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Higher order quantization conditions for two-body scattering with spin

Lucas Chandler*, Frank X. Lee†, and Andrei Alexandru‡

  • *Contact author: lchandler27@gwu.edu
  • †Contact author: fxlee@gwu.edu
  • ‡Contact author: aalexan@gwu.edu
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Phys. Rev. D 113, 114519 – Published 18 June, 2026

DOI: https://doi.org/10.1103/527n-tjbr

Abstract

We examine the Lüscher quantization condition to high order for the scattering of a spinless particle and a spin-1/2 particle in a periodic box. First, we derive the quantization conditions in a nonrelativistic framework up to total angular momentum J=11/2 in both cubic and elongated geometries, and for both rest and moving frames. Then, we introduce a method to transparently cross-check their convergence, using both quantized energy levels in the box and infinite-volume phase shifts for the same potential. We clarify how to incorporate spin-orbit coupling into the formalism and show in detail how the quantization conditions converge order by order in the various irreducible representations. In all, we validated 19 quantization conditions (12 in cubic box, 7 in elongated box). This is a necessary step in applying the method in precision studies of systems in finite volume with half-integer spin, such as meson-baryon scattering.

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References (71)

  1. Martin Lüscher, Two particle states on a torus and their relation to the scattering matrix, Nucl. Phys. B354, 531 (1991).
  2. N. Ishii, S. Aoki, and T. Hatsuda, Nuclear force from lattice QCD, Phys. Rev. Lett. 99, 022001 (2007).
  3. Takumi Iritani, Sinya Aoki, Takumi Doi, Shinya Gongyo, Tetsuo Hatsuda, Yoichi Ikeda, Takashi Inoue, Noriyoshi Ishii, Hidekatsu Nemura, and Kenji Sasaki (HAL QCD Collaboration), Systematics of the HAL QCD potential at low energies in lattice QCD, Phys. Rev. D 99, 014514 (2019).
  4. Noriyoshi Ishii, Sinya Aoki, Takumi Doi, Tetsuo Hatsuda, Yoichi Ikeda, Takashi Inoue, Keiko Murano, Hidekatsu Nemura, and Kenji Sasaki, Hadron–hadron interactions from imaginary-time Nambu–Bethe–Salpeter wave function on the lattice, Phys. Lett. B 712, 437 (2012).
  5. Peng Guo and Vladimir Gasparian, Toward extracting the scattering phase shift from integrated correlation functions, Phys. Rev. D 108, 074504 (2023).
  6. Peng Guo, Frank X. Lee, and Andrei Alexandru, Toward extracting scattering phase shifts from integrated correlation functions. V. Complex ϕ4 field model in 3+1 dimensions, Phys. Rev. D 112, 074511 (2025).
  7. Peng Guo, Paul LeVan, Frank X. Lee, and Yong Zhao, Extracting scattering phase shift in quantum mechanics on quantum computers, Phys. Rev. D 113, 054512 (2026).
  8. K. Rummukainen and Steven A. Gottlieb, Resonance scattering phase shifts on a nonrest frame lattice, Nucl. Phys. B450, 397 (1995).
  9. C. H. Kim, C. T. Sachrajda, and Stephen R. Sharpe, Finite-volume effects for two-hadron states in moving frames, Nucl. Phys. B727, 218 (2005).
  10. Ziwen Fu, Rummukainen-Gottlieb’s formula on two-particle system with different mass, Phys. Rev. D 85, 014506 (2012).
  11. Luka Leskovec and Sasa Prelovsek, Scattering phase shifts for two particles of different mass and nonzero total momentum in lattice QCD, Phys. Rev. D 85, 114507 (2012).
  12. M. Gockeler, R. Horsley, M. Lage, U. G. Meissner, P. E. L. Rakow, A. Rusetsky, G. Schierholz, and J. M. Zanotti, Scattering phases for meson and baryon resonances on general moving-frame lattices, Phys. Rev. D 86, 094513 (2012).
  13. Veronique Bernard, Michael Lage, Ulf-G. Meissner, and Akaki Rusetsky, Resonance properties from the finite-volume energy spectrum, J. High Energy Phys. 08 (2008) 024.
  14. Yan Li, Jia-jun Wu, Derek B. Leinweber, and Anthony W. Thomas, Hamiltonian effective field theory in elongated or moving finite volume, Phys. Rev. D 103, 094518 (2021).
  15. M. Doring, U. G. Meissner, E. Oset, and A. Rusetsky, Scalar mesons moving in a finite volume and the role of partial wave mixing, Eur. Phys. J. A 48, 114 (2012).
  16. Ning Li and Chuan Liu, Generalized Lüscher formula in multichannel baryon-meson scattering, Phys. Rev. D 87, 014502 (2013).
  17. Peng Guo, Jozef Dudek, Robert Edwards, and Adam P. Szczepaniak, Coupled-channel scattering on a torus, Phys. Rev. D 88, 014501 (2013).
  18. Xu Feng, Xin Li, and Chuan Liu, Two particle states in an asymmetric box and the elastic scattering phases, Phys. Rev. D 70, 014505 (2004).
  19. Frank X. Lee and Andrei Alexandru, Scattering phase-shift formulas for mesons and baryons in elongated boxes, Phys. Rev. D 96, 054508 (2017).
  20. C. Liu, X. Feng, and S. He, Two particle states in a box and the s-matrix in multi-channel scattering, Int. J. Mod. Phys. A 21, 847 (2006).
  21. Raul A. Briceno and Zohreh Davoudi, Moving multichannel systems in a finite volume with application to proton-proton fusion, Phys. Rev. D 88, 094507 (2013).
  22. Raul A. Briceno, Two-particle multichannel systems in a finite volume with arbitrary spin, Phys. Rev. D 89, 074507 (2014).
  23. Yan Li, Jia-Jun Wu, Curtis D. Abell, Derek B. Leinweber, and Anthony W. Thomas, Partial wave mixing in Hamiltonian effective field theory, Phys. Rev. D 101, 114501 (2020).
  24. M. Döring, U. G. Meißner, E. Oset, and A. Rusetsky, Unitarized chiral perturbation theory in a finite volume: Scalar meson sector, Eur. Phys. J. A 47, 139 (2011).
  25. Maxwell T. Hansen and Stephen R. Sharpe, Multiple-channel generalization of lellouch-lüscher formula, Phys. Rev. D 86, 016007 (2012).
  26. Peng Guo and Frank X. Lee, Toward extracting scattering phase shift from integrated correlation functions. III. Coupled channels, Phys. Rev. D 111, 054506 (2025).
  27. S. Aoki et al. (CP-PACS Collaboration), Lattice QCD calculation of the rho meson decay width, Phys. Rev. D 76, 094506 (2007).
  28. Xu Feng, Karl Jansen, and Dru B. Renner, Resonance parameters of the rho-meson from lattice QCD, Phys. Rev. D 83, 094505 (2011).
  29. C. B. Lang, Daniel Mohler, Sasa Prelovsek, and Matija Vidmar, Coupled channel analysis of the rho meson decay in lattice QCD, Phys. Rev. D 84, 054503 (2011); 89, 059903(E) (2014).
  30. S. Aoki et al. (CS Collaboration), ρ meson decay in 2+1 flavor lattice QCD, Phys. Rev. D 84, 094505 (2011).
  31. Dehua Guo, Andrei Alexandru, Raquel Molina, and Michael Döring, Rho resonance parameters from lattice QCD, Phys. Rev. D 94, 034501 (2016).
  32. Jozef J. Dudek, Robert G. Edwards, and David J. Wilson (Hadron Spectrum), An a0 resonance in strongly coupled πη, KK¯ scattering from lattice QCD, Phys. Rev. D 93, 094506 (2016).
  33. Thomas Luu and Martin J. Savage, Extracting scattering phase-shifts in higher partial-waves from lattice QCD calculations, Phys. Rev. D 83, 114508 (2011).
  34. John Bulava, Brendan Fahy, Ben Hörz, Keisuke J. Juge, Colin Morningstar, and Chik Him Wong, I=1 and i=2 π−π scattering phase shifts from nf=2+1 lattice QCD, Nucl. Phys. B910, 842 (2016).
  35. Gunnar S. Bali, Sara Collins, Antonio Cox, Gordon Donald, Meinulf Göckeler, C. B. Lang, and Andreas Schäfer (RQCD Collaboration), ρ and K* resonances on the lattice at nearly physical quark masses and Nf=2, Phys. Rev. D 93, 054509 (2016).
  36. Colin Morningstar, John Bulava, Bijit Singha, Ruairí Brett, Jacob Fallica, Andrew Hanlon, and Ben Hörz, Estimating the two-particle K-matrix for multiple partial waves and decay channels from finite-volume energies, Nucl. Phys. B924, 477 (2017).
  37. M. Mai, C. Culver, A. Alexandru, M. Döring, and F. X. Lee, Cross-channel study of pion scattering from lattice QCD, Phys. Rev. D 100, 114514 (2019).
  38. Zhengli Wang, Derek B. Leinweber, Chuan Liu, Liuming Liu, Peng Sun, Anthony W. Thomas, Jia jun Wu, Hanyang Xing, and Kang Yu, Spectral parameters of the ρ resonance from lattice QCD, J. High Energy Phys. 08 (2025) 064.
  39. John Bulava, Andrew D. Hanlon, Ben Hörz, Colin Morningstar, Amy Nicholson, Fernando Romero-López, Sarah Skinner, Pavlos Vranas, and André Walker-Loud, Elastic nucleon-pion scattering at mπ=200  MeV from lattice QCD, Nucl. Phys. B987, 116105 (2023).
  40. John Bulava, Bárbara Cid-Mora, Andrew D. Hanlon, Ben Hörz, Daniel Mohler, Colin Morningstar, Joseph Moscoso, Amy Nicholson, Fernando Romero-López, Sarah Skinner, and André Walker-Loud (Baryon Scattering (BaSc) Collaboration), Lattice QCD study of πΣ−K¯n scattering and the Λ(1405) resonance, Phys. Rev. D 109, 014511 (2024).
  41. Constantia Alexandrou, Simone Bacchio, Giannis Koutsou, Theodoros Leontiou, Srijit Paul, Marcus Petschlies, and Ferenc Pittler, Elastic nucleon-pion scattering amplitudes in the Δ channel at physical pion mass from lattice QCD, Phys. Rev. D 109, 034509 (2024).
  42. Adrian L. Kiratidis, Waseem Kamleh, Derek B. Leinweber, and Benjamin J. Owen, Lattice baryon spectroscopy with multi-particle interpolators, Phys. Rev. D 91, 094509 (2015).
  43. Colin Morningstar, Nucleon scattering from lattice QCD, Proc. Sci. CD2024 (2026) 003 [arXiv:2504.01950].
  44. Luka Leskovec, Christian B. Lang, M. Padmanath, and Sasa Prelovsek, A lattice QCD study of pion–nucleon scattering in the Roper channel, Few-Body Syst. 59 (2018), 10.1007/s00601-018-1419-2.
  45. Zhong-Lin Ma, Zhan-Wei Liu, and Jiong-Jiong Liu, Odd-parity strange baryons Σ(12−) below 1.8 GeV with a Hamiltonian effective field theory, Phys. Rev. D 113, 014037 (2026).
  46. Colin Morningstar, Low-lying baryon resonances from lattice QCD, Acta Phys. Pol. B 57, 2 (2026), https://www.actaphys.uj.edu.pl/fulltext?series=Reg&vol=57&aid=2-A10.
  47. Silas R. Beane, William Detmold, Thomas C. Luu, Kostas Orginos, Martin J. Savage, and Aaron Torok, Multi-pion systems in lattice QCD and the three-pion interaction, Phys. Rev. Lett. 100, 082004 (2008).
  48. Peng Guo and Michael Döring, Lattice model of heavy-light three-body system, Phys. Rev. D 101, 034501 (2020).
  49. M. Mai, M. Döring, C. Culver, and A. Alexandru, Three-body unitarity versus finite-volume π+π+π+ spectrum from lattice QCD, Phys. Rev. D 101, 054510 (2020).
  50. Ben Hörz and Andrew Hanlon, Two- and three-pion finite-volume spectra at maximal isospin from lattice QCD, Phys. Rev. Lett. 123, 142002 (2019).
  51. Fernando Romero-López, Stephen R. Sharpe, Tyler D. Blanton, Raúl A. Briceño, and Maxwell T. Hansen, Numerical exploration of three relativistic particles in a finite volume including two-particle resonances and bound states, J. High Energy Phys. 10 (2019) 007.
  52. Tyler D. Blanton, Fernando Romero-López, and Stephen R. Sharpe, Implementing the three-particle quantization condition including higher partial waves, J. High Energy Phys. 03 (2019) 106.
  53. Maxim Mai, Andrei Alexandru, Ruairí Brett, Chris Culver, Michael Döring, Frank X. Lee, and Daniel Sadasivan (GWQCD Collaboration), Three-body dynamics of the a1(1260) resonance from lattice QCD, Phys. Rev. Lett. 127, 222001 (2021).
  54. Andrei Alexandru, Ruairí Brett, Chris Culver, Michael Döring, Dehua Guo, Frank X. Lee, and Maxim Mai, Finite-volume energy spectrum of the K−K−K− system, Phys. Rev. D 102, 114523 (2020).
  55. Maxim Mai, Michael Döring, and Akaki Rusetsky, Multi-particle systems on the lattice and chiral extrapolations: A brief review, Eur. Phys. J. Special Topics 230, 1623 (2021).
  56. Haobo Yan, Maxim Mai, Marco Garofalo, Ulf-G. Meißner, Chuan Liu, Liuming Liu, and Carsten Urbach, ω meson from lattice QCD, Phys. Rev. Lett. 133, 211906 (2024).
  57. Haobo Yan, Maxim Mai, Marco Garofalo, Yuchuan Feng, Michael Döring, Chuan Liu, Liuming Liu, Ulf-G. Meißner, and Carsten Urbach, Emergence of the π(1300) Resonance from Lattice QCD, Phys. Rev. Lett. 136, 141901 (2026).
  58. Yuchuan Feng, Chris Culver, Michael Döring, Maxim Mai, Andrei Alexandru, and Frank X. Lee, Coupled-channel approach to isotensor πππ scattering from lattice QCD, arXiv:2601.16916.
  59. Rishabh Bubna, Effective field theory methods applied to two-body and three-body systems, Ph.D. thesis, U. Bonn (main) (2026).
  60. Herzallah Alharazin, André Baião Raposo, John Bulava, Sebastian Dawid, Jeremy R. Green, Colin Morningstar, Fernando Romero-López, Miguel Salg, Stephen R. Sharpe, and Andres Stump, Three-body study of the tcc(3875)+ from lattice QCD, arXiv:2602.17204.
  61. Stephen R. Sharpe, Three-particle scattering amplitudes from lattice QCD, arXiv:2601.04147.
  62. Frank X. Lee, Andrei Alexandru, and Ruairí Brett, Higher order finite volume quantization conditions for two spinless particles, Phys. Rev. D 105, 054517 (2022).
  63. See Supplemental Material at http://link.aps.org/supplemental/10.1103/527n-tjbr. The supplement consists of two parts. A pdf file named “qc-convergence.pdf” that has the phaseshift prediction at order 1 plus order-by-order convergence data for all the QCs. A folder named “qc-matrix” that contains python code to recover the full matrix on demand for any QC at any order.
  64. Frank X. Lee, Colin Morningstar, and Andrei Alexandru, Energy spectrum of two-particle scattering in a periodic box, Int. J. Mod. Phys. C 31, 2050131 (2020).
  65. F. Calogero, Variable Phase Approach to Potential Scattering (Academic Press, New York, 1967).
  66. Martin J. Klein, On a degeneracy theorem of kramers, Am. J. Phys. 20, 65 (1952).
  67. Robert S. Mulliken, Report on notation for the spectra of polyatomic molecules, J. Chem. Phys. 23, 1997 (1955).
  68. S. L. Altmann and P. Herzig, Point-Group Theory Tables, Oxford Science Publications (Clarendon Press, New York, 1994).
  69. Eugene P. Wigner, Normal form of antiunitary operators, J. Math. Phys. (N.Y.) 1, 409 (1960).
  70. J. O Dimmock and R. G Wheeler, Irreducible representations of magnetic groups, J. Phys. Chem. Solids 23, 729 (1962).
  71. J. D. Newmarch and R. M. Golding, The character table for the corepresentations of magnetic groups, J. Math. Phys. (N.Y.) 23, 695 (1982).

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Phys. Rev. D 113, 114519– Published 18 June, 2026
  • Received 6 March 2026
  • Accepted 19 May 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP3.

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