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

• DOI: 10.1007/jhep07(2020)047
• Publication Date: 2020-07-08T22:13:32Z
• Authors: Maxwell T. Hansen; Fernando Romero-L...; Stephen R. Sharpe
• Keywords: Lattice (music)

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, constrain scattering amplitudes all possible values two- and three-pion isospin. As case identical considered previously, result splits into two steps: first defines non-perturbative function with roots equal allowed $E_n(L)$, given cubic volume side-length $L$. This depends on an intermediate three-body quantity, denoted $\mathcal{K}_{\mathrm{df},3}$, which can thus be constrained from QCD input. second step is set integral equations relating $\mathcal{K}_{\mathrm{df},3}$ physical amplitude, $\mathcal M_3$. Both key relations, $E_n(L) \leftrightarrow \mathcal{K}_{\mathrm{df},3}$ $\mathcal{K}_{\mathrm{df},3}\leftrightarrow \mathcal M_3$, are shown block-diagonal basis definite isospin, $I_{\pi \pi \pi}$, so that fact recovers four independent corresponding \pi}=0,1,2,3$. also provide generalized threshold expansion channels, as well parameterizations resonances $I_{\pi\pi\pi}=0$ $I_{\pi\pi\pi}=1$. example utility formalism, we toy implementation $I_{\pi\pi\pi}=0$, focusing quantum numbers $\omega$ $h_1$ resonances.