Renormalization group and completeness of field theories. Nucleon-pion couplings

Abstract

The problem of the mathematical consistency of a field theory defined by a given local Hamiltonian is studied in terms of the propagator (Green’s function) formalism. It is necessary for the mathematical consistency of a theory that all the branching equations satisfield by its propagators be covariant under the transformations of its renormalization group (which can be explicitly written). This analysis (which differs in method, but not in principle, from the standard renormalization program) permits to find systematically and explicitly all the terms that need be added to the original Hamiltonian if this was not complete to start with,i.e. if covariance could not be secured for the set of branching equations obtained from it alone. Local non-renormalizable theories are mathematically meaningless, because they originate from only fragments of Hamiltonians which are meaningful only if taken as wholes; the missing terms (even if infinite in number) can be exactly reconstructed with the present method, which leads naturally to identify the concepts of mathematical consistency and of physical completeness. All meaningful relations among coupling constants, such as symmetry requirements, must remain invariant under the renormalization group, which plays a role as important in the search for completeness, as that of the gauge groupe in electrodynamical problems. For the sake of concereteness, and as a first example, this method is illustrated with reference to the study of the standard meson-nucleon couplings, scalar and pseudoscalar, neutral and charged; the well known \gf{su3} and \gf{su1} (scalar), \gf{su4} (pseudoscalar) terms are obtained (a precedent erroneous statement about the renormalizability of the neutral scalar coupling is corrected, so that now all results agree with the expected ones). Another example is treated in the Appendix.