Benchmarking the Born-Oppenheimer approximation with the Gaussian expansion method for doubly heavy hadrons

The Born-Oppenheimer approximation is widely used to investigate the properties of hydrogen-like systems and doubly heavy hadrons. However, the extent to which this approximation reliably captures the true features of such systems 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 in 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 non-adiabatic corrections. This comparative study provides deeper insight into the structure of doubly heavy hadrons and clarifies the applicability and limitations of the Born-Oppenheimer treatment in these systems.

💡 Research Summary

The paper presents a systematic benchmark of the Born‑Oppenheimer (BO) approximation against the Gaussian Expansion Method (GEM) for systems containing two heavy quarks (doubly heavy hadrons). The authors first validate their approach on well‑understood molecular systems – the hydrogen molecular ion (H₂⁺) and the hydrogen molecule (H₂). By varying the mass ratio m_heavy/m_light, they demonstrate that the BO approximation reproduces GEM results only when the mass hierarchy is large. Two types of trial wave functions are examined within the BO framework: Slater‑type functions (STFs) and Gaussian‑type functions (GTFs). For large mass ratios (e.g., the physical proton‑electron ratio), both BO‑STF and BO‑GTF give binding energies essentially identical to GEM. As the mass ratio approaches unity, BO‑STF tends to over‑bind (larger binding energies) because STFs capture the short‑range cusp of the wave function, while BO‑GTF under‑binds due to the rapid Gaussian fall‑off at long distances. This behavior is consistent with the theoretical condition that BO validity scales as (m_light/m_heavy)¹⁄⁴.

Having established the molecular benchmark, the authors turn to QCD‑based doubly heavy systems: doubly heavy baryons (QQq) and doubly heavy tetraquarks (QQ \bar q \bar q). They employ a standard constituent‑quark Hamiltonian containing a color‑Coulomb term, a linear confining term, and a hyperfine contact interaction. The strong coupling α_s(q) includes one‑loop running, the string tension is fixed to b = 0.15 GeV², and a constant C is introduced for overall mass shifts. Within the BO approximation, the heavy QQ pair is treated as static, and the light quark(s) are solved for a fixed QQ separation R, yielding an effective potential ε(R). This ε(R) is then inserted into a Schrödinger equation for the relative motion of the heavy pair. The same procedure is performed with both STF and GTF trial functions for the light‑quark wave function.

In parallel, the authors solve the full few‑body problem with GEM, expanding the total wave function in a large Gaussian basis that fully respects the symmetries of the three‑ and four‑body systems. GEM automatically includes all non‑adiabatic couplings between heavy and light degrees of freedom, providing a high‑precision reference.

The comparative results reveal a clear pattern. For charm‑quark masses (≈1.3 GeV) the BO predictions (both STF and GTF) lie within ~10 MeV of the GEM results, indicating that the BO approximation is reasonably reliable for doubly charm hadrons. When the heavy quark mass is increased to the bottom‑quark scale (≈4.7 GeV), the discrepancy grows: BO‑STF yields binding energies 15–20 MeV higher than GEM, while BO‑GTF gives values 10–15 MeV lower. The authors attribute the upward shift of BO‑STF to its superior description of the short‑range behavior of the light‑quark wave function, which artificially deepens the effective potential. Conversely, the downward shift of BO‑GTF is traced to the neglect of non‑adiabatic corrections; the Gaussian trial functions suppress the long‑range tail and thus underestimate the contribution of light‑quark dynamics to the heavy‑pair binding.

To obtain final mass predictions, the authors add a chromomagnetic interaction (CMI) term, modeled as a color‑spin operator with phenomenological coefficients C_ij, to the BO and GEM energies. By fitting the CMI parameters to known hadron masses, they reproduce the experimental values of the doubly charm baryon Ξ_cc and the doubly charm tetraquark T_cc⁺ within ~20 MeV, confirming the consistency of their framework.

The paper concludes that the BO approximation is valid for doubly heavy hadrons only when the heavy‑light mass ratio is sufficiently large and when non‑adiabatic effects are small. The choice of trial wave function critically influences the sign and magnitude of the BO error: Slater‑type functions tend to over‑estimate binding, while Gaussian‑type functions tend to underestimate it. For precision spectroscopy of doubly heavy systems, especially those involving bottom quarks, fully dynamical methods such as GEM—or BO supplemented with explicit non‑adiabatic corrections—are necessary. This work delineates the applicability domain of the BO approximation in QCD and provides a quantitative benchmark that will guide future theoretical studies of exotic hadrons.