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, $E_n(L)$, 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 input. The second step is a set of integral equations relating ${\mathcal{K}}_{\mathrm{df},3}$ to the physical scattering amplitude, ${\mathcal{M}}_3$. Both of the key relations, $E_n(L)\leftrightarrow {\mathcal{K}}_{\mathrm{df},3}$ and ${\mathcal{K}}_{\mathrm{df},3}\leftrightarrow {\mathcal{M}}_3$, are shown to be block-diagonal in the basis of definite three-pion isospin, $I_{\pi \pi \pi }$, so that one in fact recovers four independent relations, corresponding to $I_{\pi \pi \pi }=0,1,2,3$. 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 $I_{\pi \pi \pi }=0$ and $I_{\pi \pi \pi }=1$. As an example of the utility of the generalized formalism, we present a toy implementation of the quantization condition for $I_{\pi \pi \pi }=0$, focusing on the quantum numbers of the $\omega$ and $h_1$ resonances.

Comments: 46 pages, 4 figures. Updated to match erratum published in JHEP. Main conclusions and results unchanged