Study of 3he(e,e') Longitudinal Response Functions with the Integral-Transform Method

The method of integral transforms is first applied for studying the $^3$He longitudinal response functions. The transforms are calculated from localized bound-state-type solutions to an inhomogenous Schr\"odinger-type three-body equation. Several versions of local $s$-wave spin-dependent potentials supplemented with a singlet $p$-wave potential and with the proton-proton Coulomb interaction are used as a two-nucleon input. The conventional charge density operator is utilized. The three-body equations are solved with a high acuracy. It is found that the contribution of the $T=3/2$ final states to the problem is suppressed and it amounts about 15\%. This might be ascribed to symmetry requirements. The contributions of the $p$-wave $NN$ interaction and of the Coulomb interaction are found to amount several per cent. Uncertainty due to different choices of $s$-wave $NN$ forces is of a similar magnitude provided that the low-energy $NN$ data are properly described. The results are compared with the integral transforms of the experimental response functions. For $q=300$ MeV/c experimental and theoretical results coincide within their uncertainties. For $q=500$ MeV/c a noticeable difference is detected.