Low-Energy Theorems in the Chiral Soliton Model

Abstract

We show that low-energy theorems for pion-photoproduction and Compton scattering can be reproduced within the Skyrme-soliton model using the formulation developed for pion­ nucleon scattering. The electric polarizability of nucleon is shown to be given by a disper­ sion integral with the longitudinal-photon absorption cross section, while the magnetic one is given by using the dispersion integral. We explicitly give contributions from the L1 state as the resonance and the 1rL1 channel coupling. We have recently shown that the soft-pion theorems such as the Goldberger­ Treiman, Adler-Weisberger and Tomozawa-Weinberg relations, can be reproduced within the Skyrme-soliton model. 1' In this treatment, we developed a full-field method for the soliton model. The method gives the same amplitudes as those of the usual collective coordinate method using a constrained field theory. In the method, the same Born amplitude can be obtained by assuming the full field to be an interpolat­ ing field between the asymptotic in and out fields. Here, we do not decompose the field into the soliton and the fluctuating fields. This method is superior to the former one, because the axial-vector and vector currents are found to satisfy the current algebra. The soft-pion theorems can be straightforwardly proven such as the Adler-Weisberger (AW) and the Tomozawa-Weinberg (TW) relations. Whether low-energy theorems (LET) are reproduced is a crucial test both for models of nucleon and for approximations, because LET are based on general principles such as current algebra, partially conserving axial-vector current and gauge invariance. We tried in a previous paper 2 ' to reproduce the LET for pion­ photoproduction3' in the model: We have shown that the LET is reproduced for leading order in mn:/M through the electric and magnetic Born terms, but not com­ pletely for terms linear in mn:/M, where M(mn:) is the nucleon (pion) mass. In this work we try to reproduce this linear term following the method of Low,4' and then investigate the Compton scattering involving the electromagnetic polar­ izabilities of the nucleon. § 2. Full field method in the Skyrme model In the usual approach to the pion-nucleon scattering in the Skyrme model we