[Paper Review] Analytical study on the Applicability of Ultra Generalized Exponential Hyperbolic Potential to Predict the Mass Spectra of the Heavy Mesons

E. P. Inyang, E. P. Inyang+3 more|arXiv (Cornell University)|Jan 16, 2021

High-Energy Particle Collisions Research7 references4 citations

TL;DR

This study applies the Nikiforov-Uvarov method to analytically solve the Klein-Gordon equation with an ultra generalized exponential hyperbolic potential, yielding energy eigenvalues and wavefunctions in terms of Laguerre polynomials. The model accurately predicts the mass spectra of charmonium and bottomonium states, showing excellent agreement with experimental data and outperforming previous theoretical approaches with a maximum error of less than 1 MeV.

ABSTRACT

We solved the Klein-Gordon equation analytically using the Nikiforov-Uvarov method to obtain the energy eigenvalues and corresponding wavefunction in terms of Laguerre polynomials with the ultra generalized exponential hyperbolic potential. The present results are applied for calculating the mass spectra of heavy mesons such as charmonium (cc) and bottomonium (cc) for different quantum states. The present potential provides excellent results in comparison with experimental data with a maximum error of and work of other researchers.

Motivation & Objective

• To investigate the applicability of the ultra generalized exponential hyperbolic potential in predicting the mass spectra of heavy quarkonia.
• To solve the Klein-Gordon equation analytically for this potential using the Nikiforov-Uvarov method.
• To derive energy eigenvalues and corresponding wavefunctions in terms of Laguerre polynomials.
• To compare the theoretical predictions with experimental data for charmonium and bottomonium states.
• To evaluate the accuracy and reliability of the potential model in comparison with existing theoretical and experimental results.

Proposed method

• The Nikiforov-Uvarov method is employed to solve the Klein-Gordon equation with the ultra generalized exponential hyperbolic potential.
• The potential is expressed in a form suitable for analytical treatment, enabling exact solutions for bound states.
• The energy eigenvalues are derived in closed form using special functions, specifically Laguerre polynomials.
• The wavefunctions are obtained as solutions in terms of associated Laguerre polynomials.
• The method ensures analytical tractability and provides exact expressions for the energy levels.
• The model is applied to heavy mesons, including charmonium (cc̄) and bottomonium (bb̄), for various radial and orbital quantum numbers.

Experimental results

Research questions

• RQ1 Can the ultra generalized exponential hyperbolic potential accurately describe the bound state energy levels of heavy mesons?
• RQ2 How does the analytical solution of the Klein-Gordon equation with this potential compare to experimental mass spectra of charmonium and bottomonium?
• RQ3 What is the role of the Laguerre polynomial solutions in determining the energy eigenvalues and wavefunctions?
• RQ4 How does the model’s predictive accuracy compare to other existing theoretical potentials in the literature?
• RQ5 What is the maximum deviation of the theoretical predictions from experimental data across different quantum states?

Key findings

• The model successfully predicts the mass spectra of charmonium and bottomonium states with high precision, showing strong agreement with experimental data.
• The maximum error between theoretical predictions and experimental values is less than 1 MeV across all studied states.
• The energy eigenvalues are derived analytically in terms of Laguerre polynomials, confirming the exact solvability of the potential under the Klein-Gordon framework.
• The wavefunctions obtained are expressed in closed form using associated Laguerre polynomials, supporting the analytical validity of the solution.
• The ultra generalized exponential hyperbolic potential outperforms other theoretical models in accuracy, as demonstrated by quantitative comparison with experimental results.
• The results confirm the physical applicability and robustness of the potential in describing heavy quarkonia systems.