Empirical Partial-Wave Analysis ofπ+pElastic Scattering Above 1 GeV/c

The partial-wave equation $\frac{d\sigma }{d\Omega }=\left|\right[\frac{1}{2ik}]\Sigma \stackrel{L}{l=0\mathrm{}}\mathrm{}(2l+1)(1-a_l)P_l\left(cos\theta \right){|}^2$ has been used to fit most of the recent $\pi +p$ differential cross-section measurements above 1 GeV/c. The $a_l$ were determined by the method of weighted least squares, with the further requirement that they be real and they satisfy either constraints of the form $1\ge 1-a_l\ge 0$ (which allows the scattering to be interpreted as purely absorptive) or the more relaxed constraints $2\ge 1-a_l\ge 0$. This equation with the requirements does not allow the scattering amplitude to have a spin-flip part or a real part, but for one set of data further terms were added to allow these additional parts of the scattering amplitude. For each differential cross section at the various energies, a set of $a_l$ values was determined which in almost all cases fit the measured cross sections quite well. These sets of $a_l$ parameters have two properties in common. First, all $a_l$ except $a_0$ satisfy $1\ge 1-a_l\ge 0$. The $a_0$ parameters ($s$-wave amplitudes) required $1-a_0\ge 1$ except for the higher energies where $1\ge 1-a_0\ge 0$ was obtained. Second, graphs of $1-a_l$ versus $l$ (one graph for each different cross-section measurement) show that $1-a_l$ decreases rather smoothly with increasing $l$ and that the curve is either roughly linear or concave upward. No striking variations in the $a_l$ parameters are observed when the energy is close to one of the $\pi +p$ total cross section resonances. The $a_l$ parameters are interpreted using $1-a_l$ as a measure of the absorption of the $l\mathrm{th}$ partial wave by inelastic processes. Differential cross section measurements of ${\pi }^{-}+p$ at 2.01 GeV/c and of ${\pi }^{+}+p$ at 2.02 GeV/c, previously published only in graphical form, are given in the Appendix.