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

Abstract

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, En(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, ℳ3. Both of the key relations, En(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, Iπππ , so that one in fact recovers four independent relations, corresponding to Iπππ = 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 Iπππ = 0 and Iπππ = 1. As an example of the utility of the generalized formalism, we present a toy implementation of the quantization condition for Iπππ = 0, focusing on the quantum numbers of the ω and h1 resonances.