On Existence of Self-Tuning Solutions in Static Braneworlds without Singularities

A static self-tuning SO(3)xZ_2 symmetric and translation invariant braneworld setup with flat brane is considered. We discuss the null energy conditions (NEC) for matter on the brane and in the bulk and prove that for the static regular background with broken Lorentz invariance the NEC and positiveness of the total energy density on the brane and NEC in the bulk cannot be satisfied simultaneously. Then we give some examples and elaborate some special cases. For instance, we provide a macroscopic solution for a background with Lifshitz scaling.