Inclusive (p, Delta ++) and (p,p' pi + reactions.

Abstract

Formulas are derived for cross sections for inclusive (p,${\mathrm{\ensuremath{\Delta}}}^{++}$) and (p,p\ensuremath{'} ${\mathrm{\ensuremath{\pi}}}^{+}$) reactions; these are shown to depend on the pion self-energy in the nucleus, and the production and decay vertices of the ${\mathrm{\ensuremath{\Delta}}}^{++}$. The pion self-energy includes both nucleon excitation (${\mathrm{NN}}^{\ensuremath{-}1}$) as well as delta excitation (\ensuremath{\Delta}${\mathrm{N}}^{\ensuremath{-}1}$) in the nucleus. The pion-nucleon couplings in the transition vertex as well as in the self-energy are written using relativistic pseudovector coupling; results are compared with a nonrelativistic reduction. The distortions of the continuum particles are included in a ``distorted-wave Fermi gas'' approximation. Calculations are made for the excitation spectrum of the nucleus, d\ensuremath{\sigma}/d\ensuremath{\omega}, and the angular distribution in the (p,${\mathrm{\ensuremath{\Delta}}}^{++}$) reaction, for proton beam energies from threshold to 3 GeV. For the (p,p\ensuremath{'} ${\mathrm{\ensuremath{\pi}}}^{+}$) reaction cross sections are calculated for the proton spectrum and the distribution d\ensuremath{\sigma}/d${\mathrm{\ensuremath{\Omega}}}_{\mathrm{\ensuremath{\pi}}}$\ensuremath{\Omega}p\ensuremath{'}${\mathrm{dE}}_{\mathrm{p}\ensuremath{'}}$${\mathrm{dE}}_{\mathrm{\ensuremath{\pi}}}$ at fixed values of the proton and proton angles and a fixed value of ${\mathrm{E}}_{\mathrm{p}\ensuremath{'}}$. The latter essentially gives the excitation energy spectrum of the nucleus. Many interesting features are seen in these spectra. The differences in results using relativistic pion-nucleon coupling and its nonrelativistic reduction are shown to increase with the beam energy. Around 400 MeV, there is little difference in the cross sections but at 1 GeV and beyond the difference becomes large, changing both the magnitude and shape of the distributions. The inclusion of distortions reduces the magnitude of the cross section but leaves the shape unchanged.