Accuracy of the distorted-wave impulse approximation for breakup reactions in a three-body model

We have studied the reaction mechanisms of the ($p,2p$) reaction on ${}_{}{}^4{}_{}{}^{}\mathrm{He}$ at energies of 65 and 100 MeV. The dynamics of the reaction is taken to be that of a three-particle system, the ${}_{}{}^4{}_{}{}^{}\mathrm{He}$ target being viewed as a proton bound to an inert triton. The particles interact via separable potentials chosen to fit bound state and scattering data of the two-particle subsystems. The protons are treated as identical bosons interacting with each other only in the relative $S$ state while the proton-triton potential is limited to relative $L\le 2$. The ${}_{}{}^4{}_{}{}^{}\mathrm{He}\mathrm{}(p,2p)\mathrm{}{}_{}{}^3{}_{}{}^{}\mathrm{H}$ doubly differential cross section can be calculated exactly within this model by solving the Faddeev equations numerically using the method of deformed contours and Padé approximants. This can be compared with the distorted-wave impulse approximation (DWIA), calculated in this model by summing the appropriate subset of the multiple scattering series. This is done in momentum space and treats off-shell and finite range effects of the nucleon-nucleon $T$ matrix as well as recoil effects exactly. We find that the DWIA is an adequate approximation at 100 MeV but is almost double the exact result at 65 MeV. Its use in the extraction of spectroscopic information below 100 MeV is therefore suspect. The shapes of the exact, distorted, and plane wave cross sections agree quite well with each other (and with the experimental shape), differing only in magnitude.