He3andHe4proton-proton densities

We calculate the proton-proton densities of ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ and ${}_{}{}^4{}_{}{}^{}\mathrm{He}$ using ground-state wave functions obtained from the Faddeev and Faddeev-Yakubovsky methods, respectively. The experimental ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ proton-proton momentum space density is extracted from the electric-charge form-factor data and the longitudinal Coulomb sum rule. We use this as a testing ground for coordinate space Faddeev wave functions based on different realistic nucleon-nucleon interactions, such as the Bonn, Paris, Argonne, Urbana, and SSC-C potentials. We find that the Paris potential reproduces the proton-proton density best, although it does not have sufficient repulsion to reproduce the first minimum. Upon Fourier transformation into coordinate space, we find that the proton-proton density data are not consistent with any of the realistic nucleon-nucleon interactions. With the Faddeev-Yakubovsky amplitudes for the ${}_{}{}^4{}_{}{}^{}\mathrm{He}_{\mathrm{g}.\mathrm{s}.\mathrm{}}^{}{}_{}{}^{}$ wave function and the SSC-C potential, the calculated proton-proton density is consistent with the data, which is, however, of poor quality. Recent (e,e’) data will help to provide the required refinements.