Pion Production fromπ−−pCollisions in the Long-Range Interaction Model

Long-range interactions of negative pions with protons resulting in single and double pion production are examined in terms of a meson field theory model in which it is assumed that there is a pion-pion interaction and that the single virtual pion exchange graphs are dominant in large-impact-parameter collisions. This pion exchange leads, in single pion production, to an "excited state," ${\pi }^{*}$, which "decays" into two pions, and, in double pion production, also to an "excited state," $N^{*}$, which "decays" into a pion and a nucleon. For sufficiently high relative energy of the incident ${\pi }^{-}$ and proton, each of these processes can occur for small values of the invariant square of the virtual pion four-momentum, ${\Delta }^2$, in which case the virtual pion carries very little transverse three-momentum. For small ${\Delta }^2$ it is shown that it is reasonable to neglect final state interactions between the "decay products" of the ${\pi }^{*}$ and those of the $N^{*}$. It is found that in the limit ${\Delta }^2\to -{\mu }^2$, where $\mu$ is the pion rest mass, the virtual pion behaves kinematically, in the ${\pi }^{*}$ barycentric system, as an incoming pion which scatters elastically with the incident ${\pi }^{-}$, and also behaves, in the $N^{*}$ barycentric system, as an incoming pion which scatters elastically with the incident proton. Thus, for small ${\Delta }^2$ in the physical region the ${\pi }^{*}$ and $N^{*}$ vertices are defined as the corresponding off-the-mass-shell scattering amplitudes. A ratio of appropriately defined double-to-single pion production cross sections is obtained which is independent of the details of the assumed pion-pion interaction, and depends only on the relative strengths of $N^{*}$ and $N$ formation. This ratio is estimated by means of the $p$-wave static nucleon model, applied in appropriate coordinate systems. For incident 5-Bev/c pions this model leads to double pion production which is important compared to single pion production, both because of the $\frac{3}{2}-\frac{3}{2}$ pion-nucleon resonance and because of important phase-space factors. Kinematical considerations similar to those described above suggest that this model may also be the theoretical genesis of the "two fireballs" model proposed for ultrarelativistic nucleon-nucleon collisions.