Dynamical chiral symmetry breaking in gauge theories with extra dimensions

We investigate dynamical chiral symmetry breaking in vector-like 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 non-local gauge fixing method is adopted so as to keep the ladder approximation consistent with the Ward-Takahashi identities. Using the one-loop $\overline{\mathrm{M}\mathrm{S}}$ gauge coupling of the truncated KK effective theory which has a nontrivial ultraviolet fixed point (UV-FP) $g_{\ast }$ for the (dimensionless) bulk gauge coupling $\widehat{g}$, we find that there exists a critical number of flavors, $N_f^{\mathrm{c}\mathrm{r}\mathrm{i}\mathrm{t}}$ ($\simeq 4.2,1.8$ for $D=6,8$ for SU(3) gauge theory): For $N_f>N_f^{\mathrm{c}\mathrm{r}\mathrm{i}\mathrm{t}}$, the dynamical chiral symmetry breaking takes place not only in the strong-coupling phase'' (${\hat g} >g_*$) but also in the weak-coupling phase’’ ($\widehat{g}<g_{\ast }$) when the cutoff is large enough. For $N_f<N_f^{\mathrm{c}\mathrm{r}\mathrm{i}\mathrm{t}}$, 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 the $N_f^{\mathrm{c}\mathrm{r}\mathrm{i}\mathrm{t}}$ turns out to be a value similar to that of the $\overline{\mathrm{M}\mathrm{S}}$ case, notwithstanding the enhancement of the coupling compared with that of the $\overline{\mathrm{M}\mathrm{S}}$.