$\pi \mathrm{NNN}$-$\mathrm{NNN}$ problem: Connectedness, transition amplitudes, and quasiparticle approximation

In this paper we review the present status of the $\pi \mathrm{NNN}$-$\mathrm{NNN}$ problem. In particular, we reconsider the chain-labeled approach recently proposed by us, and identify a class of graphs, previously overlooked, which prevents the kernel of the corresponding $\pi \mathrm{NNN}$-$\mathrm{NNN}$ equations from being connected. We propose some approximate schemes, yielding connected-kernel equations. A generalization of the residue method allows us to relate the transition amplitudes for the coupled $\pi \mathrm{NNN}$-$\mathrm{NNN}$ system to the chain-labeled formalism. The quasiparticle approach is extended to the present situation, where emission/absorption of particles is allowed. The open problems for the $\pi \mathrm{NNN}$-$\mathrm{NNN}$ system in light of the present and of previous approaches are finally discussed.