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

Abstract

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," ${\ensuremath{\pi}}^{*}$, 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 ${\ensuremath{\pi}}^{\ensuremath{-}}$ and proton, each of these processes can occur for small values of the invariant square of the virtual pion four-momentum, ${\ensuremath{\Delta}}^{2}$, in which case the virtual pion carries very little transverse three-momentum. For small ${\ensuremath{\Delta}}^{2}$ it is shown that it is reasonable to neglect final state interactions between the "decay products" of the ${\ensuremath{\pi}}^{*}$ and those of the ${N}^{*}$. It is found that in the limit ${\ensuremath{\Delta}}^{2}\ensuremath{\rightarrow}\ensuremath{-}{\ensuremath{\mu}}^{2}$, where $\ensuremath{\mu}$ is the pion rest mass, the virtual pion behaves kinematically, in the ${\ensuremath{\pi}}^{*}$ barycentric system, as an incoming pion which scatters elastically with the incident ${\ensuremath{\pi}}^{\ensuremath{-}}$, and also behaves, in the ${N}^{*}$ barycentric system, as an incoming pion which scatters elastically with the incident proton. Thus, for small ${\ensuremath{\Delta}}^{2}$ in the physical region the ${\ensuremath{\pi}}^{*}$ 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 $\frac{3}{2}\ensuremath{-}\frac{3}{2}$ 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.