Thick brane in reduced Horndeski theory

Horndeski theory is the most general scalar-tensor theory retaining second-order field equations, although the action includes higher-order terms. This is achieved by a special choice of coupling constants. In this paper, we investigate the thick brane system in reduced Horndeski theory, especially the effects of the nonminimal derivative coupling and the cubic Galileon term on thick brane. First, the equations of motion are presented and a set of analytic background solutions are obtained. Then, the effects of these two terms on the thick brane are analyzed. It is found that the background solutions become asymmetric with the presence of the cubic Galileon term. However, the energy density of the thick brane is still symmetric with or without this term and splits with the presence of the nonminimal derivative coupling. The stability of the thick brane under tensor perturbation is also considered. It is shown that the tachyon is absent and the graviton zero mode can be localized on the brane. The localized graviton zero mode recovers the four-dimensional Newtonian potential and the presence of the nonminimal derivative coupling results in a splitting of its wave function. Besides, the presence of the cubic Galileon term results in the asymmetry of the effective potential and zero mode of the graviton. The correction of the massive graviton Kaluza-Klein modes to the Newtonian potential is also analyzed briefly.