Model for Low-Energy Meson-Baryon Scattering

Abstract

An effective Lagrangian, containing relativistic $\mathrm{SU}(6)$-invariant three- and four-point interaction terms, is constructed for the scattering of the 35 meson representation by the 56 baryons. The lowest-order matrix elements calculated from the Lagrangian include single-particle exchange terms that account for long-range forces, and four-point interaction terms that approximate short-range forces for low-energy scattering. Because of the imposition of free-field conditions on the scattered particles, the amplitudes maintain a coplanar $U(3)\ensuremath{\bigotimes}U(3)$ symmetry, which is then broken by the mass differences in the multiplets. There are three parameters in the model, which are fixed by the ${S}_{11}$, ${S}_{31}$, and ${P}_{11}$ scattering lengths for $\ensuremath{\pi}N$ elastic scattering. Unitarity is implemented without introducing more parameters, by equating the matrix elements to those of the reaction matrix $K$. The amplitudes for all other reactions contained in 35 \ensuremath{\bigotimes} 56 scattering are determined thereby. In particular, the cross sections and angular distributions for $\ensuremath{\pi}N\ensuremath{\rightarrow}\ensuremath{\pi}N$, $\mathrm{KN}\ensuremath{\rightarrow}\mathrm{KN}$, $\ensuremath{\pi}N\ensuremath{\rightarrow}\ensuremath{\eta}N$, $\ensuremath{\pi}N\ensuremath{\rightarrow}K\ensuremath{\Lambda}$, $\ensuremath{\pi}N\ensuremath{\rightarrow}K\ensuremath{\Sigma}$, and $\overline{K}N\ensuremath{\rightarrow}\ensuremath{\eta}\ensuremath{\Lambda}$ are calculated near their thresholds and compared with experiment. In all cases where comparisons can be made, the model is in fairly good agreement with the data, with the exception of $\ensuremath{\pi}N\ensuremath{\rightarrow}K\ensuremath{\Sigma}$. The threshold amplitudes for vector-meson production are also given as a prediction of the model.