Renormalization in the Covariant Treatment of Pion-nucleon Scattering

Abstract

481The problem of renormalization for pion-nucleon scattering is treated in the Bethe-Salpeter formalism. A method to subtract divergencies, especially, overlapping-divergencies, which appear in the solution of an integral equation, is proposed and it is shown that the Salam's prescriptions for the subtraction of overlapping-divergencies can be achieved in a closed form. Finally, the Green's function for T= 1/2 state is obtained in a form free from all the divergencies. § 1. IntroductionWhen we treat the problems of pion-nucleon scattering, nuclear forces, etc. in the jJs-jJs theory, difficulties occur in the renormalization of diverging quantities. In the case of the weak-coupling perturbation theory, the renormalization program is thoroughly executed in all orders of the· coupling constant. But it is a well-known fact that the couplingstrength between pion and nucleon (f~/47r) is about 10-20, and we are not able to resort to the perturbation theory. Many methods have been investigated to take into account higher-order effects. Of these, however, the strong coupling theory1) and the intermediate coupling theory~) are essentially based on the non-relativistic standpoint and are not accessible to the jJs-ps theory. In the Tamm-Dancoff method~), nucleon recoil is easily taken into account, but the procedure to limit the number of pions and nucleonpairs is not carried out in a covariant way and renormalization is performed only lamely. The canonical transformation method by Sawada 4 ) is also available to investigate the contributions of pion cloud around a nucleon, taking into account the effect of nucleon recoil. But, in this case, the correspondence to the Feynman graph is difficult to see and the renormalization procedure is almost impossible to carry out. In this treatment, divergencies are avoided by a cut-off, but the results thus obtained may be in some doubt for the quantitative arguments. On the other hand, in the Bethe-Salpeter (B. S.) approximation method,5) we retain the standpoint of primary interaction and the correspondence to the Feynman graph is clear. But, in this approximation, higher-order contributions which constitute the main part of pion cloud are not taken so extensively as in the treatment by Sawada.In the present paper, we stand on the B. S.-formalism and carry out the renormali-* A preliminary report of this paper was given

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