Conical singularities and the Vainshtein screening in full GLPV theories

In Gleyzes-Langlois-Piazza-Vernizzi (GLPV) theories, it is known that the conical singularity arises at the center of a spherically symmetric body ($r=0$) in the case where the parameter $\alpha_{{\rm H}4}$ characterizing the deviation from the Horndeski Lagrangian $L_4$ approaches a non-zero constant as $r \to 0$. We derive spherically symmetric solutions around the center in full GLPV theories and show that the GLPV Lagrangian $L_5$ does not modify the divergent property of the Ricci scalar $R$ induced by the non-zero $\alpha_{{\rm H}4}$. Provided that $\alpha_{{\rm H}4}=0$, curvature scalar quantities can remain finite at $r=0$ even in the presence of $L_5$ beyond the Horndeski domain. For the theories in which the scalar field $\phi$ is directly coupled to $R$, we also obtain spherically symmetric solutions inside/outside the body to study whether the fifth force mediated by $\phi$ can be screened by non-linear field self-interactions. We find that there is one specific model of GLPV theories in which the effect of $L_5$ vanishes in the equations of motion. We also show that, depending on the sign of a $L_5$-dependent term in the field equation, the model can be compatible with solar-system constraints under the Vainshtein mechanism or it is plagued by the problem of a divergence of the field derivative in high-density regions.