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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 Dcr beyond which the UV fixed point vanishes, such that renormalizability is possible for D <= Dcr. Within the framework of exact RG equations, the critical dimension for various SU (N) gauge theories can be computed to lie near five dimensions: 5 ~< Dcr < 6. Therefore, our results exclude nonperturbative renormalizability of gauge theories in D=6 and higher dimensions.
Holger Gies (Fri,) studied this question.
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