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, 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 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 non-renormalizable theories. Using the property of locality of counter-terms, we get recurrence relations connecting leading, subleading, etc., UV divergences in all orders of PT in any local theory. This allows one to get generalized RG equations that have an integro-differential form and sum up the leading logarithms. This way one can cure the problem of violation of unitarity in non-renormalizable theories by summing up the leading asymptotics. 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.