Renormalizability of gauge theories in extra dimensions

We analyze the possibility of nonperturbative renormalizability of gauge theories in D>4 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_cr beyond which the UV fixed point vanishes, such that renormalizability is possible for D<~D_cr. Within the framework of exact renormalization group equations, the critical dimension for various SU(N) gauge theories can be computed to lie near five dimensions: 5≲D_cr<6. Therefore, our results exclude nonperturbative renormalizability of gauge theories in D=6 and higher dimensions.