Comments onππPhase Shifts as Determined from the Peripheral Model

The determination of the $S$-wave $\pi \pi$ phase shifts ${{\delta }_0}^I$ at low energy from the analysis of $\pi N\to \left(2\mathrm{}\pi \right)N$ is examined critically from the standpoint of the one-pion-exchange model with absorptive corrections. It is found that: (1) The value of ${{\delta }_0}^I$ depends strongly on the $P$-wave phase shifts, which cannot be unambiguously determined, at $m_{\pi \pi }<600\mathrm{}$ MeV, by using a Breit-Wigner formula. (2) The ratio of the production density matrix elements $\rho$ (with the $\pi \pi$ elastic scattering amplitudes factored out) depends strongly on $m_{\pi \pi }$ for $m_{\pi \pi }<600\mathrm{}$ MeV. (3) The $\frac{\mathrm{}(F-B)\mathrm{}}{\mathrm{}(F+B)\mathrm{}}$ asymmetry shows a sizeable dependence on the momentum transfer $t$ to the nucleon. It is concluded that more accurate data at low $m_{\pi \pi }$ are required in order to determine ${{\delta }_0}^I$ for $m_{\pi \pi }<600\mathrm{}$ MeV. Tables of the $\rho (m_{\pi \pi }, t)$ calculated from the absorption model for an incident-pion laboratory kinetic energy of 4 BeV are included. These could be directly applied to the data to obtain the low-energy $\pi \pi$ phase shifts.