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Theoretical prediction of masses and decay constants of bottomonium and bottom-charm mesons with spin dependent Cornell potential

This paper employs a spin-dependent Cornell potential within first-order perturbation theory to accurately predict the mass spectra and decay properties of bottomonium and bottom-charm mesons, achieving high agreement with experimental data and demonstrating the framework's effectiveness through the inclusion of QCD running coupling effects.

Original authors: E. Omugbe, E. P Inyang, E. M. Khokha

Published 2026-10-26

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CC BY 4.0

Original authors: E. Omugbe, E. P Inyang, E. M. Khokha

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Deep within the fabric of the universe, matter is built from tiny, fundamental particles called quarks. These particles are never found alone; they are forever bound together by the strongest force in nature, the strong interaction, to form larger structures known as hadrons. Among these, mesons are particularly interesting because they are made of just two pieces: a quark and its antimatter partner, an antiquark. Understanding how these pairs stick together and how heavy they are is a central puzzle for physicists. The difficulty lies in the fact that the rules governing this binding are incredibly complex at the low energies where these particles exist, making it hard to predict their properties with simple math. To navigate this complexity, scientists often rely on models that act like maps, using effective forces to describe how quarks pull and push against each other. One of the most trusted maps is the Cornell potential, a mathematical description that combines a short-range pull, similar to how electric charges attract, with a long-range stretching force that prevents the quarks from ever escaping each other.

A team of researchers has now refined this map to include a crucial detail that was often left out: the way the spins of the quarks interact with each other. In this new study, the scientists applied a method called perturbation theory to a version of the Cornell potential that accounts for these spin-dependent forces. Instead of trying to solve the entire complex problem at once, they treated the spin interactions as a small adjustment to a simpler, solvable system. This allowed them to derive clear, analytical formulas for the energy levels of these meson pairs. They tested their approach by checking if it could reproduce the results of the standard Cornell potential when the spin effects were turned off, finding that their method matched well with other sophisticated numerical techniques used by the scientific community.

Using this refined framework, the team calculated the masses of bottomonium mesons, which are made of a bottom quark and a bottom antiquark, as well as bottom-charm mesons, which consist of a bottom quark and a charm quark. They adjusted the parameters of their model to fit the known experimental masses of fifteen different bottomonium states. The results were strikingly accurate. For the bottomonium states, the calculated masses deviated from the actual experimental values by an average of only 0.15 percent. This level of precision represents an improvement over several previous theoretical predictions that relied on different approximation methods. The masses for the bottom-charm mesons also aligned well with other established theoretical values found in the literature.

Beyond just weighing these particles, the researchers also explored how they decay, or break apart, into other forms of energy. They calculated the decay constants, which are numbers that describe how likely a meson is to transform into other particles, such as pairs of photons or pairs of electrons and positrons. They found that these constants generally decrease as the energy level of the meson increases, a pattern that matches what is seen in other studies. When they included corrections for the way the strong force changes strength at different energy scales, their predictions for the decay rates became even more consistent with experimental data. This sensitivity to the wave function at the center of the meson highlights how the internal structure of these particles dictates their behavior.

The study confirms that treating the spin-dependent forces as a small perturbation on the Cornell potential provides a simple yet powerful way to describe the heavy meson world. By avoiding more complicated approximations that sometimes drift away from reality, this approach offers a unified view where the mass, the internal wave structure, and the decay properties can all be calculated consistently. The findings suggest that this method is a reliable tool for understanding the heavy quark bound states, offering a clearer picture of the fundamental forces that hold matter together. The work does not claim to solve every mystery of the strong force, but it does provide a robust and effective framework that agrees closely with what nature actually shows us in the laboratory.

Technical Summary: Theoretical Prediction of Masses and Decay Constants of Bottomonium and Bottom-Charmed Mesons

Problem Statement
The accurate prediction of heavy meson mass spectra and decay rates remains a central challenge in hadron spectroscopy due to the non-perturbative nature of Quantum Chromodynamics (QCD) at low energies. While phenomenological models utilizing effective potentials, such as the Cornell potential, have been widely employed, standard approaches often neglect significant spin-dependent forces (spin-spin, spin-orbit, and tensor interactions) required for precise mass splitting calculations. Furthermore, recent studies utilizing the spin-dependent Cornell potential have frequently relied on the Pekeris approximation or Taylor expansions of Gaussian functions to achieve analytical solutions, a process the authors note can lead to deviations from experimental data. Additionally, previous works often fail to simultaneously address both mass spectra and decay properties (decay constants and widths) within a unified framework that incorporates QCD running coupling effects.

Methodology
This study employs first-order perturbation theory to solve the bound-state problem for heavy mesons using a spin-dependent Cornell potential. The authors decompose the Hamiltonian into an unperturbed "parent" system and a perturbation:

Unperturbed System: The parent potential consists of the Coulomb-like term ( $-\alpha_s/r$ ) and the specific $1/r$ term ( $A_0/r$ ) appearing in the general potential equation (Eq. 14), which admits exact analytical solutions for energy eigenvalues and wave functions.
Perturbation: The remaining terms of the spin-dependent potential, including the linear confinement term ( $Br$ ), the exponential screening term, and other spin-dependent components, are treated as perturbations.
Calculations:
- Analytical expressions for energy eigenvalues are derived up to the first-order correction.
- Potential parameters ( $B$ , $\alpha_s$ , $\sigma$ ) and quark masses are determined by fitting the analytical mass-spectrum equation to 15 established experimental data points for bottomonium mesons using the NLPSolve function in Maple.
- These fitted parameters are subsequently applied to calculate the mass spectra and decay properties of bottomonium ( $b\bar{b}$ ) and bottom-charm ( $b\bar{c}$ ) mesons.
- Decay constants (vector and pseudoscalar) are calculated using the wave functions at the origin derived from the mass difference between vector and pseudoscalar states, utilizing the Van Royen–Weisskopf formula.
- Radiative corrections are incorporated by calculating the QCD running strong coupling constant, $\alpha_s(\mu)$ , at appropriate renormalization scales (leading and first-order approximations).

Key Contributions

Analytical Framework: The paper provides analytical expressions for energy eigenvalues and wave functions for the spin-dependent Cornell potential without resorting to the Pekeris approximation or Taylor expansions of the Gaussian spin-spin term, which are common in prior literature.
Unified Calculation: The study simultaneously predicts mass spectra, decay constants, and decay widths (di-photon for pseudoscalars and di-leptonic for vectors) for both bottomonium and bottom-charm systems within a single theoretical framework.
Inclusion of Running Coupling: The authors explicitly incorporate the QCD running strong coupling constant into the decay width calculations, demonstrating its impact on improving the description of decay properties.
Validation: The method is validated by showing that in the spin-independent limit, the results reproduce the standard Cornell potential spectrum and align closely with results from the Matrix Numerov method, Artificial Neural Networks, and the Asymptotic Iteration Method.

Results

Mass Spectra: The model yields an average percentage absolute deviation (APAD) of approximately 0.15% for bottomonium states compared to experimental masses. This represents an improvement over several previous theoretical predictions (e.g., those in references [4, 27, 34, 40]) and shows comparable accuracy to recent studies. The calculated masses for bottom-charm mesons align with other theoretical values in the literature.
Decay Constants: The calculated vector and pseudoscalar decay constants exhibit a decreasing trend with increasing principal quantum number. The results generally satisfy the inequality $M_V f_V \geq M_P f_P$ .
Decay Widths:
- For the bottomonium $1^1S_0$ state, the predicted di-photon decay width is 0.3286 eV (leading order) and 0.5199 eV (with radiative corrections), which is consistent with experimental data and other theoretical models.
- For the $2^1S_0$ state, the decay widths are noted to be underestimated but remain comparable to specific references.
- Vector di-leptonic decay widths for the $1^3S_1$ state are close to experimental observations, while the $2^3S_1$ state shows underestimation relative to other theoretical values.
Parameter Fitting: The study determines specific potential parameters for bottomonium and bottom-charm systems, including quark masses ( $m_b = 4.9148$ GeV, $m_c = 1.80$ GeV) and coupling constants, which are consistent with literature ranges.

Significance and Claims
The authors claim that the perturbative treatment of the spin-dependent Cornell potential offers a simple and effective phenomenological framework for describing the mass spectra and decay properties of heavy and heavy-light mesons. The study emphasizes that:

The approach avoids the limitations of the Pekeris approximation, leading to improved agreement with experimental mass data.
Incorporating the QCD running coupling at appropriate renormalization scales significantly improves the description of decay properties, highlighting the sensitivity of these observables to the wave function at the origin and the effective coupling strength.
The framework provides a unified treatment where mass spectra, wave functions, and decay properties are calculated consistently.
The results support the applicability of this model to bottomonium and bottom-charm mesons, extending the treatment of decay properties beyond previous approximation methods.

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