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

TL;DR

This study evaluates the accuracy of the Born-Oppenheimer approximation in potential models for doubly heavy hadrons by comparing it with Gaussian expansion method results, revealing its dependence on trial wave functions and heavy-quark mass.

Contribution

It provides a comprehensive assessment of the Born-Oppenheimer approximation's validity and limitations in modeling doubly heavy hadrons, considering different trial wave functions.

Findings

Born-Oppenheimer results are close to Gaussian expansion for small heavy-quark masses.

Slater-type functions tend to overestimate binding energy at larger quark masses.

Gaussian-type functions underestimate binding energy due to neglecting non-adiabatic corrections.

Abstract

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 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 non-adiabatic 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.