Trinucleon system in a two-body model: Coulomb effect on bound and scattering states

Recently it has been suggested that as in the neutron deuteron problem the theoretically calculated spin-doublet S wave proton deuteron scattering length ${}_{}{}^2{}_{}{}^{}\mathrm{a}_{\mathrm{pd}}^{}{}_{}{}^{}$ is correlated with ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ binding energy. This strongly suggests that ${}_{}{}^2{}_{}{}^{}\mathrm{a}_{\mathrm{pd}\mathrm{\simeq }0}^{}{}_{}{}^{}$. In this paper we study various properties of the low energy trinucleon system including this correlation among scattering length and binding energy for the spin-doublet neutron deuteron and the proton deuteron systems in an ad hoc two-body model. The two-body potential models we employ have a long range tail of $r^{\mathrm{-}2}$ type and have been parametrized to produce the correct binding energies of ${}_{}{}^3{}_{}{}^{}\mathrm{H}$ and ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ and the correct s wave spin-doublet neutron deuteron scattering lengths ${}_{}{}^2{}_{}{}^{}\mathrm{a}_{\mathrm{nd}}^{}{}_{}{}^{}$. The present study predicts the low energy nucleon deuteron phase shifts and predicts the proton deuteron scattering length to be ${}_{}{}^2{}_{}{}^{}\mathrm{a}_{\mathrm{pd}\mathrm{\simeq }(0.15\mathrm{}\pm 0.1}^{}{}_{}{}^{}$) fm.