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Binding energy of the Tbb tetraquark from lattice QCD with relativistic and nonrelativistic heavy-quark actions

Jakob Hoffmann1 and Stefan Meinel2

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Phys. Rev. D 113, 114516 – Published 15 June, 2026

DOI: https://doi.org/10.1103/j29k-6rx5

Abstract

We present a new determination of the b¯b¯ud (JP=1+, I=0) tetraquark binding energy using lattice quantum chromodynamics (QCD) with domain-wall light quarks and a nonperturbatively tuned three-parameter anisotropic-clover “relativistic” action for the b quarks. We also perform a direct comparison with a reanalysis of data generated in prior work using a lattice-nonrelativistic QCD (NRQCD) action for the b quarks and otherwise identical parameters. Using the new data with relativistic b quarks from seven different ensembles with multiple lattice spacings and pion masses, we perform combined chiral and continuum extrapolations and obtain (mTbb−mB−mB*)RHQ=(−76±23)  MeV. For the NRQCD data from five ensembles, we perform chiral-only extrapolations and obtain (mTbb−mB−mB*)NRQCD=(−74±17±10)  MeV. The lower magnitude of the results obtained here, compared to the original analysis in [Phys. Rev. D 100, 014503 (2019)], is due to the use of the symmetric parts of the correlation matrices with local four-quark operators only.

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References (121)

  1. J. P. Ader, J. M. Richard, and P. Taxil, Do narrow heavy multiquark states exist?, Phys. Rev. D 25, 2370 (1982).
  2. J. Carlson, L. Heller, and J. A. Tjon, Stability of dimesons, Phys. Rev. D 37, 744 (1988).
  3. A. V. Manohar and M. B. Wise, Exotic QQq¯q¯ states in QCD, Nucl. Phys. B399, 17 (1993).
  4. B. Silvestre-Brac and C. Semay, Systematics of L=0 q2q¯2 systems, Z. Phys. C 57, 273 (1993).
  5. D. M. Brink and F. Stancu, Tetraquarks with heavy flavors, Phys. Rev. D 57, 6778 (1998).
  6. J. Vijande, F. Fernandez, A. Valcarce, and B. Silvestre-Brac, Tetraquarks in a chiral constituent quark model, Eur. Phys. J. A 19, 383 (2004).
  7. D. Janc and M. Rosina, The Tcc=DD* molecular state, Few-Body Syst. 35, 175 (2004).
  8. J. Vijande, A. Valcarce, and K. Tsushima, Dynamical study of of QQu¯d¯ mesons, Phys. Rev. D 74, 054018 (2006).
  9. F. S. Navarra, M. Nielsen, and S. H. Lee, QCD sum rules study of QQu¯d¯ mesons, Phys. Lett. B 649, 166 (2007).
  10. D. Ebert, R. N. Faustov, V. O. Galkin, and W. Lucha, Masses of tetraquarks with two heavy quarks in the relativistic quark model, Phys. Rev. D 76, 114015 (2007).
  11. M. Zhang, H. X. Zhang, and Z. Y. Zhang, QQq¯q¯ four-quark bound states in chiral SU(3) quark model, Commun. Theor. Phys. 50, 437 (2008).
  12. S. H. Lee and S. Yasui, Stable multiquark states with heavy quarks in a diquark model, Eur. Phys. J. C 64, 283 (2009).
  13. P. Bicudo and M. Wagner, Lattice QCD signal for a bottom-bottom tetraquark, Phys. Rev. D 87, 114511 (2013).
  14. Z. S. Brown and K. Orginos, Tetraquark bound states in the heavy-light heavy-light system, Phys. Rev. D 86, 114506 (2012).
  15. P. Bicudo, K. Cichy, A. Peters, and M. Wagner, BB interactions with static bottom quarks from Lattice QCD, Phys. Rev. D 93, 034501 (2016).
  16. P. Bicudo, K. Cichy, A. Peters, B. Wagenbach, and M. Wagner, Evidence for the existence of udb¯b¯ and the non-existence of ssb¯b¯ and ccb¯b¯ tetraquarks from lattice QCD, Phys. Rev. D 92, 014507 (2015).
  17. P. Bicudo, J. Scheunert, and M. Wagner, Including heavy spin effects in the prediction of a b¯b¯ud tetraquark with lattice QCD potentials, Phys. Rev. D 95, 034502 (2017).
  18. A. Francis, R. J. Hudspith, R. Lewis, and K. Maltman, Lattice prediction for deeply bound doubly heavy tetraquarks, Phys. Rev. Lett. 118, 142001 (2017).
  19. M. Karliner and J. L. Rosner, Discovery of doubly-charmed Ξcc baryon implies a stable (bbu¯d¯) tetraquark, Phys. Rev. Lett. 119, 202001 (2017).
  20. E. J. Eichten and C. Quigg, Heavy-quark symmetry implies stable heavy tetraquark mesons QiQjq¯kq¯l, Phys. Rev. Lett. 119, 202002 (2017).
  21. Z.-G. Wang, Analysis of the axialvector doubly heavy tetraquark states with QCD sum rules, Acta Phys. Pol. B 49, 1781 (2018).
  22. P. Junnarkar, N. Mathur, and M. Padmanath, Study of doubly heavy tetraquarks in lattice QCD, Phys. Rev. D 99, 034507 (2019).
  23. W. Park, S. Noh, and S. H. Lee, Masses of the doubly heavy tetraquarks in a constituent quark model, Nucl. Phys. A983, 1 (2019).
  24. C. Deng, H. Chen, and J. Ping, Systematical investigation on the stability of doubly heavy tetraquark states, Eur. Phys. J. A 56, 9 (2020).
  25. B. Wang, Z.-W. Liu, and X. Liu, B¯(*)B¯(*) interactions in chiral effective field theory, Phys. Rev. D 99, 036007 (2019).
  26. L. Leskovec, S. Meinel, M. Pflaumer, and M. Wagner, Lattice QCD investigation of a doubly-bottom b¯b¯ud tetraquark with quantum numbers I(JP)=0(1+), Phys. Rev. D 100, 014503 (2019).
  27. M.-Z. Liu, T.-W. Wu, M. Pavon Valderrama, J.-J. Xie, and L.-S. Geng, Heavy-quark spin and flavor symmetry partners of the X(3872) revisited: What can we learn from the one boson exchange model?, Phys. Rev. D 99, 094018 (2019).
  28. E. Hernández, J. Vijande, A. Valcarce, and J.-M. Richard, Spectroscopy, lifetime and decay modes of the Tbb− tetraquark, Phys. Lett. B 800, 135073 (2020).
  29. R. J. Hudspith, B. Colquhoun, A. Francis, R. Lewis, and K. Maltman, A lattice investigation of exotic tetraquark channels, Phys. Rev. D 102, 114506 (2020).
  30. P. Mohanta and S. Basak, Construction of bbu¯d¯ tetraquark states on lattice with NRQCD bottom and HISQ up and down quarks, Phys. Rev. D 102, 094516 (2020).
  31. Y. Tan, W. Lu, and J. Ping, Systematics of QQq¯q¯ in a chiral constituent quark model, Eur. Phys. J. Plus 135, 716 (2020).
  32. Q.-F. Lü, D.-Y. Chen, and Y.-B. Dong, Masses of doubly heavy tetraquarks TQQ′ in a relativized quark model, Phys. Rev. D 102, 034012 (2020).
  33. E. Braaten, L.-P. He, and A. Mohapatra, Masses of doubly heavy tetraquarks with error bars, Phys. Rev. D 103, 016001 (2021).
  34. R. N. Faustov, V. O. Galkin, and E. M. Savchenko, Heavy tetraquarks in the relativistic quark model, Universe 7, 94 (2021).
  35. T. Guo, J. Li, J. Zhao, and L. He, Mass spectra of doubly heavy tetraquarks in an improved chromomagnetic interaction model, Phys. Rev. D 105, 014021 (2022).
  36. L. R. Dai, E. Oset, A. Feijoo, R. Molina, L. Roca, A. M. Torres, and K. P. Khemchandani, Masses and widths of the exotic molecular B(s)(*)B(s)(*) states, Phys. Rev. D 105, 074017 (2022).
  37. Y. Kim, M. Oka, and K. Suzuki, Doubly heavy tetraquarks in a chiral-diquark picture, Phys. Rev. D 105, 074021 (2022).
  38. X. Chen, F.-L. Wang, Y. Tan, and Y. Yang, Double-heavy tetraquarks with strangeness in the chiral quark model, Chin. Phys. C 47, 023102 (2023).
  39. M. Praszalowicz, Doubly heavy tetraquarks in the chiral quark soliton model, Phys. Rev. D 106, 114005 (2022).
  40. J.-M. Richard, A. Valcarce, and J. Vijande, Doubly-heavy tetraquark bound states and resonances, Nucl. Part. Phys. Proc. 324–329, 64 (2023).
  41. T.-W. Wu and Y.-L. Ma, Doubly heavy tetraquark multiplets as heavy antiquark-diquark symmetry partners of heavy baryons, Phys. Rev. D 107, L071501 (2023).
  42. L. Maiani, A. Pilloni, A. D. Polosa, and V. Riquer, Doubly heavy tetraquarks in the Born-Oppenheimer approximation, Phys. Lett. B 836, 137624 (2023).
  43. Y. Song and D. Jia, Mass spectra of doubly heavy tetraquarks in diquark−antidiquark picture, Commun. Theor. Phys. 75, 055201 (2023).
  44. X.-Y. Liu, W.-X. Zhang, and D. Jia, Doubly heavy tetraquarks: Heavy quark bindings and chromomagnetically mixings, Phys. Rev. D 108, 054019 (2023).
  45. R. J. Hudspith and D. Mohler, Exotic tetraquark states with two b¯ quarks and JP=0+ and 1+ Bs states in a nonperturbatively tuned lattice NRQCD setup, Phys. Rev. D 107, 114510 (2023).
  46. T. Aoki, S. Aoki, and T. Inoue, Lattice study on a tetraquark state Tbb in the HAL QCD method, Phys. Rev. D 108, 054502 (2023).
  47. L. Meng, Y.-K. Chen, Y. Ma, and S.-L. Zhu, Tetraquark bound states in constituent quark models: Benchmark test calculations, Phys. Rev. D 108, 114016 (2023).
  48. A. Feijoo, L. R. Dai, L. M. Abreu, and E. Oset, Correlation function for the Tbb state: Determination of the binding, scattering lengths, effective ranges, and molecular probabilities, Phys. Rev. D 109, 016014 (2024).
  49. Y.-S. Ren, G.-J. Wang, Z. Yang, and J.-J. Wu, Investigation on the bottom analogs Tbb− of Tcc+, Phys. Rev. D 110, 074007 (2024).
  50. C. Alexandrou, J. Finkenrath, T. Leontiou, S. Meinel, M. Pflaumer, and M. Wagner, b¯b¯ud and b¯b¯us tetraquarks from lattice QCD using symmetric correlation matrices with both local and scattering interpolating operators, Phys. Rev. D 110, 054510 (2024).
  51. B. Colquhoun, A. Francis, R. J. Hudspith, R. Lewis, K. Maltman, and W. G. Parrott, Improved analysis of strong-interaction-stable doubly bottom tetraquarks on the lattice, Phys. Rev. D 110, 094503 (2024).
  52. A. Francis, Lattice perspectives on doubly heavy tetraquarks, Prog. Part. Nucl. Phys. 140, 104143 (2025).
  53. S.-Y. Li, Y.-R. Liu, Z.-L. Man, Z.-G. Si, and J. Wu, Doubly heavy tetraquark states in a mass splitting model, Phys. Rev. D 110, 094044 (2024).
  54. Q. Meng, G.-J. Wang, and M. Oka, Mass spectra of full-heavy and double-heavy tetraquark states in the conventional quark model, Phys. Rev. D 111, 014018 (2025).
  55. M. Berwein, N. Brambilla, A. Mohapatra, and A. Vairo, Hybrids, tetraquarks, pentaquarks, doubly heavy baryons, and quarkonia in Born-Oppenheimer effective theory, Phys. Rev. D 110, 094040 (2024).
  56. J. Hoffer, G. Eichmann, and C. S. Fischer, Structure of open-flavor four-quark states in the charm and bottom region, Phys. Rev. D 111, 054028 (2025).
  57. L.-J. Jiang, C.-S. An, C.-R. Deng, G. Li, and J.-J. Xie, Theoretical study on low-lying hidden-bottom and double-bottom tetraquark states, Eur. Phys. J. C 84, 1214 (2024).
  58. N. Brambilla, A. Mohapatra, T. Scirpa, and A. Vairo, Nature of χc1(3872) and Tcc+(3875), Phys. Rev. Lett. 135, 131902 (2025).
  59. K.-K. Zhang, W.-X. Zhang, and D. Jia, Systematics of doubly heavy strange and nonstrange tetraquarks, Phys. Rev. D 112, 054008 (2025).
  60. S. Prelovsek, E. Ortiz-Pacheco, S. Collins, L. Leskovec, M. Padmanath, and I. Vujmilovic, Doubly heavy tetraquarks from lattice QCD: Incorporating diquark-antidiquark operators and the left-hand cut, Phys. Rev. D 112, 014507 (2025).
  61. M. Nagatsuka and S. Sasaki, Lattice study of scattering phase shifts for DD* and BB* systems using twisted boundary conditions: Search for bound state formation, Phys. Rev. D 112, 114510 (2025).
  62. B. S. Tripathy, N. Mathur, and M. Padmanath, bbu¯d¯ and bsu¯d¯ tetraquarks from lattice QCD using two-meson and diquark-antidiquark variational basis, Phys. Rev. D 111, 114504 (2025).
  63. I. Vujmilovic, S. Collins, L. Leskovec, and S. Prelovsek, Electromagnetic form factors and structure of the Tbb tetraquark from lattice QCD, Phys. Rev. Lett. 136, 161901 (2026).
  64. B. A. Thacker and G. P. Lepage, Heavy quark bound states in lattice QCD, Phys. Rev. D 43, 196 (1991).
  65. G. P. Lepage, L. Magnea, C. Nakhleh, U. Magnea, and K. Hornbostel, Improved nonrelativistic QCD for heavy quark physics, Phys. Rev. D 46, 4052 (1992).
  66. G. P. Lepage and P. B. Mackenzie, On the viability of lattice perturbation theory, Phys. Rev. D 48, 2250 (1993).
  67. N. H. Shakespeare and H. D. Trottier, Tadpole renormalization and relativistic corrections in lattice NRQCD, Phys. Rev. D 58, 034502 (1998).
  68. T. C. Hammant, A. G. Hart, G. M. von Hippel, R. R. Horgan, and C. J. Monahan, Radiative improvement of the lattice NRQCD action using the background field method and application to the hyperfine splitting of quarkonium states, Phys. Rev. Lett. 107, 112002 (2011); 115, 039901(E) (2015).
  69. R. J. Dowdall et al. (HPQCD Collaboration), The Upsilon spectrum and the determination of the lattice spacing from lattice QCD including charm quarks in the sea, Phys. Rev. D 85, 054509 (2012).
  70. A. X. El-Khadra, A. S. Kronfeld, and P. B. Mackenzie, Massive fermions in lattice gauge theory, Phys. Rev. D 55, 3933 (1997).
  71. P. Chen, Heavy quarks on anisotropic lattices: The charmonium spectrum, Phys. Rev. D 64, 034509 (2001).
  72. S. Aoki, Y. Kuramashi, and S.-i. Tominaga, Relativistic heavy quarks on the lattice, Prog. Theor. Phys. 109, 383 (2003).
  73. S. Aoki, Y. Kayaba, and Y. Kuramashi, A perturbative determination of mass dependent O(a) improvement coefficients in a relativistic heavy quark action, Nucl. Phys. B697, 271 (2004).
  74. N. H. Christ, M. Li, and H.-W. Lin, Relativistic heavy quark effective action, Phys. Rev. D 76, 074505 (2007).
  75. H.-W. Lin and N. Christ, Non-perturbatively determined relativistic heavy quark action, Phys. Rev. D 76, 074506 (2007).
  76. Y. Aoki, N. H. Christ, J. M. Flynn, T. Izubuchi, C. Lehner, M. Li, H. Peng, A. Soni, R. S. Van de Water, and O. Witzel (RBC and UKQCD Collaborations), Nonperturbative tuning of an improved relativistic heavy-quark action with application to bottom spectroscopy, Phys. Rev. D 86, 116003 (2012).
  77. S. Meinel, M. Pflaumer, and M. Wagner, Search for b¯b¯us and b¯c¯ud tetraquark bound states using lattice QCD, Phys. Rev. D 106, 034507 (2022).
  78. W. Parrott, B. Colquhoun, A. Francis, R. Hudspith, R. Lewis, and K. Maltman, Quark mass dependence of doubly heavy tetraquark binding, Proc. Sci., LATTICE2024 (2025) 084.
  79. T. Iritani et al., Mirage in temporal correlation functions for baryon-baryon interactions in lattice QCD, J. High Energy Phys. 10 (2016) 101.
  80. T. Yamazaki, K.-i. Ishikawa, and Y. Kuramashi (PACS Collaboration), Comparison of different source calculations in two-nucleon channel at large quark mass, Eur. Phys. J. Web Conf. 175, 05019 (2018).
  81. B. Hörz et al., Two-nucleon S-wave interactions at the SU(3) flavor-symmetric point with mud≃msphys: A first lattice QCD calculation with the stochastic Laplacian Heaviside method, Phys. Rev. C 103, 014003 (2021).
  82. A. Nicholson et al., Toward a resolution of the NN controversy, Proc. Sci., LATTICE2021 (2022) 098.
  83. J. R. Green, Status of two-baryon scattering in lattice QCD, Proc. Sci., CD2024 (2026) 019.
  84. J. Bulava et al. (Baryon Scattering Collaboration), Di-nucleons do not form bound states at heavy pion mass, Phys. Rev. C 113, 024002 (2026).
  85. W. Detmold, A. V. Grebe, D. C. Hackett, M. Illa, R. J. Perry, P. E. Shanahan, and M. L. Wagman (NPLQCD Collaboration), Excited-state uncertainties in lattice-QCD calculations of multi-hadron systems, arXiv:2601.22272.
  86. B. Blossier, M. Della Morte, G. von Hippel, T. Mendes, and R. Sommer, On the generalized eigenvalue method for energies and matrix elements in lattice field theory, J. High Energy Phys. 04 (2009) 094.
  87. Y. Aoki et al. (RBC and UKQCD Collaborations), Continuum limit physics from 2+1 flavor domain wall QCD, Phys. Rev. D 83, 074508 (2011).
  88. T. Blum et al. (RBC and UKQCD Collaborations), Domain wall QCD with physical quark masses, Phys. Rev. D 93, 074505 (2016).
  89. P. A. Boyle, L. Del Debbio, N. Garron, A. Juttner, A. Soni, J. T. Tsang, and O. Witzel (RBC/UKQCD Collaboration), SU(3)-breaking ratios for D(s) and B(s) mesons, arXiv:1812.08791.
  90. D. B. Kaplan, A method for simulating chiral fermions on the lattice, Phys. Lett. B 288, 342 (1992).
  91. V. Furman and Y. Shamir, Axial symmetries in lattice QCD with Kaplan fermions, Nucl. Phys. B439, 54 (1995).
  92. Y. Shamir, Chiral fermions from lattice boundaries, Nucl. Phys. B406, 90 (1993).
  93. R. C. Brower, H. Neff, and K. Orginos, The Möbius domain wall fermion algorithm, Comput. Phys. Commun. 220, 1 (2017).
  94. Y. Iwasaki and T. Yoshie, Renormalization group improved action for SU(3) lattice gauge theory and the string tension, Phys. Lett. B 143B, 449 (1984).
  95. S. Meinel, Status of next-generation Λb→p,Λ,Λc form-factor calculations, Proc. Sci., LATTICE2023 (2024) 275.
  96. S. Meinel and G. Rendon, Λb→Λ*(1520)ℓ+ℓ− form factors from lattice QCD, Phys. Rev. D 103, 074505 (2021).
  97. S. Meinel and G. Rendon, Λb→Λc*(2595,2625)ℓ−ν¯ form factors from lattice QCD, Phys. Rev. D 103, 094516 (2021).
  98. S. Meinel and G. Rendon, Λc→Λ*(1520) form factors from lattice QCD and improved analysis of the Λb→Λ*(1520) and Λb→Λc*(2595,2625) form factors, Phys. Rev. D 105, 054511 (2022).
  99. T. Blum, T. Izubuchi, and E. Shintani, New class of variance-reduction techniques using lattice symmetries, Phys. Rev. D 88, 094503 (2013).
  100. E. Shintani, R. Arthur, T. Blum, T. Izubuchi, C. Jung, and C. Lehner, Covariant approximation averaging, Phys. Rev. D 91, 114511 (2015).
  101. W. E. Caswell and G. P. Lepage, Effective Lagrangians for bound state problems in QED, QCD, and other field theories, Phys. Lett. B 167, 437 (1986).
  102. E. Eichten and B. R. Hill, An effective field theory for the calculation of matrix elements involving heavy quarks, Phys. Lett. B 234, 511 (1990).
  103. E. Eichten and B. R. Hill, Renormalization of heavy—light bilinears and fB for Wilson fermions, Phys. Lett. B 240, 193 (1990).
  104. J. Heitger and R. Sommer (ALPHA Collaboration), Nonperturbative heavy quark effective theory, J. High Energy Phys. 02 (2004) 022.
  105. R. Sommer, Introduction to non-perturbative heavy quark effective theory, in Les Houches Summer School: Session 93: Modern Perspectives in Lattice QCD: Quantum Field Theory and High Performance Computing (2010), pp. 517–590, arXiv:1008.0710.
  106. Z. S. Brown, W. Detmold, S. Meinel, and K. Orginos, Charmed bottom baryon spectroscopy from lattice QCD, Phys. Rev. D 90, 094507 (2014).
  107. S. Meinel, Bottomonium spectrum at order v6 from domain-wall lattice QCD: Precise results for hyperfine splittings, Phys. Rev. D 82, 114502 (2010).
  108. G. P. Lepage, lsqfit: Least-squares data fitting for Python, https://github.com/gplepage/lsqfit (2023).
  109. W. I. Jay and E. T. Neil, Bayesian model averaging for analysis of lattice field theory results, Phys. Rev. D 103, 114502 (2021).
  110. See Supplemental Material at http://link.aps.org/supplemental/10.1103/j29k-6rx5 for files containing, for each ensemble, the B, B*, and Tbb energies in lattice units, the B*−B hyperfine splitting in MeV, and the Tbb binding energy in MeV.
  111. B. Blok, J. G. Korner, D. Pirjol, and J. C. Rojas, Spectator effects in the heavy quark effective theory, Nucl. Phys. B496, 358 (1997).
  112. E. E. Jenkins, Heavy meson masses in chiral perturbation theory with heavy quark symmetry, Nucl. Phys. B412, 181 (1994).
  113. Y. Aoki et al. (Flavour Lattice Averaging Group (FLAG), FLAG review 2024, Phys. Rev. D 113, 014508 (2026).
  114. W. Detmold, C. J. D. Lin, and S. Meinel, Axial couplings and strong decay widths of heavy hadrons, Phys. Rev. Lett. 108, 172003 (2012).
  115. F. Bernardoni, J. Bulava, M. Donnellan, and R. Sommer (ALPHA Collaboration), Precision lattice QCD computation of the B*Bπ coupling, Phys. Lett. B 740, 278 (2015).
  116. J. M. Flynn, P. Fritzsch, T. Kawanai, C. Lehner, B. Samways, C. T. Sachrajda, R. S. Van de Water, and O. Witzel (RBC and UKQCD Collaborations), The B*Bπ Coupling Using Relativistic Heavy Quarks, Phys. Rev. D 93, 014510 (2016).
  117. A. Gérardin, J. Heitger, S. Kuberski, H. Simma, and R. Sommer (ALPHA Collaboration), Precision B*Bπ coupling from three-flavor lattice QCD, Proc. Sci., LATTICE2021 (2022) 540.
  118. S. Navas et al. (Particle Data Group), Review of particle physics, Phys. Rev. D 110, 030001 (2024).
  119. P. Boyle, G. Cossu, A. Yamaguchi, and A. Portelli, Grid: A next generation data parallel C++ QCD library, Proc. Sci., LATTICE2015 (2016) 023.
  120. C. Lehner, M. Bruno, D. Richtmann, M. Schlemmer, R. Lehner, D. Knüttel, T. Wurm, L. Jin, S. Bürger, A. Hackl, and A. Klein, Grid Python toolkit (GPT), 2024-10, 10.5281/zenodo.14017415 (2024).
  121. A. Gray, I. Allison, C. T. H. Davies, E. Dalgic, G. P. Lepage, J. Shigemitsu, and M. Wingate, The Upsilon spectrum and m(b) from full lattice QCD, Phys. Rev. D 72, 094507 (2005).

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Phys. Rev. D 113, 114516– Published 15 June, 2026
  • Received 19 March 2026
  • Accepted 26 May 2026
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