Dynamical symmetry breaking on a brane with bulk gauge theory

We analyze a structure of dynamical chiral symmetry breaking in orbifold gauge theories with matter fields (fermions) on the fixed point. We find that the boundary chiral phase structure of QED and QCD on the orbifold is quite nontrivial depending on the bulk constitution, and we claim that particular attention should be given to the dynamically generated masses in various kinds of phenomenological orbifold models. It is revealed recently that the orbifold field theory can provide, e.g., the weak and Planck hierarchy [1], a fermion mass hierarchy (via localization [2]), a gauge symmetry and a supersymmetry (SUSY) breaking [3, 4] (by the Wilson line [5]), SUSY breaking mediation mechanisms (e.g., [6]), and the proton stability (e.g., [7]), based on perturbative (or tree) analyses. However an effective higher-dimensional theory such as the orbifold model means Mc < Λ by definition, i.e., the compactification scale should be less than the cut-off scale of the theory. This implies an existence of Kaluza-Klein particles below Λ and we have a question about their effect on a nonperturbative dynamics of the theory. One simple example for such a nonperturbative dynamics is a chiral symmetry breaking structure in a SU(N) Yang-Milles and matter (fermion) theory on the orbifold. Assuming only a kink-type mass ǫ(y)Mkink, we have a chiral symmetry for the fermion zero mode. The fermion zero mode couples to KK gauge bosons as well as a gauge boson zero mode. So the chiral phase structure may be different from the usual 4D case (e.g., a four-fermion approximation, Ref. [8]). The simplest case for the analysis is |Mkink| → Λ that means all the KK fermions decouple and the fermion becomes a brane field effectively. Also a lot of orbifold models introduces intrinsic brane fields (fermions) which couples to a bulk gauge boson. Therefore we analyze a dynamical chiral symmetry breaking (DSB) on a boundary with a bulk gauge theory. We consider QCD (QED) on M4×S/Z2 with quarks (electrons) on the fixed point. For generality, the bulk geometry is assumed as, ds = GMNdx dx = eημνdx dx − dy, where k is a AdS curvature scale. We use R, Λ and Λ5D as a radius of the fifth dimension, a 4D (brane) effective cut-off scale and a 5D cut-off scale respectively. The 4D effective Lagrangian is given by