Charmonium states in QCD-inspired quark potential model using Gaussian expansion method

Charmonium states in QCD-inspired quark potential model using Gaussian expansion method

Lu Cao Lu Cao School of Physical Science and Technology, Southwest University, Chongqing 400700, China
22email: physicaolu@126.comYou-Chang Yang Department of Physics, Zunyi Normal College, Zunyi 563002, China
44email: yangyouchang@yahoo.com.cnHong ChenSchool of Physical Science and Technology, Southwest University, Chongqing 400700, China
Corresponding Author. 66email: chenh@swu.edu.cn
   You-Chang Yang Lu Cao School of Physical Science and Technology, Southwest University, Chongqing 400700, China
22email: physicaolu@126.comYou-Chang Yang Department of Physics, Zunyi Normal College, Zunyi 563002, China
44email: yangyouchang@yahoo.com.cnHong ChenSchool of Physical Science and Technology, Southwest University, Chongqing 400700, China
Corresponding Author. 66email: chenh@swu.edu.cn
   Hong Chen Lu Cao School of Physical Science and Technology, Southwest University, Chongqing 400700, China
22email: physicaolu@126.comYou-Chang Yang Department of Physics, Zunyi Normal College, Zunyi 563002, China
44email: yangyouchang@yahoo.com.cnHong ChenSchool of Physical Science and Technology, Southwest University, Chongqing 400700, China
Corresponding Author. 66email: chenh@swu.edu.cn
Received: date / Accepted: date
Abstract

We study the mass spectrum and electromagnetic processes of charmonium system with the spin-dependent potentials fully taking into account in the solution of the Schroedinger equation and the results for the pure scalar and scalar-vector mixing linear confining potentials are compared. It is revealed that the scalar-vector mixing confinement is important for reproducing the mass spectrum and decay widths and the vector component is found to be around , the long-standing discrepancy in M1 radiative transitions of and is alleviated by means of the state wave functions obtained via the Hamiltonian with the full spin-dependent potential. This work also intends to identify few of the copious higher charmonium-like states as the ones. Particularly, the newly observed and are assigned as MeV and MeV, which strongly favor the assignments respectively. The corresponding radiative transitions, leptonic and two-photon decay widths have been also calculated for the further experimental study.

Keywords:
charmonium potential model Gaussian expansion method XYZ mesons
journal: Few-Body Systems

1 Introduction

Due to the impressive increase of experimental results, charmonium () spectroscopy has renewed great interest recently, coming along with the striking disagreement with theoretical expectations [1, 2, 3]. The unexpected and still-fascinating has been joined by more than a dozen other charmonium-like states, while the series of vacancy have been left on the list. It is urgent to identify the possible new members of charmonium family from the abundant observations.

The QCD inspired potential models have been playing an important role in investigating heavy quarkonium, owning to the presence of large nonperturbative effects in this energy region. Most quark potential models [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16] have common ingredients under the non-relativistic limit, despite some differences in the detailed corrections for relativistic and coupled channel effects, which typically are the Coulomb-like term induced by one-gluon exchange plus the long-range confining potential expected from nonperturbative QCD. Anyway, the nature of confining mechanism has been veiled so far. In the original Cornell model [17, 18], it was assumed as Lorentz scalar, which gives a vanishing long-range magnetic contribution and agrees with the flux tube picture of quark confinement [19]. Another possibility [9, 20] is that confinement may be a more complicated mixture of scalar and timelike vector, while the vector potential is anticonfinning. In pure models, the Lorentz nature of confinement is tested by the multiplet splitting of orbitally excited charmonium states.

In addition, numerical precision of the calculation method is important in testing different models. As several numerical methods fail in the potentials with and even higher negative power, the corrections to the quark-antiquark potential has to be usually treated as perturbation [21, 22, 20]. However, the accuracy of perturbation expansion has been alerted recently [23], which indicates the most significant effect of the different treatments is on the wave functions. The exact solution of the full Hamiltonian provides every state with its own wave function, while the perturbative treatment leads to the same angular momentum multiplets sharing the identical radial wave function. It is known that the radiative transitions, leptonic and double-photon decay widths are quite sensitive to the shape of wave function and its behavior at the origin. Each physical particle with different quantum numbers should have different state wave function. Thus, it is interesting to study the still-puzzling confining mechanism and the numerical precision of the calculation method.

In our calculation framework, the spin-dependent and -independent interactions have been totally taken into account in the Hamiltonian, where the different confining assumptions are compared from the mass spectrum and decay properties. The precise wave functions are inspected through the electromagnetic processes, i.e. radiative transitions and leptonic decays, which are considered to be a nichetargeting test for the overlap of radial integration and the subtle information at the origin. Since the relativistic reconstruction of the static confining potential is not unique, it complicates the nature of confinement. Hence special concerns are focused on the minimal but relatively well-understood models, with the aim of gleaning the actual influences of different forms of confinement potential and the difference between the exact and the perturbative solutions.

In the following section, the two potential models are described in detail, along with the adopted variational approach and the optimization of parameters. The numerical results are assembled in Sec.3, a discussion related to the latest experimental results are put in Sec.4. Finally, Sec.5 summaries the remarks and conclusions.

2 Potential models and calculational approach

The confinement of quarks is assumed to be purely scalar linear type in NR model[21], and the scalar-vector mixed one in MNR [9].Once the Lorentz structure of central part are fixed in the two forms,

(1)
(2)

the spin-orbit term and the tensor term can be directly derived from the standard Breit-Fermi expression to order with the charmed quark mass . Summarily, the interaction potentials are

(3)
(4)

where is the orbital momentum and is the spin of charmonium. In the mixed-confining model, stands for the vector exchange scale. The singularity of contact hyperfine interaction within the spin-spin term has been smeared by Gaussian as in Ref.[21], . The involved operators are diagonal in a basis with the matrix elements,

(5)
(6)
(7)

Instead of separating the spin-dependent interactions into leading order portions, we solved the Schroedinger equation of the unperturbed Hamiltonian with full potentials,

(8)

where is the reduced mass. Here, is or which includes the spin-independent interactions as well as the spin-dependent ones. The Hamiltonian includes the full potential, enable us to maintain the subtleness of wave function. With the help of a well-chosen set of Gaussian basis functions, namely Gaussian Expansion Method [24], the singular behavior of in spin-dependent terms at short distance can be refined variationally. Then, the wave functions are expanded in terms of a set of Gaussian basis functions as

(9)
(10)

The basis-related parameters are determined by the variational principle, resulting the reasonably stable eigen solutions and differ slightly according to the explicit forms of potential models. Table 1 shows the utilized basis for the two potential models.

Parameter NR MNR

Basis Space 10 8
[GeV] 0.4 0.4
[GeV] 15.8 6.8
Potential Model [GeV] 1.4786 1.5216
0.5761 0.6344
[GeV] 0.1468 0.1361
[GeV] 1.1384 1.2058
- -0.2193
Table 1: Basis-related parameters for the potential models

To determine the parameters () appearing in the potentials, a merit function has been defined to search the best-fit parameters by its minimization,

(10)

where denotes the number of targeted data, the are the associated errors and represents the experimental and theoretical values. Given a trial set of model-depended parameters, a procedure calculating the are developed to improve the trial solution with the increments and repeated until effectively stops decreasing. The increments are solved by the set of linear equations

(11)

with

(12)
(13)

The iteration has been taken in the numerical analysis of nonlinear systems. Here, the partial derivative has to be solved numerically because the is not analytic.

3 Numerical Results

3.1 Mass Spectrum

Combining the nonrelativistic kinetic term and the interactions terms, we diagonalize the full Hamiltonian, which leads to a generalized eigenvalue problem. After the fitting procedure, the energy levels and corresponding eign functions of the two potential models are obtained with its optimized parameter set (shown in Table 1).

The vector component of linear confining are fitted to be for the MNR model. This result consists with the one-fifth vector exchange given by Ref. [23]. If defined an additional -independent adjustable parameter in the vector linear confinement, the value of mixing coefficient should be for some decay considerations [20]. In addition, the scalar-vector mixing model requires less linear potential slope, but more splitting scale from spin-spin interaction.

Totally 60 states have been calculated with the two potential models, and summarized in Table 2. We fit the mass of well-established states marked by . Except somewhat low at , both of the two potential models are overall good to reproduce the spectrum.

State Expt. [25] Ref. [21] Ours
NR GI NR MNR

2982 2975 2990.4 2978.4
3090 3098 3085.1 3087.7
3630 3623 3646.5 3646.9
3672 3676 3682.1 3684.7
4043 4064 4071.9 4058.0
4072 4100 4100.2 4087.0
4384 4425 4420.9 4391.4
4406 4450 4439.4 4411.4
3556 3550 3551.4 3559.3
3505 3510 3500.4 3517.7
3424 3445 3351.9 3366.3
3516 3517 3514.6 3526.9
3972 3979 3979.8 3973.1
3925 3953 3933.5 3935.0
3852 3916 3835.7 3842.7
3934 3956 3944.6 3941.9
4317 4337 4383.4 4352.4
4271 4371 4317.9 4298.7
4202 4292 4216.7 4207.6
4279 4318 4333.9 4309.7
4736.7 4703.1
4620.3 4590.5
4551.8 4521.7
4639.5 4606.7
3806 3849 3814.6 3812.6
3800 3838 3807.7 3820.1
3785 3819 3785.3 3808.8
3799 3837 3807.3 3815.1
4167 4217 4182.9 4166.1
4158 4208 4173.7 4168.7
4142 4194 4150.4 4154.4
4158 4208 4173.7 4164.9
4572.5 4526.5
4558.8 4523.6
4525.8 4502.2
4559.7 4521.4
4021 4095 4037.4 4024.7
4029 4097 4044.0 4047.6
4029 4092 4042.4 4059.7
4026 4094 4041.1 4040.8
4348 4425 4371.1 4344.7
4352 4426 4374.4 4362.4
4351 4422 4369.9 4369.8
4350 4424 4372.3 4356.8
4744.4 4684.8
4747.7 4698.5
4743.9 4704.2
4745.9 4694.3
4214 4312 4236.9 4213.5
4228 4320 4250.6 4244.7
4237 4323 4258.2 4267.4
4225 4317 4247.1 4237.9
Table 2: Experimental and theoretical Charmonium mass spectrum. The masses are in units of MeV, and the denotes the states used in the optimization of potential parameters. Our full-potential calculation results are listed in comparison with the perturbative results of NR model and its relativized extension GI model [21].

3.2 Leptonic Decays

The lowest-order expressions of electronic decay width the first-order QCD corrections [26] are

(14)
(15)

where is the c-quark charge in units of , is the fine-structure constant, and are the S-wave and D-wave state mass respectively with the radial excitation number . Note that here and the are essentially the strong coupling constants of different mass scales, and we adopt as in Ref. [27, 28]. is the radial wave function at the origin, and is the second derivative of the radial -wave function at the origin. Within the Gaussian basis space, the analytic formula for is explicitly presented in the APPENDIX. Table 3 compares the leptonic decay given in Ref. [27] with our results in NR and MNR models.

Particle State Ref.[27] Ref.[29] NR MNR Expt.[25]

11.8 6.6 12.13 5.6 3.1 6.0 3.3 5.60.140.02
4.29 2.4 5.03 2.3 1.3 2.2 1.2 2.40.07
2.53 1.42 3.48 1.9 1.0 1.8 0.98 0.860.07
1.25 0.7 2.63 1.3 0.70 1.3 0.70 0.580.07
0.055 0.031 0.056 0.089 0.050 0.079 0.044 0.270.018
0.066 0.037 0.096 0.15 0.084 0.13 0.073 0.830.07
Table 3: Leptonic decay width, in units of keV.

3.3 Two-photon Decay

The two-photon decay widths are important to identify the potential charmonium states, i.e. , etc. and the latest , . With the first-order QCD radiative corrections [26], the two-photon decay widths of , and explicitly are

(15)
(16)
(17)

where is the first derivative of the radial -wave function at the origin. Our numerical results of two-photon decay widths are shown in Table 4 in comparison with Refs.[28, 30].

State Ref. [28] Ref. [30] NR MNR Expt. [25]

5.5 10.94 7.4 7.5 6.7
1.8 3.2 2.9
2.9 2.5
2.0 1.8
2.9 6.38 11 10.8 2.290.18
1.9 7.7 6.7
7.9 6.5
0.50 0.57 0.29 0.27 0.500.03
0.52 0.43 0.39
0.81 0.66
Table 4: Two-photon decay width, in units of keV.

3.4 Radiative Transition

Because radiative transition is sensitively dependent on the detailed features of the wave functions, it is of great interest to have careful inspection. Besides, it has been noticed that the known M1 rates showing serious disagreement between the previous theoretical calculation [21] and experiment. In our full-potential calculation framework, the wave functions are directly associated with the eigenvectors of Hamiltonian corresponding to the masses. The E1 transition rate between an initial charmonium state of radial quantum number , orbital angular momentum , spin , and total angular momentum , and a final state is given by Ref. [31] as

(17)
(18)

In the above formulas, is the charge of c-quark in unit of , and , represent the eigen mass of initial state and the total energy of final state respectively. The momentum of the final photon equals in the nonrelativistic approximation [32]. The angular matrix element is

(19)

The Gaussian expanded wave functions give rise to the analytic formulas for the overlap integral, i.e. Eq. (A.8), and the transition matrix elements Eq. (A.10). The stands for the dimension of Gaussian basis as defined above. The numerical results of E1 transition rates are presented in Table 5 and Table 6, as well as those of M1 transition in Table 7.

Ref.[21] Ours [25] [keV]
Initial meson Final meson [MeV] [keV] [MeV] [keV]
NR GI NR GI NR MNR NR MNR

128 128 38 24 128 123 43 39
171 171 54 29 177 163 48 38
261 261 63 26 315 305 34 29
111 119 49 36 130 118 72 56
455 508 0.70 12.7 512 494 6 6.44
494 547 0.53 0.85 556 530 0.76 0.44
577 628 0.27 0.63 680 657 8.9 8.1
485 511 9.1 28 519 496 6.1 7.7
429 429 424 313 436 440 421 405
390 389 314 239 391 404 330 341
303 303 152 114 256 267 97 104
504 496 498 352 485 506 465 473
276 282 304 207 287 278 300 264
232 258 183 183 243 242 243 234
162 223 64 135 151 155 77 83
285 305 280 218 287 284 297 274
779 784 81 53 794 787 108 111
741 763 71 14 757 756 27 33
681 733 56 1.3 677 681 30 28
839 856 140 85 839 846 104 116
163 128 88 29 162 157 81 76
168 139 17 5.6 168 150 14 10
197 204 1.9 1.0 190 161 0.98 0.64
123 113 35 18 124 113 37 30
152 179 22 21 145 124 16 11
81 143 13 51 50 34 4.2 1.4
133 117 60 27 135 125 61 51
Table 5: 2S, 3S, 1P and 2P E1 radiative transitions
Ref. [21] Ours [25]
Initial meson Final meson [MeV] [keV] [MeV] [keV] [keV]
NR GI NR GI NR MNR NR MNR

268 231 509 199 273 257 434 360
225 212 303 181 212 206 292 272
159 188 109 145 115 119 63 72
229 246 276 208 254 244 377 332
585 602 55 30 645 617 112 97
545 585 45 8.9 589 570 30 31
484 563 32 0.045 501 490 25 19
593 627 75 43 633 612 95 90
1048 1063 34 19 1106 1081 62 59
1013 1048 31 2.2 1057 1040 10 11
960 1029 27 1.5 980 971 23 22
1103 1131 72 38 1135 1126 73 76
147 118 148 51 196 182 279 231
156 128 31 9.9 205 180 48 34
155 141 2.1 0.77 227 194 3.2 2.1
112 108 58 35 142 128 121 92
111 121 19 15 164 142 50 34
43 97 4.4 35 66 53 21 11
119 109 99 48 157 142 205 158
481 461 0.049 6.8 532 506 1.6 1.4
486 470 0.0091 0.13 538 500 0.76 0.42
512 530 0.00071 0.001 557 510 0.11 0.062
445 452 0.035 4.6 480 452 1.2 1.8
472 512 0.014 0.39 500 462 0.11 0.00029
410 490 0.037 9.7 409 380 28 30
453 454 0.16 5.7 495 466 0.21 0.56
242 282 272 296 254 245 340 302
236 272 64 66 248 252 79 82
278 314 307 268 295 290 321 301
208 208 4.9 3.3 227 241 6.8 8.1
250 251 125 77 274 280 146 153
338 338 403 213 409 417 367 362
264 307 339 344 281 277 398 374
566 609 29 16 584 563 23 23
558 602 7.1 0.62 576 565 0.85 1.5
597 640 26 23 619 600 38 41
559 590 0.79 0.027 556 552 0.18 0.080
598 628 14 3.4 599 588 3.6 5.5
677 707 27 35 722 713 99 109
585 634 40 25 607 589 38 41
Table 6: 3P, 1D and 2D E1 radiative transitions
Initial Final Ref. [21] Ours [keV]
meson meson [MeV] [keV] [MeV] [keV]
NR GI NR GI NR MNR NR MNR

116 115 2.9 2.4 93 107 1.5 2.2 [25]
48 48 0.21 0.17 35 38 0.086 0.096 [33]
639 638 4.6 9.6 627 639 3.1 3.8 [34]
501 501 7.9 5.6 518 516 6.1 6.9
29 35 0.046 0.067 28 29 0.043 0.044
382 436 0.61 2.6 429 416 0.70 0.71
922 967 3.5 9.0 960 958 3.2 3.7
312 361 1.3 0.84 371 356 1.7 1.6
810 856 6.3 6.9 867 854 5.9 6.5
360 380 0.071 0.11 374 364 1.3 1.2
400 420 0.058 0.36 419 401 0.16 0.13
485 504 0.033 1.5 548 534 5.6 5.3
430 435 0.067 1.3 438 421 1.0 0.89
388 412 0.050 0.045 397 387 0.15 0.13
321 379 0.029 0.50 308 303 4.8 4.8
340 323 1.7 1.5
382 358 0.17 0.12
470 442 3.5 2.7
712 685 2.3 2.1
753 719 0.27 0.22
871 840 8.9 8.2
417 391 1.2 0.99
783 747 1.8 1.5
357 342 0.16 0.12
729 703 0.22 0.20
263 257 3.7 3.3
644 626 5.3 5.1
Table 7: M1 radiative partial widths

4 Discussion

4.1 , ,

The world-average is keV [25]. The significant leptonic width implies that there is a sizeable S-D mixing between the and state, since it is expected to be highly suppressed if is a pure D-wave state (shown in Table 3). The mixing arises both from the usual relativistic correction terms and coupling to strong decay channels, and will affect the E1 transition rates of and [32]. We investigate the mixing under the present calculation framework from two scenarios respectively: electronic annihilation and dipole transition. Assuming the and to be a mixture of a and a state as

(19)

Then, Eqs. (14, 15) can be expressed as

(20)
(21)

Through the fitting of the ratio , combining the experimental data [25] and the radial wave function values at the origin, i.e. GeV and GeV, we find two solutions for the mixing angle:

(22)

These solutions are smaller than or determined by leptonic decay widths in [35, 27], partially due to more precise measurements. Impending, we illustrate the mixing-dependency of E1 transitions of our model in Fig. 1 and Fig. 2. The decay widths of , obviously favor , while the matching angle for locates around and