- Physical Review Journals
All JournalsPhysics Magazine

Physical Review D

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Structure of the lightest nucleus in the visible Universe

Satvir Kaur1,2,*, Chandan Mondal1,2,†, Xingbo Zhao1,2,3,‡, and Chueng-Ryong Ji4,§

  • *Contact author: satvir@impcas.ac.cn
  • †Contact author: mondal@impcas.ac.cn
  • ‡Contact author: xbzhao@impcas.ac.cn
  • §Contact author: crji@ncsu.edu
PDF

Phys. Rev. D 113, 054008 – Published 4 March, 2026

DOI: https://doi.org/10.1103/xkdk-ymn6

Abstract

The simplest atomic nucleus, the deuteron, provides key insights into the strong nuclear interactions among quarks and gluons that shape the visible Universe. We present the first attempt to calculate the internal structure of the deuteron by incorporating hidden-color degrees of freedom, modeling it as an effective mixture of singlet-singlet and octet-octet color clusters beyond the traditional proton-neutron picture. By employing the separation of variables for the light-front two-cluster bound-state equation, we explore how these hidden color correlations shape both its spin and electromagnetic structure. We incorporate the transverse and longitudinal dynamics by two Schrödinger-like equations, namely, the light-front holography and the ’t Hooft equation, respectively. Our results for the electromagnetic form factors and structure functions, including tensor-polarized function, align well with experimental data, offering insights into the partonic structure of the deuteron. Its tensor property could pave the way for a new era in spin physics, guiding future experimental investigations.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (77)

  1. D. P. Anderle et al., Front. Phys. (Beijing) 16, 64701 (2021).
  2. C. G. Callan, Jr., R. F. Dashen, and D. J. Gross, Phys. Rev. D 17, 2717 (1978).
  3. C. R. Ji, J. Phys. Conf. Ser. 543, 012004 (2014).
  4. B. L. G. Bakker and C.-R. Ji, Prog. Part. Nucl. Phys. 74, 1 (2014).
  5. V. A. Matveev and P. Sorba, Lett. Nuovo Cimento 20, 435 (1977).
  6. V. A. Matveev and P. Sorba, Nuovo Cimento Soc. Ital. Fis. 45A, 257 (1978).
  7. H. Hogaasen, P. Sorba, and R. Viollier, Z. Phys. C 4, 131 (1980).
  8. S. J. Brodsky, C.-R. Ji, and G. P. Lepage, Phys. Rev. Lett. 51, 83 (1983).
  9. G. R. Farrar, K. Huleihel, and H.-y. Zhang, Phys. Rev. Lett. 74, 650 (1995).
  10. M. Bashkanov, S. J. Brodsky, and H. Clement, Phys. Lett. B 727, 438 (2013).
  11. G. A. Miller, Phys. Rev. C 89, 045203 (2014).
  12. C. Mondal, S. Kaur, J. Wu, S. Xu, X. Zhao, and J. P. Vary (BLFQ Collaboration), J. Subatomic Part. Cosmol. 3, 100070 (2025).
  13. S. Kaur, J. Wu, S. Xu, C. Mondal, X. Zhao, and J. P. Vary, in Proceedings of the 16th Conference on Quark Confinement and the Hadron Spectrum (2025), arXiv:2503.21371.
  14. C. Mondal, D. Chakrabarti, and X. Zhao, Eur. Phys. J. A 53, 106 (2017).
  15. T. Gutsche, V. E. Lyubovitskij, and I. Schmidt, Phys. Rev. D 94, 116006 (2016).
  16. T. Gutsche, V. E. Lyubovitskij, I. Schmidt, and A. Vega, Phys. Rev. D 91, 114001 (2015).
  17. B.-d. Sun and Y.-b. Dong, Chin. Phys. C 41, 013102 (2017).
  18. W. Shi, R. Peng, T.-X. Liu, S. Lyu, and B. Long, Phys. Rev. C 106, 015505 (2022).
  19. N. Huseynova, S. Mamedov, and J. Samadov, Chin. Phys. C 47, 013104 (2023).
  20. L. E. Marcucci, F. Gross, M. T. Pena, M. Piarulli, R. Schiavilla, I. Sick, A. Stadler, J. W. Van Orden, and M. Viviani, J. Phys. G 43, 023002 (2016).
  21. A. Khachi, L. Kumar, M. R. G. Kumar, and O. S. K. S. Sastri, Phys. Rev. C 107, 064002 (2023).
  22. M. Chemtob, E. J. Moniz, and M. Rho, Phys. Rev. C 10, 344 (1974).
  23. R. Blankenbecler and J. F. Gunion, Phys. Rev. D 4, 718 (1971).
  24. G. Piller, W. Melnitchouk, and A. W. Thomas, Phys. Rev. C 54, 894 (1996).
  25. V. A. Karmanov and A. V. Smirnov, Nucl. Phys. A575, 520 (1994).
  26. V. A. Karmanov and A. V. Smirnov, Nucl. Phys. A546, 691 (1992).
  27. D. Abbott et al. (JLAB t20 Collaboration), Eur. Phys. J. A 7, 421 (2000).
  28. I. Sick and D. Trautmann, Nucl. Phys. A637, 559 (1998).
  29. C. Zhang et al., Phys. Rev. Lett. 107, 252501 (2011).
  30. D. M. Nikolenko et al., Phys. Rev. Lett. 90, 072501 (2003).
  31. M. Garcon and J. W. Van Orden, Adv. Nucl. Phys. 26, 293 (2001).
  32. S. Kumano, Eur. Phys. J. A 60, 205 (2024).
  33. A. Airapetian et al. (HERMES Collaboration), Phys. Rev. Lett. 95, 242001 (2005).
  34. K. Slifer et al., The Deuteron Tensor Structure Function b1, Report No. E12-13-011, Thomas Jefferson National Accelerator Facility, https://www.jlab.org/sites/default/files/PAC/PAC51/PAC51_REPORT_2023.pdf.
  35. J. Poudel, A. Bacchetta, J.-P. Chen, D. Keller, I. Fernando, E. Long, D. Ruth, N. Santiesteban, and K. Slifer, arXiv:2502.20044.
  36. M. Brooks et al., SeaQuest with a Transversely Polarized Target (E1039), https://twist.phys.virginia.edu/work/E1039proposal_final.pdf.
  37. D. Keller (SpinQuest Collaboration), arXiv:2205.01249.
  38. P. Hoodbhoy, R. L. Jaffe, and A. Manohar, Nucl. Phys. B312, 571 (1989).
  39. H. Khan and P. Hoodbhoy, Phys. Rev. C 44, 1219 (1991).
  40. J. Edelmann, G. Piller, and W. Weise, Phys. Rev. C 57, 3392 (1998).
  41. W. Cosyn, Y.-B. Dong, S. Kumano, and M. Sargsian, Phys. Rev. D 95, 074036 (2017).
  42. A. Accardi et al., Eur. Phys. J. A 52, 268 (2016).
  43. C.-R. Ji and S. J. Brodsky, Phys. Rev. D 34, 1460 (1986).
  44. S. J. Brodsky, G. F. de Teramond, H. G. Dosch, and J. Erlich, Phys. Rep. 584, 1 (2015).
  45. G. ’t Hooft, Nucl. Phys. B75, 461 (1974).
  46. M. Ahmady, H. Dahiya, S. Kaur, C. Mondal, R. Sandapen, and N. Sharma, Phys. Lett. B 823, 136754 (2021).
  47. S. S. Chabysheva and J. R. Hiller, Ann. Phys. (Amsterdam) 337, 143 (2013).
  48. N. Sharma, Phys. Rev. D 109, 014019 (2024).
  49. M. Ahmady, S. Kaur, S. L. MacKay, C. Mondal, and R. Sandapen, Phys. Rev. D 104, 074013 (2021).
  50. M. Ahmady, S. Kaur, C. Mondal, and R. Sandapen, Phys. Lett. B 836, 137628 (2023).
  51. B. Gurjar, C. Mondal, and S. Kaur, Phys. Rev. D 109, 094017 (2024).
  52. H. J. Melosh, Phys. Rev. D 9, 1095 (1974).
  53. C. R. Ji, P. L. Chung, and S. R. Cotanch, Phys. Rev. D 45, 4214 (1992).
  54. J. R. Forshaw and R. Sandapen, Phys. Rev. Lett. 109, 081601 (2012).
  55. H.-M. Choi and C.-R. Ji, Phys. Rev. D 70, 053015 (2004).
  56. S. J. Brodsky and B. T. Chertok, Phys. Rev. Lett. 37, 269 (1976).
  57. A. P. Kobushkin and A. I. Syamtomov, Phys. At. Nucl. 58, 1477 (1995); arXiv:hep-ph/9409411.
  58. S. J. Brodsky and C.-R. Ji, Phys. Rev. D 33, 2653 (1986).
  59. I. The et al., Phys. Rev. Lett. 67, 173 (1991).
  60. D. Benaksas, D. Drickey, and D. Frerejacque, Phys. Rev. 148, 1327 (1966).
  61. T. E. O. Ericson and M. Rosa-Clot, Annu. Rev. Nucl. Part. Sci. 35, 271 (1985).
  62. A. Afanasev, V. D. Afanasev, and S. V. Trubnikov, arXiv:nucl-th/9808047.
  63. A. Bacchetta and P. J. Mulders, Phys. Rev. D 62, 114004 (2000).
  64. A. Bacchetta and P. J. Mulders, Phys. Lett. B 518, 85 (2001).
  65. P. Mulders, Transverse momentum dependence in structure functions in hard scattering processes (2001), https://www.nat.vu.nl/∼mulders/correlations-0.pdf.
  66. S. Kaur, C. Mondal, and H. Dahiya, J. High Energy Phys. 01 (2021) 136.
  67. R. D. Ball et al. (NNPDF Collaboration), Eur. Phys. J. C 77, 663 (2017).
  68. M. Arneodo et al. (New Muon Collaboration), Phys. Lett. B 364, 107 (1995).
  69. M. Arneodo et al. (New Muon Collaboration), Nucl. Phys. B483, 3 (1997).
  70. D. D. Gabbert, Determination of the structure function F2 at HERMES, Ph.D. thesis, Hamburg University, 2008.
  71. B. Adeva et al. (Spin Muon Collaboration), Phys. Rev. D 58, 112001 (1998).
  72. B. Adeva et al. (Spin Muon Collaboration), Phys. Lett. B 302, 533 (1993).
  73. D. Adams et al. (Spin Muon Collaboration), Phys. Lett. B 357, 248 (1995).
  74. B. Adeva et al. (Spin Muon Collaboration), Phys. Rev. D 60, 072004 (1999); 62, 079902(E) (2000).
  75. A. Airapetian et al. (HERMES Collaboration), Phys. Rev. D 75, 012007 (2007).
  76. K. Abe et al. (E143 Collaboration), Phys. Rev. D 58, 112003 (1998).
  77. E. S. Ageev et al. (COMPASS Collaboration), Phys. Lett. B 612, 154 (2005).

Outline

Information

Phys. Rev. D 113, 054008– Published 4 March, 2026
  • Received 22 July 2025
  • Accepted 11 February 2026
Crossmark - Check for updates button
Creative Commons Logo

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.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation