Processπ−p→π+π−nnear Threshold

Abstract

A phenomenological analysis has been made of the process ${\pi }^{-}p\to {\pi }^{+}{\pi }^{-}n$ near the pion-production threshold. The basis for this analysis consisted of treating formally the two-pion system in the final state as though it were a single system $\tilde{\sigma }$ with $T=J=0\mathrm{}$ but with a continuous mass spectrum. A partial-wave decomposition was made of the $\pi N\to \tilde{\sigma }N$ pseudo-two-to-two-particle scattering amplitude, and all partial-wave amplitudes except that for which the $\tilde{\sigma }$ $n$ state is in the $s$ wave were given the values obtained from the single-pion-exchange graph. The remaining amplitude with an $s$-wave $\tilde{\sigma }$ $n$ final state was calculated separately. By assuming that of the three final-state particles only the two pions provided an important interaction, a phenomenological form for this partial wave was derived in terms of the $D$ function arising in an $\frac{N}{D}$ solution for the $T=J=0\mathrm{}$ pion-pion amplitude. All the partial waves were then combined to calculate the low-energy differential cross sections, and the values of the various parameters were adjusted to give an optimum fit to the experimental data. The most satisfactory fit to the data was obtained with a $T=J=0\mathrm{}$ phase shift whose scattering length was negative but soon turned over and became positive.