Abstract
The correlation function observed in high-energy collision experiments encodes critical information about the emitted source and hadronic interactions. While the proton-proton interaction potential is well constrained by nucleon-nucleon scattering data, these measurements offer a unique avenue to investigate the proton-emitting source, reflecting the dynamical properties of the collisions. In this context, the understanding of other hadronic interactions such as hyperon-nucleon remains limited. In this work, we present an unbiased approach to reconstruct proton-emitting sources from experimental correlation functions. Within an automatic differentiation framework, we parameterize the source functions with deep neural networks, to compute correlation functions. This approach achieves a lower chi-squared value compared to conventional Gaussian source functions and captures the long-tail behavior, in qualitative agreement with simulation predictions. We finally apply our method to extract hyperon-nucleon correlations.
Similar content being viewed by others
Data availability
The data that support the findings of this study are available from the corresponding author upon request.
Code availability
The open codes can be found in a public GitHub repository, https://github.com/Anguswlx/InferSFs.
References
Yukawa, H. On the interaction of elementary particles. I. Proc. Phys. Math. Soc. Jpn. 17, 48–57 (1935).
Bryan, R. A. & Scott, B. L. Nucleon-nucleon scattering from one-boson-exchange potentials. Phys. Rev. 135, B434–B450 (1964).
Kiang, D., Preston, M. A. & Yip, P. One-boson-exchange potential and nuclear matter. Phys. Rev. 170, 907–915 (1968).
Lacombe, M. et al. Parametrization of the Paris n n potential. Phys. Rev. C 21, 861–873 (1980).
Wiringa, R. B., Stoks, V. G. J. & Schiavilla, R. An accurate nucleon-nucleon potential with charge independence breaking. Phys. Rev. C 51, 38–51 (1995).
Reid Jr, R. V. Local phenomenological nucleon-nucleon potentials. Ann. Phys. 50, 411–448 (1968).
Stoks, V. G. J., Klomp, R. A. M., Terheggen, C. P. F. & de Swart, J. J. Construction of high quality N N potential models. Phys. Rev. C 49, 2950–2962 (1994).
Nagels, M. M., Rijken, T. A. & Yamamoto, Y. Extended-soft-core baryon-baryon model ESC16. I. Nucleon-nucleon scattering. Phys. Rev. C 99, 044002 (2019).
Entem, D. R. & Machleidt, R. Accurate charge dependent nucleon nucleon potential at fourth order of chiral perturbation theory. Phys. Rev. C 68, 041001 (2003).
Machleidt, R. & Entem, D. R. Chiral effective field theory and nuclear forces. Phys. Rep. 503, 1–75 (2011).
Epelbaum, E. Few-nucleon forces and systems in chiral effective field theory. Prog. Part. Nucl. Phys. 57, 654–741 (2006).
Drischler, C., Hebeler, K. & Schwenk, A. Chiral interactions up to next-to-next-to-next-to-leading order and nuclear saturation. Phys. Rev. Lett. 122, 042501 (2019).
Hammer, H.-W., Nogga, A. & Schwenk, A. Three-body forces: from cold atoms to nuclei. Rev. Mod. Phys. 85, 197 (2013).
Gazit, D., Quaglioni, S. & Navratil, P. Three-nucleon low-energy constants from the consistency of interactions and currents in chiral effective field theory. Phys. Rev. Lett. 103, 102502 (2009).
Aoki, S. Hadron interactions in lattice QCD. Prog. Part. Nucl. Phys. 66, 687–726 (2011).
Iritani, T. et al. Systematics of the HAL QCD potential at low energies in lattice QCD. Phys. Rev. D. 99, 014514 (2019).
Yamazaki, T., Ishikawa, K. I., Kuramashi, Y. & Ukawa, A. Study of quark mass dependence of binding energy for light nuclei in 2+1 flavor lattice QCD. Phys. Rev. D 92, 014501 (2015).
Wagman, M. L. et al. Baryon-baryon Interactions and spin-flavor symmetry from lattice quantum chromodynamics. Phys. Rev. D 96, 114510 (2017).
Aoki, S. & Doi, T. Lattice QCD and Baryon-Baryon Interactions, 1–31 (Springer, 2023).
Baym, G. et al. From hadrons to quarks in neutron stars: a review. Rep. Prog. Phys. 81, 056902 (2018).
Brown, R. H. & Twiss, R. Lxxiv. A new type of interferometer for use in radio astronomy. Lond. Edinb. Dublin Philos. Mag. J. Sci. 45, 663–682 (1954).
Pratt, S. Pion Interferometry for exploding sources. Phys. Rev. Lett. 53, 1219–1221 (1984).
Lisa, M. A., Pratt, S., Soltz, R. & Wiedemann, U. Femtoscopy in relativistic heavy ion collisions. Ann. Rev. Nucl. Part. Sci. 55, 357–402 (2005).
Alt, C. et al. Bose-Einstein correlations of pi-pi- pairs in central Pb+Pb collisions at A-20, A-30, A-40, A-80, and A-158 GeV. Phys. Rev. C 77, 064908 (2008).
Li, Q. -f, Steinheimer, J., Petersen, H., Bleicher, M. & Stocker, H. Effects of a phase transition on HBT correlations in an integrated Boltzmann+Hydrodynamics approach. Phys. Lett. B 674, 111–116 (2009).
Wiedemann, U. A. & Heinz, U. W. Resonance contributions to HBT correlation radii. Phys. Rev. C 56, 3265–3286 (1997).
Kisiel, A., Florkowski, W. & Broniowski, W. Femtoscopy in hydro-inspired models with resonances. Phys. Rev. C 73, 064902 (2006).
Xu, J. et al. Imaging freeze-out sources and extracting strong interaction parameters in relativistic heavy-ion collisions. Chin. Phys. Lett. 42, 031401 (2025).
Si, D. et al. Extracting neutron-neutron interaction strength and spatiotemporal dynamics of neutron emission from the two-particle correlation function. Phys. Rev. Lett. 134, 222301 (2025).
Pratt, S. CRAB V3.0 https://web.pa.msu.edu/people/pratts/freecodes/crab/home.html.
Mihaylov, D. L. et al. A femtoscopic correlation analysis tool using the Schrödinger equation (CATS). Eur. Phys. J. C 78, 394 (2018).
Fabbietti, L., Mantovani Sarti, V. & Vazquez Doce, O. Study of the strong interaction among hadrons with correlations at the LHC. Ann. Rev. Nucl. Part. Sci. 71, 377–402 (2021).
Adams, J. et al. Proton-lambda correlations in central Au+Au collisions at S(NN)**(1/2) = 200-GeV. Phys. Rev. C 74, 064906 (2006).
Adam, J. et al. The proton-Ω correlation function in Au+Au collisions at \(\sqrt{{s}_{NN}}\)=200 GeV. Phys. Lett. B 790, 490–497 (2019).
Adamczyk, L. et al. ΛΛ correlation function in Au+Au collisions at \(\sqrt{{s}_{NN}}=\) 200 GeV. Phys. Rev. Lett. 114, 022301 (2015).
Adamczyk, L. et al. Measurement of Interaction between antiprotons. Nature 527, 345–348 (2015).
Acharya, S. et al. p-p, p-Λ and Λ-Λ correlations studied via femtoscopy in pp reactions at \(\sqrt{s}\) = 7 TeV. Phys. Rev. C 99, 024001 (2019).
Acharya, S. et al. Experimental evidence for an attractive p-ϕ interaction. Phys. Rev. Lett. 127, 172301 (2021).
Acharya, S. et al. Investigation of the p–Σ0 interaction via femtoscopy in pp collisions. Phys. Lett. B 805, 135419 (2020).
Acharya, S. et al. First observation of an attractive interaction between a proton and a cascade Baryon. Phys. Rev. Lett. 123, 112002 (2019).
Acharya, S. et al. First measurement of the Λ–Ξ interaction in proton–proton collisions at the LHC. Phys. Lett. B 844, 137223 (2023).
Lednicky, R. & Lyuboshits, V. L. Final state interaction effect on pairing correlations between particles with small relative momenta. Yad. Fiz. 35, 1316–1330 (1981).
Ohnishi, A., Morita, K., Miyahara, K. & Hyodo, T. Hadron–hadron correlation and interaction from heavy–ion collisions. Nucl. Phys. A 954, 294–307 (2016).
Kisiel, A. Non-identical particle femtoscopy at s(NN)**(1/2) = 200-AGeV in hydrodynamics with statistical hadronization. Phys. Rev. C 81, 064906 (2010).
Sinyukov, Y. M., Shapoval, V. M. & Naboka, V. Y. On mT dependence of femtoscopy scales for meson and baryon pairs. Nucl. Phys. A 946, 227–239 (2016).
Csorgo, T., Hegyi, S. & Zajc, W. A. Bose-Einstein correlations for Levy stable source distributions. Eur. Phys. J. C 36, 67–78 (2004).
Aamodt, K. et al. Femtoscopy of pp collisions at \(\sqrt{s}=0.9\) and 7 TeV at the LHC with two-pion Bose-Einstein correlations. Phys. Rev. D 84, 112004 (2011).
Adam, J. et al. Two-pion femtoscopy in p-Pb collisions at \(\sqrt{{s}_{NN}}=5.02\) TeV. Phys. Rev. C 91, 034906 (2015).
Acharya, S. et al. Search for a common baryon source in high-multiplicity pp collisions at the LHC. Phys. Lett. B 811, 135849 (2020).
Acharya, S. et al. Common femtoscopic hadron-emission source in pp collisions at the LHC. Eur. Phys. J. C 85, 198 (2025).
Sirunyan, A. M. et al. Bose-Einstein correlations in pp, pPb, and PbPb collisions at \(\sqrt{{s}_{NN}}=0.9-7\) TeV. Phys. Rev. C 97, 064912 (2018).
Sirunyan, A. M. et al. Bose-Einstein correlations of charged hadrons in proton-proton collisions at \(\sqrt{s}=\) 13 TeV. JHEP 03, 014 (2020).
Aad, G. et al. Two-particle Bose–Einstein correlations in pp collisions at \(\sqrt{{{{\bf{s}}}}}=\) 0.9 and 7 TeV measured with the ATLAS detector. Eur. Phys. J. C 75, 466 (2015).
Day, B. D. Three-body correlations in nuclear matter. Phys. Rev. C 24, 1203–1271 (1981).
Ohnishi, A. Hadrons, Quark-Gluon Plasma, and Neutron Stars, 1–58 (Springer, 2023).
Aarts, G. et al. Physics-driven learning for inverse problems in quantum chromodynamics. Nat. Rev. Phys. 7, 154–163 (2025).
Bishop, C. M. & Bishop, H. Deep learning: Foundations and concepts (Springer Nature, 2023).
Bodmer, A. R., Usmani, Q. N. & Carlson, J. Binding energies of hypernuclei and three-body lambda N N forces. Phys. Rev. C 29, 684–687 (1984).
Rijken, T. A. & Yamamoto, Y. Extended-soft-core baryon-baryon model. II. Hyperon-nucleon interaction. Phys. Rev. C 73, 044008 (2006).
Sasaki, K., Oset, E. & Vicente Vacas, M. J. Scalar lambda N and lambda lambda interaction in a chiral unitary approach. Phys. Rev. C 74, 064002 (2006).
Polinder, H., Haidenbauer, J. & Meissner, U.-G. Hyperon-nucleon interactions: a chiral effective field theory approach. Nucl. Phys. A 779, 244–266 (2006).
Nagels, M. M., Rijken, T. A. & Yamamoto, Y. Extended-soft-core baryon-baryon model ESC16. II. Hyperon-nucleon interactions. Phys. Rev. C 99, 044003 (2019).
Haidenbauer, J. et al. Hyperon-nucleon interaction at next-to-leading order in chiral effective field theory. Nucl. Phys. A 915, 24–58 (2013).
Nemura, H. Lambda-nucleon and sigma-nucleon potentials from space-time correlation function on the lattice. PoS LATTICE2021, 272 (2022).
Wang, F. -q. & Pratt, S. Lambda proton correlations in relativistic heavy ion collisions. Phys. Rev. Lett. 83, 3138–3141 (1999).
Wu, L., Zhu, Z. & Weinan, E. Towards understanding generalization of deep learning: perspective of loss landscapes. Preprint at https://doi.org/10.48550/arXiv.1706.10239 (2017).
Rosca, M., Weber, T., Gretton, A. & Mohamed, S. A case for new neural network smoothness constraints. In Proc. on “I Can’t Believe It’s Not Better!” At NeurIPS Workshops (eds Zosa Forde, J., Ruiz, F., Pradier, M. F. & Schein, A.) Vol. 137, 21–32 https://proceedings.mlr.press/v137/rosca20a.html (PMLR, 2020).
Shi, S., Wang, L. & Zhou, K. Rethinking the ill-posedness of the spectral function reconstruction—why is it fundamentally hard and how artificial neural networks can help. Comput. Phys. Commun. 282, 108547 (2023).
Kingma, D. P. & Ba, J. Adam: a method for stochastic optimization. Preprint at https://doi.org/10.48550/arXiv.1412.6980 (2014).
Wang, L., Shi, S. & Zhou, K. Automatic differentiation approach for reconstructing spectral functions with neural networks. In Proc. 35th Conference on Neural Information Processing Systems (Curran Associates, Inc., 2021).
Shi, S., Zhou, K., Zhao, J., Mukherjee, S. & Zhuang, P. Heavy quark potential in the quark-gluon plasma: deep neural network meets lattice quantum chromodynamics. Phys. Rev. D 105, 014017 (2022).
Wang, L., Shi, S. & Zhou, K. Reconstructing spectral functions via automatic differentiation. Phys. Rev. D 106, L051502 (2022).
Soma, S., Wang, L., Shi, S., Stöcker, H. & Zhou, K. Reconstructing the neutron star equation of state from observational data via automatic differentiation. Phys. Rev. D 107, 083028 (2023).
Acknowledgements
We thank Drs. Takumi Doi, Tetsuo Hatsuda, and Zhigang Xiao for helpful discussions. We thank the DEEP-IN working group at RIKEN-iTHEMS for support in the preparation of this paper. LW is supported by the RIKEN TRIP initiative (RIKEN Quantum), JSPS KAKENHI Grant No. 25H01560, and JST-BOOST Grant No. JPMJBY24H9. J.X. is support by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) through the grant CRC-TR 211 “Strong-interaction matter under extreme conditions”-Project number 315477589-TRR 211.
Funding
Open Access funding enabled and organized by Projekt DEAL.
Author information
Authors and Affiliations
Contributions
Both Jiaxing Zhao and Lingxiao Wang contributed equally to all aspects of the article.
Corresponding author
Ethics declarations
Competing interests
The authors declare no competing interests.
Peer review
Peer review information
Communications Physics thanks the anonymous reviewers for their contribution to the peer review of this work. [A peer review file is available].
Additional information
Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Supplementary information
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Wang, L., Zhao, J. Learning hadron emitting sources with deep neural networks. Commun Phys (2026). https://doi.org/10.1038/s42005-026-02530-w
Received:
Accepted:
Published:
DOI: https://doi.org/10.1038/s42005-026-02530-w
