Renormalization of Derivative Coupling Theories

Abstract

The method of functional integrals is applied to the problem of meson theories with derivative couplings. In the static limit, solutions in closed form can be exhibited. The infinities occurring in the theory are found to be removable in terms of $Z_2$ and mass renormalizations, contrary to the conclusions of perturbation analysis. The divergences occurring here have the form of essential singularities, in contradistinction to the branch-point behavior of the usual "renormalizable" theories. The lack of validity of the perturbation expansion is thereby accounted for. These techniques can be extended to treat the full recoil neutral $\mathrm{ps}\left(\mathrm{pv}\right)\mathrm{}$ problem omitting closed loops. The theory is represented in terms of an exponential coupling which permits a nonperturbation series solution for the various propagators. Two infinite renormalizations are again required. The resultant functions are given meaning by analytic continuation procedures which are adapted to the four-dimensional nature of the problem. The form of the effective coupling suggests a rearrangement of the answer in terms of exponentials of the meson propagator. As a result mass operator-like structures can be defined. These explicitly exhibit the transcendental nature of the coupling and generalized equivalence theorems with $\mathrm{ps}\left(\mathrm{ps}\right)\mathrm{}$ theory can be generated. In a similar fashion, effective interaction operators for the two-nucleon and meson-nucleon Green's functions are derived. The possible applicability of these quantities to questions of physical interest such as nuclear potentials and multiple meson production is briefly mentioned.