1 Introduction

The applicability of any nonrelativistic potential model during studies of heavy hadrons spectra can be well checked by reproducing the heavy quarkonium states. The best candidate for this role is the charmonium consisting of one heavy quark Q and one heavy antiquark \({\bar{Q}}\)  [1, 2]. Analyzing the charmonium spectra, one can establish some interesting features. For that purpose, one keeps in mind some facts in formulating a better approach. First of all, one can note the large value of the current quark mass in the characteristic energy scale \(m_Q\gg \Lambda _{\mathrm{QCD}}\). Secondly, one can pay attensition to the smallness of the heavy-quark velocity \(v_Q\ll c\) inside the charmonia, which can be crudely estimated from the radial excitation energy differences corresponding to the given quantum numbers. These two facts indicate that the relativistic effects can be taken into account as systematic corrections (for example, see the review in Ref.  [3]).

On the other hand, one can also note the smallness of nonperturbative effects in the spectra of charmonia. The authors of Ref.  [4] estimated the contribution to the spin-independent heavy quark potential due to the nonperturbative dynamics in the framework of the instanton vacuum model of quantum chromodynamics (QCD)  [5, 6]. They also found the relatively small value of difference between the current and constituent quark masses in comparison with the constituent or current quark mass  [4]. Recent calculations  [7, 8] based on the estimations above showed that the contributions due to the nonperturbative dynamics can also be considered as small corrections. In particular, the authors of Ref.  [8] showed that the instanton effects are first-order perturbative corrections. Nevertheless, in the Ref. [7] the authors pointed out that the instantons may shed some light on the origins of the parameters used in the phenomenological potential approaches [9, 10]. The studies performed in Refs. [4, 7] lead to the conclusion that, although the nonperturbative dynamics is very nontrivial at low energies, at the high energies, it seems to be tightly hidden behind the confinement mechanism, which is not yet fully understood.

All said, the above more or less explains the success of the phenomenological potential model [11] where complicated and unknown dynamics is expressed in terms of the effective values of phenomenological parameters. Therefore, one has readily a nonrelativistic Schrödinger approach for describing the energy spectra of heavy quarkonium. Technically, the quarkonium is very similar to the hydrogen atom; i.e., one can solve the one-body problem in the given external potential field instead of considering the two-body relativistic system. The difference from the hydrogen atom problem is only due to the nature of the interaction and its corresponding range. Consequently, due to the strong nature of the interactions the excitation energies of charmonia will be much lager than the electron excitation energies in the hydrogen atom. The size of a charmonium also will be much less in comparison with the size of the hydrogen due to the short-range nature of the interactions. One can use these obvious facts in applying a numerical method to the charmonium problem and can easily find the appropriate variational parameters of the model.

Starting the discussions of the heavy quark potential, one can note that the basic spin-independent central interaction between the quark and the antiquark can be well separated into two parts. The first scalar exchange part is fully phenomenological because of an unknown confinement mechanism. The most popular choice for this interaction is expressed as a linearly increasing potential due to the area law of the Wilson loop [12] for the heavy-quark potential. The second, the vector exchange part, is due to the perturbative one-gluon exchange mechanism at short distances and in the lowest order, has a Coulomb-interaction-like form with the corresponding running coupling constant. The spin-dependent parts of the interactions can be reproduced from the central potential in the framework of nonrelativistic expansions [13]. The corresponding model is called a nonrelativistic constituent quark model.

In the present work, we discuss the interesting features of the nonrelativistic constituent quark model and propose a new set of parameters for describing the charmonium states on a basis of updated experimental data [14]. While we are doing that, as an input, we concentrate only on the minimal part of spectrum instead of considering the whole spectrum. After fitting the parameters in the most compact way we concentrate on the whole spectrum and analyze the applicability of the nonrelativistic potential model approach.

The paper is organized in the following way: In the next section,Section 2, we briefly repeat the main features of the model and give a short description of a variational approach to the problem. In Section 3, the results from calculations will be presented and discussed. In the last section, Section 4, we summarize our results and draw the corresponding conclusions.

2 \(Q\bar{Q}\) potential and variational method

In the simple constituent quark model, the total \(Q\bar{Q}\) potential has the following standard form:

$$\begin{aligned} V_{Q\bar{Q}}(\varvec{r})&= V_C(r) +V_{SS}(r) (\varvec{S}_Q\!\cdot \!\varvec{S}_{{\bar{Q}}})\nonumber \\&+V_{LS}(r)(\varvec{L}\cdot \varvec{S}) \nonumber \\&+V_{T}(r)\left[ 3(\varvec{S}_Q\!\cdot \!\varvec{n})(\varvec{S}_{{\bar{Q}}}\!\cdot \! \varvec{n})-\varvec{S}_Q\cdot \varvec{S}_{{\bar{Q}}}\right] , \end{aligned}$$
(1)

where \({\varvec{S}}_Q\) (\({\varvec{S}}_{{\bar{Q}}}\)) is the spin of the quark (antiquark), \({\varvec{L}}\) is relative orbital momentum, and \({\varvec{S}}= {\varvec{S}}_Q+{\varvec{S}}_{{\bar{Q}}}\) is total spin of the quarkonium system. We work in the center-of-mass frame, therefore, the radius vector \({\varvec{r}}\) is given in terms of the relative coordinates \({\varvec{r}}={\varvec{r}}_Q-{\varvec{r}}_{{\bar{Q}}}\), and \({\varvec{n}}={\varvec{r}}/r\) defines the unit vector in the direction of the radius vector. In Eq. (1), \(V_C(r)\), \(V_{SS}(r)\), \(V_{LS}(r)\) and \(V_T(r)\) are the central, spin-spin, spin-orbit and tensor potentials, depending on the relative distance between the quark and antiquark.

The central part of the potential in a nonrelativistic reduction employs the following “Coulomb+linear” form:

$$\begin{aligned} V_C(r)=\kappa r-\frac{4\alpha _s}{3r}, \end{aligned}$$
(2)

where \(\kappa \) is a parameter defining the string tension and

$$\begin{aligned} \alpha _{\mathrm {s}}(\mu ) = \frac{1}{\beta _0\ln (\mu ^2/\Lambda _{\mathrm {QCD}}^2)} \end{aligned}$$
(3)

is the strong running coupling constant at the one-loop level. Its value is determined from the characteristic energy scale \(\mu \) corresponding to the problem. Further, \(\beta _0=(33-2N_f)/(12\pi )\) is the beta function at the one-loop level, and \(\Lambda _{\mathrm {QCD}}\) is the dimensional transmutation parameter. The nonrelativistic expansion of \(Q\bar{Q}\) interactions allows us to relate the spin-dependent parts of the potential to the central part [13]. Thus, the spin-dependent interactions corresponding to the “vector one-gluon-exchange+scalar confinement” are given as

$$\begin{aligned} V_{SS}^{\mathrm{(P)}}(r)= & {} \frac{32\pi \alpha _s}{9m_Q^2}\delta _\sigma (r), \end{aligned}$$
(4)
$$\begin{aligned} V_{LS}^{\mathrm{(P)}}(r)= & {} \frac{1}{2m_Q^2}\left( \frac{4\alpha _s}{r^3}-\frac{\kappa }{r}\right) , \end{aligned}$$
(5)
$$\begin{aligned} V_{T}^{\mathrm{(P)}}(r)= & {} \frac{4\alpha _s}{m_Q^2r^3}, \end{aligned}$$
(6)

where \(m_Q\) is the heavy quark mass. In practical calculations, the pointlike spin-spin interaction in Eq. (4) can be “smeared” by using an exponential function of the form

$$\begin{aligned} \delta _\sigma (r)=\left( \frac{\sigma }{\sqrt{\pi }}\right) ^3 e^{-\sigma ^2r^2}, \end{aligned}$$
(7)

where \(\sigma \) is a smearing parameter. In such a way, the \(Q\bar{Q}\) potential can be described in terms of only four parameters, \(\kappa \), \(\alpha _s\), \(m_Q\) and \(\sigma \). Usually, these parameters are found by fitting the whole charmonium spectrum. In the present work, we will follow the phenomenological approach, but find those parameters by fitting only some minimal part of the S-wave spectrum instead of considering the whole spectrum.

After fitting the form of potential, to evaluate the energy states of quarkonia in a nonrelativistic potential approach, one needs to solve the Schrödinger equation

$$\begin{aligned} ({\hat{H}}-E)|\Psi _{JJ_3}\rangle =0. \end{aligned}$$
(8)

Here \({\hat{H}}\) is the Hamilton operator and \(|\Psi _{JJ_3}\rangle \) represents the state vector with total angular momentum J and its third component \(J_3\). The coordinate space projection of the state vector \(\langle \varvec{r}|\Psi _{JJ_3}\rangle \) will reproduce the coordinate space representation of the Hamiltonian as

$$\begin{aligned} {\hat{H}}(\varvec{r})=-\frac{\hbar ^2}{m_Q}\nabla ^2+V_{Q\bar{Q}}(\varvec{r}), \end{aligned}$$
(9)

where \(m_Q\) arises from the doubled reduced mass of the quarkonium system. The matrix elements of the \(Q\bar{Q}\) potential in the standard basis \( | {}^{2S+1}L_J\rangle \), which is given in terms of the total spin S, the orbital angular momentum L, and the total angular momentum J satisfying the relation \(\varvec{J}= \varvec{L}+\varvec{S}\), has the following form:

$$\begin{aligned} V_{Q\bar{Q}}(r)&=\langle {}^{2S+1}L_J| V_{Q\bar{Q}}(\varvec{r}) | {}^{2S+1}L_J\rangle \nonumber \\&= V(r)+ \left[ \frac{1}{2}S(S+1)-\frac{3}{4}\right] V_{SS}(r) \nonumber \\&+ \langle {\varvec{L}}\cdot {\varvec{S}}\rangle V_{LS}(r)\ +\left\{ -\frac{\langle {\varvec{L}}\cdot {\varvec{S}}\rangle \left( \langle {\varvec{L}}\cdot {\varvec{S}}\rangle +2\right) }{(2L-1)(2L+3)}\right. \nonumber \\&\left. +\frac{S(S+1)L(L+1)}{3(2L-1)(2L+3)}\right\} V_T(r), \end{aligned}$$
(10)

where \(\langle {\varvec{L}}\cdot {\varvec{S}}\rangle \) is defined as

$$\begin{aligned} \langle {\varvec{L}}\cdot {\varvec{S}}\rangle = \frac{1}{2}\left[ J(J+1) -L(L+1) -S(S+1)\right] . \end{aligned}$$

The corresponding radial part of the wave function for a given orbital momentum L is a solution of the Schrödinger equation

$$\begin{aligned} \left( -\frac{\hbar ^2}{m_Q}\nabla ^2+V_{Q\bar{Q}}(r)-E\right) \psi _{LL_3}(\varvec{r})=0, \end{aligned}$$
(11)

where the angular part of the wave function \(\psi _{LL_3}(\varvec{r})\) is represented in terms of the standard spherical harmonics \(Y_{LL_3}(\hat{\varvec{r}})\). In order to solve Eq. (11) numerically, we will follow the Gaussian expansion method (for details, see for review in Ref. [15]), where the state vector \(|\psi _{LL_3}\rangle \) is expanded in terms of a set of basis vectors \(\{|\phi _{nLL_3}\rangle ;\,n=1,2,\dots ,n_{\mathrm {max}}\}\) as

$$\begin{aligned} |\psi _{LL_3}\rangle =\sum _{n=1}^{n_{\mathrm{max}}}C_{n}^{(L)}|\phi _{nLL_3}\rangle . \end{aligned}$$
(12)

Here, n is a radial quantum number. Thus, the radial excitations corresponding to the given angular momentum value will be reproduced naturally. In the Gaussian expansion method, the radial part \(\phi ^G_{nL}(r)\) of the total eigenfunction in the spherical coordinate basis

$$\begin{aligned} \phi _{nLL_3}(\varvec{r})=\phi ^G_{nL}(r)Y_{LL_3}(\hat{\varvec{r}}) \end{aligned}$$
(13)

can be expressed in terms of Gaussian trial functions as

$$\begin{aligned} \phi ^G_{nL}(r)=\left( \frac{2^{2L+{7}/{2}}r_{n}^{-2L-3}}{\sqrt{\pi }(2L+1)!!}\right) ^{1/2}r^{L}e^{-(r/r_{n})^2}. \end{aligned}$$
(14)

For the given set, n runs the values \(n=1,2,\dots ,n_{\mathrm{max}}\), and the corresponding \(r_n\)’s play the roles of variational parameters. The variational parameters can be optimized using a geometric progression [15]

$$\begin{aligned} r_n=r_1a^{n-1},\quad n=1,2,\dots ,n_{\mathrm{max}}. \end{aligned}$$
(15)

Therefore, the actual number of parameters is reduced to three (e.g., \(r_1\), \(r_{n_{\mathrm{max}}}\) and \(n_{\mathrm{max}}\)) for the given values of the orbital quantum number L, the spin S and the total angular momentum J.

The expansion coefficients \(C_n^{(L)}\) in Eq. (12) and the eigenenergies \(E_n^{(L)}\) are determined by employing the Rayleigh-Ritz variational principle. This leads to a generalized matrix eigenvalue problem

$$\begin{aligned} \sum _{n=1}^{n_{\mathrm{max}}}\left( K_{mn}^{(L)}+V_{mn}^{(L)} -E_n^{(L)}N_{mn}^{(L)}\right)&C_{n}^{(L)}=0,\nonumber \\ m=1,2,\dots ,n_{\mathrm{max}}, \end{aligned}$$
(16)

where the corresponding matrix elements are defined in the following way:

$$\begin{aligned} K_{mn}^{(L)}&=\langle \phi _{mLL_3}\left| \frac{\hat{p}^2}{m_Q}\right| \phi _{nLL_3}\rangle , \end{aligned}$$
(17)
$$\begin{aligned} V_{mn}^{(L)}&=\langle \phi _{mLL_3}|V_{Q\bar{Q}}|\phi _{nLL_3}\rangle , \end{aligned}$$
(18)
$$\begin{aligned} N_{mn}^{(L)}&=\langle \phi _{mLL_3}|\phi _{nLL_3}\rangle . \end{aligned}$$
(19)

3 Results and discussions

As we mentioned above in the phenomenological approaches, the parameters of the model, \(\kappa \), \(\alpha _s\), \(\sigma \) and \(m_c\), are fitted to the spectra of experimentally known charmonium states. For example, the authors of Ref. [11] proposed the set of potential parameters given in Table 1 (see the model referred as NR).

Table 1 Parameters of the nonrelativistic potential models
Full size table

Using that set of parameters, they calculated all allowed E1 radiative partial widths and some important M1 widths. As an input for the fitting of parameters, they used 11 meson states: 6 states corresponding to the S-wave, 3 states corresponding to the P-wave and 2 states corresponding to the D-wave. The input values of these energy states are given in Table 2 (see the 2nd column). The results of their calculations showed that the nonrelativistic potential model with a certain set of parameters describes the charmonium spectrum very well.

Table 2 Experimental and calculated spectra of \(c\bar{c}\) states
Full size table

However, nowadays, more experiments are performed, and some new states are fixed in the particle data  [14]. The values of the charmonium states extracted from the current experimental data are given in the last column of the Table 2. A natural question arises – “How will the parameters of the nonrelativistic potential model change if one concentrates on the updated experimental data?” Partially, our aim in the present work is to answer this question. However, our main aim in the present work is not only fitting the updated experimental data by means of the new set of parameters. In addition to the fitting process, we want to check, “How well the nonrelativistic expansion works in the potential approaches.”

As we said above, the authors of Ref. [11] fitted the parameters of the potential to all eleven, experimentally known at that time, states of the charmonium spectrum. Therefore, the beauty of the nonrelativistic expansion seems to be hidden. We want to emphasize that, in principle, one can concentrate on part of spectrum to fit the parameters of the model. For example, one can concentrate on the S-wave part of spectrum for fitting the parameters of model. During this process, the spin-orbit and the tensor interactions do not contribute to the total interaction. One can also act in an opposite way by concentrating on the part of the spectrum where the spin-orbit and the tensor interactions are important. The reason for the possibility of such choices is the fact that the central and the spin-dependent parts of the potential are related to each other in the nonrelativistic expansion and can be described by using the same set of parameters. In an ideal case, only four input states are needed to fit four “arbitrary” parameters of the potential model.

Consequently, as a possible test of the nonrelativistic expansion, in the present work, we consider the “ideal case” and fit the parameters of model according to some part of S-wave charmonia. In such a way, we ignore the spin-orbit and the tensor interactions during the fitting process. More specifically, we propose a potential model in which the parameters are fitted using the four lowest S-wave spin 0 and spin 1 states. On top of that, we will fit the lowest spin zero \(1^1\mathrm{S}_0\) and \(2^1\mathrm{S}_0\) states exactly. The parameters of the corresponding interaction potential are given in Table 1 (see the model referred as NR4), and the values of corresponding input states are given in Table 2 (see the 4\(^{\mathrm{th}}\) column).

To keep good accuracy of numerical calculations, during the fitting process, we used a basis set with 30 to 50 Gaussian functions corresponding to the given values of L, S and J. Thus, the value of the first parameter \(n_{\mathrm{max}}\) from the three variational parameters is a free input and equals to the definite integer number belonging to the interval \(n_{\mathrm{max}}\in [30,50]\). The values of other two corresponding variational parameters, \(r_1\) and \(r_{n_{\mathrm{max}}}\), are found by minimizing not only the ground state energy \(E_1\) but also minimizing simultaneously the lowest 10 to 20 radial excitation energies \(\sum _{i=n}^{n_{\mathrm{min}}}E_n\) (i.e. \(n_{\mathrm{min}}\in [5,20]\)) from the possible 30 to 50 energy states. The convergences of the results are checked by increasing the number of total radial excitations starting from \(n_{\mathrm{max}}\sim 10\) with \(n_{\mathrm{min}}\sim 5\) to the above-mentioned final values, \(n_{\mathrm{max}}\in [30,50]\) and \(n_{\mathrm{min}}\in [10,20]\).

Now, let us discuss our results. Analyzing the S-wave results of the NR4 model, one can note that the values of the first two spin 0 and the first two spin 1 S-wave states are obviously reproduced very well. Naturally, these energy states are input and reproduced better in comparison with the results of the NR model. A quick look at the calculated \(3^3\mathrm{S}_1\)-state energy value for NR4 and comparing it with the corresponding experimental value shows a large difference of around 45 MeV. Nevertheless, this problem seams to be unavoidable if one fits the parameters to the whole spectrum, e.g., compare the corresponding NR result where the difference from the experimental value is around 33 MeV. However, the next excited \(4^3\mathrm{S}_1\)-state energy for NR4 is reproduced at almost its averaged experimental value while the NR model gives a relatively different result. One can conclude that, in general, S-wave states are reproduced better in the NR4 model that they are in the NR model.

The power of a nonrelativistic expansion becomes more obvious when we include the spin-orbit and the tensor interactions for the analysis of the whole spectrum. While the input parameters are already fixed, we do not need to play with them anymore. Therefore, the calculations of \(L\ge 1\) states are straightforward and does not require any fitting process.

The beauty of the nonrelativistic expansion is realized when we analyze the P-wave states. One can see that, although we are not fitting them, among the six experimentally known states three of them, \(1^3\mathrm{P}_0\), \(1^1\mathrm{P}_1\), and \(2^3\mathrm{P}_0\) are reproduced at their experimental value for the NR4 model. Two states, \(1^3\mathrm{P}_2\) and \(1^3\mathrm{P}_1\), among the remaining three P-wave states in the table are also reproduced quite well, with differences of 3 MeV and 6 MeV from the experimental values, respectively. Only one state \(2^3\mathrm{P}_2\) is far from its experimental value, the difference being 51 MeV. For comparison, the general fit using the NR model reproduces only one state \(1^3\mathrm{P}_2\) at its experimental value. Another state, \(1^3\mathrm{P}_1\), is the same as in the case of the NR4 model, and three (\(1^3\mathrm{P}_0\), \(1^1\mathrm{P}_1\) and \(2^3\mathrm{P}_0\)) from the remaining four states are reproduced approximately with 10 MeV differences from the experimental values, respectively. The last state \(2^3\mathrm{P}_2\) is very far from the experimental value, and difference of 47 MeV is almost the same as it is in the NR4 case. One could also conclude, that concentration, respectively, on the S-wave (e.g., \(1^1\mathrm{S}_0\) and \(2^1\mathrm{S}_0\)) and the P-wave (e.g., \(1^3\mathrm{P}_0\) and \(1^1\mathrm{P}_1\)) states as an input will lead to more or less similar values of the potential parameters in comparison with the values in Table 2. Summarizing the analysis of P-wave states, one can conclude that the NR4 model much better reproduces the experimental data in comparison with the NR model.

Analyzing higher energy states corresponding to the NR and the NR4 models and comparing them with the available experimental data is also interesting. Coming to the D-wave states, one can note that the \(1^3\mathrm{D}_1\)-state energy value is reproduced relatively better in the NR model. However, the situation is reversed if we analyze the \(1^3\mathrm{D}_2\)-state energy. Here, the NR4 model gives a relatively better result in comparison with the NR model. Finally, the experimentally available highest energy state \(2^3\mathrm{D}_1\) is again reproduced relatively better in the NR4 model. Again and in general, D-wave states are also better reproduced in NR4 model.

Consequently, one can conclude that, although the number of input parameters in the NR4 model are chosen in a maximum compact form, it gives better result than the NR model. From Table 2 one can also draw a general conclusion that the fitting to the S-wave part of the spectrum is completely satisfactory. In a such way, we see that the nonrelativistic expansion in the potential models for describing the fine structure of charmonium states, indeed, works pretty well.

4 Summary

In the present work, we aimed to test the nonrelativistic potential model and reparametrization of the nonrelatistic potential based on the currently available experimental data. In particular, we investigated the applicability of a nonrelativistic expansion in the potential approaches to the charmonium spectrum. For that purpose, we concentrated only on the four lowest S-wave states of the charmonium spectrum. By doing that we demonstrated, that the concentration to the some minimal part (four S-wave or P-wave states) of charmonium spectrum was enough during the fitting process for the values of the potential parameters. The model quite satisfactorily described the entire spectrum of charmonia. From our studies, one can make a general conclusion that the nonrelativistic potential approach is indeed a good approximation for describing the spectrum, including the fine-structure of charmonium states.