Production Mechanisms and Relations between Spin Density Matrix Elements

Abstract

The spin structure of amplitudes for the production of a single baryon or meson resonance of arbitrary spin in a quasi-two-body final state has been investigated. If the dominant mechanism is the exchange of a single set of quantum numbers in the crossed channel, the spin density matrix elements are shown to satisfy bilinear relations. Typical relations are ${\left(\mathrm{Re}{\rho }_{m,n}\right)}^2+{\left(\mathrm{Re}{\rho }_{m,-n}\right)}^2={\rho }_{\mathrm{mm}}{\rho }_{\mathrm{nn}}$ for baryon production, and ${(\mathrm{Re}{\rho }_{m,n}\pm \mathrm{Re}{\rho }_{m,-n})}^2=({\rho }_{\mathrm{mm}}\pm {\rho }_{m,-m})({\rho }_{\mathrm{nn}}\pm {\rho }_{n,-n})$ for meson production. It is suggested that the use of these relations in the analysis of data is a good test for the presence of two or more exchange contributions. The relations are used to analyze vector- and tensor-meson production from $\pi N$ and $\mathrm{KN}$ initial states, and $\Delta \mathrm{}\left(1236\right)\mathrm{}$ production from $\pi N$ and $\mathrm{NN}$ initial states. Two of the most interesting results concern $\Delta$ production. We find that the small-angle data on $\mathrm{NN}\to N\Delta$ cannot be dominated by $\pi$ exchange alone, and analysis of the data on $\pi N\to \pi \Delta$ at $8 \frac{\mathrm{GeV}}{c}$ indicates the existence of contributions in addition to $\rho$ exchange.