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

###### 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 |

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 |

### 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 |

### 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

### 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 |

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 |

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 |

## 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) |

(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