Skip to main content
SpringerLink
Log in

Heavy Quark Mesons: Mass Spectrum and Mass Relations

  • Published:
Few-Body Systems Aims and scope Submit manuscript

Abstract

We derive an assumption-free mass relation that connects heavy-quark mesons (\(c\bar{c}\), \(b\bar{b}\) and \(B_c\) mesons) with different quantum numbers, but same orbital angular momentum. The obtained formula is exact, provided the hyperfine mass splitting, \(\delta _{hf}\), and \(\delta _\xi :=M(n^1 L_\ell )/M(n^3 L_\ell )-1\) are determined independently. For completeness, we obtain the heavy-quark mesons spectrum following a Schrödinger wave equation (SWE) approach. The Song-Lin potential is employed along with the corresponding relativistic corrections, namely: spin-spin, spin-orbit and tensor interactions. Our numerical strategy is based upon the Numerov method, which enables us to cast our SWE approach into a matrix eigenvalue problem, for which high numerical accuracy can be achieved. Notably, only a single set of parameters is required to produce the known spectrum of the heavy-quark mesons.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Murray Gell-Mann. The Eightfold Way: A Theory of strong interaction symmetry. 3 1961

  2. Susumu Okubo, Note on unitary symmetry in strong interactions. Prog. Theor. Phys. 27, 949–966 (1962)

    Article  ADS  Google Scholar 

  3. Craig D. Roberts, Empirical consequences of emergent mass. Symmetry 12(9), 1468 (2020)

    Article  Google Scholar 

  4. A. Bazavov et al., Equation of state in (2+1)-flavor QCD. Phys. Rev. D 90, 064503 (2014)

    Article  Google Scholar 

  5. Miguel Angel Escobedo, Effective field theory and lattice QCD approaches for hard probes in QCD matter. PoS, HardProbes 2018, 026 (2018)

    Google Scholar 

  6. Rasmus Larsen, Stefan Meinel, Swagato Mukherjee, Peter Petreczky, Bethe-Salpeter amplitudes of Upsilons. Phys. Rev. D 102, 114508 (2020)

    Article  ADS  Google Scholar 

  7. N. Brambilla et al., Heavy quarkonium: progress, puzzles, and opportunities. Eur. Phys. J. C 71, 1534 (2011)

    Article  ADS  Google Scholar 

  8. Khepani Raya, Minghui Ding, Adnan Bashir, Lei Chang, Craig D. Roberts, Partonic structure of neutral pseudoscalars via two photon transition form factors. Phys. Rev. D 95(7), 074014 (2017)

    Article  ADS  Google Scholar 

  9. Minghui Ding, Fei Gao, Lei Chang, Y.-X. Liu, Roberts, Leading-twist parton distribution amplitudes of S-wave heavy-quarkonia. Phys. Lett. B 753, 330–335 (2016)

    Article  ADS  Google Scholar 

  10. Taichi Kawanai, Shoichi Sasaki, Potential description of charmonium and charmed-strange mesons from lattice QCD. Phys. Rev. D 92(9), 094503 (2015)

    Article  ADS  Google Scholar 

  11. E. Eichten, K. Gottfried, T. Kinoshita, John B. Kogut, K.D. Lane, and Tung-Mow Yan. The Spectrum of Charmonium. Phys. Rev. Lett., 34: 369–372, 1975. [Erratum: Phys.Rev.Lett. 36, 1276 (1976)]

  12. E. Eichten, K. Gottfried, T. Kinoshita, K.D. Lane, and Tung-Mow Yan. Charmonium: The Model. Phys. Rev. D, 17:3090, 1978. [Erratum: Phys.Rev.D 21, 313 (1980)]

  13. E. Eichten, K. Gottfried, T. Kinoshita, K.D. Lane, Tung-Mow. Yan, Charmonium: comparison with experiment. Phys. Rev. D 21, 203 (1980)

    Article  ADS  Google Scholar 

  14. Adedayo O. Adelakun, Abajingin D. Dele, Solution of quantum anharmonic oscillator with quartic perturbation. Adv. Phys. Theories Appl. 27(10), 38–43 (2014)

    Google Scholar 

  15. Jamil Ahmed, Rahila Manzoor, Alfredo Raya, Variational constraints of masses and radii of \(c\bar{c}\)-mesons. Quantum Phys. Lett. 6(08), 99–103 (2017)

    Article  Google Scholar 

  16. Jami Ahmed, Rahila Manzoor, Alfredo Raya, A new variational technique and its applications to heavy quarkonia. Revista Mexicana de Física 67(1), 33–53 (2021)

    MathSciNet  Google Scholar 

  17. Xiao-tong Song, H.-F. Lin, A new potential model for heavy quarkonium. Z. Phys. C 34, 223 (1987)

    Article  Google Scholar 

  18. Mohamed Chabab, Latifa Sanhaji, Confining interquark potentials from nonAbelian gauge theories coupled to dilaton. Int. J. Mod. Phys. A 20, 1863–1866 (2005)

    Article  ADS  Google Scholar 

  19. E. Leader and E. Predazzi (1982) An introduction to gauge theories and the ‘New Physics’. 1

  20. Myron Bander, Theories of quark confinement. Phys. Rept. 75, 205 (1981)

    Article  ADS  Google Scholar 

  21. William H. Press, Saul A. Teukolsky, William T. Vetterling, Brian P. Flannery, Numerical Recipes 3rd Edition: The Art of Scientific Computing, 3rd edn. (Cambridge University Press, USA, 2007)

    MATH  Google Scholar 

  22. Halil Mutuk, Spin averaged mass spectrum of heavy quarkonium via asymptotic iteration method. Can. J. Phys. 97(12), 1342–1348 (2019)

    Article  ADS  MathSciNet  Google Scholar 

  23. Mohandas Pillai, Josh Goglio, Thad Walker, Matrix numerov method for solving schrödingers equation. Am. J. Phys. 80(11), 1017–1019 (2012)

    Article  ADS  Google Scholar 

  24. Yasser Mustafa, Galal Hassan, and Tarek Nahool (2014) Numerical study of heavy mesons spectra using matrix numerovs method. Int. J. New. Hor. Phys, 01

  25. S. Titard, F.J. Yndurain, Rigorous QCD evaluation of spectrum and other properties of heavy q anti-q systems. Phys. Rev. D 51, 6348–6363 (1995)

    Article  ADS  Google Scholar 

  26. Richard F. Lebed, Eric S. Swanson, Quarkonium \(h\) states as arbiters of exoticity. Phys. Rev. D 96(5), 056015 (2017)

    Article  ADS  Google Scholar 

  27. Clara Peset, Antonio Pineda, Jorge Segovia, P-wave heavy quarkonium spectrum with next-to-next-to-next-to-leading logarithmic accuracy. Phys. Rev. D 98(9), 094003 (2018)

    Article  ADS  MathSciNet  Google Scholar 

  28. Lei Chang, Muyang Chen, Xue-qian Li, Y.-X. Liu, K. Raya, Can the hyperfine mass splitting formula in heavy Quarkonia be applied to the \(B_c\) system? Few Body Syst. 62(1), 4 (2021)

    Article  ADS  Google Scholar 

  29. S. Dobbs et al., Precision measurement of the mass of the h(c)(P-1(1)) state of charmonium. Phys. Rev. Lett. 101, 182003 (2008)

    Article  ADS  Google Scholar 

  30. I. Adachi et al., First observation of the \(P\)-wave spin-singlet bottomonium states \(h_b(1P)\) and \(h_b(2P)\). Phys. Rev. Lett. 108, 032001 (2012)

    Article  ADS  Google Scholar 

  31. P.A. Zyla et al., Review of particle physics. PTEP 2020(8), 083C01 (2020)

    Google Scholar 

  32. Henry Lamm, \(P-\)state positronium for precision physics: an ultrafine splitting at \(\alpha ^6\). Phys. Rev. A 96(2), 022515 (2017)

    Article  ADS  Google Scholar 

  33. J. Segovia, C. Albertus, D.R. Entem, F. Fernandez, E. Hernandez, M.A. Perez-Garcia, Semileptonic \(B\) and \(B_{s}\) decays into orbitally excited charmed mesons. Phys. Rev. D 84, 094029 (2011)

    Article  ADS  Google Scholar 

  34. X.-H. Guo, K.-W. Wei, X.-H. Wu, Some mass relations for mesons and baryons in Regge phenomenology. Phys. Rev. D 78, 056005 (2008)

    Article  ADS  Google Scholar 

  35. Wolfgang Lucha and Franz F. Schoberl. Effective potential models for hadrons. In International Summer School for Students on Development in Nuclear Theory and Particle Physics, 12 1995

  36. A. De, Howard Georgi Rujula, S.L. Glashow, Hadron masses in a Gauge theory. Phys. Rev. D 12, 147–162 (1975)

    ADS  Google Scholar 

  37. Pablo G. Ortega, Jorge Segovia, David R. Entem, Francisco Fernandez, Spectroscopy of \(\mathbf{B_c}\) mesons and the possibility of finding exotic \(\mathbf{B_c}\)-like structures. Eur. Phys. J. C 80(3), 223 (2020)

    Article  ADS  Google Scholar 

  38. Rudolf N. Faustov, Dietmar Ebert, and Vladimir O. Galkin. Spectroscopy and Regge trajectories of heavy quarkonia and \(B_c\) mesons. PoS, Baldin-ISHEPP-XXI:058, 2012

  39. W.-J. Deng, H. Liu, L.-C. Gui, X.-H. Zhong, Spectrum and electromagnetic transitions of bottomonium. Phys. Rev. D 95(7), 074002 (2017)

    Article  ADS  Google Scholar 

  40. N.R. Soni, B.R. Joshi, R.P. Shah, H.R. Chauhan, J.N. Pandya, \(Q\bar{Q}\) ( \(Q\in \{b, c\}\) ) spectroscopy using the Cornell potential. Eur. Phys. J. C 78(7), 592 (2018)

    Article  ADS  Google Scholar 

  41. Raghav Chaturvedi, N.R. Soni, J.N. Pandya, and A.K. Rai. (2018) Bottominium spectroscopy in effective field theory formalism. 10

  42. Matthias Neubert, Effective field theory and heavy quark physics. Theor. Adv. Stud. Inst. Elem. Part. Phys.: Phys. D 4, 12 (2005)

    Google Scholar 

  43. Jorge Segovia, Pablo G. Ortega, David R. Entem, Francisco Fernández, Bottomonium spectrum revisited. Phys. Rev. D 93(7), 074027 (2016)

    Article  ADS  Google Scholar 

  44. Nora Brambilla, Antonio Pineda, Joan Soto, Antonio Vairo, Potential NRQCD: An Effective theory for heavy quarkonium. Nucl. Phys. B 566, 275 (2000)

    Article  ADS  Google Scholar 

  45. Xurong Chen, F.-K. Guo, C.D. Roberts, R. Wang, Selected Science Opportunities for the EicC. Few Body Syst. 61(4), 43 (2020)

    Article  ADS  Google Scholar 

  46. M. Yu, Barabanov, Diquark correlations in hadron physics: Origin, impact and evidence. Prog. Part. Nucl. Phys. 116, 103835 (2021)

    Article  Google Scholar 

  47. M.A. Bedolla, J. Ferretti, C.D. Roberts, E. Santopinto, Spectrum of fully-heavy tetraquarks from a diquark+antidiquark perspective. Eur. Phys. J. C 80(11), 1004 (2020)

    Article  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Centre For High Energy Physics, Punjab University, Lahore, 54590, Pakistan

    J. Ahmed & R. Manzoor

  2. School of Physics, Nankai University, 300071, Tianjin, China

    L. Chang & K. Raya

  3. Instituto de Física y Matemáticas, Universidad Michoacana de San Nicolás de Hidalgo. Edificio C-3, Ciudad Universitaria. Fco. J. Míjica s/n, Col. Felícitas del Río, C.P. 58040, Morelia, Michoacan, Mexico

    A. Raya

  4. Centro de Ciencias Exactas, Universidad del Bío-Bío, Avda. Andrés Bello 720, Casilla 447, 3800708, Chillán, Chile

    A. Raya

  5. Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México, Apartado Postal 70-543, C.P. 04510, CDMX, Mexico City, México

    K. Raya

Authors
  1. J. Ahmed
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. R. Manzoor
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. L. Chang
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. A. Raya
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. K. Raya
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to K. Raya.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ahmed, J., Manzoor, R., Chang, L. et al. Heavy Quark Mesons: Mass Spectrum and Mass Relations. Few-Body Syst 62, 39 (2021). https://doi.org/10.1007/s00601-021-01624-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00601-021-01624-1

Search

Navigation