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

Authors: M. Radin
Journal: Advances in High Energy Physics
 (2017)

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## Abstract

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 charmoniumψcc¯, and finally we have compared the results with the other theoretical results and experimental data.


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## Body

## 1. Introduction

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 [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-quarkc,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 Section2, 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 quark-antiquark potential model is described in the momentum-helicity 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(1)ΦjMj=1E-p2/mVΦjMj,where V denotes the quark-antiquark interaction, m is mass of the quark or antiquark, and ΦjMj is the meson bound state with the total angular momentum j. Mj 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](2)ΦSjMjp=2πE-p2/m∑Λ′∫0∞dp′p′2VMjΛ′Sp,p′ΦSjΛ′p′,with(3)VMjΛ′Sp,p′=∫-11dcos⁡θ′VMjΛ′Sp,p′,θ′dMjΛ′jθ′,where p is the magnitude of the relative momentum of the quark and antiquark, S is the total spin of meson, Λ is the spin projection along the relative momentum, and dMjΛ′jθ′ are the rotation matrices. For an arbitrary total angular momentum j, and singlet case of the total spin state, (2) leads to one equation: (4)Φ0jMjp=2πE-p2/m∫0∞dp′p′2VMj00p,p′Φ0j0p′.Also for j=0 and triplet case of the total spin state, (2) leads to one equation as follows:(5)Φ1j0p=2πE-p2/m∫0∞dp′p′2V001p,p′Φ1j0p′.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:(6)Ψ111p=2πE-p2/m∫0∞dp′p′2V111p,p′-V-111p,p′Ψ111p′,(7)Ψ011p=2πE-p2/m13∫0∞dp′p′22V111p,p′+2V011p,p′+V001p,p′+2V101p,p′+2V-111p,p′Ψ011p′+2V111p,p′+V011p,p′-V001p,p′-2V101p,p′+V-111p,p′Ψ211p′,(8)Ψ211p=2πE-p2/m13∫0∞dp′p′22V111p,p′-2V011p,p′-V001p,p′+V101p,p′+V-111p,p′Ψ011p′+V111p,p′-2V011p,p′+2V001p,p′-2V101p,p′+V-111p,p′Ψ211p′,where ΨlSjp is the partial wave component of the wave function which is connected to the momentum-helicity component of the wave function as [3](9)ΦjSΛp=∑l2l+14πClSj;0ΛΛΨlSjp.The inverse relation is written as(10)ΨlSjp=4π2l+12j+1∑ΛClSj;0ΛΛΦjSΛp.


## 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](11)Vr,p=σr+fcαs1r-πm2δr+1m2p·pr-34m2L·σ1+σ2r3-2π3m2δrσ1·σ2-14m23σ1·r^σ1·r^-σ1·σ2r3,where σ is the string tension, αs is the strong-interaction fine-structure constant, fc is the color factor which is -4/3 for quark-antiquark and -2/3 for quark-quark, σ1 and σ2 are the Pauli matrices, and L is the total orbital angular momentum operator. Fourier transformation of this potential to momentum space yields(12)
p
V
p
′
=
σ
δ
q
r
c
+
1
2
π
2
q
4
2
cos
⁡
q
r
c
-
2
+
q
r
c
sin
⁡
q
r
c
+
f
c
α
s
e
-
λ
2
q
2
δ
q
r
c
+
1
2
π
2
q
2
1
-
sin
⁡
q
r
c
q
r
c
1
+
p
2
m
2
-
1
8
π
2
m
2
+
3
8
π
2
m
2
q
2
i
σ
1
+
σ
2
·
p
×
p
′
-
1
12
π
2
m
2
σ
1
·
σ
2
+
1
24
π
2
m
2
3
σ
1
·
q
^
σ
2
·
q
^
-
σ
1
·
σ
2
,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⁡-1/2λ2q2 at the quark-gluon vertex as in [6] to remove singularity of the kernels due to existence of one-gluon exchange potential. The variable λ can be interpreted as size of the quark. In [7] the point-like quark-gluon vertex is replaced by a form factor, 1/(q2+β2) in which β-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 fcαsp2/(m2r) 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(13)VΛΛ′Sp,p′≡pSΛVp′SΛ′=σp^SΛ∣p^′SΛ′δqrc+12π2q42cos⁡qrc-2+qrcsinqrc+fcαse-λ2q2p^SΛ∣p^′SΛ′1rcδq+12π2q21-sin⁡qrcqrc1+p2m2-18π2m2-112π2m22SS+1-3+38π2m2pp′q2γSS+1-2ΛΛ′-1γSS+1-2Λ′2-2Λ2-2Λ′2Λ2-124π2m2q26pp′ΛΛ′+2p′2SS+1-3Λ′2+2p2SS+1-3Λ2-pp′γSS+1-3pp′γSS+1-2Λ′2-2Λ2-2Λ′2Λ2,where γ=p^′·p^=cos⁡θcos⁡θ′+sin⁡θsin⁡θ′cos⁡φ-φ′ and p;p^SΛ is the momentum-helicity basis state which is eigenstate of the helicity operator S·p^ as(14)S·p^p;p^SΛ=Λp;p^SΛ.Also we have [1](15)p^SΛ∣p^′SΛ′=∑N=-SSeiNφ-φ′dNΛSθdNΛ′Sθ′.If the vectorp is along z-direction, it is clear that (15) is reduced to(16)z^SΛ∣p^′SΛ′=e-iΛφ′dΛΛ′Sθ′.For numerical calculations we need the matrix elements of the potential VΛΛ′Sp,p′,θ′. These matrix elements are related to the matrix elements of (13) as follows:(17)VΛΛ′Sp,p′,θ′=eiΛφ′pz;z^SΛVp′;p^′SΛ′.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(18)VΛΛ′Sp,p′,θ′=σdΛΛ′Sθ′δqrc+12π2q42cos⁡qrc-2+qrcsin⁡qrc+fcαse-λ2q2dΛΛ′Sθ′1rcδq+12π2q21-sin⁡qrcqrc1+p2m2-18π2m2-112π2m22SS+1-3+38π2m2pp′q2γSS+1-2ΛΛ′-1γSS+1-2Λ′2-2Λ2-2Λ′2Λ2-124π2m2q26pp′ΛΛ′+2p′2SS+1-3Λ′2+2p2SS+1-3Λ2-pp′γSS+1-3pp′γSS+1-2Λ′2-2Λ2-2Λ′2Λ2,with γ=p^′·z^=cos⁡θ′.


## 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,p2∪p2,pmax⁡, respectively, as follows:(19)p=1+x1/p1+2/p2-1/p1x,p=pmax⁡-p22x+pmax⁡+p22.Then we have calculated the matrix elements of the potential VΛΛ′(p,p′,θ′), from (18). According to (3) integration over the spherical angle variable θ′ 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 qmax⁡ = 10 GeV. This selection is carried out so that the numerical results do not depend on this choice. The typical values for p1 and p2 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 Table1 are fixed by a fit to the masses of the states ηc, J/ψ, and hc, 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.Table 1
Parameters of the model.
σ   G e V / f m 1.222 λ   G e V - 1 0.3154 m   G e V 1.269 α s 0.2863 rcfm 10Table 2
Comparison of the obtained charmonium mass spectrum with the experimental data and another work.
n 2 S + 1 L J Candidate Exp. [8] Ref. [9] Mass [MeV] 1 1 S 0 η c 2980.4 ± 1.2 2980 2980.4 1 3 S 1 J / ψ 3096.916 ± 0.011 3097 3096.9 1 1 P 1 h c 3526.21 ± 0.25 3527 3526.2 1 3 P 0 χ c 0 3415.16 ± 0.35 3430 3397.4 1 3 P 1 χ c 1 3510.59 ± 0.10 3503 3503.5 2 1 S 0 η c ′ 3638 ± 5 3674 3683.1 2 3 S 1 ψ ′ 3686.093 ± 0.034 3765 3760.8 1 3 D 1 ψ ′ ′ 3770 ± 2.4 3855 3850.6 3 3 S 1 ψ ′ ′ ′ 4040 ± 10 4291 4285.4Table 3
Percent of each partial wave in mixed charmonium states.
n 2 S + 1 L J c c - P S % P D % 1 3 S 1 1 3 S 1 - 1 3 D 1 J / ψ 99.93 0.07 2 3 S 1 2 3 S 1 - 2 3 D 1 ψ ′ 99.90 0.10 3 3 S 1 3 3 S 1 - 3 3 D 1 ψ ′ ′ ′ 99.88 0.12 1 3 D 1 ( 1 3 D 1 - 1 3 S 1 ) ψ ′ ′ 99.88 0.12As a test of our numerical calculations we have shown convergence of the results as a function of number of grid pointsNP1, NP2, and Nθ for the momentum and angle variables in Table 4. NP1 and NP2 are the number of grid points for the intervals [0, p2] and [p2, pmax⁡], respectively. Nθ is corresponding to number of grid points for spherical angle variable. In our calculations we have chosen NP1=100, NP2=100 and Nθ=200 grid points for to achieve an acceptable accuracy.Table 4
The calculated charmonium mass spectrum as function of the number of grid pointsNP1, NP1, and Nθ.
N p 1 N p 2 N θ η c J / ψ h c χ c 0 χ c 1 η c ′ ψ ′ ψ ′ ′ ψ ′ ′ ′ 100 100 140 2980.602 3096.942 3526.244 3397.517 3503.552 368.3357 3760.778 3850.573 4285.415 100 100 160 2980.420 3096.951 3526.226 3397.449 3503.541 3683.153 3760.782 3850.563 4285.410 100 100 180 2980.370 3096.954 3526.221 3397.434 3503.539 3683.095 3760.784 3850.561 4285.409 100 100 200 2980.356 3096.954 3526.219 3397.431 3503.538 3683.080 3760.784 3850.560 4285.409 100 100 220 2980.353 3096.954 3526.219 3397.430 3503.538 3683.075 3760.784 3850.560 4285.409 100 100 240 2980.352 3.096954 3526.219 3397.430 3503.538 3683.074 3760.784 3850.560 4285.409 100 100 260 2980.352 3.096954 3526.219 3397.430 3503.538 3683.074 3760.784 3850.560 4285.409  100 60 200 2975.716 3097.131 3525.746 3397.449 3394.900 3503.182 3677.378 3850.232 4284.965 100 80 200 2980.218 3096.961 3526.205 3397.376 3503.530 3682.920 3760.787 3850.553 4285.405 100 100 200 2980.356 3096.954 3526.219 3397.431 3503.541 3683.538 3760.080 3850.560 4285.409 100 120 200 2980.356 3096.954 3526.219 3397.431 3503.541 3503.538 3683.080 3850.560 4285.409  50 100 200 2980.356 3096.954 3526.219 3397.429 3503.538 3683.080 3760.784 3850.558 4285.409 80 100 200 2980.356 3096.954 3526.219 3397.430 3503.538 3683.080 3760.784 3850.560 4285.409 100 100 200 2980.356 3096.954 3526.219 3397.431 3503.541 3683.080 3760.784 3850.560 4285.409


## 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 meson-nucleon potentials which is underway.



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