Renormalizable 1/N_f Expansion for Field Theories in Extra Dimensions

TL;DR

This paper introduces a $1/N_f$ expansion method to construct renormalizable perturbation theories in higher-dimensional, traditionally nonrenormalizable, field theories, enabling consistent quantum descriptions in extra dimensions.

Contribution

It develops a novel $1/N_f$ expansion approach that renders higher-dimensional field theories renormalizable and analyzes their renormalization group behavior and unitarity properties.

Findings

Effective coupling becomes dimensionless and runs with RG equations.

Beta function is nonpolynomial and shows UV or IR behavior depending on dimension.

Theory generally contains ghost states but can be made unitary through coupling adjustments.

Abstract

We demonstrate how one can construct renormalizable perturbative expansion in formally nonrenormalizable higher dimensional field theories. It is based on $1/N_f$-expansion and results in a logarithmically divergent perturbation theory in arbitrary high space-time dimension. First, we consider a simple example of $N$-component scalar filed theory and then extend this approach to Abelian 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 some modification which, however, does not change the main results. The resulting effective coupling is dimensionless and is running in accordance with the usual RG equations. The corresponding beta function is calculated in the leading order and is nonpolynomial in effective coupling. It exhibits either UV asymptotically free or IR free behaviour depending on the dimension of space-time. The original dimensionful coupling plays a role of a mass and is also logarithmically renormalized. We analyze also the analytical properties of a resulting theory and demonstrate that in general it acquires several ghost states with negative and/or complex masses. In the former case, the ghost state can be removed by a proper choice of the coupling. As for the states with complex conjugated masses, their contribution to physical amplitudes cancels so that the theory appears to be unitary.