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(H4) characterizing the deviation from the Horndeski Lagrangian L-4 approaches a non-zero constant as r -> 0. We derive spherically symmetric solutions around the center in full GLPV theories and show that the GLPV Lagrangian does not modify the divergent property of the Ricci scalar R induced by the non-zero alpha(H4). Provided that alpha(H4) = 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 L5 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.