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  • Open Access

Learning pole structures of hadronic states using predictive uncertainty estimation

Felix Frohnert1,2,*, Denny Lane B. Sombillo3,*,†, Evert van Nieuwenburg1,2, and Patrick Emonts4,5,6,2,‡

  • *These authors contributed equally to this work.
  • †Contact author: dbsombillo@up.edu.ph
  • ‡Contact author: patrick.emonts@uni-ulm.de
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Phys. Rev. D 113, 056019 – Published 20 March, 2026

DOI: https://doi.org/10.1103/1jn6-l5vq

Abstract

Matching theoretical predictions to experimental data remains a central challenge in hadron spectroscopy. In particular, the identification of new hadronic states is difficult, as exotic signals near threshold can arise from a variety of physical mechanisms. A key diagnostic in this context is the pole structure of the scattering amplitude, but different configurations can produce similar signatures. The mapping between pole configurations and line shapes is especially ambiguous near the mass threshold, where analytic control is limited. In this work, we introduce an uncertainty-aware machine learning approach for classifying pole structures in S-matrix elements. Our method is based on an ensemble of classifier chains that provide both epistemic and aleatoric uncertainty estimates. We apply a rejection criterion based on predictive uncertainty, achieving a validation accuracy of nearly 95% while discarding only a small fraction of high-uncertainty predictions. Trained on synthetic data with known pole structures, the model generalizes to previously unseen experimental data, including enhancements associated with the Pcc¯(4312)+ state observed by LHCb. In this, we infer a four-pole structure, representing the presence of a genuine compact pentaquark in the presence of a higher channel virtual state pole with nonvanishing width. While evaluated on this particular state, our framework is broadly applicable to other candidate hadronic states and offers a scalable tool for pole structure inference in scattering amplitudes.

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References (73)

  1. S. L. Olsen, T. Skwarnicki, and D. Zieminska, Nonstandard heavy mesons and baryons: Experimental evidence, Rev. Mod. Phys. 90, 015003 (2018).
  2. P. Koppenburg (LHCb Collaboration), List of hadrons observed at the LHC, Report No. LHCb-FIGURE-2021-001, 2021, see 2022 update online.
  3. Z. Liu and R. E. Mitchell, New hadrons discovered at BESIII, Sci. Bull. 68, 2148 (2023).
  4. M. Mai, U.-G. Meißner, and C. Urbach, Towards a theory of hadron resonances, Phys. Rep. 1001, 1 (2023).
  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. A. Ali, L. Maiani, and A. D. Polosa, Multiquark Hadrons (Cambridge University Press, Cambridge, England, 2019).
  8. F.-K. Guo, X.-H. Liu, and S. Sakai, Threshold cusps and triangle singularities in hadronic reactions, Prog. Part. Nucl. Phys. 112, 103757 (2020).
  9. D. Morgan and M. R. Pennington, fo(S*): Molecule or quark state?, Phys. Lett. B 258, 444 (1991); 269, 477(E) (1991).
  10. D. Morgan, Pole counting and resonance classification, Nucl. Phys. A543, 632 (1992).
  11. A. M. Badalian, L. P. Kok, M. I. Polikarpov, and Y. A. Simonov, Resonances in coupled channels in nuclear and particle physics, Phys. Rep. 82, 31 (1982).
  12. D. L. B. Sombillo, Y. Ikeda, T. Sato, and A. Hosaka, Model independent analysis of coupled-channel scattering: A deep learning approach, Phys. Rev. D 104, 036001 (2021).
  13. Z. Zhang, J. Liu, J. Hu, Q. Wang, and U.-G. Meißner, Revealing the nature of hidden charm pentaquarks with machine learning, Sci. Bull. 68, 981 (2023).
  14. D. A. O. Co and D. L. B. Sombillo, Analysis of hidden-charm pentaquarks as triangle singularities via deep learning, Proc. Sci. QNP2024 (2025) 031 [arXiv:2411.14044].
  15. A. Natekin and A. Knoll, Gradient boosting machines, a tutorial, Front. Neurorobotics 7, 21 (2013).
  16. A. V. Dorogush, V. Ershov, and A. Gulin, CatBoost: Gradient boosting with categorical features support, arXiv:1810.11363.
  17. J. Read, B. Pfahringer, G. Holmes, and E. Frank, Classifier chains for multi-label classification, Mach. Learn. 85, 333 (2011).
  18. J. Read, B. Pfahringer, G. Holmes, and E. Frank, Classifier chains: A review and perspectives, J. Artif. Intell. Res. 70 (2021).
  19. M.-L. Zhang and Z.-H. Zhou, A review on multi-label learning algorithms, IEEE Trans. Knowledge Data Eng. 26, 1819 (2014).
  20. S. Adhikari et al. (GlueX Collaboration), Measurement of the J/ψ photoproduction cross section over the full near-threshold kinematic region, Phys. Rev. C 108, 025201 (2023).
  21. I. Strakovsky, W. J. Briscoe, E. Chudakov, I. Larin, L. Pentchev, A. Schmidt, and R. L. Workman, Plausibility of the LHCb Pc(4312)+ in the GlueX γp→J/ψp total cross sections, Phys. Rev. C 108, 015202 (2023).
  22. L. Ng, L. Bibrzycki, J. Nys, C. Fernández-Ramírez, A. Pilloni, V. Mathieu, A. J. Rasmusson, and A. P. Szczepaniak (Joint Physics Analysis Center Collaboration), Deep learning exotic hadrons, Phys. Rev. D 105, L091501 (2022).
  23. N. G. van Kampen, S-matrix and causality condition. I. Maxwell field, Phys. Rev. 89, 1072 (1953).
  24. N. G. van Kampen, S-matrix and causality condition. II. Nonrelativistic particles, Phys. Rev. 91, 1267 (1953).
  25. Y. Ikeda, T. Hyodo, D. Jido, H. Kamano, T. Sato, and K. Yazaki, Structure of Λ(1405) and threshold behavior of πΣ scattering, Prog. Theor. Phys. 125, 1205 (2011).
  26. S. Rakityansky, Jost Functions in Quantum Mechanics: A Unified Approach to Scattering, Bound, and Resonant State Problems (Springer International Publishing, New York, 2022).
  27. L. M. Santos and D. L. B. Sombillo, Interpretation of near-threshold peaks using the method of independent s-matrix poles, Phys. Rev. C 108, 045204 (2023).
  28. M. Kato, Analytical properties of two-channel S-matrix, Ann. Phys. (N.Y.) 31, 130 (1965).
  29. D. Morgan, Pole counting and resonance classification, Nucl. Phys. A543, 632 (1992).
  30. D. Morgan and M. R. Pennington, f0(s*): Molecule or quark state?, Phys. Lett. B 258, 444 (1991).
  31. D. Morgan and M. R. Pennington, Decay Jψ→φ(MM) demands the f0(s*) be narrow, Phys. Rev. D 48, 5422 (1993).
  32. R. G. Newton, Structure of the many-channel S matrix, J. Math. Phys. (N.Y.) 2, 188 (1961).
  33. K. J. L. Couteur and R. E. Peierls, The structure of a non-relativistic S-matrix, Proc. R. Soc. A 256, 115 (1997).
  34. LHCb Collaboration, Observation of a narrow pentaquark state, pc(4312)+, and of the two-peak structure of the pc(4450)+, Phys. Rev. Lett. 122, 222001 (2019).
  35. LHC-collaboration, Observation of j/ψp resonances consistent with pentaquark states in λb0→j/ψk−p decays, Phys. Rev. Lett. 115, 072001 (2015).
  36. F.-K. Guo, C. Hanhart, U.-G. Meißner, Q. Wang, Q. Zhao, and B.-S. Zou, Hadronic molecules, Rev. Mod. Phys. 90, 015004 (2018).
  37. C.-J. Xiao, Y. Huang, Y.-B. Dong, L.-S. Geng, and D.-Y. Chen, Exploring the molecular scenario of pc(4312), pc(4440), and pc(4457), Phys. Rev. D 100, 014022 (2019).
  38. M.-L. Du, V. Baru, F.-K. Guo, C. Hanhart, U.-G. Meißner, J. A. Oller, and Q. Wang, Interpretation of the lhcb pc states as hadronic molecules and hints of a narrow pc(4380), Phys. Rev. Lett. 124, 072001 (2020).
  39. D. A. O. Co, V. A. A. Chavez, and D. L. B. Sombillo, Deep learning framework for disentangling triangle singularity and pole-based enhancements, Phys. Rev. D 110, 114034 (2024).
  40. Y. LeCun, Y. Bengio, and G. Hinton, Deep learning, Nature (London) 521, 436 (2015).
  41. I. Goodfellow, Y. Bengio, and A. Courville, Deep Learning (MIT Press, Cambridge, MA, 2016), http://www.deeplearningbook.org.
  42. S. J. D. Prince, Understanding Deep Learning (MIT Press, Cambridge, MA, 2023).
  43. Y. Bengio, A. Courville, and P. Vincent, Representation learning: A review and new perspectives, IEEE Trans. Pattern Anal. Mach. Intell. 35, 1798 (2013).
  44. A. Zheng and A. Casari, Feature Engineering for Machine Learning: Principles and Techniques for Data Scientists (O’Reilly Media, Inc., Sebastopol, 2018).
  45. G. Dong and H. Liu, Feature Engineering for Machine Learning and Data Analytics (CRC Press, Boca Raton, 2018).
  46. A. Krizhevsky, I. Sutskever, and G. E. Hinton, ImageNet classification with deep convolutional neural networks, in Advances in Neural Information Processing Systems (Curran Associates, Inc., Red Hook, 2012), Vol. 25.
  47. K. He, X. Zhang, S. Ren, and J. Sun, Deep residual learning for image recognition, in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (IEEE, Piscataway, 2016), pp. 770–778.
  48. J. Devlin, M.-W. Chang, K. Lee, and K. Toutanova, BERT: Pre-training of deep bidirectional transformers for language understanding, in Proceedings of the 2019 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, Volume 1 (Long and Short Papers), edited by J. Burstein, C. Doran, and T. Solorio (Association for Computational Linguistics, Minneapolis, 2019), pp. 4171–4186.
  49. A. Vaswani, N. Shazeer, N. Parmar, J. Uszkoreit, L. Jones, A. N. Gomez, Ł. ukasz Kaiser, and I. Polosukhin, Attention is all you need, in Advances in Neural Information Processing Systems (Curran Associates, Inc., Red Hook, 2017), Vol. 30.
  50. M. Kuhn and K. Johnson, Feature Engineering and Selection: A Practical Approach for Predictive Models (Chapman and Hall/CRC, New York, 2019).
  51. D. Jannach, G. de Souza P. Moreira, and E. Oldridge, Why are deep learning models not consistently winning recommender systems competitions yet? A position paper, in Proceedings of the Recommender Systems Challenge 2020, RecSysChallenge ’20 (Association for Computing Machinery, New York, 2020), pp. 44–49.
  52. M. Christ, N. Braun, J. Neuffer, and A. W. Kempa-Liehr, Time series feature extraction on basis of scalable hypothesis tests (tsfresh—A python package), Neurocomputing;Variable Star Bulletin 307, 72 (2018).
  53. C. H. Lubba, S. S. Sethi, P. Knaute, S. R. Schultz, B. D. Fulcher, and N. S. Jones, Catch22: CAnonical Time-series CHaracteristics, arXiv:1901.10200.
  54. B. D. Fulcher and N. S. Jones, Highly comparative feature-based time-series classification, IEEE Trans. Knowledge Data Eng. 26, 3026 (2014).
  55. A. Bagnall, J. Lines, A. Bostrom, J. Large, and E. Keogh, The great time series classification bake off: A review and experimental evaluation of recent algorithmic advances, Data Min. Knowl. Discov. 31, 606 (2017).
  56. F. Frohnert, pole-learning: Uncertainty-aware classification of s-matrix pole structures, https://github.com/FelixFrohnertDB/pole-learning (2025).
  57. V. Borisov, T. Leemann, K. Seßler, J. Haug, M. Pawelczyk, and G. Kasneci, Deep neural networks and tabular data: A survey, IEEE Trans. Neural Networks Learn. Syst. 35, 7499 (2024).
  58. R. Shwartz-Ziv and A. Armon, Tabular data: Deep learning is not all you need, arXiv:2106.03253.
  59. L. Prokhorenkova, G. Gusev, A. Vorobev, A. V. Dorogush, and A. Gulin, CatBoost: Unbiased boosting with categorical features, arXiv:1706.09516.
  60. D. L. B. Sombillo, Y. Ikeda, T. Sato, and A. Hosaka, Classifying the pole of an amplitude using a deep neural network, Phys. Rev. D 102, 016024 (2020).
  61. D. L. B. Sombillo, Y. Ikeda, T. Sato, and A. Hosaka, Classifying near-threshold enhancement using deep neural network, Few-Body Syst. 62, 52 (2021).
  62. D. Amodei, C. Olah, J. Steinhardt, P. Christiano, J. Schulman, and D. Mané, Concrete problems in AI safety, arXiv:1606.06565.
  63. C. Guo, G. Pleiss, Y. Sun, and K. Q. Weinberger, On calibration of modern neural networks, arXiv:1706.04599.
  64. A. Kendall and Y. Gal, What uncertainties do we need in Bayesian deep learning for computer vision?, arXiv:1703.04977.
  65. Y. Gal, Uncertainty in Deep Learning (University of Cambridge, Cambridge, England, 2016).
  66. M. Abdar, F. Pourpanah, S. Hussain, D. Rezazadegan, L. Liu, M. Ghavamzadeh, P. Fieguth, X. Cao, A. Khosravi, U. R. Acharya, V. Makarenkov, and S. Nahavandi, A review of uncertainty quantification in deep learning: Techniques, applications and challenges, Inf. Fusion 76, 243 (2021).
  67. B. Lakshminarayanan, A. Pritzel, and C. Blundell, Simple and scalable predictive uncertainty estimation using deep ensembles, arXiv:1612.01474.
  68. S. Depeweg, J. M. Hernández-Lobato, F. Doshi-Velez, and S. Udluft, Decomposition of uncertainty in Bayesian deep learning for efficient and risk-sensitive learning, arXiv:1710.07283v4.
  69. A. Malinin, L. Prokhorenkova, and A. Ustimenko, Uncertainty in gradient boosting via ensembles, arXiv:2006.10562.
  70. L. M. Santos, V. A. A. Chavez, and D. L. B. Sombillo, Pole structure of pψn(4312)+ via machine learning and uniformized S-matrix, J. Phys. G 52, 015104 (2025).
  71. P. C. Bruns, M. Mai, and U. G. Meissner, Chiral dynamics of the S11(1535) and S11(1650) resonances revisited, Phys. Lett. B 697, 254 (2011).
  72. F. Frohnert and P. Emonts, FelixFrohnertDB/Pole-Learning, https://zenodo.org/records/18878202 (2026).
  73. P. A. Zyla et al. (Particle Data Group), Review of particle physics, Prog. Theor. Exp. Phys. 2020, 083C01 (2020).

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Phys. Rev. D 113, 056019– Published 20 March, 2026
  • Received 30 July 2025
  • Accepted 3 March 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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