In extra dimensional theories, the four-dimensional field theory is reduced
from a fundamental field theory in the bulk spacetime by integrating the extra
dimensional part. In this paper we investigate the effective action of a
self-interacting scalar field on a brane in the five-dimensional thick
braneworld scenario. We consider two typical thick brane solutions and obtain
the P\”{o}schl-Teller and harmonic potentials of the Kaluza-Klein (KK) modes,
respectively. The analytical mass spectra and wave functions along extra
dimension of the KK modes are obtained. Further, the effective coupling
constant between different KK particles, cross section, and decay rate for some
processes of the KK particles are related to the fundamental coupling in five
dimensions and the new physics energy scale. Some interesting properties of
these interactions are found with these calculations. The KK particles with
higher mode have longer lifetime, and they almost do not interact with ordinary
matter on the brane if their mode numbers are large enough. Thus, these KK
particles with higher modes might be a candidate of dark matter
