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We demonstrate how one can construct renormalizable perturbative expansion informally nonrenormalizable higher dimensional field theories. It is based on$1/N_f$-expansion and results in a logarithmically divergent perturbationtheory in arbitrary high space-time dimension. First, we consider a simpleexample of $N$-component scalar filed theory and then extend this approach toAbelian and non-Abelian gauge theories with $N_f$ fermions. In the latter case,due to self-interaction of non-Abelian fields the proposed recipe requires somemodification which, however, does not change the main results. The resultingeffective coupling is dimensionless and is running in accordance with the usualRG equations. The corresponding beta function is calculated in the leadingorder and is nonpolynomial in effective coupling. It exhibits either UVasymptotically free or IR free behaviour depending on the dimension ofspace-time. The original dimensionful coupling plays a role of a mass and isalso logarithmically renormalized. We analyze also the analytical properties ofa resulting theory and demonstrate that in general it acquires several ghoststates with negative and/or complex masses. In the former case, the ghost statecan be removed by a proper choice of the coupling. As for the states withcomplex conjugated masses, their contribution to physical amplitudes cancels sothat the theory appears to be unitary.

Renormalizable 1/N_f Expansion for Field Theories in Extra Dimensions