^{1}

^{3}.

In this paper we have solved the nonrelativistic form of the Lippmann-Schwinger equation in the momentum-helicity space by inserting a spin-dependent quark-antiquark potential model numerically. To this end, we have used the momentum-helicity basis states for describing a nonrelativistic reduction of one-gluon exchange potential. Then we have calculated the mass spectrum of the charmonium

During the past years, several models and methodological approaches based on solving the relativistic and nonrelativistic form of the Schrödinger or Lippmann-Schwinger equation have been developed for studying the light and heavy mesons in the coordinate and momentum spaces, respectively.

Recently, the three-dimensional approach based on momentum-helicity basis states for studding the nucleon-nucleon scattering and deuteron state has been developed [

In the heavy-quark

This article is organized as follows. In Section

The nonrelativistic form of the homogenous Lippmann-Schwinger equation for describing the heavy meson bound state is given by

The spin-dependent potential model that we have used in our calculations is sum of the Linear and a simple nonrelativistic reduction of an effective one-gluon exchange potential without retardation. This potential in the coordinate space is given in terms of [

For numerical calculations as a first step we have used the Gaussian quadrature grid points to discretize the momentum and the angle variables. The integration interval for the momentum is covered by two different hyperbolic and linear mappings of the Gauss-Legendre points from the interval

The parameters of the potential model which are shown in Table

Parameters of the model.

| 1.222 |

| 0.3154 |

| 1.269 |

| 0.2863 |

| 10 |

Comparison of the obtained charmonium mass spectrum with the experimental data and another work.

| Candidate | Exp. [ | Ref. [ | Mass [MeV] |
---|---|---|---|---|

| | | 2980 | 2980.4 |

| | | 3097 | 3096.9 |

| | | 3527 | 3526.2 |

| | | 3430 | 3397.4 |

| | | 3503 | 3503.5 |

| | | 3674 | 3683.1 |

| | | 3765 | 3760.8 |

| | | 3855 | 3850.6 |

| | | 4291 | 4285.4 |

Percent of each partial wave in mixed charmonium states.

| | | |
---|---|---|---|

| | | 0.07 |

| | | 0.10 |

| | | 0.12 |

| | | 0.12 |

As a test of our numerical calculations we have shown convergence of the results as a function of number of grid points

The calculated charmonium mass spectrum as function of the number of grid points

| | | | | | | | | | | |
---|---|---|---|---|---|---|---|---|---|---|---|

100 | 100 | | | 3096.942 | 3526.244 | 3397.517 | 3503.552 | 368.3357 | 3760.778 | 3850.573 | 4285.415 |

100 | 100 | | | 3096.951 | 3526.226 | 3397.449 | 3503.541 | 3683.153 | 3760.782 | 3850.563 | 4285.410 |

100 | 100 | | | 3096.954 | 3526.221 | 3397.434 | 3503.539 | 3683.095 | 3760.784 | 3850.561 | 4285.409 |

100 | 100 | | | 3096.954 | 3526.219 | 3397.431 | 3503.538 | 3683.080 | 3760.784 | 3850.560 | 4285.409 |

100 | 100 | | | 3096.954 | 3526.219 | 3397.430 | 3503.538 | 3683.075 | 3760.784 | 3850.560 | 4285.409 |

100 | 100 | | | 3.096954 | 3526.219 | 3397.430 | 3503.538 | 3683.074 | 3760.784 | 3850.560 | 4285.409 |

100 | 100 | | | 3.096954 | 3526.219 | 3397.430 | 3503.538 | 3683.074 | 3760.784 | 3850.560 | 4285.409 |

| |||||||||||

100 | 60 | | | 3097.131 | 3525.746 | 3397.449 | 3394.900 | 3503.182 | 3677.378 | 3850.232 | 4284.965 |

100 | 80 | | | 3096.961 | 3526.205 | 3397.376 | 3503.530 | 3682.920 | 3760.787 | 3850.553 | 4285.405 |

100 | 100 | | | 3096.954 | 3526.219 | 3397.431 | 3503.541 | 3683.538 | 3760.080 | 3850.560 | 4285.409 |

100 | 120 | | | 3096.954 | 3526.219 | 3397.431 | 3503.541 | 3503.538 | 3683.080 | 3850.560 | 4285.409 |

| |||||||||||

50 | 100 | | | 3096.954 | 3526.219 | 3397.429 | 3503.538 | 3683.080 | 3760.784 | 3850.558 | 4285.409 |

80 | 100 | | | 3096.954 | 3526.219 | 3397.430 | 3503.538 | 3683.080 | 3760.784 | 3850.560 | 4285.409 |

100 | 100 | | | 3096.954 | 3526.219 | 3397.431 | 3503.541 | 3683.080 | 3760.784 | 3850.560 | 4285.409 |

In this paper we have extended an approach based on momentum-helicity basis states for calculation of mass spectrum of heavy mesons by solving nonrelativistic form of the Lippmann-Schwinger equation. As an application we have used this approach to obtain the mass spectrum of charmonium. The advantage of working with helicity states is that states are the eigenstates of the helicity operator appearing in the quark-antiquark potential. Thus, using the helicity representation is less complicated than using the spin representation with a fixed quantization axis for representation of spin-dependent potentials. This work is the first step toward studying single, double, and triple heavy-flavor baryons in the framework of the nonrelativistic quark model by formulation of the Faddeev equation in the 3D momentum-helicity representation. Furthermore, we can apply this formalism straightforwardly for investigation of heavy pentaquark systems, which can be considered as two-body (heavy meson and baryon) systems with meson-nucleon potentials which is underway.

The three-dimensional Fourier transformation of the potential

The author declares that there is no conflict of interests regarding the publication of this paper.