Existence of the Trineutron

The possibility of the existence of a trineutron bound state has been investigated by using a separable $N-N$ interaction in the $p$ wave. Two different sets of central and spin-orbit forces of the (L·S) and ${\left(\mathbf{L}\mathbf{·}\mathbf{S}\right)}^2$ types have been used to obtain a detailed fit to the ${}_{}{}^3{}_{}{}^{}P_j^{}{}_{}{}^{}$ scattering phase shifts. The calculations have been performed for the $J^P={\frac{1}{2}}^{+}$ state, which was shown by Mitra and Bhasin to be the best possible candidate for such a bound state. The solution of the coupled one-dimensional integral equations, as an eigenvalue problem in the "inverse strength parameter" of the $n-n$ interaction, shows that the requisite eigenvalue for zero binding energy for the $n^3$ system has a sufficiently wide margin over the corresponding quantity obtained from the $p-p$ phase shifts. This confirms the earlier conclusion of Mitra and Bhasin based on effective central $p$-wave forces, but is contradictory to the results of variational approaches with conventional potentials given by Okamoto and Davies, and by Barbi. The possible significance of the difference is discussed.