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.
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02 February 2021
An Erratum to this paper has been published: https://doi.org/10.1007/JHEP02(2021)014
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Hansen, M.T., Romero-López, F. & Sharpe, S.R. Generalizing the relativistic quantization condition to include all three-pion isospin channels. J. High Energ. Phys. 2020, 47 (2020). https://doi.org/10.1007/JHEP07(2020)047
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DOI: https://doi.org/10.1007/JHEP07(2020)047