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Halil Mutuk, "S-Wave Heavy Quarkonium Spectra: Mass, Decays, and Transitions", Advances in High Energy Physics, vol. 2018, Article ID 5961031, 12 pages, 2018. https://doi.org/10.1155/2018/5961031
S-Wave Heavy Quarkonium Spectra: Mass, Decays, and Transitions
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 and and one of the most important playgrounds for our understanding of the strong interactions of quarks and gluons. Quantum chromodynamics (QCD) is thought to be the true theory of these strong interactions. QCD is a nonabelian local gauge field theory with the symmetry group . In principle, one should be able to calculate hadronic properties such as mass spectrum and transitions by using QCD principles. But QCD does not readily supply us these hadronic properties. This challenge can be attributed to the several features that are not present in other local gauge field theories.
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: asymptotic freedom, confinement, and dynamical breaking of chiral symmetry. Asymptotic freedom says that strong interaction coupling constant, , is a function of momentum transfer. When the momentum transfer in a quark-quark collision increases (at short distances), the coupling constant becomes weaker whereas it becomes larger when momentum transfer decreases (at large distances). The idea behind confinement is that, there are no free quarks outside of a hadron; i.e., color charged particles (quarks and gluons) cannot be isolated out of hadrons. Flux tube model gives a reasonable explanation of confinement. When the distance between quark-antiquark (or quarks) pair increases, the gluon field between a pair of color charges forms a flux tube (or string) between them resulting a potential energy which depends linearly on the distance, where is the string constant. As distance increases between quarks, the potential energy can create new quark-antiquark pairs in colorless forms instead of a free quark. Up to now, nobody has been able to prove that confinement from QCD. Lattice QCD calculations simulate this confinement well and give a value for the string tension . The last feature of QCD is the dynamical breaking of chiral symmetry. The QCD Lagrangian with quark flavor has an exact chiral symmetry but breaks down to symmetry because of the nonvanishing expectation value of QCD vacuum [2, 3]. The Goldstone bosons corresponding to this symmetry breaking are the pseudoscalar mesons.
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 () states, potential models have played a key role in understanding of heavy quarkonium spectroscopy [4, 5]. These potentials were in type of Coulomb plus linear confining potential with spin dependent interactions. The discovery of bottomonium () states were well described by the potential model picture which was used in the charmonium case. Heavy quarkonium spectroscopy was studied since that era with fruitful works [6–18]. A general review about potential models can be found in [19, 20] and references therein.
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 . These potentials can be classified as QCD motivated potentials [21–25] and phenomenological potentials [26–31]. The most commonly used phenomenological potentials are power-law potentials, for example  and logarithmic potentials, for example . The detailed properties of these type potentials are studied extensively in . All the potentials which are mentioned here have almost similar behaviour in the range of which is characteristic region of charmonium and bottomonium systems [32, 33]. Outside the range, the behaviour of potentials differ. Up to now, no one was able to obtain a potential which is compatible at the whole range of distances by using QCD principles.
The potential model calculations have been quite successful in describing the hadron spectrum. Most of the phenomenological potentials must satisfy the following conditions:It means that static potential is a monotone nondecreasing and concave function of which is a general property of gauge theories .
The great success of quarkonium phenomenology was somehow cracked at 2003 after the observation of . The properties of this exotic particle are not compatible with the conventional quark model, the reason why it is named exotic. For example in , the authors studied near threshold zero in the S-wave. There are other exotic states, , and the exotic particle zoo is growing. In this paper we will present some exotic states in the framework of quark model.
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  and a logarithmic potential  to study S-wave heavy quarkonium. These potentials satisfy Eqn. (1), i.e. having nonsingular behaviour for . For our purposes, it must be mentioned that power-law and logarithmic potentials have nice scaling properties when used with a nonrelativistic Schrödinger equation . We generated S-wave charmonium and bottomonium mass spectrum with the decays and M1 transitions. At Section 2 we give out theoretical model. In Sections 3 and 4, we generate S-wave heavy quarkonium spectrum, decays and transitions. In Section 5 we discuss our results and in Section 6 we conclude our results.
2. Formulation of the Model
When quark model was proposed, many authors treated baryons in detail with the harmonic oscillator quark model by using harmonic oscillator wave functions [37–39]. Mesons comparing to baryons are simpler objects since they are composites of two quarks. The reason for using harmonic oscillator wave function is that they form a complete set for a confining potential .
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 [15, 41, 42]. The procedure for this method is calculating expectation value of the Hamiltonian via the trial wave function:The mass of the meson is found by adding two times the mass of quark to the eigenenergyThe Hamiltonian we consider iswhere , is the relative momentum, is the reduced mass, and is the potential between quarks. The spectrum can be obtained via solving Schrödinger equationwith the harmonic oscillator wave function defined asHere is the radial wave function given aswith the associated Laguerre polynomials and the normalization constant is the well-known spherical harmonics.
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 (7), is treated as a variational parameter and it is determined for each state by minimizing the expectation value of the Hamiltonian.
In the following sections we study power-law and logarithmic potentials in order to obtain full spectrum.
3. Mass Spectra of Power-Law and Logarithmic Potentials
Power-law potential is given by They showed that upsilon and charmonium spectra can be fitted with that potential. The small power of refers to a situation in which the spacing of energy levels is independent of the quark masses. This situation is also valid for the purely logarithmic potential 
At first step we obtained spin averaged mass spectrum for and systems, respectively. The constituent quark masses are and for power-law potential and and for logarithmic potential. Table 1 shows the charmonium spectrum and Table 2 shows the bottomonium spectrum.
Since the interquark potential does not contain the spin dependent part, (2) gives the spin averaged mass for the corresponding states. The calculated masses agree well with the available experimental data and with the values obtained from other theoretical studies. A general potential usually includes spin-spin interaction, spin-orbit interaction, and tensor force terms. To obtain whole picture, it is necessary to consider spin dependent terms within the potential. For , there are spin-orbit and tensor force terms which contribute to the fine structure. For equal mass , the spin-orbit interaction is given by and is responsible for the wave splittings. Again for equal mass , the tensor potential is given byFor , there is spin-spin term which we will consider in the present work. In the model of the spin averaged mass spectra discussion, all the spin dependent effects are ignored and hence it fails to take into account the splittings due to spin. For example, such splitting exist between the (1S) and mesons by . These mesons occupy the level. The in the (1S) have , while in the , . As a result of this, the mass difference should be related to spin dependent interaction.
3.1. Spin-Spin Interaction
Mass splitting is closely connected with the Lorentz-structure of the quark potential . The origin of the spin-spin interaction term lies in the one-gluon exchange term which is related to . Spin is proportional of the magnetic moment of a particle. Magnetic moments generate short range fields . In the case of heavy quarkonium systems which are nonrelativistic, wave functions of two particles overlap in a significant amount. This means that particles are very close to each other. So spin-spin interactions play a significant role in the dynamics. The spin-spin interaction term of two particles can be written asThis term can explain wave splittings and has no contribution to states. Putting this term into Schrödinger equation we getImplementing Dirac-delta function property we obtainThe matrix element of spin products can be obtained via so thatTherefore we obtain hyperfine splittings energy asHere is the wave function at the origin and can be obtained by the following relation:Expectation value is obtained by the wave function given in (6). S-wave charmonium and bottomonium masses can be seen in Tables 3 and 4. In this calculation, is taken to be 0.37 for charmonium and 0.26 for bottomonium .
As can be seen from figures, Regge trajectories show nonlinear behaviour.
4. Dynamical Properties
4.1. Decay Constants
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 () and the vector () decay constants are defined, respectively, through the matrix elements andIn the first relation, is meson momentum and is pseudoscalar meson state. In the second relation, is mass, is the polarization vector, and is the state vector of meson.
The matrix elements can be calculated by quark model wave function in the momentum space. The result isfor pseudoscalar meson andfor the vector meson .
In the nonrelativistic limit, these two equations take a simple form which is known to be Van Royen and Weisskopf relation  for the meson decay constants
The first-order correction which is also known as QCD correction factor is given bywhere is given by Here for pseudoscalar mesons and for vector mesons. Decay constants are given in Tables 7 and 8 for pseudoscalar and vector mesons, respectively.
4.2. Leptonic Decay Widths
Leptonic decay of a vector meson with quantum numbers can be pictured by the following annihilation via a virtual photonThis state is neutral and in principle can decay into a different lepton pair rather than electron-positron pair. The above amplitude can be calculated by the Van Royen and Weisskopf relation where is the fine structure constant, is the quark charge, is the mass of state, and is the wave function at the origin of initial state. The term in the parenthesis is the first-order QCD correction factor while represents higher corrections. The obtained values for leptonic decay widths can be found in Tables 9 and 10 for charmonium and bottomonium, respectively.
4.3. Two-Photon Decay Width
states with quantum number of charmonium and bottomonium can decay into two photons. In the nonrelativistic limit, the decay width for state can be written as The term in the parenthesis is the first-order QCD radiative correction. The results are listed in Table 11.
4.4. Two-Gluon Decay Width
Two-gluon decay width is given by 
The terms in the parenthesis refer to QCD corrections. The obtained results are given in Table 12.
4.5. M1 Transitions
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 . Magnetic transitions conserve both parity and orbital angular momentum of the initial and final states but in the M1 transitions the spin of the state changes. M1 width between two S-wave states is given by where is the photon energy and is the zeroth-order spherical Bessel function. In the case of small , spherical Bessel function tends to 1, . Thus transitions between the same principal quantum numbers, , are favored and usually known to be allowed. In the other case, when the overlap integral between initial and final state is 0 and generally designated as forbidden transitions. The obtained transition rates for the allowed ones of S-wave charmonium and bottomonium states are given in Tables 13 and 14, respectively.
5. Results and Discussion
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  they used standard color Coulomb plus linear scalar form, and also included a Gaussian smeared contact hyperfine interaction in the zeroth-order potential. In , the authors used a nonrelativistic potential model with screening effect. In  nonrelativistic linear potential and screened potential, in [14, 16, 44] a modified of nonrelativistic potential models and in  Hulthen potential are used. Potential models give reliable results with the appropriate parameters in the model. Therefore, the shape of the potential at the limits and have similar behaviours.
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 [52, 53]. For pseudoscalar mesons, the corrected value of power and logarithmic potentials are a few MeV above than the available experimental data. For the rest of the pseudoscalar mesons, obtained results are compatible with other studies. In the case of vector mesons, logarithmic potential gave higher values than power potential. In the meson, when the radially states are excited, both two potential gave similar results within the error of experimental value. Computations of the vector decay constant beyond the weak binding limit can be important in the quark potential model frame and need more elaboration .
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.
levels of charmonium and bottomonium states can decay into two photons or gluons. Especially two- photon decays of these levels are important for understanding the accuracy of theoretical models. Obtained results are smaller than the nonrelativistic widths including the one-loop QCD correction factor. For example, results of power and logarithmic potentials in (1S) are not in accord with experimental data. The reason of these differences can be due to the static potential between quarks that we used in the solution of two-body Schrödinger equation. For higher states, power and logarithmic potentials results are comparable with other studies. Two-photon decays are complicated processes such as pseudoscalar meson decay to two photons is governed by an intermediate vector meson followed by a meson dominance transition to a photon . These schematic diagrams must be added to calculations to obtain a whole picture. For two-gluon decay widths, two phenomenological potentials gave comparable results with the available experimental data. Notice that in some cases QCD corrected factor is in accord with the experimental data whereas in some cases it is not. The reason for this can be that, there are significant radiative corrections from three-gluon decays so computing only two-gluon decay width could not explain the mechanism in all details.
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 . The nonrelativistic predictions for its rate are more than two times larger than the experimental data . In the charmonium sector, the available experimental data for is comparable with the power potential result, while logarithmic potential result is 1 eV higher. In the bottomonium sector, there is no experimental data available on M1 transitions. Since photon energies and transition rates are very small, the detection of these transitions is an objection. And this can be a reason why no spin-singlet S-wave levels have been observed yet . The obtained values for M1 transitions are comparable with the references.
Some states in the charmonium and bottomonium sector show properties different from the conventional quarkonium state. Some examples are , , and . For , there is not much available experimental data and more is needed. Wang et al. studied two-body open charm OZI-allowed strong decays of and considered as and , respectively, by the improved Bethe-Salpeter method combined with the . They calculated strong decay width of as and as where the experimental values are for and for . They concluded that is a good candidate of and is a not good candidate of due to larger decay width of comparing to experimental data. We give our results comparing to these exotic states in Table 15.
Looking at Table 15, we can deduce that, according to our model and results, we can assign as , as , and as . To be more accurate, more data is needed to corroborate whether these states are conventional quarkonium or not.
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 where nonperturbative methods are need to be used. The improvements on the potentials can be made and spin-spin, spin-orbit type interactions can be added to model to arrive high accuracy. The potential model approach is a valuable task, which has given to us many insights into the nature of both heavy and light quarkonium physics. Using a relativistic approach together with a model in which and thresholds are taken into account, detailed analysis can be made on various aspects of heavy quarkonium.
No data were used to support this study.
Conflicts of Interest
The author declares that they have no conflicts of interest.
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Copyright © 2018 Halil Mutuk. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP3.