Renormalizability of gauge theories in extra dimensions

We analyze the possibility of nonperturbative renormalizability of gauge theories in $D>\mathrm{}4\mathrm{}$ dimensions. We develop a scenario, based on Weinberg’s idea of asymptotic safety, that allows for renormalizability in extra dimensions owing to a non-Gaussian ultraviolet stable fixed point. Our scenario predicts a critical dimension $D_{\mathrm{cr}}$ beyond which the UV fixed point vanishes, such that renormalizability is possible for $D<~D_{\mathrm{cr}}.$ Within the framework of exact renormalization group equations, the critical dimension for various $\mathrm{SU}\mathrm{}\left(N\right)\mathrm{}$ gauge theories can be computed to lie near five dimensions: $5\lesssim D_{\mathrm{cr}}<6.\mathrm{}$ Therefore, our results exclude nonperturbative renormalizability of gauge theories in $D=6\mathrm{}$ and higher dimensions.