ΛNTensor Forces for Scattering and for theΛ-Particle Binding in Nuclear Matter

The effect of $\Lambda N$ tensor forces on the $\Lambda$-particle binding in nuclear matter is studied with the use of second-order perturbation theory and the Brueckner-Bethe reaction-matrix approach in the $g$-matrix approximation. The $g$ matrix is calculated self-consistently by use of the Kallio-Day version of the reference-spectrum method. The free kinetic energies are assumed for the unoccupied states. One-boson-exchange (OBE) models indicate that the $\Lambda N$ tensor force is expected to be of short range and moderate strength. For short-range tensor forces the dominant momentum components are very large, and the effects of such forces are only slightly modified by the nuclear medium. On the other hand, if the $\Lambda N$ tensor forces were of rather long range, they would be quite strongly suppressed in nuclear matter. These features are very clearly exhibited by consideration of the effective nonlocal central potentials that represent the ($s$-state) effect of tensor forces for nuclear matter and for scattering. The ratio of the (nuclear-matter) expectation values of these two effective potentials is a good measure of the suppression. The expectation value of the effective potential for nuclear matter is just the second-order perturbation-theory energy. Reaction-matrix calculations show that higher-order effects may become quite important for shorter ranges. Such calculations have, in particular, been made for various mixtures of central and tensor forces chosen to give a constant $s$-wave scattering length. Yukawa shapes corresponding to the kaon and one- and two-pion masses were used, as well as "realistic" OBE potentials with a hard core and a tensor component due to kaon exchange (and also approximately due to $\eta$ exchange). For a particular mixture, the suppression is measured by the reduction in the well depth relative to the depth for a purely central potential which has the same hard core and the same scattering length and effective range as the mixture. For the short-range tensor forces there is rather little suppression even for very strong tensor forces which account for all the triplet scattering. Different assumptions about the $d$-state interaction have an almost negligible effect on the $s$-state well depth if the same assumption is made for both scattering and nuclear matter. Similar considerations are made for the effect of tensor forces in the $p$ wave, for which we find very little suppression ($\lesssim 1$ MeV). We conclude that if central and short-range tensor forces are chosen to compensate each other for low-energy scattering, they will also compensate each other quite closely for nuclear matter. In particular, for the OBE potentials with strengths consistent with the phenomenological values of the $\Lambda N\overline{K}$ coupling constants, the reduction in the well depth is at most about 4 MeV. The conclusions about the $\Lambda N$ interaction obtained from a comparison of the calculated and phenomenological well depths are, therefore, effectively unchanged by the presence of a $\Lambda N$ tensor force. Consequently, in order to bring the two numbers into agreement, it is necessary to invoke a substantial short-range repulsion, a rather weak $p$-state interaction, and suppression of the $\Lambda N-\Sigma N$ coupling and/or repulsive $\Lambda \mathrm{NN}$ three-body forces.