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

Journal Article · Wed Jul 08 00:00:00 EDT 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.
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 Washington, Seattle, WA (United States); Univ. of Valencia (Spain)
• Sponsoring Organization: USDOE Office of Science (SC), High Energy Physics (HEP); Generalitat Valenciana; European Union (EU)
• Grant/Contract Number: SC0011637; PROMETEO/2019/083; 713673
• OSTI ID: 1646656
• Alternate ID(s): OSTI ID: 1774010
• Journal Information: Journal of High Energy Physics (Online), Vol. 2020, Issue 7; ISSN 1029-8479
• Publisher: Springer Berlin
• Country of Publication: United States
• Language: English

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72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS