$p$-Wave Theory of Three-Nucleon States

A formalism is developed for the structure of a three-fermion system through $p$-wave interactions in all pairs. It is mainly designed for the analysis of three-nucleon systems of highest seniority, in particular the tri-neutron, which has been the subject of experimental interest in recent times. The basic ($p$-wave) interaction is taken to be the separable type, as an extension of the formalism for previous three-body investigations with $s$-wave interactions in pairs. A detailed classification of the various three-body states involved is given in the ($\mathrm{LSJ}$) coupling scheme. A generous application of the permutation group for three objects separately in the spin and spatial degrees of freedom dispenses with the need for formal use of fractional parentage coefficients in the construction of various states. The allowed structures of the different wave functions compatible with the assumption of factorable interactions are written down in terms of one-dimensional "spectator functions" for both even and odd parity cases and their integral equations are obtained. The effect of a $p$-wave spin-orbit force on the mixing of possible three-body states of given $J$ is analyzed and, in a reasonable approximation, shown to be expressible in terms of a simple modification of the strength parameter of the two-body interaction in the integral equations for the spectator functions. The state of strongest attraction is found to be ${\left(1\mathrm{} \frac{3}{2} \frac{1}{2}\right)}^{+}$, and offers the best chances, for a bound trineutron state. The states of odd parity are found to be much less attractive than, and hence not of as much interest as, those of even parity for three-nucleon systems. It is suggested that those could be of greater interest in a possible quark model of baryon resonances with $p$-wave forces, to which the present formalism can be mathematically adapted, assuming its physical validity.