We investigate the dynamical chiral symmetry breaking (D$\chi$SB) in vector-like gauge theories with extra dimensions, based on the analysis of the gap equation (the Schwinger-Dyson equation of the bulk fermion mass) with use of the one-loop \bar{MS} gauge coupling in the truncated Kaluza-Klein (KK) effective theory. In our analysis, we assume the existence of the non-perturbative ultraviolet fixed point (UV-FP) $g_*$ for the dimensionless bulk gauge coupling, which is defined by the bulk gauge coupling times some powers of the renormalization scale $\mu$. Within the one-loop analysis, the UV-FP can be found actually.
One might expect that the D$\chi$SB never occur in the weakly coupled regime where the dimensionless bulk gauge coupling is smaller than g_*. However, it turns out not true: We find that there exists a critical point $g_{crit}$ for $g_*$. For $g_* > g_{crit}$, the bulk fermion acquires its dynamical mass even in the weakly coupled regime. Here, we note that the dimensionless bulk gauge coupling approaches very quickly the UV-FP $g_*$ due to its power-like running. Namely, the dimensionless bulk gauge coupling is nearly equal to $g_*$ over a wide range of the integrand in the gap equation. It means that the dimensionless bulk gauge coupling for a large value of $g_*$ is not so small even in the weakly coupled regime. The value of $g_{crit}$ gives a constraint for models of DEWSB in the bulk based on this weakly interacting phase.
On the other hand, the D$\chi$SB occurs in the whole strong coupled regime due to the infrared-dynamics like as the four dimensional QCD. This phase may be applied to the Technicolor model in the bulk.
We also perform a similar analysis using the one-loop effective gauge coupling including finite renormalization effects of many KK modes, which break down the decoupling theorem.