Generalizing the relativistic quantization condition to include all three-pion isospin channels

We present a generalization of the relativistic, finite-volume, three-particle quantization condition for non-identical pions in isosymmetric QCD. The resulting formalism allows one to use discrete finite-volume energies, determined using lattice QCD, to constrain scattering amplitudes for all possible values of two- and three-pion isospin. As for the case of identical pions considered previously, the result splits into two steps: the first defines a non-perturbative function with roots equal to the allowed energies, E$_{n}$(L), in a given cubic volume with side-length L. This function depends on an intermediate three-body quantity, denoted $ {\mathcal{K}}_{\mathrm{df},3,} $ which can thus be constrained from lattice QCD in- put. The second step is a set of integral equations relating $ {\mathcal{K}}_{\mathrm{df},3} $ to the physical scattering amplitude, ℳ$_{3}$. Both of the key relations, E$_{n}$(L) ↔$ {\mathcal{K}}_{\mathrm{df},3} $ and $ {\mathcal{K}}_{\mathrm{df},3}\leftrightarrow {\mathrm{\mathcal{M}}}_3, $ are shown to be block-diagonal in the basis of definite three-pion isospin, I$_{πππ}$ , so that one in fact recovers four independent relations, corresponding to I$_{πππ}$ = 0, 1, 2, 3. We also provide the generalized threshold expansion of $ {\mathcal{K}}_{\mathrm{df},3} $ for all channels, as well as parameterizations for all three-pion resonances present for I$_{πππ}$ = 0 and I$_{πππ}$ = 1. As an example of the utility of the generalized formalism, we present a toy implementation of the quantization condition for I$_{πππ}$ = 0, focusing on the quantum numbers of the ω and h$_{1}$ resonances.