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Baryon-baryon, meson-meson, and meson-baryon interactions in nonrelativistic QCD

Benoît Assi1,2, Anthony V. Grebe1, and Michael L. Wagman1

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Phys. Rev. D 113, 016013 – Published 13 January, 2026

DOI: https://doi.org/10.1103/h1br-lydt

Abstract

Van der Waals potentials describing interactions between color-singlet mesons and/or baryons vanish at leading order in potential nonrelativistic quantum chromodynamics (pNRQCD). This result and constraints from Gauss’s law are used to prove that weakly coupled pNRQCD van der Waals potentials in generic non-Abelian gauge theories with only heavy quarks are too weak to form bound states whose color state is a product of color singlets. Quantum Monte Carlo calculations of four, five, and six quarks with equal masses provide numerical evidence that exotic color configurations are higher energy than products of color-singlet hadrons, suggesting that equal-mass fully heavy tetraquark, pentaquark, and hexaquark bound states do not exist at next-to-leading order in pNRQCD and at all orders in QCD-like theories in which all quark masses are asymptotically large. Mechanisms for generating hadron-hadron bound states are identified, which necessarily involve large quark-mass hierarchies, relativistic effects arising from the presence of sufficiently light quarks, or nonperturbative effects outside the scope of weakly coupled pNRQCD.

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References (122)

  1. V. Bernard, N. Kaiser, and U.-G. Meissner, Int. J. Mod. Phys. E 04, 193 (1995).
  2. S. Scherer, Adv. Nucl. Phys. 27, 277 (2003).
  3. V. Bernard and U.-G. Meissner, Annu. Rev. Nucl. Part. Sci. 57, 33 (2007).
  4. S. Weinberg, Phys. Rev. Lett. 17, 616 (1966).
  5. Y. Tomozawa, Nuovo Cimento A 46, 707 (1966).
  6. S. R. Beane and M. J. Savage, Nucl. Phys. A713, 148 (2003).
  7. E. Epelbaum, U.-G. Meissner, and W. Gloeckle, Nucl. Phys. A714, 535 (2003).
  8. S. R. Beane and M. J. Savage, Nucl. Phys. A717, 91 (2003).
  9. J. C. Berengut, E. Epelbaum, V. V. Flambaum, C. Hanhart, U. G. Meissner, J. Nebreda, and J. R. Pelaez, Phys. Rev. D 87, 085018 (2013).
  10. M. Fukugita, Y. Kuramashi, H. Mino, M. Okawa, and A. Ukawa, Phys. Rev. Lett. 73, 2176 (1994).
  11. M. Fukugita, Y. Kuramashi, M. Okawa, H. Mino, and A. Ukawa, Phys. Rev. D 52, 3003 (1995).
  12. S. R. Beane, P. F. Bedaque, K. Orginos, and M. J. Savage, Phys. Rev. Lett. 97, 012001 (2006).
  13. S. R. Beane, P. F. Bedaque, T. C. Luu, K. Orginos, E. Pallante, A. Parreño, and M. J. Savage (NPLQCD Collaboration), Nucl. Phys. A794, 62 (2007).
  14. S. R. Beane, W. Detmold, H.-W. Lin, T. C. Luu, K. Orginos, M. J. Savage, A. Torok, and A. Walker-Loud (NPLQCD Collaboration), Phys. Rev. D 81, 054505 (2010).
  15. T. Yamazaki, Y. Kuramashi, and A. Ukawa (PACS-CS Collaboration), Phys. Rev. D 81, 111504 (2010).
  16. T. Inoue, N. Ishii, S. Aoki, T. Doi, T. Hatsuda, Y. Ikeda, K. Murano, H. Nemura, and K. Sasaki (HAL QCD Collaboration), Prog. Theor. Phys. 124, 591 (2010).
  17. T. Inoue, N. Ishii, S. Aoki, T. Doi, T. Hatsuda, Y. Ikeda, K. Murano, H. Nemura, and K. Sasaki (HAL QCD Collaboration), Phys. Rev. Lett. 106, 162002 (2011).
  18. S. R. Beane et al. (NPLQCD Collaboration), Phys. Rev. Lett. 106, 162001 (2011).
  19. T. Inoue, S. Aoki, T. Doi, T. Hatsuda, Y. Ikeda, N. Ishii, K. Murano, H. Nemura, and K. Sasaki (HAL QCD Collaboration), Nucl. Phys. A881, 28 (2012).
  20. S. R. Beane, E. Chang, W. Detmold, H. W. Lin, T. C. Luu, K. Orginos, A. Parreño, M. J. Savage, A. Torok, and A. Walker-Loud (NPLQCD Collaboration), Phys. Rev. D 85, 054511 (2012).
  21. S. R. Beane, E. Chang, S. D. Cohen, W. Detmold, H. W. Lin, T. C. Luu, K. Orginos, A. Parreño, M. J. Savage, and A. Walker-Loud (NPLQCD Collaboration), Phys. Rev. D 87, 034506 (2013).
  22. T. Yamazaki, K.-i. Ishikawa, Y. Kuramashi, and A. Ukawa, Phys. Rev. D 86, 074514 (2012).
  23. S. R. Beane et al. (NPLQCD Collaboration), Phys. Rev. C 88, 024003 (2013).
  24. N. Ishii, Few Body Syst. 54, 1071 (2013).
  25. T. Yamazaki, K.-i. Ishikawa, Y. Kuramashi, and A. Ukawa, Phys. Rev. D 92, 014501 (2015).
  26. E. Berkowitz, T. Kurth, A. Nicholson, B. Joó, E. Rinaldi, M. Strother, P. M. Vranas, and A. Walker-Loud, Phys. Lett. B 765, 285 (2017).
  27. M. L. Wagman, F. Winter, E. Chang, Z. Davoudi, W. Detmold, K. Orginos, M. J. Savage, and P. E. Shanahan, Phys. Rev. D 96, 114510 (2017).
  28. S. Gongyo et al., Phys. Rev. Lett. 120, 212001 (2018).
  29. A. Francis, J. R. Green, P. M. Junnarkar, C. Miao, T. D. Rae, and H. Wittig, Phys. Rev. D 99, 074505 (2019).
  30. B. Hörz et al., Phys. Rev. C 103, 014003 (2021).
  31. J. R. Green, A. D. Hanlon, P. M. Junnarkar, and H. Wittig, Phys. Rev. Lett. 127, 242003 (2021).
  32. S. Amarasinghe, R. Baghdadi, Z. Davoudi, W. Detmold, M. Illa, A. Parreño, A. V. Pochinsky, P. E. Shanahan, and M. L. Wagman, Phys. Rev. D 107, 094508 (2023).
  33. W. Detmold, M. Illa, W. I. Jay, A. Parreño, R. J. Perry, P. E. Shanahan, and M. L. Wagman (NPLQCD Collaboration), Phys. Rev. D 111, 114501 (2025).
  34. J. Bulava et al. (BaSc Collaboration), arXiv:2505.05547.
  35. P. Junnarkar and N. Mathur, Phys. Rev. Lett. 123, 162003 (2019).
  36. Y. Lyu, H. Tong, T. Sugiura, S. Aoki, T. Doi, T. Hatsuda, J. Meng, and T. Miyamoto, Phys. Rev. Lett. 127, 072003 (2021).
  37. N. Mathur, M. Padmanath, and D. Chakraborty, Phys. Rev. Lett. 130, 111901 (2023).
  38. P. M. Junnarkar and N. Mathur, Phys. Rev. D 106, 054511 (2022).
  39. N. S. Dhindsa, N. Mathur, and M. Padmanath, Phys. Rev. D 112, 054501 (2025).
  40. A. Mitridate, M. Redi, J. Smirnov, and A. Strumia, J. High Energy Phys. 10 (2017) 210.
  41. J. M. Cline, W. Huang, and G. D. Moore, Phys. Rev. D 94, 055029 (2016).
  42. T. DeGrand and E. T. Neil, Phys. Rev. D 101, 034504 (2020).
  43. J. M. Cline, SciPost Phys. Lect. Notes 52, 1 (2022).
  44. P. Asadi, E. D. Kramer, E. Kuflik, G. W. Ridgway, T. R. Slatyer, and J. Smirnov, Phys. Rev. Lett. 127, 211101 (2021).
  45. P. Asadi, E. D. Kramer, E. Kuflik, G. W. Ridgway, T. R. Slatyer, and J. Smirnov, Phys. Rev. D 104, 095013 (2021).
  46. P. Asadi, T. R. Slatyer, and J. Smirnov, Phys. Rev. D 106, 015012 (2022).
  47. P. Asadi, A. Batz, and G. D. Kribs, Phys. Rev. D 111, 095025 (2025).
  48. P. Asadi, G. D. Kribs, and C. J. H. Mantel, Phys. Rev. D 111, 095030 (2025).
  49. A. Bodas, M. A. Buen-Abad, A. Hook, and R. Sundrum, J. High Energy Phys. 06 (2024) 052.
  50. Y. Gouttenoire, E. Kuflik, and D. Liu, Phys. Rev. D 109, 035002 (2024).
  51. V. V. Dixit and M. Sher, Phys. Rev. D 37, 1097 (1988).
  52. J. Yoo and R. J. Scherrer, Phys. Rev. D 67, 043517 (2003).
  53. V. F. Dmitriev, V. V. Flambaum, and J. K. Webb, Phys. Rev. D 69, 063506 (2004).
  54. C. M. Muller, G. Schafer, and C. Wetterich, Phys. Rev. D 70, 083504 (2004).
  55. N. Chamoun, S. J. Landau, M. E. Mosquera, and H. Vucetich, J. Phys. G 34, 163 (2007).
  56. A. Coc, N. J. Nunes, K. A. Olive, J.-P. Uzan, and E. Vangioni, Phys. Rev. D 76, 023511 (2007).
  57. S. J. Landau, M. E. Mosquera, C. G. Scoccola, and H. Vucetich, Phys. Rev. D 78, 083527 (2008).
  58. P. F. Bedaque, T. Luu, and L. Platter, Phys. Rev. C 83, 045803 (2011).
  59. A.-K. Burns, V. Keus, M. Sher, and T. M. P. Tait, Phys. Rev. D 109, 123506 (2024).
  60. H. Meyer and U.-G. Meißner, J. High Energy Phys. 06 (2024) 074; 01 (2025) 33.
  61. U.-G. Meißner, B. C. Metsch, and H. Meyer, Eur. Phys. J. A 61, 122 (2025).
  62. G. T. Bodwin, E. Braaten, and G. P. Lepage, Phys. Rev. D 51, 1125 (1995); 55, 5853(E) (1997).
  63. W. E. Caswell and G. P. Lepage, Phys. Lett. 167B, 437 (1986).
  64. A. Pineda and J. Soto, Nucl. Phys. B, Proc. Suppl. 64, 428 (1998).
  65. A. Pineda and J. Soto, Phys. Rev. D 58, 114011 (1998).
  66. N. Brambilla, A. Pineda, J. Soto, and A. Vairo, Nucl. Phys. B566, 275 (2000).
  67. N. Brambilla, A. Pineda, J. Soto, and A. Vairo, Rev. Mod. Phys. 77, 1423 (2005).
  68. A. Pineda, Prog. Part. Nucl. Phys. 67, 735 (2012).
  69. N. Brambilla, A. Pineda, J. Soto, and A. Vairo, Phys. Lett. B 470, 215 (1999).
  70. B. A. Kniehl, A. A. Penin, V. A. Smirnov, and M. Steinhauser, Nucl. Phys. B635, 357 (2002).
  71. A. Pineda and F. J. Yndurain, Phys. Rev. D 58, 094022 (1998).
  72. N. Brambilla, A. Vairo, X. Garcia i Tormo, and J. Soto, Phys. Rev. D 80, 034016 (2009).
  73. N. Brambilla, Y. Jia, and A. Vairo, Phys. Rev. D 73, 054005 (2006).
  74. N. Brambilla and A. Vairo, Phys. Rev. D 62, 094019 (2000).
  75. N. Brambilla, P. Pietrulewicz, and A. Vairo, Phys. Rev. D 85, 094005 (2012).
  76. M. J. Savage and M. B. Wise, Phys. Lett. B 248, 177 (1990).
  77. N. Brambilla, A. Vairo, and T. Rosch, Phys. Rev. D 72, 034021 (2005).
  78. N. Brambilla, J. Ghiglieri, and A. Vairo, Phys. Rev. D 81, 054031 (2010).
  79. N. Brambilla, F. Karbstein, and A. Vairo, Phys. Rev. D 87, 074014 (2013).
  80. B. Assi and M. L. Wagman, Phys. Rev. D 108, 096004 (2023).
  81. M. E. Luke, A. V. Manohar, and I. Z. Rothstein, Phys. Rev. D 61, 074025 (2000).
  82. A. V. Manohar and I. W. Stewart, Phys. Rev. D 63, 054004 (2001).
  83. A. H. Hoang, arXiv:hep-ph/0204299.
  84. Y. Jia, J. High Energy Phys. 10 (2006) 073.
  85. F. J. Llanes-Estrada, O. I. Pavlova, and R. Williams, Eur. Phys. J. C 72, 2019 (2012).
  86. J. Carlson, S. Gandolfi, F. Pederiva, S. C. Pieper, R. Schiavilla, K. E. Schmidt, and R. B. Wiringa, Rev. Mod. Phys. 87, 1067 (2015).
  87. Y. Yan and D. Blume, J. Phys. B 50, 223001 (2017).
  88. S. Gandolfi, D. Lonardoni, A. Lovato, and M. Piarulli, Front. Phys. 8, 117 (2020).
  89. B. Assi and M. L. Wagman, Phys. Rev. D 110, 094001 (2024).
  90. A. Czarnecki, B. Leng, and M. B. Voloshin, Phys. Lett. B 778, 233 (2018).
  91. N. Brambilla, V. Shtabovenko, J. Tarrús Castellà, and A. Vairo, Phys. Rev. D 95, 116004 (2017).
  92. S. J. Brodsky, I. A. Schmidt, and G. F. de Teramond, Phys. Rev. Lett. 64, 1011 (1990).
  93. M. E. Luke, A. V. Manohar, and M. J. Savage, Phys. Lett. B 288, 355 (1992).
  94. S. J. Brodsky and F. S. Navarra, Phys. Lett. B 411, 152 (1997).
  95. S. J. Brodsky and G. A. Miller, Phys. Lett. B 412, 125 (1997).
  96. M. Peskin, Nucl. Phys. B156, 365 (1979).
  97. G. Bhanot and M. Peskin, Nucl. Phys. B156, 391 (1979).
  98. N. Brambilla, G. a. Krein, J. Tarrús Castellà, and A. Vairo, Phys. Rev. D 93, 054002 (2016).
  99. N. Brambilla, S. Eidelman, C. Hanhart, A. Nefediev, C.-P. Shen, C. E. Thomas, A. Vairo, and C.-Z. Yuan, Phys. Rep. 873, 1 (2020).
  100. X.-K. Dong, F.-K. Guo, A. Nefediev, and J. T. Castellà, Phys. Rev. D 107, 034020 (2023).
  101. G. Huang, J. Zhao, and P. Zhuang, Phys. Rev. D 103, 054014 (2021).
  102. D. Zwanziger, Nucl. Phys. B518, 237 (1998).
  103. J. Greensite, An Introduction to the Confinement Problem (Springer, New York, 2011).
  104. S. J. Brodsky, M. Melles, and J. Rathsman, Phys. Rev. D 60, 096006 (1999).
  105. Y. Schroder, Phys. Lett. B 447, 321 (1999).
  106. C. Anzai, M. Prausa, A. V. Smirnov, V. A. Smirnov, and M. Steinhauser, Phys. Rev. D 88, 054030 (2013).
  107. R. N. Lee, A. V. Smirnov, V. A. Smirnov, and M. Steinhauser, Phys. Rev. D 94, 054029 (2016).
  108. A. S. Kronfeld, in Proceedings of the 40th International Symposium on Lattice Field Theory, (2024), arXiv:2401.07380.
  109. N. Brambilla, J. Komijani, A. S. Kronfeld, and A. Vairo (TUMQCD Collaboration), Phys. Rev. D 97, 034503 (2018).
  110. B. Assi, A. Kronfeld, S. Vaiva, and M. Wagman, Baryons and multi-hadron systems in renormalon subtracted non-relativistic QCD (to be published).
  111. T. Collet and M. Steinhauser, Phys. Lett. B 704, 163 (2011).
  112. B. A. Kniehl, A. A. Penin, Y. Schroder, V. A. Smirnov, and M. Steinhauser, Phys. Lett. B 607, 96 (2005).
  113. N. Brambilla, A. Pineda, J. Soto, and A. Vairo, Phys. Rev. D 63, 014023 (2001).
  114. B. Assi, B. A. Kniehl, and J. Soto, Nucl. Phys. B992 116173 (2023).
  115. N. Brambilla, A. Pineda, J. Soto, and A. Vairo, Phys. Rev. D 60, 091502 (1999).
  116. B. A. Kniehl and A. A. Penin, Nucl. Phys. B563, 200 (1999).
  117. J. Toulouse and C. J. Umrigar, J. Chem. Phys. 126, 084102 (2007).
  118. M. N. Anwar, J. Ferretti, F.-K. Guo, E. Santopinto, and B.-S. Zou, Eur. Phys. J. C 78, 647 (2018).
  119. E. A. Hylleraas and A. Ore, Phys. Rev. 71, 493 (1947).
  120. Rashmi and A. Upadhyay, arXiv:2410.00633.
  121. W. Kummer, W. Modritsch, and A. Vairo, Z. Phys. C 72, 653 (1996).
  122. C. Hughes, E. Eichten, and C. T. H. Davies, Phys. Rev. D 97, 054505 (2018).

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Phys. Rev. D 113, 016013– Published 13 January, 2026
  • Received 25 August 2025
  • Accepted 12 December 2025
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