Schwinger-Dyson analysis of dynamical symmetry breaking on a brane with bulk Yang-Mills theory

The dynamically generated fermion mass is investigated in a flat brane world with (4+δ)-dimensional bulk space-time, and in the Randall-Sundrum (RS) brane world. We consider the bulk Yang-Mills theory interacting with a fermion confined on a four-dimensional brane. Based on the effective theory below a reduced cutoff scale on the brane, we formulate the Schwinger-Dyson equation of the brane fermion propagator. By using the improved ladder approximation we numerically solve the Schwinger-Dyson equation and find that the dynamical fermion mass is near the reduced cutoff scale on the brane for the flat brane world with $\delta >~3$ and for the RS brane world. In the RS brane world Kaluza-Klein excited modes of the bulk gauge field localize around the $y=\pi R$ brane and this enhances dynamical symmetry breaking on the brane. The decay constant of the fermion and the antifermion composite operator can be taken to be of the order of the electroweak scale, much smaller than the Planck scale. Therefore the electroweak mass scale can be realized from only the Planck scale in the RS brane world due to fermion and antifermion pair condensation. That is a dynamical realization of the Randall-Sundrum model which solves the weak-Planck hierarchy problem.