Research Article  Open Access
M. Radin, "Charmonium Mass Spectrum with SpinDependent Interaction in MomentumHelicity Space", Advances in High Energy Physics, vol. 2017, Article ID 8321513, 5 pages, 2017. https://doi.org/10.1155/2017/8321513
Charmonium Mass Spectrum with SpinDependent Interaction in MomentumHelicity Space
Abstract
In this paper we have solved the nonrelativistic form of the LippmannSchwinger equation in the momentumhelicity space by inserting a spindependent quarkantiquark potential model numerically. To this end, we have used the momentumhelicity basis states for describing a nonrelativistic reduction of onegluon exchange potential. Then we have calculated the mass spectrum of the charmonium , and finally we have compared the results with the other theoretical results and experimental data.
1. Introduction
During the past years, several models and methodological approaches based on solving the relativistic and nonrelativistic form of the Schrödinger or LippmannSchwinger equation have been developed for studying the light and heavy mesons in the coordinate and momentum spaces, respectively.
Recently, the threedimensional approach based on momentumhelicity basis states for studding the nucleonnucleon scattering and deuteron state has been developed [1, 2]. We extend this approach to particle physics problems by solving the nonrelativistic form of the LippmannSchwinger equation to obtain the mass spectrum of the heavy mesons using the nonrelativistic quarkantiquark interaction in terms of a linear confinement, a Coulomb, and various spindependent pieces.
In the heavyquark mesons the differences between energy levels are small compared to the particle masses. Hence, the nonrelativistic LippmannSchwinger equation can be used to study their quantum behavior. To this end, we have used the nonrelativistic form of the LippmannSchwinger equation in the momentumhelicity representation to study the charmonium as a heavy meson. For this purpose, we have used a nonrelativistic quarkantiquark potential based on onegluon exchange in the momentumhelicity representation.
This article is organized as follows. In Section 2, the nonrelativistic LippmannSchwinger equation in the momentumhelicity basis states which leads to coupled and uncoupled integral equations for various quantum numbers is presented briefly. In Section 3, a spindependent quarkantiquark potential model is described in the momentumhelicity basis states. The details of the numerical calculations and the results obtained for the charmonium are presented in Section 4. Finally, a summary and an outlook are provided in Section 5.
2. LippmannSchwinger Equation in MomentumHelicity Basis States
The nonrelativistic form of the homogenous LippmannSchwinger equation for describing the heavy meson bound state is given bywhere denotes the quarkantiquark interaction, is mass of the quark or antiquark, and is the meson bound state with the total angular momentum . is projection of the total angular momentum along the quantization axis. The integral form of this equation in the momentumhelicity basis states is written as [3]withwhere is the magnitude of the relative momentum of the quark and antiquark, is the total spin of meson, is the spin projection along the relative momentum, and are the rotation matrices. For an arbitrary total angular momentum , and singlet case of the total spin state, (2) leads to one equation: Also for and triplet case of the total spin state, (2) leads to one equation as follows:For and it is more complicated. For example, for , (2) leads to one equation for channel and two coupled equations for channels and as follows:where is the partial wave component of the wave function which is connected to the momentumhelicity component of the wave function as [3]The inverse relation is written as
3. QuarkAntiquark Potential in MomentumHelicity Basis States
The spindependent potential model that we have used in our calculations is sum of the Linear and a simple nonrelativistic reduction of an effective onegluon exchange potential without retardation. This potential in the coordinate space is given in terms of [4]where is the string tension, is the stronginteraction finestructure constant, is the color factor which is for quarkantiquark and for quarkquark, and are the Pauli matrices, and is the total orbital angular momentum operator. Fourier transformation of this potential to momentum space yields
where is the momentum transfer. The kernels of integral equations have singularity. To overcome this problem we have used the regularized form of linear confining and Coulomb parts of the potential [5]. Details of Fourier transformation of regularized parts of the potential are given in the appendix. Also we have used a Gaussian form factor, at the quarkgluon vertex as in [6] to remove singularity of the kernels due to existence of onegluon exchange potential. The variable can be interpreted as size of the quark. In [7] the pointlike quarkgluon vertex is replaced by a form factor, in which is the effective quark size to eliminate the singularity. In this work we have used both regularized form and Gaussian form factor for Coulomb and parts of the potential which cause the convergence of numerical results faster. Therefore, the final form of the potential in the momentumhelicity space is written aswhere and is the momentumhelicity basis state which is eigenstate of the helicity operator asAlso we have [1]If the vector p is along direction, it is clear that (15) is reduced toFor numerical calculations we need the matrix elements of the potential . These matrix elements are related to the matrix elements of (13) as follows:By considering (13), (16), and (17), the final form of the matrix elements of the potential which is inserted in the numerical calculations is written aswith .
4. Discussion and Numerical Results
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 GaussLegendre points from the interval to the intervals , respectively, as follows:Then we have calculated the matrix elements of the potential , from (18). According to (3) integration over the spherical angle variable has been done independently. Finally, we have solved the integral equations (4)–(8) as eigenvalue equations. The integration over momentum variable is cut off at = 10 GeV. This selection is carried out so that the numerical results do not depend on this choice. The typical values for and are 1 GeV and 3 GeV, respectively. These selections are done till the total number of grid points for momentum intervals is decreased. Other selections can be done but by different grid points for momentum variables.
The parameters of the potential model which are shown in Table 1 are fixed by a fit to the masses of the states , , and , similar to what is done in [9]. The results of charmonium mass spectrum are shown in Table 2. They are compared with the experimental data and another theoretical work. As it is clear from (7) and (8), the existence of the tensor term in the potential causes the coupling of the  and partial waves and this coupling as is shown in Table 3 is so weak. I show the mixed charmonium states in Table 2 by their dominant partial wave.


As a test of our numerical calculations we have shown convergence of the results as a function of number of grid points , , and for the momentum and angle variables in Table 4. and are the number of grid points for the intervals [0, ] and [, ], respectively. is corresponding to number of grid points for spherical angle variable. In our calculations we have chosen , and grid points for to achieve an acceptable accuracy.

5. Summary and Outlook
In this paper we have extended an approach based on momentumhelicity basis states for calculation of mass spectrum of heavy mesons by solving nonrelativistic form of the LippmannSchwinger 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 quarkantiquark potential. Thus, using the helicity representation is less complicated than using the spin representation with a fixed quantization axis for representation of spindependent potentials. This work is the first step toward studying single, double, and triple heavyflavor baryons in the framework of the nonrelativistic quark model by formulation of the Faddeev equation in the 3D momentumhelicity representation. Furthermore, we can apply this formalism straightforwardly for investigation of heavy pentaquark systems, which can be considered as twobody (heavy meson and baryon) systems with mesonnucleon potentials which is underway.
Appendix
Fourier Transformation of the Regularized Linear Confining and Coulomb Parts of the Potential
The threedimensional Fourier transformation of the potential is defined aswhere . Fourier transformation of the regularized linear confining and Coulomb parts of the quarkantiquark potential is written aswhere potential is kept fixed at cutoff . Therefore inserting the linear and Coulomb , parts of quarkantiquark potential, in the above equation and calculation of corresponding integrals analytically yield
Competing Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
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Copyright
Copyright © 2017 M. Radin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP^{3}.