Parametrization of the elastic sector of the nucleon-nucleon scattering matrix. I.

We present a method for parametrizing the 2×2 $S$ matrix for coupled-channel nucleon-nucleon elastic scattering above the pion-production threshold, namely, we introduce two dummy nucleon-Roper-resonance channels and unitarize the consequent 4×4 $S$ matrix. The elastic nonunitary 2×2 nucleon-nucleon $S$ matrix is then recovered as a submatrix of the 4×4 matrix. We parametrize the unitary 4×4 $S$ matrix by generalizing the prescription of Stapp, Ypsilantis, and Metropolis for 2×2 matrices. The six phases parametrizing the 2×2 submatrix are well behaved as the interaction vanishes, and automatically satisfy a unitarity condition which we derive, viz., det $(1-S_{\mathrm{a}\mathrm{}\mathrm{a}}^{†}{S}_{\mathrm{aa}})\ge 0$, where $S_{\mathrm{aa}}$ is the 2×2 submatrix. Of the six phases, three represent the absorption and parametrize a real, symmetric 2×2 matrix $N$, analogous to the absorption parameter $\eta$ in the uncoupled case. A method is given for recovering the phase parameters from the $S$ matrix or other parametrizations.