Top mode standard model with extra dimensions

We critically examine a version of the top mode standard model recently cast in extra dimensions by Arkani-Hamed, Cheng, Dobrescu, and Hall, based on the (improved) ladder Schwinger-Dyson (SD) equation for the D- $\mathrm{}(=6,8-)\mathrm{}$ dimensional gauge theories. We find that the bulk QCD cannot have larger coupling beyond the nontrivial ultraviolet (UV) fixed point, the existence of which is supported by a recent lattice analysis. The coupling strength at the fixed point is evaluated by using the one-loop renormalization group equation. It is then found that, in a version with only the third family (as well as the gauge bosons) living in the D-dimensional bulk, the critical (dimensionless) coupling for dynamical chiral symmetry breaking to occur is larger than the UV fixed point of the bulk QCD coupling for $D=6,$ while smaller for $D=8.\mathrm{}$ We further find that the improved ladder SD equation in D dimensions has an approximate scale invariance due to the running of the coupling and hence has an essential-singularity scaling of the “conformal phase transition,” similar to Miransky scaling in the four-dimensional ladder SD equation with a nonrunning coupling. This essential-singularity scaling can resolve the fine-tuning even when the cutoff (“string scale”) is large. Such a theory has a large anomalous dimension ${\gamma }_m=D/2\mathrm{}-1$ and is expected to be free from the flavor-changing-neutral-current problem as in walking technicolor for $D=4.\mathrm{}$ Furthermore, the induced bulk Yukawa coupling becomes finite even at infinite cutoff limit (in the formal sense), similar to the renormalizability of the gauged Nambu–Jona-Lasinio model. Comments are made on the use of the “effective” coupling, which includes finite renormalization effects, instead of the $\overline{\mathrm{MS}}$ running coupling in the improved ladder SD equation.