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Register a new userRenormalizable Expansion for Nonrenormalizable Theories: I. Scalar Higher Dimensional Theories
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We demonstrate how one can construct renormalizable perturbative expansion in formally nonrenormalizable higher dimensional scalar theories. It is based on 1/N-expansion and results in a logarithmically divergent perturbation theory in arbitrary high odd space-time dimension. 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.
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The previously developed renormalizable perturbative 1/N-expansion in higher dimensional scalar field theories is extended to gauge theories with fermions. It is based on the $1/N_f$-expansion and results in a logarithmically divergent perturbation theory in arbitrary high odd space-time dimension. Due to the self-interaction of non-Abelian fields the proposed recipe requires some modification which, however, does not change the main results. The new 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. The original dimensionful gauge coupling plays a role of mass and is also logarithmically renormalized. Comments on the unitarity of the resulting theory are given.
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.
Some nonrenormalizable theories are less singular than all renormalizable theories, and one can use lattice simulations to extract physical information from them. This paper discusses four nonrenormalizable theories that have finite euclidian and minkowskian Greens functions. Two of them have finite euclidian action densities and describe scalar bosons of finite mass. The space of nonsingular nonrenormalizable theories is vast.
This paper has been withdrawn to address an omission. It will be resubmitted in the near future.
We classify a large set of melonic theories with arbitrary $q$-fold interactions, demonstrating that the interaction vertices exhibit a range of symmetries, always of the form $mathbb{Z}_2^n$ for some $n$, which may be $0$. The number of different theories proliferates quickly as $q$ increases above $8$ and is related to the problem of counting one-factorizations of complete graphs. The symmetries of the interaction vertex lead to an effective interaction strength that enters into the Schwinger-Dyson equation for the two-point function as well as the kernel used for constructing higher-point functions.
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