The Cornell potential remains one of the most widely used frameworks for describing quark–antiquark bound states, as it naturally incorporates two essential features of Quantum Chromodynamics (QCD): short-distance Coulombic interaction and long-distance confinement. However, a persistent issue in such models is the ambiguity in choosing the values of key parameters—particularly the strong coupling constant (αs) and the constant shift (c).

In this work, we systematically explore the parameterisation space of the Cornell potential and establish quantitative constraints on αs and c within a QCD-inspired potential model. The goal is to identify the conditions under which the linear confinement term can be treated consistently as a perturbation.

A central aspect of the analysis is the role of perturbation theory. Since the Cornell potential contains both Coulombic and linear terms, the choice of which part acts as the “parent” Hamiltonian is non-trivial. Using the Dalgarno method, we examine the case where the Coulombic term is taken as the parent and the linear term as the perturbation, deriving analytic wavefunctions that retain physical interpretability.

To ensure the validity of this perturbative approach, two key constraints are imposed:

First, an expectation-value condition based on the physical size of the bound state. The average radius ⟨r⟩ must remain smaller than the critical distance r₀ at which the potential vanishes. This ensures that the Coulombic interaction dominates within the relevant spatial region.

Second, a convergence condition on the perturbation series. The perturbative expansion must converge sufficiently fast, which imposes bounds on αs and the confinement parameter. This analysis reveals that the allowed parameter range is not arbitrary but tightly constrained by both physical and mathematical consistency.

The results show that a consistent parameterisation space exists within:
- 0.20 ≤ αs ≤ 0.64  
- −1.2 ≤ c ≤ −0.66  

Within this domain, the perturbative treatment remains valid across heavy–light meson systems such as D, Ds, B, Bs, and Bc. Notably, the allowed parameter region varies across systems, reflecting the role of reduced mass and quark composition.

An important observation is that the parameter space for the Bc system is significantly broader than for other mesons, indicating richer dynamics due to the presence of two heavy quarks. This suggests that Bc mesons can serve as a sensitive testing ground for potential models.

Overall, this work provides a structured and physically grounded framework for selecting parameters in QCD potential models. By constraining αs and c through both expectation-value and convergence criteria, it establishes a reliable foundation for subsequent studies of meson spectroscopy, decay properties, and dynamical modeling.

Journal link: https://doi.org/10.1140/epjc/s10052-022-11061-x (European Physical Journal C)