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

Abstract

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 K df , 3 , $$ {\mathcal{K}}_{\mathrm{df},3,} $$ which can thus be constrained from lattice QCD in- put. The second step is a set of integral equations relating K df , 3 $$ {\mathcal{K}}_{\mathrm{df},3} $$ to the physical scattering amplitude, ℳ3. Both of the key relations, E n (L) ↔ K df , 3 $$ {\mathcal{K}}_{\mathrm{df},3} $$ and K df , 3 ↔ ℳ 3 , $$ {\mathcal{K}}_{\mathrm{df},3}\leftrightarrow {\mathrm{\mathcal{M}}}_3, $$ 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 K df , 3 $$ {\mathcal{K}}_{\mathrm{df},3} $$ 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.

Keywords

• Lattice QCD
• Scattering Amplitudes