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

Understand

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.