Three-particle systems with resonant subprocesses in a finite volume

In previous work, we have developed a relativistic, model-independent three-particle quantization condition, but only under the assumption that no poles are present in the two-particle K matrices that appear as scattering subprocesses M. T. Hansen and S. R. Sharpe, Phys. Rev. D 90, 116003 (2014) ; M. T. Hansen and S. R. Sharpe, Phys. Rev. D 92, 114509 (2015) ; R. A. Brice\~no et al. , Phys. Rev. D 95, 074510 (2017). . Here we lift this restriction, by deriving the quantization condition for identical scalar particles with a G-parity symmetry, in the case that the two-particle K matrix has a pole in the kinematic regime of interest. As in earlier work, our result involves intermediate infinite-volume quantities with no direct physical interpretation, and we show how these are related to the physical three-to-three scattering amplitude by integral equations. This work opens the door to study processes such as a₂, in which the is rigorously treated as a resonance state.