Non-Renormalizable Interactions: A Self-Consistency Manifesto
Seminar “Theory of Fundamental Interactions”
Date and Time: Thursday, 16 January 2020, at 4:00 PM
Venue: Blokhintsev Hall (4th floor), Bogoliubov Laboratory of Theoretical Physics
Seminar topic: “Non-Renormalizable Interactions: A Self-Consistency Manifesto”
Speaker: D. I. Kazakov
The renormalization procedure is proved to be a rigorous way to get finite answers in renormalizable class of field theories. We claim, however, that it is redundant if one reduces the requirement of finiteness to the S-matrix elements only and do not require the finiteness of the intermediate quantities like the off-shell Green functions. This approach being applied to quantum field theories does not distinguish between renormalizable and non-renormalizable interactions and provides the basis to get the finite scattering amplitudes in both cases with controllable arbitrariness of subtraction 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 which have to be integrated inside the subgraphs. Then, locality of the counter terms, precisely like in renormalizable theories, leads to recurrence relations connecting the leading, subleading, etc., UV divergences in all orders of perturbation theory. This allows one to get the generalized RG equations which have 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 the non-renormalizable interactions are much more complicated. We illustrate the basic features of our approach on several examples.