Dynamical chiral symmetry breaking in gauge theories with extra dimensions

We investigate dynamical chiral symmetry breaking in vectorlike gauge theories in D dimensions with D-4 compactified extra dimensions, based on the gap equation (Schwinger-Dyson equation) and the effective potential for the bulk gauge theories within the improved ladder approximation. The nonlocal gauge fixing method is adopted so as to keep the ladder approximation consistent with the Ward-Takahashi identities. Using the one-loop modified minimal subtraction scheme ((MS) over bar) gauge coupling of the truncated Kaluza-Klein effective theory, which has a nontrivial ultraviolet fixed point g(*) for the (dimensionless) bulk gauge coupling (g) over cap, we find that there exists a critical number of flavors N-f(crit) [similar or equal to4.2,1.8 for D=6,8 for SU(3) gauge theory]. For N-f>N-f(crit), the dynamical chiral symmetry breaking takes place not only in the "strong-coupling phase" ((g) over cap >g(*)) but also in the "weak-coupling phase" (g<g(*)) when the cutoff is large enough. For N-f<N-f(crit), on the other hand, only the strong-coupling phase is a broken phase and we can formally define a continuum (infinite cutoff) limit, so that the physics is insensitive to the cutoff in this case. We also perform a similar analysis using the one-loop "effective gauge coupling." We find that N-f(crit) turns out to be a value similar to that of the (MS) over bar case, notwithstanding the enhancement of the coupling compared with that of the (MS) over bar.