A relativistic optical model for pion-nucleon scattering

Abstract

A theoretical study of pion-nucleon scattering for pion laboratory momentum between 2 and 5 BeV/c was made by generalizing the nuclear optical model. The Bethe-Salpeter equation for pion-nucleon scattering via the simultaneous exchange of a scalar-isoscalar particle and a vector-isovector particle (ϱ) was derived in the ladder approximation. The corresponding Blankenbecler-Sugar equation was then derived and a relativistic optical potential was obtained by allowing the coupling constants to become complex. Of the 6 parameters in the optical potential (the masses of the two particles and the real and imaginary parts of the product of the coupling constants for the two particles) only the imaginary part of the scalar coupling was energy dependent. The other coupling parameters were fixed by fitting the data at a pion laboratory momentum of 4 BeV/c. The ϱ mass was fixed at its experimental value of 765 MeV. To fit the forward diffraction peak we needed an isoscalar mass of 450 MeV. Using these parameters good fits to the total and elastic cross sections and the π ±− p elastic differential cross sections for t ≲ 1.5 (BeV/c) 2 were obtained. While fits to the differential charge exchange cross sections and the ratio of the real to imaginary part of the forward scattering amplitude were in reasonable agreement with experiment, the polarizations only agreed qualitatively. A discussion of the significance of the results and of possible extensions to the above theory are presented.