Non-renormalizable Interactions: A Self-Consistency Manifesto

Abstract

The renormalization procedure is proved to be a rigorous way to get finite answers in a renormalizable class of field theories. We claim, however, that it is redundant if one reduces the requirement of finiteness to S-matrix elements only and does not require finiteness of intermediate quantities like the off-shell Green functions. We suggest a novel view on the renormalization procedure. It is based on the usual BPHZ R-operation, which is equally applicable to any local QFT independently of whether it is renormalizable or not. The key point is the replacement of the multiplicative renormalization, used in renormalizable theories, by an operation when the renormalization constants depend on the fields and momenta that have to be integrated inside the subgraphs. This approach being applied to quantum field theories does not distinguish between renormalizable and non-renormalizable interactions and provides the basis for getting finite scattering amplitudes in both cases. The arbitrariness of the subtraction procedure is fixed by imposing a normalization condition on the scattering amplitude as a whole rather than on an infinite series of new operators appearing in the process of subtraction of UV divergences in non-renormalizable theories.We show that using the property of locality of counter-terms, precisely as in renormalizable theories, one can get recurrence relations connecting leading, subleading, etc., UV divergences in all orders of perturbation theory in any local theory. This allows one to get generalized RG equations that have an integro-differential form and sum up the leading logarithms in all orders of PT in full analogy with the renormalizable case. This way one can cure the problem of violation of unitarity in non-renormalizable theories by summing up the leading asymptotics. The approach can be applied to any theory though technically non-renormalizable interactions are much more complicated than renormalizable ones. We illustrate the basic features of our approach by several examples.Our main statement is that non-renormalizable theories are self-consistent, they can be well treated within the usual BPHZ R-operation, and the arbitrariness can be fixed to a finite number of parameters just as in the renormalizable case.

Exact summation of leading logs around $T\bar T$ deformation of $O(N+1)$-symmetric 2D QFTs

“…It is worth mentioning that similar equations, although in a different context, were derived in Refs. [21][22][23][24][25][26].…”

Renormalization group equations and the recurrence pole relations in pure quantum gravity

“…It was suggested in [7] (see also [8] and for some later developments [9]) that the Renormalization Group methods should be equally applicable to the non-renormalizable theories. This program if successful would make the non-renormalizable theories to look quite similar to the renormalizable ones.…”