Generalizing the relativistic quantization condition to include all three-pion isospin channels
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...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
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SpringerOpen
2020-07-01
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| Series: | Journal of High Energy Physics |
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| Online Access: | http://link.springer.com/article/10.1007/JHEP07(2020)047 |
| _version_ | 1819116048530538496 |
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| author | Maxwell T. Hansen Fernando Romero-López Stephen R. Sharpe |
| author_facet | Maxwell T. Hansen Fernando Romero-López Stephen R. Sharpe |
| author_sort | Maxwell T. Hansen |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-12-22T05:10:54Z |
| format | Article |
| id | doaj.art-aafb4e9100df40978fc85e6b4774c701 |
| institution | Directory Open Access Journal |
| issn | 1029-8479 |
| language | English |
| last_indexed | 2024-12-22T05:10:54Z |
| publishDate | 2020-07-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Journal of High Energy Physics |
| spelling | doaj.art-aafb4e9100df40978fc85e6b4774c7012022-12-21T18:37:59ZengSpringerOpenJournal of High Energy Physics1029-84792020-07-012020714910.1007/JHEP07(2020)047Generalizing the relativistic quantization condition to include all three-pion isospin channelsMaxwell T. Hansen0Fernando Romero-López1Stephen R. Sharpe2Theoretical Physics Department, CERNIFIC, CSIC-Universitat de ValènciaPhysics Department, University of WashingtonAbstract 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.http://link.springer.com/article/10.1007/JHEP07(2020)047Lattice QCDScattering Amplitudes |
| spellingShingle | Maxwell T. Hansen Fernando Romero-López Stephen R. Sharpe Generalizing the relativistic quantization condition to include all three-pion isospin channels Journal of High Energy Physics Lattice QCD Scattering Amplitudes |
| title | Generalizing the relativistic quantization condition to include all three-pion isospin channels |
| title_full | Generalizing the relativistic quantization condition to include all three-pion isospin channels |
| title_fullStr | Generalizing the relativistic quantization condition to include all three-pion isospin channels |
| title_full_unstemmed | Generalizing the relativistic quantization condition to include all three-pion isospin channels |
| title_short | Generalizing the relativistic quantization condition to include all three-pion isospin channels |
| title_sort | generalizing the relativistic quantization condition to include all three pion isospin channels |
| topic | Lattice QCD Scattering Amplitudes |
| url | http://link.springer.com/article/10.1007/JHEP07(2020)047 |
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