^{1}

^{3}.

In this paper we revisited phenomenological potentials. We studied S-wave heavy quarkonium spectra by two potential models. The first one is power potential and the second one is logarithmic potential. We calculated spin averaged masses, hyperfine splittings, Regge trajectories of pseudoscalar and vector mesons, decay constants, leptonic decay widths, two-photon and two-gluon decay widths, and some allowed M1 transitions. We studied ground and 4 radially excited S-wave charmonium and bottomonium states via solving nonrelativistic Schrödinger equation. Although the potentials which were studied in this paper are not directly QCD motivated potential, obtained results agree well with experimental data and other theoretical studies.

Heavy quarkonium is the bound state of

Foremost, being a nonabelian gauge theory, gluons which are gauge bosons, have color charge and interact among themselves. Unlike from quantum electrodynamics (QED), where a photon does not interact with other photon, in QCD one must consider interactions among gluons. This nonabelian nature of the theory makes some calculations complicated, for example, loops in propagators.

There are three other important features of QCD:

The present aspects of the QCD caused other approaches to deal with these challenges. QCD sum rules, Lattice QCD, and potential models (quark models) are examples of these approaches. These approaches are nonperturbative since the strong interaction coupling constant, which should be the perturbation parameter of QCD is of the order one in low energies, hence the truncation of the perturbative expansion cannot be carried out. Since perturbation theory is not applicable, a nonperturbative approach has to be used to study systems that involve strong interactions. QCD sum rules and lattice QCD are based on QCD itself whereas in potential models, one assumes an interquark potential and solves a Schrödinger-like equation. The advantage of potential model is that, excited states can be studied in the framework of potential models whereas in QCD sum rules and lattice QCD, only the ground state or in some exceptional cases excited states can be studied.

After the discovery of charmonium (

In the potential models, many features such as mass spectra and decay properties of heavy quarkonium could be described by an interquark potential in two-body Schrödinger equation. Interquark potentials are obtained both from phenomenology and theory. In the phenomenological method, it is assumed that a potential exist with some parameters to be determined by fits to the data. In the theory side, one can use perturbative QCD to determine the potential form at short distances and use lattice QCD at long distances [

The potential model calculations have been quite successful in describing the hadron spectrum. Most of the phenomenological potentials must satisfy the following conditions:

The great success of quarkonium phenomenology was somehow cracked at 2003 after the observation of

Energy spectra of heavy quarkonium are a rich source of the information on the nature of interquark forces and decay mechanisms. The prediction of mass spectrum in accordance with the experimental data does not verify the validity of a model for explaining hadronic interactions. Different potentials can produce reliable spectra with the experimental data. Thus other physical properties such as decay constants, leptonic decay widths, radiative decay widths, etc. need to be calculated.

A specific form of the QCD potential in the whole range of distances is not known. Therefore one needs to use potential models. In this work we revisited a power-law potential [

When quark model was proposed, many authors treated baryons in detail with the harmonic oscillator quark model by using harmonic oscillator wave functions [

In order to obtain mass spectra, we solved Schrödinger equation by variational method. The variational method by using harmonic oscillator wave function gave successful results for heavy and light meson spectrum [

Armed with these, the expectation value of the given Hamiltonian can be calculated. In the variational method, one chooses a trial wave function depending on one or more parameters and then finds the values of these parameters by minimizing the expectation value of the Hamiltonian. It is a good tool for finding ground state energies but as well as energies of excited states. The condition for obtaining excited states energies is that the trial wave function should be orthogonal to all the energy eigenfunctions corresponding to states having a lower energy than the energy level considered. In (

In the following sections we study power-law and logarithmic potentials in order to obtain full spectrum.

Power-law potential is given by [

At first step we obtained spin averaged mass spectrum for

Spin-averaged mass spectrum of charmonium (in MeV).

State | Power | Logarithmic | [ | [ |
---|---|---|---|---|

1S | 3067 | 3067 | 3067 | 3117 |

2S | 3701 | 3655 | 3667 | 3684 |

3S | 4054 | 3980 | 4121 | 4078 |

4S | 4306 | 4204 | 4513 | 4407 |

5S | 4504 | 4376 | 4866 |

Spin-averaged mass spectrum of Bottomonium (in MeV).

State | Power | Logarithmic | [ | [ |
---|---|---|---|---|

1S | 9473 | 9444 | 9443 | 9523 |

2S | 10049 | 10033 | 9729 | 10035 |

3S | 10384 | 10357 | 10312 | 10373 |

4S | 10624 | 10581 | 10593 | |

5S | 10813 | 10753 | 10840 | |

6S | 10986 | 10964 | 11065 |

Since the interquark potential does not contain the spin dependent part, (

Mass splitting is closely connected with the Lorentz-structure of the quark potential [

Charmonium mass spectrum (in MeV). In [

State | Exp. [ | Power | Logarithmic | [ | [ | [ | [ |
---|---|---|---|---|---|---|---|

| 2984 | 2980 | 2954 | 2979 | 2982 | 2983 | 2984 |

| 3639 | 3624 | 3555 | 3623 | 3630 | 3635 | 3637 |

| 3983 | 3887 | 3991 | 4043 | 4048 | 4004 | |

| 4240 | 4117 | 4250 | 4384 | 4388 | 4264 | |

| 4441 | 4294 | 4446 | 4690 | 4459 | ||

| 3097 | 3096 | 3104 | 3097 | 3090 | 3097 | 3097 |

| 3686 | 3727 | 3689 | 3673 | 3672 | 3679 | 3679 |

| 4040 | 4078 | 4011 | 4022 | 4072 | 4078 | 4030 |

| 4328 | 4233 | 4273 | 4406 | 4412 | 4281 | |

| 4525 | 4403 | 4463 | 4711 | 4472 |

Bottomonium mass spectrum (in MeV).

State | Exp. [ | Power | Logarithmic | [ | [ | [ | [ |
---|---|---|---|---|---|---|---|

| 9399 | 9452 | 9420 | 9389 | 9390 | 9402 | 9455 |

| 9999 | 10030 | 10011 | 9987 | 9990 | 9976 | 9990 |

| 10367 | 10338 | 10330 | 10326 | 10336 | 10330 | |

| 10608 | 10562 | 10595 | 10584 | 10623 | ||

| 10798 | 10735 | 10817 | 10800 | 10869 | ||

| 11005 | 10990 | 11011 | 10988 | 11097 | ||

| 9460 | 9480 | 9452 | 9460 | 9460 | 9465 | 9502 |

| 10023 | 10055 | 10040 | 10016 | 10015 | 10003 | 10015 |

| 10355 | 10393 | 10364 | 10351 | 10343 | 10354 | 10349 |

| 10579 | 10629 | 10588 | 10611 | 10597 | 10635 | 10607 |

| 10865 | 10818 | 10759 | 10831 | 10811 | 10878 | 10818 |

| 11019 | 11019 | 11006 | 11023 | 10997 | 11102 | 10995 |

The mass differences are shown in Tables

Mass differences of S-wave charmonium states (in MeV).

State | Exp. [ | Power | Logarithmic | [ | [ | [ | [ |
---|---|---|---|---|---|---|---|

| 113 | 116 | 150 | 118 | 108 | 114 | 113 |

| 47 | 103 | 134 | 50 | 42 | 44 | 42 |

| 95 | 124 | 31 | 29 | 30 | 26 | |

| 88 | 116 | 23 | 22 | 24 | 17 | |

| 84 | 109 | 17 | 21 | 13 |

Mass differences of S-wave bottomonium states (in MeV).

State | Exp. [ | Power | Log | [ | [ | [ | [ |
---|---|---|---|---|---|---|---|

| 61 | 28 | 32 | 71 | 70 | 63 | 47 |

| 24 | 25 | 29 | 29 | 25 | 27 | 25 |

| 26 | 26 | 21 | 17 | 18 | 19 | |

| 21 | 26 | 16 | 13 | 12 | ||

| 20 | 24 | 14 | 11 | 9 | ||

| 14 | 16 | 12 | 9 |

As can be seen from Tables

The Regge trajectories for pseudoscalar and vector mesons are shown in Figures

Regge trajectories of pseudoscalar charmonium in

Regge trajectories of vector charmonium in

Regge trajectories of pseudoscalar bottomonium in

Regge trajectories of vector bottomonium in

As can be seen from figures, Regge trajectories show nonlinear behaviour.

Leptonic decay constants give information about short distance structure of hadrons. In the experiments this regime is testable since the momentum transfer is very large. The pseudoscalar (

The matrix elements can be calculated by quark model wave function in the momentum space. The result is

In the nonrelativistic limit, these two equations take a simple form which is known to be Van Royen and Weisskopf relation [

The first-order correction which is also known as QCD correction factor is given by

Pseudoscalar decay constants (in MeV).

State | Exp. [ | Power | Power | Logarithmic | Logarithmic | [ | [ | [ |
---|---|---|---|---|---|---|---|---|

| 335 | 543 | 415 | 578 | 442 | 471 | 360 | 402 |

| 473 | 362 | 497 | 380 | 374 | 286 | 240 | |

| 330 | 252 | 442 | 338 | 332 | 254 | 193 | |

| 325 | 248 | 412 | 315 | 312 | 239 | ||

| 253 | 193 | 387 | 304 | ||||

| 517 | 431 | 585 | 488 | 834 | 694 | 599 | |

| 479 | 400 | 535 | 447 | 567 | 472 | 411 | |

| 345 | 288 | 504 | 421 | 508 | 422 | 354 | |

| 313 | 261 | 482 | 402 | 481 | 401 | ||

| 283 | 236 | 465 | 388 | ||||

| 208 | 186 | 434 | 374 |

Vector decay constants (in MeV).

State | Exp. [ | Power | Power | Logarithmic | Logarithmic | [ | [ | [ |
---|---|---|---|---|---|---|---|---|

| 335 | 529 | 363 | 563 | 386 | 462 | 317 | 393 |

| 279 | 463 | 318 | 487 | 334 | 369 | 253 | 293 |

| 174 | 324 | 222 | 436 | 299 | 329 | 226 | 258 |

| 319 | 219 | 406 | 279 | 310 | 212 | ||

| 248 | 170 | 382 | 262 | 290 | 199 | ||

| 708 | 516 | 402 | 584 | 455 | 831 | 645 | 665 |

| 482 | 482 | 373 | 535 | 416 | 566 | 439 | 475 |

| 346 | 350 | 269 | 504 | 393 | 507 | 393 | 418 |

| 325 | 316 | 243 | 482 | 375 | 481 | 373 | 388 |

| 369 | 285 | 222 | 464 | 362 | 458 | 356 | 367 |

| 241 | 203 | 442 | 354 | 439 | 341 |

Leptonic decay of a vector meson with

Charmonium leptonic decay widths (in keV). The widths calculated with and without QCD corrections are denoted by

Power | Logarithmic | [ | [ | Exp. [ | |||||
---|---|---|---|---|---|---|---|---|---|

State | | | | | | | | | |

| 3.435 | 1.277 | 3.154 | 1.173 | 11.8 | 6.60 | 6.847 | 2.536 | 5.55 |

| 2.880 | 1.071 | 2.362 | 0.878 | 4.29 | 2.40 | 3.666 | 1.358 | 2.33 |

| 2.153 | 0.800 | 1.888 | 0.702 | 2.53 | 1.42 | 2.597 | 0.962 | 0.86 |

| 1.839 | 0.684 | 1.642 | 0.610 | 1.73 | 0.97 | 2.101 | 0.778 | 0.58 |

| 1.590 | 0.591 | 1.551 | 0.576 | 1.25 | 0.70 | 1.710 | 0.633 |

Bottomonium leptonic decay widths (in keV). The widths calculated with and without QCD corrections are denoted by

Power | Logarithmic | [ | [ | Exp. [ | |||||
---|---|---|---|---|---|---|---|---|---|

State | | | | | | | | | |

| 0.817 | 0.456 | 0.847 | 0.473 | 2.31 | 1.60 | 1.809 | 0.998 | 1.340 |

| 0.686 | 0.383 | 0.709 | 0.396 | 0.92 | 0.64 | 0.797 | 0.439 | 0.612 |

| 0.610 | 0.340 | 0.630 | 0.352 | 0.64 | 0.44 | 0.618 | 0.341 | 0.443 |

| 0.557 | 0.311 | 0.576 | 0.322 | 0.51 | 0.35 | 0.541 | 0.298 | 0.272 |

| 0.526 | 0.294 | 0.535 | 0.299 | 0.42 | 0.29 | 0.481 | 0.265 | 0.31 |

| 0.492 | 0.278 | 0.501 | 0.282 | 0.31 | 0.22 | 0.432 | 0.238 | 0.130 |

Two-photon decay widths (in keV). The widths calculated with and without QCD corrections are denoted by

Power | Logarithmic | [ | [ | [ | Exp. [ | ||||
---|---|---|---|---|---|---|---|---|---|

State | | | | | | | | | |

| 1.10 | 0.664 | 1.450 | 0.869 | 11.17 | 6.668 | 3.69 | 7.18 | 7.2 |

| 0.987 | 0.592 | 1.291 | 0.774 | 8.48 | 5.08 | 1.4 | 1.71 | |

| 0.907 | 0.543 | 1.184 | 0.710 | 7.57 | 4.53 | 0.930 | 1.21 | |

| 0.847 | 0.508 | 1.105 | 0.662 | 0.720 | ||||

| 0.801 | 0.480 | 1.044 | 0.620 | |||||

| 0.277 | 0.199 | 0.277 | 0.199 | 0.58 | 0.42 | 0.214 | 0.45 | |

| 0.212 | 0.153 | 0.246 | 0.177 | 0.29 | 0.20 | 0.121 | 0.11 | |

| 0.195 | 0.142 | 0.226 | 0.162 | 0.24 | 0.17 | 0.09 | 0.063 | |

| 0.188 | 0.136 | 0.211 | 0.151 | 0.07 | ||||

| 0.176 | 0.129 | 0.199 | 0.143 | |||||

| 0.164 | 0.116 | 0.182 | 0.134 |

Two-gluon decay width is given by [

The terms in the parenthesis refer to QCD corrections. The obtained results are given in Table

Two-gluon decay widths (in MeV). The widths calculated with and without QCD corrections are denoted by

Power | Logarithmic | [ | [ | Exp. [ | ||||
---|---|---|---|---|---|---|---|---|

State | | | | | | | | |

| 32.04 | 50.15 | 41.93 | 32.44 | 50.82 | 66.68 | 15.70 | 26.7 |

| 28.55 | 44.70 | 37.32 | 24.64 | 38.61 | 5.08 | 8.10 | 14 |

| 26.22 | 41.04 | 34.23 | 53.59 | 21.99 | |||

| 24.50 | 38.35 | 31.96 | 50.03 | ||||

| 23.15 | 36.24 | 30.18 | 47.24 | ||||

| 5.50 | 7.50 | 12.82 | 17.49 | 13.72 | 18.80 | 11.49 | |

| 4.90 | 6.69 | 11.41 | 15.56 | 6.73 | 9.22 | 5.16 | |

| 4.50 | 6.14 | 10.46 | 14.28 | 5.58 | 7.64 | 3.80 | |

| 4.20 | 5.74 | 9.77 | 13.33 | ||||

| 3.97 | 5.42 | 9.22 | 12.58 | ||||

| 3.62 | 5.18 | 8.68 | 10.86 |

M1 (magnetic dipole transition) decay widths can give more information about spin-singlet states. Moreover M1 transition rates show the validity of theory against experiment [

Radiative M1 decay widths of charmonium. In [

Initial | Final | Power | Logarithmic | [ | [ | Exp. [ | |||
---|---|---|---|---|---|---|---|---|---|

| | | | | | | | ||

| | 114.9 | 1.96 | 113.8 | 2.83 | 3.28 | 2.39 | 2.44 | 1.13 |

| | 111.5 | 1.39 | 101.5 | 2.01 | 1.45 | 0.19 | 0.19 | |

| | 93.8 | 1.10 | 93.8 | 1.59 | 0.051 | 0.088 | ||

| | 87.1 | 0.88 | 87.1 | 1.27 | ||||

| | 83.2 | 0.74 | 83.2 | 1.10 |

Radiative M1 decay widths of bottomonium.

Initial | Final | Power | Logarithmic | [ | [ | [ | Exp. [ | ||
---|---|---|---|---|---|---|---|---|---|

| | | | | | | | ||

| | 27.9 | 0.88 | 31.9 | 1.46 | 5.8 | 10 | 9.34 | |

| | 24.9 | 0.62 | 28.9 | 1.09 | 1.4 | 0.59 | 0.58 | |

| | 25.9 | 0.54 | 25.9 | 0.78 | 0.8 | 0.25 | 0.66 | |

| | 20.9 | 0.37 | 20.9 | 0.41 | ||||

| | 19.9 | 0.32 | 19.9 | 0.35 | ||||

| | 14.3 | 0.29 | 14.4 | 0.27 |

In the present paper we studied S-wave heavy quarkonium spectra with two phenomenological potentials. We have computed spin averaged masses, hyperfine splittings, Regge trajectories for pseudoscalar and vector mesons, decay constants, leptonic decay widths, two-photon and gluon decay widths, and allowed M1 partial widths of S-wave heavy quarkonium states.

In general, most of the quark model potentials tend to be similar, having a Coulomb term and a linear term. For example, in [

Spin averaged mass spectra give idea about the formulation of model since the results are close to experimental values due to contributions from spin dependent interactions are small compared to contribution from potential part. If one ignores all spin dependent interactions, obtained results under this assumption are thought to be averages over related spin states for principal quantum number. Including hyperfine interaction, we obtained the mass spectra for pseudoscalar and vector mesons. The obtained spectra for both charmonium and bottomonium are in good agreement with the experimentally observed spectra and other theoretical studies.

Both power and logarithmic potentials produced approximately same mass differences and are in agreement with experiment for the lowest state in charmonium sector. But for the highest states, the shift is not compatible with the references. The reason for this should be the behaviour of linear part of the potential. In the case of bottomonium sector, mass differences of both power and logarithmic potentials are in accord with the given studies except the lowest state.

The fundamental point in the Regge trajectories is that they can predict masses of unobserved states. For the hadrons constituting of light quarks, the Regge trajectories are approximately linear but for the heavy quarkonium case Regge trajectories can be nonlinear. In the present work, we found that all Regge trajectories show nonlinear properties.

The decay constants are calculated for pseudoscalar and vector mesons by equating their field theoretical definition with the analogous quark model potential definition. This is valid in the nonrelativistic and weak binding limits where quark model state vectors form good representations of the Lorentz group [

Obtained leptonic decay widths are comparable with the experimental values and other theoretical studies. The QCD corrected factors are more close to experimental values for power and logarithmic potential and this can be referred as the importance of the QCD correction factor in calculating the decay constants and other short range phenomena using potential models.

Finally M1 transitions are calculated. The M1 radiative decay rates are very sensitive to relativistic effects. Even for allowed transitions relativistic and nonrelativistic results differ significantly. An important example is the decay of

Some states in the charmonium and bottomonium sector show properties different from the conventional quarkonium state. Some examples are

Exotic states. Experimental data are taken from [

Mass | Strong decay | Two-photon decay | |||||||
---|---|---|---|---|---|---|---|---|---|

Power | Logarithmic | Experiment | Power | Logarithmic | Experiment | Power | Logarithmic | Experiment | |

| | | |||||||

| 3983 | 3887 | 41.04 | 53.59 | 0.543 | 0.710 | |||

| |||||||||

| | | | ||||||

| 4240 | 4117 | 38.35 | 50.03 | 0.508 | 0.612 | |||

| |||||||||

| | | | ||||||

| 4441 | 4297 | 36.24 | 47.24 | 0.480 | 0.620 |

Looking at Table

Quark potential models have been very successful to study on various properties of mesons. The short distance behaviour of interquark potential appears to be similar where QCD perturbation theory can be applied where at large distance the potential is linear in

No data were used to support this study.

The author declares that they have no conflicts of interest.

_{s}

_{MS}dependence

_{MS}dependence”

^{*}and N

^{*}resonances in the quark model

_{c}meson