^{1}

^{1}

^{3}.

In the present work, the mass spectra of the bound states of heavy quarks

The study of quarkonium systems provides a good understating of the quantitative description of quantum chromodynamics (QCD) theory, the standard model and particle physics [

In the present work, an interaction potential in the quark-antiquark bound system is taken as a general polynomial to get the general eigenvalue solution. In the next step, we chose a specific potential according to the physical properties of the system

The paper is organized as follows: In Section

The Nikiforov-Uvarov (NU) method is based on solving the hypergeometric-type second-order differential equation.

By taking

Also we one can take

Applying the same systematic way for equation (

By comparing equations (

The radial Schrödinger equation of a quark and antiquark system is

A special case of the above potential was chosen to describe the

The third term is a harmonic term and the fourth is an anharmonic term and they are responsible also for quark confinement.

For the above chosen potential, we put

In this section, we will calculate the spectra for the bound states of heavy quarks such as charmonium, bottomonium, and

In the case of charmonium

Mass spectra of charmonium in comparison with other works

Type | present work | [ | [ | [ | [ | [ | EXP [ |
---|---|---|---|---|---|---|---|

1s | 3.0969 | 3.078 | 3.096 | 3.096 | 3.096 | 3.078 | 3.0969 |

1p | - | 3.415 | 3.433 | 3.433 | 3.255 | 3.415 | - |

2s | - | 4.187 | 3.686 | 3.686 | 3.686 | 3.581 | - |

1d | 3.6861 | 3.752 | 3.676 | 3.770 | 3.504 | 3.749 | 3.6861 |

2p | 3.7702 | 4.143 | 3.910 | 4.023 | 3.779 | 3.917 | 3.773 |

3s | - | 5.297 | 3.984 | 4.040 | 4.040 | 4.085 | - |

4s | - | 6.407 | 4.150 | 4.358 | 4.269 | 4.589 | - |

2d | - | - | - | 4.096 | - | 3.078 | - |

4d | 4.160 | - | - | - | - | - | 4.159 |

1g | 4.039 | - | - | - | - | - | 4.039 |

6d | 4.263 | - | - | - | - | - | 4.263 |

Mass spectra of bottomonium in comparison with other works

Type | present work | [ | [ | [ | [ | [ | EXP [ |
---|---|---|---|---|---|---|---|

1s | 9.4600 | 9.510 | 9.460 | 9.460 | 9.460 | 9.510 | 9.4601 |

1p | - | 9.612 | 9.840 | 9.811 | 9.916 | 9.862 | - |

2s | - | 10.627 | 10.023 | 10.023 | 10.023 | 10.038 | - |

1d | - | 10.214 | 10.140 | 10.161 | 9.864 | 10.214 | - |

2p | - | 10.944 | 10.160 | 10.374 | 10.114 | 10.390 | - |

3s | - | 11.726 | 10.280 | 10.355 | 10.355 | 10.655 | - |

2d | 10.02 | - | - | - | - | - | 10.023 |

4s | 10.571 | 12.834 | 10.420 | 10.655 | 10.567 | 11.094 | 10.579 |

3d | 10.358 | - | - | - | - | - | 10.355 |

6S | 11.0198 | - | - | - | - | - | 11.019 |

5d | 10.873 | - | - | - | - | - | 10.876 |

Mass spectra of

Type | present work | [ | [ | [ | [ | EXP [ |
---|---|---|---|---|---|---|

1s | 6.227 | 6.349 | 6.264 | 6.270 | 6.278 | 6.277 |

1p | 6.287 | 6.715 | 6.700 | 6.699 | 6.486 | - |

2s | 6.714 | 6.821 | 6.856 | 6.853 | 6.866 | - |

1d | 6.398 | - | - | - | 6.772 | - |

2p | 6.759 | 7.102 | 7.108 | 7.091 | 6.973 | - |

3s | 7.09 | 7.175 | 7.244 | 7.193 | 7.181 | - |

4s | 7.386 | - | - | - | 7.369 | - |

2d | 6.85 | - | - | - | 7.128 | - |

4d | 7.469 | - | - | - | - | - |

1g | 6.728 | - | - | - | - | - |

The mass spectra of the quarkonia (charmonium, bottomonium, and

The information given in our tables is available for readers in the original references listed in our work.

Hesham Mansour is a Fellow of the Institute of Physics (FInstP).

The authors declare that they have no conflicts of interest.

_{c}meson