Three-particle formalism for multiple channels: the $\eta \pi \pi + K \overline K \pi$ system in isosymmetric QCD

We generalize previous three-particle finite-volume formalisms to allow for multiple three-particle channels. For definiteness, we focus on the two-channel $\eta \pi \pi$ and $K \overline K \pi$ system in isosymmetric QCD, considering the positive $G$ parity sector of the latter channel, and neglecting the coupling to modes with four or more particles. The formalism we obtain is thus appropriate to study the $b_1(1235)$ and $\eta(1295)$ resonances. The derivation is made in the generic relativistic field theory approach using the time-ordered perturbation theory method. We study how the resulting quantization condition reduces to that for a single three-particle channel when one drops below the upper ($K\overline K \pi$) threshold. We also present parametrizations of the three-particle K matrices that enter into the formalism.