. In this case, both the propagation of a fifth force and the deviation from Horndeski theories are suppressed outside a compact body in such a way that the model is compatible with local gravity experiments inside the solar system.