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Extraction of the pion-nucleon coupling constant using the effective-range expansion with the left-hand cut

Bo-Yang Liu1,2, Bing Wu3,*, Ji-Wei Fu3, Meng-Lin Du3,†, Feng-Kun Guo1,2,4,‡, and Ulf-G. Meißner5,6,7,§

  • *Contact author: wu.bing@uestc.edu.cn
  • †Contact author: du.ml@uestc.edu.cn
  • ‡Contact author: fkguo@itp.ac.cn
  • §Contact author: meissner@hiskp.uni-bonn.de
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Phys. Rev. D 114, 074030 – Published 13 October, 2026

DOI: https://doi.org/10.1103/184t-bpbg

Abstract

We apply the generalized effective-range expansion of Du et al. [Phys. Rev. Lett. 135, 011903 (2025)], which incorporates the left-hand cut from one-pion exchange to low-energy neutron–proton scattering in the S01 and S13 channels. The amplitude zero for the center-of-mass momentum near 0.35 GeV in the S01 channel is naturally accommodated within this framework. We extract the pole position, scattering length, effective range, shape parameter, and the pseudoscalar pion–nucleon coupling constant gπN2/(4π) at different expansion orders. The low-energy parameters are stable and consistent with established values, while gπN2/(4π) exhibits larger uncertainties. The extraction of gπN2/(4π) is data driven, relying on the analytic constraints from the left-hand cut and phase-shift data within the one-pion-exchange approximation. Despite larger uncertainties compared to high-precision extractions, the consistency with established values demonstrates that this framework can probe the left-hand-cut singularity.

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References (79)

  1. H. A. Bethe, Theory of the effective range in nuclear scattering, Phys. Rev. 76, 38 (1949).
  2. J. M. Blatt and J. D. Jackson, On the interpretation of neutron-proton scattering data by the Schwinger variational method, Phys. Rev. 76, 18 (1949).
  3. S. Navas et al. (Particle Data Group), Review of particle physics, Phys. Rev. D 110, 030001 (2024).
  4. M.-Z. Liu, Y.-W. Pan, Z.-W. Liu, T.-W. Wu, J.-X. Lu, and L.-S. Geng, Three ways to decipher the nature of exotic hadrons: Multiplets, three-body hadronic molecules, and correlation functions, Phys. Rep. 1108, 1 (2025).
  5. F.-K. Guo, C. Hanhart, U.-G. Meißner, Q. Wang, Q. Zhao, and B.-S. Zou, Hadronic molecules, Rev. Mod. Phys. 90, 015004 (2018); 94, 029901(E) (2022).
  6. H.-X. Chen, W. Chen, X. Liu, Y.-R. Liu, and S.-L. Zhu, An updated review of the new hadron states, Rep. Prog. Phys. 86, 026201 (2023).
  7. L. Meng, B. Wang, G.-J. Wang, and S.-L. Zhu, Chiral perturbation theory for heavy hadrons and chiral effective field theory for heavy hadronic molecules, Phys. Rep. 1019, 1 (2023).
  8. N. Brambilla, S. Eidelman, C. Hanhart, A. Nefediev, C.-P. Shen, C. E. Thomas, A. Vairo, and C.-Z. Yuan, The XYZ states: Experimental and theoretical status and perspectives, Phys. Rep. 873, 1 (2020).
  9. V. Baru, X.-K. Dong, M.-L. Du, A. Filin, F.-K. Guo, C. Hanhart, A. Nefediev, J. Nieves, and Q. Wang, Effective range expansion for narrow near-threshold resonances, Phys. Lett. B 833, 137290 (2022).
  10. X.-K. Dong, F.-K. Guo, and B.-S. Zou, A survey of heavy-heavy hadronic molecules, Commun. Theor. Phys. 73, 125201 (2021).
  11. X.-K. Dong, F.-K. Guo, and B.-S. Zou, A survey of heavy-antiheavy hadronic molecules, Prog. Phys. 41, 65 (2021).
  12. M.-L. Du, F.-K. Guo, and B. Wu, Effective-range expansion with a long-range force, Phys. Rev. Lett. 135, 011903 (2025).
  13. M.-L. Du, F.-K. Guo, and B. Wu, Effective range expansion with the left-hand cut and its application to the Tcc(3875), Proc. Sci. CD2024 (2026) 062 [arXiv:2502.19774].
  14. E. Epelbaum and U.-G. Meißner, Chiral dynamics of few- and many-nucleon systems, Annu. Rev. Nucl. Part. Sci. 62, 159 (2012).
  15. W.-J. Wang, B. Wu, M.-L. Du, and F.-K. Guo, Effective range expansion with the left-hand cut: Higher order improvements, Phys. Rev. D 114, 014003 (2026).
  16. A. B. Raposo and M. T. Hansen, Finite-volume scattering on the left-hand cut, J. High Energy Phys. 08 (2024) 075.
  17. R. Bubna, H.-W. Hammer, F. Müller, J.-Y. Pang, A. Rusetsky, and J.-J. Wu, Lüscher equation with long-range forces, J. High Energy Phys. 05 (2024) 168.
  18. R. Bubna, H.-W. Hammer, B.-L. Hoid, J.-Y. Pang, A. Rusetsky, and J.-J. Wu, Modified lüscher zeta-function and the modified effective range expansion in the presence of a long-range force, J. High Energy Phys. 10 (2025) 197.
  19. A. B. Raposo, R. A. Briceño, M. T. Hansen, and A. W. Jackura, Extracting scattering amplitudes for arbitrary two-particle systems with one-particle left-hand cuts via lattice QCD, J. High Energy Phys. 06 (2025) 186.
  20. L. Meng and E. Epelbaum, Two-particle scattering from finite-volume quantization conditions using the plane wave basis, J. High Energy Phys. 10 (2021) 051.
  21. L. Meng, V. Baru, E. Epelbaum, A. A. Filin, and A. M. Gasparyan, Solving the left-hand cut problem in lattice QCD: Tcc(3875)+ from finite volume energy levels, Phys. Rev. D 109, L071506 (2024).
  22. J. V. Steele and R. J. Furnstahl, Removing pions from two nucleon effective field theory, Nucl. Phys. A645, 439 (1999).
  23. V. Baru, E. Epelbaum, A. A. Filin, and J. Gegelia, Low-energy theorems for nucleon-nucleon scattering at unphysical pion masses, Phys. Rev. C 92, 014001 (2015).
  24. V. Baru, E. Epelbaum, and A. A. Filin, Low-energy theorems for nucleon-nucleon scattering at Mπ=450  MeV, Phys. Rev. C 94, 014001 (2016).
  25. T. A. Lähde, U.-G. Meißner, and E. Epelbaum, An update on fine-tunings in the triple-alpha process, Eur. Phys. J. A 56, 89 (2020).
  26. H. Meyer and U.-G. Meißner, Improved constraints on the variation of the weak scale from Big Bang nucleosynthesis, J. High Energy Phys. 06 (2024) 074; 01 (2025) 33.
  27. X. Zhang, H.-L. Fu, F.-K. Guo, and H.-W. Hammer, Neutron scattering off one-neutron halo nuclei in halo effective field theory, Phys. Rev. C 108, 044304 (2023).
  28. M.-L. Du, A. Filin, V. Baru, X.-K. Dong, E. Epelbaum, F.-K. Guo, C. Hanhart, A. Nefediev, J. Nieves, and Q. Wang, Role of left-hand cut contributions on pole extractions from lattice data: Case study for Tcc(3875)+, Phys. Rev. Lett. 131, 131903 (2023).
  29. R. Navarro Pérez, J. E. Amaro, and E. Ruiz Arriola, The low-energy structure of the nucleon-nucleon interaction: Statistical versus systematic uncertainties, J. Phys. G 43, 114001 (2016).
  30. A. G. Escalante, R. Navarro Pérez, and E. Ruiz Arriola, Partial-wave correlations with uncertainties in the two-nucleon system, Phys. Rev. C 104, 054002 (2021).
  31. M. C. M. Rentmeester, R. G. E. Timmermans, J. L. Friar, and J. J. de Swart, Chiral two pion exchange and proton proton partial wave analysis, Phys. Rev. Lett. 82, 4992 (1999).
  32. R. A. Arndt, L. D. Roper, R. A. Bryan, R. B. Clark, B. J. VerWest, and P. Signell, Nucleon-nucleon partial wave analysis to 1 GeV, Phys. Rev. D 28, 97 (1983).
  33. V. G. J. Stoks, R. A. M. Klomp, C. P. F. Terheggen, and J. J. de Swart, Construction of high quality NN potential models, Phys. Rev. C 49, 2950 (1994).
  34. T. A. Rijken and V. G. J. Stoks, Soft two meson exchange nucleon-nucleon potentials. II. One pair and two pair diagrams, Phys. Rev. C 54, 2869 (1996).
  35. P. Reinert, H. Krebs, and E. Epelbaum, Semilocal momentum-space regularized chiral two-nucleon potentials up to fifth order, Eur. Phys. J. A 54, 86 (2018).
  36. R. A. Arndt, I. I. Strakovsky, and R. L. Workman, Nucleon nucleon elastic scattering to 3 GeV, Phys. Rev. C 62, 034005 (2000).
  37. V. G. J. Stoks, R. A. M. Klomp, M. C. M. Rentmeester, and J. J. de Swart, Partial wave analaysis of all nucleon-nucleon scattering data below 350 MeV, Phys. Rev. C 48, 792 (1993).
  38. G. F. Chew and S. Mandelstam, Theory of low-energy pion pion interactions, Phys. Rev. 119, 467 (1960).
  39. J. A. Oller, A Brief Introduction to Dispersion Relations, SpringerBriefs in Physics (Springer, New York, 2019).
  40. M. L. Goldberger and S. B. Treiman, Conserved currents in the theory of fermi interactions, Phys. Rev. 110, 1478 (1958).
  41. N. Fettes, U.-G. Meißner, and S. Steininger, Pion-nucleon scattering in chiral perturbation theory. 1. Isospin symmetric case, Nucl. Phys. A640, 199 (1998).
  42. R. B. Wiringa, V. G. J. Stoks, and R. Schiavilla, An Accurate nucleon-nucleon potential with charge independence breaking, Phys. Rev. C 51, 38 (1995).
  43. E. Epelbaum, H. Krebs, and U.-G. Meißner, Precision nucleon-nucleon potential at fifth order in the chiral expansion, Phys. Rev. Lett. 115, 122301 (2015).
  44. E. Epelbaum, H. Krebs, and U.-G. Meißner, Improved chiral nucleon-nucleon potential up to next-to-next-to-next-to-leading order, Eur. Phys. J. A 51, 53 (2015).
  45. J.-X. Lu, C.-X. Wang, Y. Xiao, L.-S. Geng, J. Meng, and P. Ring, Accurate relativistic chiral nucleon-nucleon interaction up to next-to-next-to-leading order, Phys. Rev. Lett. 128, 142002 (2022).
  46. R. Machleidt, K. Holinde, and C. Elster, The Bonn meson exchange model for the nucleon nucleon interaction, Phys. Rep. 149, 1 (1987).
  47. K. Schoen, D. L. Jacobson, M. Arif, P. R. Huffman, T. C. Black, W. M. Snow, S. K. Lamoreaux, H. Kaiser, and S. A. Werner, Precision neutron interferometric measurements and updated evaluations of the n−p and n−d coherent neutron scattering lengths, Phys. Rev. C 67, 044005 (2003).
  48. V. A. Babenko and N. M. Petrov, Determination of low-energy parameters of neutron-proton scattering on the basis of modern experimental data from partial-wave analyses, Phys. At. Nucl. 70, 669 (2007).
  49. O. Dumbrajs, R. Koch, H. Pilkuhn, G. Oades, H. Behrens, J. De Swart, and P. Kroll, Compilation of coupling constants and low-energy parameters, Nucl. Phys. B216, 277 (1983).
  50. R. Machleidt, High-precision, charge-dependent Bonn nucleon-nucleon potential, Phys. Rev. C 63, 024001 (2001).
  51. E. Epelbaum, H.-W. Hammer, and U.-G. Meißner, Modern theory of nuclear forces, Rev. Mod. Phys. 81, 1773 (2009).
  52. P. Reinert, H. Krebs, and E. Epelbaum, Precision determination of pion-nucleon coupling constants using effective field theory, Phys. Rev. Lett. 126, 092501 (2021).
  53. R. Arndt, W. Briscoe, I. Strakovsky, and R. Workman, Extended partial-wave analysis of πN scattering data, Phys. Rev. C 74, 045205 (2006).
  54. V. Baru, C. Hanhart, M. Hoferichter, B. Kubis, A. Nogga, and D. Phillips, Precision calculation of threshold π−d scattering, πN scattering lengths, and the GMO sum rule, Nucl. Phys. A872, 69 (2011).
  55. E. Ruiz Arriola and P. Sanchez-Puertas, Pseudoscalar meson dominance, the pion-nucleon coupling constant and the Goldberger-Treiman discrepancy, Eur. Phys. J. Special Topics 235, 2063 (2026).
  56. R. Timmermans, T. Rijken, and J. de Swart, Determination of the charged pion coupling constant from data on the charge exchange reaction p¯p→n¯n, Phys. Rev. Lett. 67, 1074 (1991).
  57. R. Navarro Pérez, J. Amaro, and E. Ruiz Arriola, Coarse-grained potential analysis of neutron-proton and proton-proton scattering below the pion production threshold, Phys. Rev. C 88, 064002 (2013).
  58. H. A. Bethe, The meson theory of nuclear forces 1. General theory, Phys. Rev. 57, 260 (1940).
  59. H. A. Bethe, The meson theory of nuclear forces. Part 2. Theory of the deuteron, Phys. Rev. 57, 390 (1940).
  60. Y. Nambu, Axial vector current conservation in weak interactions, Phys. Rev. Lett. 4, 380 (1960).
  61. M. L. Goldberger and S. B. Treiman, Decay of the π meson, Phys. Rev. 110, 1178 (1958).
  62. G. F. Chew, Proposal for determining the pion-nucleon coupling constant from the angular distribution for nucleon-nucleon scattering, Phys. Rev. 112, 1380 (1958).
  63. P. Cziffra, M. H. MacGregor, M. J. Moravcsik, and H. P. Stapp, Modified analysis of nucleon-nucleon scattering. 1. Theory and p−p scattering at 310 Mev, Phys. Rev. 114, 880 (1959).
  64. M. H. MacGregor, M. J. Moravcsik, and H. P. Stapp, Modified analysis of nucleon-nucleon scattering. 2. Completed analysis of p−p scattering at 310 Mev, Phys. Rev. 116, 1248 (1959).
  65. P. S. Signell, Quantitative evidence of one-pion exchange effects in p−p scattering, Phys. Rev. Lett. 5, 474 (1960).
  66. M. H. MacGregor, R. A. Arndt, and A. A. Dubow, Determination of the nucleon-nucleon elastic scattering matrix. 1. Phase-shift analysis of experiments near 140 MeV, Phys. Rev. 135, B628 (1964).
  67. R. A. M. Klomp, V. G. J. Stoks, and J. J. de Swart, Determination of the NNπ coupling constants in NN partial wave analyses, Phys. Rev. C 44, R1258 (1991).
  68. V. G. Stoks, R. Timmermans, and J. J. de Swart, On the pion-nucleon coupling constant, Phys. Rev. C 47, 512 (1993).
  69. R. Navarro Pérez, J. Amaro, and E. Ruiz Arriola, Precise determination of charge dependent pion-nucleon-nucleon coupling constants, Phys. Rev. C 95, 064001 (2017).
  70. J. de Swart, M. Rentmeester, and R. Timmermans, The Status of the pion-nucleon coupling constant, PiN Newslett. 13, 96 (1997).
  71. M. Sainio, Pion nucleon coupling constant: Working group summary, PiN Newslett 15, 156 (1999).
  72. D. Bugg, The pion nucleon coupling constant, Eur. Phys. J. C 33, 505 (2004).
  73. E. Matsinos, A brief history of the pion-nucleon coupling constant, arXiv:1901.01204.
  74. E. Epelbaum, A. M. Gasparyan, J. Gegelia, and H. Krebs, S10 nucleon-nucleon scattering in the modified Weinberg approach, Eur. Phys. J. A 51, 71 (2015).
  75. O. Thim, Low-energy theorems for neutron–proton scattering in χEFT using a perturbative power counting, Few Body Syst. 65, 69 (2024).
  76. J. J. de Swart, C. P. F. Terheggen, and V. G. J. Stoks, The Low-energy N P scattering parameters and the deuteron, in 3rd International Symposium on Dubna Deuteron 95 (1995).
  77. M. Pavon Valderrama and E. Ruiz Arriola, Low-energy NN scattering at next-to-next-to-next-to-next-to-leading order for partial waves with j≤5, Phys. Rev. C 72, 044007 (2005).
  78. E. Epelbaum and J. Gegelia, Weinberg’s approach to nucleon–nucleon scattering revisited, Phys. Lett. B 716, 338 (2012).
  79. T. D. Cohen and J. M. Hansen, Low-energy theorems for nucleon-nucleon scattering, Phys. Rev. C 59, 13 (1999).

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Phys. Rev. D 114, 074030– Published 13 October, 2026
  • Received 11 February 2026
  • Accepted 21 September 2026
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