On Renormalizations in Nonrenormalizable Theories

A new view of the procedure of renormalizations in nonrenormalizable theories is proposed. This view is based on the standard procedure of the BPHZ $\mathcal R$-operation, which is equally applicable to any local quantum field theory irrespective of renormalizability. The key point is that the multiplicative renormalization used in renormalizable theories is replaced by an operation in which the renormalization constant depends on the momenta over which integration in subgraphs is performed. In this case, the requirement for the counterterms to be local (precisely as in renormalizable theories) leads to recurrence relations between leading, subleading, etc., ultraviolet divergences in all orders of perturbation theory. This allows one to obtain generalized renormalization group equations for scattering amplitudes, which have an integro-differential form and lead to the summation of the leading asymptotics, just as in renormalizable theories.