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Charmonium Mass Spectrum with Spin-Dependent Interaction in Momentum-Helicity Space.

1. Introduction

During the past years, several models and methodological approaches based on solving the relativistic and nonrelativistic form of the Schrodinger or Lippmann-Schwinge requation 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 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 Lippmann-Schwinger equation to obtain the mass spectrum of the heavy mesons using the nonrelativistic quark-antiquark interaction in terms of a linear confinement, a Coulomb, and various spin-dependent pieces.

In the heavy-quark (c, b) mesons the differences between energy levels are small compared to the particle masses. Hence, the nonrelativistic Lippmann-Schwinger equation can be used to study their quantum behavior. To this end, we have used the nonrelativistic form of the Lippmann-Schwinger equation in the momentum-helicity representation to study the charmonium as a heavy meson. For this purpose, we have used a nonrelativistic quark-antiquark potential based on one-gluon exchange in the momentum-helicity representation.

This article is organized as follows. In Section 2, the nonrelativistic Lippmann-Schwinger equation in the momentum-helicity basis states which leads to coupled and uncoupled integral equations for various quantum numbers is presented briefly. In Section 3, a spin-dependent 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. Lippmann-Schwinger Equation in Momentum-Helicity Basis States

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

[mathematical expression not reproducible], (1)

where V denotes the quark-antiquark interaction, m is mass of the quark or antiquark, and [mathematical expression not reproducible] is the mesonbound state with the total angular momentum j. [M.sub.j] is projection of the total angular momentum j along the quantization axis. The integral form of this equation in the momentum-helicity basis states is written as [3]

[mathematical expression not reproducible], (2)

with

[mathematical expression not reproducible], (3)

where p is the magnitude of the relative momentum of the quark and antiquark, S is the total spin of meson, [LAMBDA] is the spin projection along the relative momentum, and [mathematical expression not reproducible] are the rotation matrices. For an arbitrary total angular momentum j, and singlet case of the total spin state, (2) leads to one equation:

[mathematical expression not reproducible]. (4)

Also for j = 0 and triplet case of the total spin state, (2) leads to one equation as follows:

[[PHI].sup.0.sub.1j](p) = [2[pi]/E - [p.sup.2]/m][[integral].sup.[infinity].sub.0]dp'[p'.sup.2][V.sup.1.sub.00](p,p') [[PHI].sup.0.sub.1j](p'). (5)

For S =1 and j > 0 it is more complicated. For example, for j = 1, (2) leads to one equation for channel P and two coupled equations for channels S and D as follows:

[mathematical expression not reproducible], (6)

[mathematical expression not reproducible], (7)

[mathematical expression not reproducible], (8)

where [[psi].sub.lSj](p) is the partial wave component of the wave function which is connected to the momentum-helicity component of the wave function as [3]

[[PHI].sup.[LAMBDA].sub.jS](p) = [summation over l] [square root of (2l + 1/4[pi])] C(lSj;0[LAMBDA][LAMBDA])[[psi].sub.lSj](p). (9)

The inverse relation is written as

[[PHI].sub.lSj](p) = [square root of (4[pi](2l + 1)/2j + 1)] [summation over [LAMBDA]]C (lSj;0[LAMBDA][LAMBDA])[[psi].sup.[LAMBDA].sub.jS](p). (10)

3. Quark-Antiquark Potential in Momentum-Helicity Basis States

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 [4]

[mathematical expression not reproducible], (11)

where [sigma] is the string tension, [[alpha].sub.s] is the strong-interaction fine-structure constant, [f.sub.c] is the color factor which is -4/3 for quark-antiquark and -2/3 for quark-quark, [[sigma].sub.1] and [[sigma].sub.2] are the Pauli matrices, and L is the total orbital angular momentum operator. Fourier transformation of this potential to momentum space yields

[mathematical expression not reproducible], (12)

where q = p' - p 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, exp((-1/2)[[lambda].sup.2][q.sup.2]) at the quark-gluon vertex as in [6] to remove singularity of the kernels due to existence of one-gluon exchange potential. The variable [lambda] can be interpreted as size of the quark. In [7] the point-like quark-gluon vertex is replaced by a form factor, 1/([q.sup.2] + [[beta].sup.2]) in which [[beta].sup.- 1] 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 [f.sub.c][[alpha].sub.s][p.sup.2]/([m.sup.2]r) parts of the potential which cause the convergence of numerical results faster. Therefore, the final form of the potential in the momentum-helicity space is written as

[mathematical expression not reproducible], (13)

where [gamma] = [??]' x [??] = cos [theta] cos [theta]' + sin [theta] sin [theta]' cos([phi] - [phi]') and |p; [??]S[LAMBDA]> is the momentum-helicity basis state which is eigenstate of the helicity operator S x [??] as

[mathematical expression not reproducible]. (14)

Also we have [1]

[mathematical expression not reproducible]. (15)

If the vector p is along z-direction, it is clear that (15) is reduced to

[mathematical expression not reproducible]. (16)

For numerical calculations we need the matrix elements of the potential [V.sup.S.sub.[LAMBDA][LAMBDA]'] (p, p', [theta]'). These matrix elements are related to the matrix elements of (13) as follows:

[mathematical expression not reproducible]. (17)

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 as

[mathematical expression not reproducible], (18)

with [mathematical expression not reproducible].

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 Gauss-Legendre points from the interval [-1, +1] to the intervals [0, [p.sub.2]] [union] [[p.sub.2], [p.sub.max]], respectively, as follows:

P = 1 + x/1/[p.sub.1] + (2/[p.sub.2] - 1/[p.sub.1])x,

p = [[p.sub.max] - [p.sub.2]/2] x + [[p.sub.max] + [p.sub.2]/2]. (19)

Then we have calculated the matrix elements of the potential [V.sub.[lambda][lambda]'](p, p', [theta]'), from (18). According to (3) integration over the spherical angle variable [theta]' 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 [q.sub.max] = 10 GeV This selection is carried out so that the numerical results do not depend on this choice. The typical values for [p.sub.1] and [p.sub.2] 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 [[eta].sub.c], J/[psi], and [h.sub.c], 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 S- and D-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 [N.sub.P1], [N.sub.P2], and [N.sub.[theta]] for the momentum and angle variables in Table 4. [N.sub.P1] and [N.sub.P2] are the number of grid points for the intervals [0, [p.sub.2]] and [[p.sub.2], [p.sub.max]], respectively. Ng is corresponding to number of grid points for spherical angle variable. In our calculations we have chosen [N.sub.P1] = 100, [N.sub.P2] = 100 and [N.sub.[theta]] = 200 grid points for to achieve an acceptable accuracy.

5. Summary and Outlook

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 mesonnucleon potentials which is underway.

Appendix

Fourier Transformation of the Regularized Linear Confining and Coulomb Parts of the Potential

The three-dimensional Fourier transformation of the potential V(r) is defined as

V(p, p') = 1/2[[pi].sup.2][q.sup.2] [[integral].sup.[infinity].sub.0] drrV(r) sin qr, (A.1)

where q = [absolute value of p - p']. Fourier transformation of the regularized linear confining and Coulomb parts of the quark-antiquark potential is written as

[mathematical expression not reproducible], (A.2)

where potential is kept fixed at cutoff [r.sub.c]. Therefore inserting the linear V(r) = [sigma]r and Coulomb V(r) = [f.sub.c][[alpha].sub.s]/r, parts of quark-antiquark potential, in the above equation and calculation of corresponding integrals analytically yield

[mathematical expression not reproducible]. (A.3)

http://dx.doi.org/10.1155/2017/8321513

Competing Interests

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

References

[1] I. Fachruddin, C. Elster, and W. Glockle, "Nucleon-nucleon scattering in a three dimensional approach," Physical Review C, vol. 62, no. 4, Article ID 044002, 2000.

[2] I. Fachruddin, C. Elster, and W. Glockle, "New forms of deuteron equations and wave function representations," Physical Review C--Nuclear Physics, vol. 63, no. 5, Article ID 054003, 2001.

[3] M. Radin and N. Tazimi, "Quark-antiquark bound state in momentum-helicity representation," Physical Review D, vol. 90, no. 8, Article ID 085020, 2014.

[4] S. N. Gupta, S. F. Radford, and W. W. Repko, "Quarkonium spectra and quantum chromodynamics," Physical Review D, vol. 26, no. 11, pp. 3305-3308, 1982.

[5] M. R. Hadizadeh and L. Tomio, "Solution of two-body bound state problems with confining potentials," in Proceedings of the International Conference "1st Stars and Galaxies: Challenges for the Next Decade", pp. 334-337, AIP, Sao Paulo, Brazil, March 2010.

[6] J. Carlson, J. B. Kogut, and V. R. Pandharipande, "Hadron spectroscopy in a flux-tube quark model," Physical Review D, vol. 28, no. 11, article 2807, 1983.

[7] D. P. Stanley and D. Robson, "Nonperturbative potential model for light and heavy quark-antiquark systems," Physical Review D, vol. 21, no. 11, pp. 3180-3196, 1980.

[8] S. Eidelman, K. G. Hayes, K. A. Olive et al., "Review of particle physics," Physics Letters B, vol. 592, no. 1-4, pp. 1-5, 2004.

[9] J. Eiglsperger, Quarkonium spectroscopy: beyond one-gluon exchange [Diploma thesis], Technische Universitot Munchen, Munich, Germany, 2007.

M. Radin

Department of Physics, K. N. Toosi University of Technology, P.O. Box 16315- 1618, Tehran, Iran

Correspondence should be addressed to M. Radin; radin@kntu.ac.ir

Received 28 August 2016; Revised 27 October 2016; Accepted 29 November 2016; Published 21 February 2017

Academic Editor: Mohammadreza Hadizadeh
Table 1: Parameters of the model.

[sigma] [GeV/fm]          1.222
[lambda] [Ge[V.sup.-1]]   0.3154
m [GeV]                   1.269
[[alpha].sub.s]           0.2863
[r.sub.c][fm]               10

Table 2: Comparison of the obtained charmonium mass spectrum
with the experimental data and another work.

[n.sup.2S+1][L.sub.j]     Candidate             Exp. [8]

[1.sup.1][S.sub.0]      [[eta].sub.c]      2980.4 [+ or -] 1.2
[1.sup.3][S.sub.1]         J/[psi]       3096.916 [+ or -]0.011
[1.sup.1][P.sub.1]        [h.sub.c]       3526.21 [+ or -]0.25
[1.sup.3][P.sub.0]      [[chi].sub.c0]    3415.16 [+ or -]0.35
[1.sup.3][P.sub.1]      [[chi].sub.c1]    3510.59 [+ or -]0.10
[2.sup.1][S.sub.0]      [[eta]'.sub.c]       3638 [+ or -] 5
[2.sup.3][S.sub.1]          [psi]'       3686.093 [+ or -] 0.034
[1.sup.3][D.sub.1]          [psi]"          3770 [+ or -] 2.4
[3.sup.3][S.sub.1]         [psi]"'          4040 [+ or -] 10

[n.sup.2S+1][L.sub.j]   Ref. [9]   Mass [MeV]

[1.sup.1][S.sub.0]        2980       2980.4
[1.sup.3][S.sub.1]        3097       3096.9
[1.sup.1][P.sub.1]        3527       3526.2
[1.sup.3][P.sub.0]        3430       3397.4
[1.sup.3][P.sub.1]        3503       3503.5
[2.sup.1][S.sub.0]        3674       3683.1
[2.sup.3][S.sub.1]        3765       3760.8
[1.sup.3][D.sub.1]        3855       3850.6
[3.sup.3][S.sub.1]        4291       4285.4

Table 3: Percent of each partial wave in mixed charmonium states.

[n.sup.2S+1][L.sub.j]                        c[bar.c]   [P.sub.S]%

[1.sup.3][S.sub.1]                           J/[psi]      99.93
([1.sup.3][S.sub.1] - [1.sup.3][D.sub.1])

[2.sup.3][S.sub.1]                            [psi]'      99.90
([2.sup.3][S.sub.1] - [2.sup.3][D.sub.1])

[3.sup.3][S.sub.1]                           [psi]'"      99.88
([3.sup.3][S.sub.1] - [3.sup.3][D.sub.1])

[1.sup.3][D.sub.1]                            [psi]"      99.88
([1.sup.3][D.sub.1] - [1.sup.3][S.sub.1])

[n.sup.2S+1][L.sub.j]                        [P.sub.D]%

[1.sup.3][S.sub.1]                              0.07
([1.sup.3][S.sub.1] - [1.sup.3][D.sub.1])

[2.sup.3][S.sub.1]                              0.10
([2.sup.3][S.sub.1] - [2.sup.3][D.sub.1])

[3.sup.3][S.sub.1]                              0.12
([3.sup.3][S.sub.1] - [3.sup.3][D.sub.1])

[1.sup.3][D.sub.1]                              0.12
([1.sup.3][D.sub.1] - [1.sup.3][S.sub.1])

Table 4: The calculated charmonium mass spectrum as function of the
number of grid points MP1, MP1, and Ne.

[N.sub.p1]   [N.sub.p2]   [N.sub.[theta]]   [[eta].sub.c]   J/[psi]

100             100             140           2980.602      3096.942
100             100             160           2980.420      3096.951
100             100             180           2980.370      3096.954
100             100             200           2980.356      3096.954
100             100             220           2980.353      3096.954
100             100             240           2980.352      3.096954
100             100             260           2980.352      3.096954
100              60             200           2975.716      3097.131
100              80             200           2980.218      3096.961
100             100             200           2980.356      3096.954
100             120             200           2980.356      3096.954
50              100             200           2980.356      3096.954
80              100             200           2980.356      3096.954
100             100             200           2980.356      3096.954

[N.sub.p1]   [h.sub.c]   [[chi].sub.c0]   [[chi].sub.c1]

100          3526.244       3397.517         3503.552
100          3526.226       3397.449         3503.541
100          3526.221       3397.434         3503.539
100          3526.219       3397.431         3503.538
100          3526.219       3397.430         3503.538
100          3526.219       3397.430         3503.538
100          3526.219       3397.430         3503.538
100          3525.746       3397.449         3394.900
100          3526.205       3397.376         3503.530
100          3526.219       3397.431         3503.541
100          3526.219       3397.431         3503.541
50           3526.219       3397.429         3503.538
80           3526.219       3397.430         3503.538
100          3526.219       3397.431         3503.541

[N.sub.p1]   [[eta].sub.'c]    [psi]'     [psi]"    [psi]"'

100             368.3357      3760.778   3850.573   4285.415
100             3683.153      3760.782   3850.563   4285.410
100             3683.095      3760.784   3850.561   4285.409
100             3683.080      3760.784   3850.560   4285.409
100             3683.075      3760.784   3850.560   4285.409
100             3683.074      3760.784   3850.560   4285.409
100             3683.074      3760.784   3850.560   4285.409
100             3503.182      3677.378   3850.232   4284.965
100             3682.920      3760.787   3850.553   4285.405
100             3683.538      3760.080   3850.560   4285.409
100             3503.538      3683.080   3850.560   4285.409
50              3683.080      3760.784   3850.558   4285.409
80              3683.080      3760.784   3850.560   4285.409
100             3683.080      3760.784   3850.560   4285.409
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Title Annotation:Research Article
Author:Radin, M.
Publication:Advances in High Energy Physics
Date:Jan 1, 2017
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