Relativistic correction on bottomia within the gaussian basis function method

Authors

  • Arezu Jahanshir
    Department of Physics and Engineering Sciences, Buein Zahra Technical University, Qazvin 34518-66391, Iran
  • Jalil Naji
    Department of Physics, Ilam University, Ilam 69391-77111, Iran

Keywords:

Bottomia, Bound state, Hyperbolic potential

Abstract

In this theoretical research to describe the latest experimental results from the Large Hadron Collider, Belle II, and heavy-ion collisions obtained in high-energy hadronic physics, we include relativistic corrections to improve predictions of the mass spectra of Bottomia resonance states using the Gaussian basis function within the tanh-shaped hyperbolic plus a linear confinement potential in the framework of nonrelativistic and relativistic quantum mechanics under the Schrödinger equation. The relativistic effects of bound states in high energy physics must be described within the framework of approximations or perturbation methods and specific relativistic equations, but in this paper, we provided a mathematically relativistic correction on the bound states mass spectrum; and considering the applied relativistic corrections to the Schrödinger equation using quantum mechanics and quantum field theory principles and high energy approximation. Also, we have helped predict and solve one of the most critical issues of resonance states in particle physics. Current theoretical work is focused on studying this problem for the mass spectra of Bottomia within the refined hybrid potential and based on the modified Schrödinger equation. The mass spectra results agree closely with the experimental and other theoretical data.

Dimensions

[1] J. Yoh, “The discovery of the b quark at Fermilab in 1977: The experiment coordinator’s story”, in Proceedings of AIP, AIP, New York, United States of America, 1998, pp. 29–42. https://doi.org/10.1063/1.55114. DOI: https://doi.org/10.1063/1.55114

[2] D. M. Asner, “Belle II executive summary”, arXiv 2203.10203v2 [hep-ex] (2022). https://doi.org/10.48550/arXiv.2203.10203.

[3] N. Brambilla, S. Eidelman, C. Hanhart & A. Vairo, “The XYZ states: experimental and theoretical status and perspectives”, Physics Reports 873 (2020) 154. https://doi.org/10.1016/j.physrep.2020.05.001. DOI: https://doi.org/10.1016/j.physrep.2020.05.001

[4] C. Bokade & A. Bhaghyesh, “Predictions for bottomonia from a relativistic screened potential model”, arXiv 2501.03147v1 [hep-ph] (2025). https://arxiv.org/pdf/2501.03147v1.

[5] Z. Tang, B. Wu, A. Hanlon & A. Rothkopf, “Quarkonium spectroscopy in the quark-gluon plasma”, arXiv 2502.09044v1 [nucl-th] (2025). https://doi.org/10.48550/arXiv.2502.09044.

[6] D. Fujiwara, Rigorous time slicing approach to Feynman path integrals, Springer International Publishing, Berlin, Germany, 2017, pp. 39–164. https://doi.org/10.1007/978-4-431-56553-6. DOI: https://doi.org/10.1007/978-4-431-56553-6_3

[7] J. Struckmeier, “Relativistic generalization of Feynman’s path integral on the basis of extended Lagrangians”, arXiv 2406.06530v1 [quant-ph] (2024). https://doi.org/10.48550/arXiv.2406.06530.

[8] G. Semenoff, Quantum field theory, Springer International Publishing, Berlin, Germany, 2023, pp. 307–355. https://doi.org/10.1007/978-981-99-5410-0. DOI: https://doi.org/10.1007/978-981-99-5410-0

[9] V. Badalov & S. Badalov, “Generalised tanh-shaped hyperbolic potential: Klein–Gordon equation’s bound state solution”, Communications in Theoretical Physics 75 (2023) 075003. https://doi.org/10.1088/1572-9494/acd441. DOI: https://doi.org/10.1088/1572-9494/acd441

[10] V. Kher, R. Chaturvedi, N. Devlani & A. K. Rai, “Bottomonia spectroscopy using Coulomb plus linear (Cornell) potential”, European Physical Journal Plus 137 (2022) 357. https://doi.org/10.1140/epjp/s13360-022-02538-5. DOI: https://doi.org/10.1140/epjp/s13360-022-02538-5

[11] R. L. Workman, “Particle Data Group”, Progress of Theoretical and Experimental Physics 2022 (2022) 083C01. https://doi.org/10.1093/ptep/ptac097. DOI: https://doi.org/10.1093/ptep/ptac097

[12] M. Dienykhan, G. Efimov, G. Ganbolds & A. N. Sissakian, Oscillator representation in quantum physics, Springer International Publishing, Berlin, Germany, 1995, pp. 196–225. https://doi.org/10.1007/978-3-540-49186-6. DOI: https://doi.org/10.1007/978-3-540-49186-6

[13] W. Greiner, S. Schramm & W. Stein, Quantum chromodynamics, Springer International Publishing, Berlin, Germany, 2007, pp. 230–441. https://doi.org/10.1007/978-3-540-48535-3. DOI: https://doi.org/10.1007/978-3-540-48535-3

[14] R. Gould, Quantum electrodynamics: Electromagnetic processes, Springer International Publishing, Berlin, Germany, 2020, pp. 75–130. https://doi.org/10.1515/9780691215846. DOI: https://doi.org/10.2307/j.ctv131bv37

[15] S. Albeverio, R. Høegh-Krohn & S. Mazzucchi, Mathematical theory of Feynman path integrals, Springer International Publishing, Berlin, Germany, 2008, pp. 19–51. https://doi.org/10.1007/978-3-540-76956-9. DOI: https://doi.org/10.1007/978-3-540-76956-9

[16] E. Copson, Asymptotic expansions: The saddle-point method, Cambridge University Press, Cambridge, United Kingdom, 2009, pp. 63–99. https://doi.org/10.1017/CBO9780511526121. DOI: https://doi.org/10.1017/CBO9780511526121

[17] K. Barley, J. Vega-Guzmán, A. Ruffing & A. D. Alhaidari, “Discovery of the relativistic Schrödinger equation”, Journal Uspekhi Fizicheskikh Nauk Russian Academy of Sciences 65 (2022) 90. https://doi.org/10.3367/ufne.2021.06.039000. DOI: https://doi.org/10.3367/UFNe.2021.06.039000

[18] W. Kutzelnigg, “Theory of the expansion of wave functions in a Gaussian basis”, Quantum Chemistry Journal 51 (1994) 447. https://doi.org/10.1002/qua.560510612. DOI: https://doi.org/10.1002/qua.560510612

[19] M. Rushka & J. Freericks, “A completely algebraic solution of the simple harmonic oscillator”, American Journal of Physics 88 (2019) 976. https://doi.org/10.1119/10.0001702. DOI: https://doi.org/10.1119/10.0001702

[20] J. Kelley & J. Leventhal, “Ladder operators for the harmonic oscillator”, in Problems in classical and quantum mechanics, Springer International Publishing, Berlin, Germany, 2017, pp. 149–188. https://doi.org/10.1007/978-3-319-46664-4_7. DOI: https://doi.org/10.1007/978-3-319-46664-4_7

[21] D. H. Lehmer, “On the maxima and minima of Bernoulli polynomials”, American Mathematical Monthly 47 (1940) 533. https://doi.org/10.1080/00029890.1940.11991015. DOI: https://doi.org/10.1080/00029890.1940.11991015

[22] Z. Sun & H. Pan, “Identities concerning Bernoulli and Euler polynomials”, arXiv:math/0409035 [math.NT] (2004). https://doi.org/10.48550/arXiv.math/0409035.

[23] A. Ahmadov, K. Abasova & M. Orucova, “Bound state solution of Schrödinger equation for extended Cornell”, Advances in High Energy Physics 2021 (2021) 1861946. https://doi.org/10.1155/2021/1861946. DOI: https://doi.org/10.1155/2021/1861946

[24] A. Zyla, “Particle Data Group”, Progress of Theoretical and Experimental Physics 2020 (2020) 083C01. https://doi.org/10.1093/ptep/ptaa104. DOI: https://doi.org/10.1093/ptep/ptaa104

[25] V. Badalov, A. I. Ahmadov, E. A. Dadashov & A. N. Gadzhiev, “Dirac equation solution with generalized tanh-shaped hyperbolic potential: Application to charmonium and bottomonia mass spectra”, arXiv2409.15538v3 [hep-ph] (2025). https://doi.org/10.48550/arXiv.2409.15538.

[26] M. Abu-Shady, “Heavy quarkonia and bc* mesons in the Cornell potential with harmonic oscillator potential in the N-dimensional”, International Journal of Applied Mathematics and Theoretical Physics 2 (2016) 16. https://doi.org/10.11648/j.ijamtp.20160202.11.

[27] B. Bukor & J. Tekel, “On quarkonium masses in 3D noncommutative space”, European Physical Journal Plus 138 (2023) 499. https://doi.org/10.1140/epjp/s13360-023-04049-3. DOI: https://doi.org/10.1140/epjp/s13360-023-04049-3

[28] K. Purohit & A. Rai, “Quarkonium spectroscopy of the linear plus modified Yukawa potential”, Physica Scripta 97 (2022) 044002. https://doi.org/10.1088/1402-4896/ac5bc2. DOI: https://doi.org/10.1088/1402-4896/ac5bc2

[29] Q. Li, M. Allosh, G. S. Hassan & A. Upadhyay, “Canonical interpretation of Y (10750) and Y (10860)in the Y family”, European Physical Journal C 80 (2020) 59. https://doi.org/10.1140/epjc/s10052-020-7626-2. DOI: https://doi.org/10.1140/epjc/s10052-020-7626-2

[30] J. Wang & X. Liu, “Identifying a characterized energy level structure of higher charmonium well matched to the peak structures in e+ e− → π + D0D∗ ”, Physics Letters B 849 (2024) 138456. https://doi.org/10.48550/arXiv.2306.14695. DOI: https://doi.org/10.1016/j.physletb.2024.138456

Published

2025-08-18

How to Cite

Relativistic correction on bottomia within the gaussian basis function method. (2025). Journal of the Nigerian Society of Physical Sciences, 7(4), 2987. https://doi.org/10.46481/jnsps.2025.2987

Issue

Volume 7, Issue 4, November 2025 (In Progress)

Section

Physics & Astronomy
Copyright & Licensing

Copyright (c) 2025 Arezu Jahanshir, Jalil Naji (Author)

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

How to Cite

Relativistic correction on bottomia within the gaussian basis function method. (2025). Journal of the Nigerian Society of Physical Sciences, 7(4), 2987. https://doi.org/10.46481/jnsps.2025.2987