Two-Pion-Exchange Contribution to the Three-BodyΛ-Nucleon Interaction

The two-pion-exchange contribution to the three-body $\Lambda$-nucleon interaction is derived from a static model and also from covariant perturbation theory. It is found that the local part of the potential calculated by the latter method is similar to that part of the static-model potential which corresponds to the formation of lambda-antisigma pairs in intermediate states. This potential is noncentral and has the form $\left({\tau }^1·{\tau }^2\right)\left({\sigma }^1·{\mathrm{r}}_1\right)\left({\sigma }^2·{\mathrm{r}}_2\right)f(r_1, r_2)$, where ${\sigma }^i$ and ${\tau }^i$ are the spin and isotopic-spin operators for the two nucleons, and ${\mathbf{r}}_1$ and ${\mathbf{r}}_2$ are the $\Lambda$-nucleon separation vectors. An estimate is made of the importance of this potential in the binding of the hypertriton by calculating its expectation value with respect to hypertriton wave functions corresponding to two-body interactions with hard cores. In these calculations, the three-body potential is found to contribute less than 5% of the expectation value of the total $\Lambda$-nucleon interaction.