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

Journal Article · 07 July 2020 · Journal of High Energy Physics (Online)

DOI: https://doi.org/10.1007/JHEP07(2020)047 · OSTI ID:1646656

Hansen, Maxwell T. ^[1]; Romero-López, Fernando  ^[2]; Sharpe, Stephen R. ^[3]

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1. European Organization for Nuclear Research (CERN), Geneva (Switzerland). Theoretical Physics Dept.; OSTI
2. Univ. of Valencia (Spain). Inst. for Corpuscular Physics (IFIC)
3. Univ. of Washington, Seattle, WA (United States). Physics Dept.

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

• Research Organization: Univ. of Valencia (Spain); Univ. of Washington, Seattle, WA (United States)
• Sponsoring Organization: European Union (EU); Generalitat Valenciana; USDOE Office of Science (SC), High Energy Physics (HEP)
• Grant/Contract Number: SC0011637
• OSTI ID: 1646656
• Journal Information: Journal of High Energy Physics (Online), Journal Name: Journal of High Energy Physics (Online) Journal Issue: 7 Vol. 2020; ISSN 1029-8479
• Publisher: Springer BerlinCopyright Statement
• Country of Publication: United States
• Language: English