How one can obtain unambiguous predictions for the S-matrix in non-renormalizable theories

Abstract: The usual Bogolyubov R-operation works in non-renormalizable theories in the same way as in renormalizable ones. However, in the non-renormalizable case, the counter-terms eliminating ultraviolet divergences do not repeat the structure of the original Lagrangian but contain new terms with a higher degree of fields and derivatives increasing from order to order of PT. If one does not aim to obtain finite off-shell Green functions but limits oneself only to the finiteness of the S-matrix, then one can use the equations of motion and drastically reduce the number of independent counter-terms. For example, it is possible to reduce all counter-terms to a form containing only operators with four fields and an arbitrary number of derivatives. And although there will still be infinitely many such counter-terms, in order to fix the arbitrariness of the subtraction procedure, one can normalize the on-shell 4-point amplitude, which must be known for arbitrary kinematics, plus the 6-point amplitude at one point. All other multiparticle amplitudes will be calculated unambiguously. Within the framework of perturbation theory, the number of independent counter-terms in a given order is limited, so does the number of normalization conditions. The constructed counter-terms are not absorbed into the normalization of a single coupling constant, the Lagrangian contains an infinite number of terms, but after fixing the arbitrariness, it allows one to obtain unambiguous predictions for observables.