The discovery of the J/ψ, the first bound state of c and c ¯ quarks, known as charmonium, was published in Ref. [1]. Reference [2] describes the first observation of the ψ(2S), marking the field of hadron spectroscopy with the beginning of an important testing ground for the properties of the strong interaction using QCD. The charmonium system allows the prediction of some of the parameters of the states using non-relativistic and relativistic potential models, lattice QCD, NRQCD and sum rules [3]. Although the first charmonium state was discovered in 1974, there are still many puzzles in charmonium physics. Charmonium spectroscopy below the open charm threshold has been well measured and agrees with the theoretical expectations. However, there is still a lack of adequate experimental information and solid theoretical predictions for the charmonium states above the open charm threshold [4]. Recently many other new resonances, named the XYZ particles, have been discovered and are still under examination, as these states do not match the predictions of the non-relativistic or semi-relativistic q q ¯ potential models.

In 1976, Siegrist and others in the MARK-I Collaboration (SLAC) observed the resonance ψ(4415) with mass 4415±7 MeV [5]. In 1978, the DASP Collaboration observed peaks for the ψ(4040), ψ(4160) and ψ(4415) resonances with masses 4040±10, 4159±20 and 4417±10 MeV respectively using a non-magnetic detector [6]. Ablikim and others in the BES Collaboration and Mo and others at the Institute of High Energy Physics, Beijing, determined the resonance parameters for ψ(4040), ψ(4160) and ψ(4415) charmonium. Eichten identified these three resonances as 3³S₁, 2³D₁ and 4³S₁ with a linear plus Coulomb potential model [7], and most later potential model calculations agree with their identification. Recently, the LHCb Collaboration measured the mass 4191 − 8 + 9 MeV of the resonance ψ(4160) with J^PC = 1⁻⁻ [8]. In 2007, a resonant structure was observed by the Belle Collaboration with mass 4664±11±5 MeV [9]. A year later the same collaboration observed a clear peak in the e + e − → Λ c + Λ c − invariant mass distribution and assumed the observed peak to be a resonance of mass 4634 − 7 + 8 − 8 + 5 MeV with the possibility of a 5³S₁ charmonium state [10].

Rapidis and others at SLAC, the LGW Collaboration, observed a resonance with mass 3772±6 MeV, just above the threshold for the production of charmed particles [11]. In a parallel observation, W. Bacino and others at SLAC discovered and confirmed the ψ(3770) resonance with mass 3770±6 MeV [12] and the parameters were determined by the SLAC and LBL Collaborations [13]. In 2006 the BES Collaboration measured the mass of the ψ(3770) resonance precisely [14], and recently its parameters have been measured using the data collected with the KEDR detector [15]. The Belle Collaboration reported the first observation of a new charmonium-like state with mass 3943±6±6 MeV in the spectrum of masses recoiling from the J/ψ in the inclusive process e⁺e⁻ → J/ψ + anything, and denoted it as X(3940) [16]. Later on, a new measurement for the X(3940) was performed by the same collaboration and the mass 3942 − 6 + 7 ± 6 MeV was reported [17]. The 3¹S₀ state is a good candidate for the X(3940) resonance [18, 19].

Evidence of a new narrow resonance X(3823) was found by Belle [20], with its mass near to potential model expectations for the centroid of the 1³D_J states. Recently, the BESIII Collaboration [21] observed a narrow resonance X(3823) through the process e⁺e⁻ → π⁺π⁻X(3823) and confirmed that it is a good candidate for the ψ(1³D₂) charmonium state.

In 2003, the Belle Collaboration observed a charmonium-like state in the decay process B^± → K^± π⁺ π⁻ J/ψ with mass 3872±0.6±0.5 MeV [22], which was confirmed by the CDF, D0 and BaBar experiments [23–25]. Several properties of the X(3872) have been determined [26–28], and the CDF Collaboration explained the X(3872) particle as a conventional charmonium c c ¯ state with J^PC being either 1⁺⁺ or 2⁻⁺ [29]. Recently the BESIII Collaboration reported the first observation of process e⁻e⁻ → γX(3872) with mass 3871±0.7±0.2 MeV [30]. In 2003, Barnes and Godfrey evaluated the strong and electromagnetic decays and considered all possible 1D and 2P charmonium assignments for X(3872) [31].

The X(3915) was observed by S.K. Choi and his team at the Belle Collaboration [32] and later on the BaBar Collaboration confirmed the existence of the charmonium-like resonance X(3915) and measured its mass 3919.4±2.2±1.6 MeV with the J^PC = 0⁺⁺ option [33, 34]. This state is conventionally identified as the χ_c0(2P) charmonium [35, 36]. In 2005, the Belle Collaboration observed the Z(3930) resonance in the γ γ → D D ¯ process [37] with mass 3929±5±2 MeV and considered it a strong candidate for the χ_c2(2P) state. The BaBar Collaboration confirmed the Z(3930) resonance as the χ_c2 (2P) state with mass 3926.7±2.7±1.1 MeV and quantum numbers J^PC = 2⁺⁺ [38].

In 2013, the BESIII Collaboration observed a new structure with mass 3899±3.6±4.9 MeV in the π^±J/ψ mass spectrum (referred as Z_c(3900)) [39] and around the same time the Belle Collaboration observed a structure with mass 3894.5±6.6±4.5 MeV in the π^±J/ψ mass spectrum [40]. The observations of Xiao and his team, based on e⁺e⁻ annihilations at s = 4170 MeV, provide independent confirmation of the existence of the Z c ± ( 3900 ) state and provide new evidence for the existence of the neutral state Z c 0 (3900) [41]. Recently the BESIII Collaboration performed an analysis which favors the assignment of the J^P = 1⁺ quantum numbers [42].

In 2009, the CDF Collaboration reported evidence for a narrow structure near the J/ψϕ threshold in B⁺ → J/ψϕ K⁺ decays with mass 4143±2.9±1.2 MeV [43], which was recently observed by the CMS [44] and D0 [45, 46] Collaborations. It has been suggested that the X(4140) resonance could be a molecular state [47–50], a tetra-quark state [51–53] or a hybrid state [54, 55]. Searches for the narrow X(4140) were negative in the LHCb [56] and BaBar [57] experiments. In 2011, the CDF Collaboration observed the X(4140) structure with a statistical significance greater than 5 standard deviations and also found evidence for a second structure X(4274) with a mass of 4274.4 − 6.7 + 8.4 ± 1.9 MeV [58]. Very recently the LHCb Collaboration confirmed the resonance X(4140) with mass 4146.5 ± 4.5 − 2.8 + 4.6 MeV and X(4274) with mass 4273.3 ± 8 − 3.6 + 17.2 MeV in the J/ψϕ invariant mass distribution, and determined their spin-parity quantum numbers to be J^PC = 1⁺⁺ for both [59]. They also investigated two new structures, named the X(4500) and X(4700), in the high J/ψϕ mass region. Reference [60] suggests that X(4274) can be a good candidate for the conventional χ_c1 (3³P₁) state. The study of charmonium in the relativistic Dirac formalism with a linear confinement potential indicates that the X(4140) state can be an admixture of two P states whereas X(4630) and X(4660) are admixtures of the S-D wave state [61].

Different theoretical models which have been used to study the charmonium spectrum include the recently developed generalized screened potential model (GSPM) [62], the non-relativistic Coulomb gauge QCD approach [63], the light front quark model (LFQM) [64], the relativistic quark model [65], the effective field theory framework of potential non-relativistic QCD (pNRQCD) approach [66], the effective Lagrangian approach [67], lattice QCD [68, 69], LCQCD and QCD sum rules [70, 71], and the widely used potential models [72–78]. The Cornell potential model is well known among the many phenomenologically successful potential models, and describes the charmonium system quite well.

The recent experimental results for the new charmonium-like X Y Z states indicate that they can be interpreted as above-threshold charmonium levels and cannot be assigned to any charmonium states in the conventional quark model. These experimental results motivate renewed theoretical interest in studies of the spectroscopy and decay properties of charmonium.

In this article, to calculate the mass spectrum of charmonium, we use Gaussian wave functions both in position space as well as momentum space with a potential model, incorporating corrections to the kinetic energy of quarks as well as incorporating the relativistic correction of O ( 1 m ) to the potential energy part of the Hamiltonian. We also investigate the Regge trajectories in both the (M² → J) and (M² → n) planes (where J is the spin and n is the principal quantum number) using our predicted masses for the charmonium, as the Regge trajectories play a significant role in identifying the nature of current and future experimentally observed charmonium states. We also obtain the pseudoscalar and vector decay constants for charmonium as well as the radiative (electric and magnetic dipole) transition rates and the annihilation decay.

The article is organized as follows. Section 2.1 presents the theoretical framework for the mass spectra, Section 2.2 presents the decay constants (f_P/V), Section 2.3 presents the radiative (E1 and M1) transitions, and Section 2.4 presents the annihilation decays. In Section 3, we discuss results for the mass spectra, (f_P/V) decays, E1 and M1 transition width, and annihilation decays. The Regge trajectories from estimated masses in the (J,M²) and (n_r,M²) planes are given in Section 3.1. Finally, we draw our conclusion in Section 4.

In the framework of the Cornell potential with a Gaussian wave function and relativistic correction of the Hamiltonian, comprised of a O ( 1 / m ) rectification in the potential energy term and elaboration of the kinetic energy term up to O ( p 10 ) , we have studied the mass spectra of charmonium states. We have calculated the center of weight masses (value of Hamiltonian yields) for the nS (n ⩽ 6), nP and nD (n ⩽ 3) charmonium states, as shown in Table 2. We observe that the Hamiltonian yields for nS (n ⩽ 3) and nP and nD (n ⩽ 3) are in accordance with experimental measurements as well as the values predicted by other theoretical models, whereas the results for nS (4 ⩽ n ⩽ 6) are underestimated or overestimated compared to the results of other theoretical models.

The calculated masses of the charmonium states are graphically represented in Fig. 1 and listed in Table 4 with the experimentally observed results. After addition of the spin hyperfine interaction to the fixed spin average mass for the ground state, we obtain the pseudoscalar state mass η_c (2995 MeV) and vector state mass J/ψ (3094 MeV). The estimated mass of 2¹S₀ (3606 MeV) is 33 MeV lower than the experimentally observed mass, whereas the mass of 3³S₁(4036) is in accordance with the mass given by the PDG [4] and by other model estimates [75, 79, 114]. Our calculated mass for 5³S₁ (4654 MeV) is 11 MeV higher than the value quoted by the PDG [4] and in accordance with the mass estimated by other models [110, 112]. We have assigned X(4660) to the 5³S₁ state of charmonium. The estimated masses of the 6³S₀ (4893 MeV) and 6³S₁ (4925 MeV) states agree with the masses estimated by other models [112].

The P-wave states, 1³P₁ with predicted mass 3511 MeV, 1¹P₁ with predicted mass 3525 MeV and 2³P₂ with predicted mass 3556 MeV, are in good agreement with the experimentally observed values [4].

We have assigned the newly observed charmonium-like state X(3900) to the 2¹P₁ (3936 MeV) and the state X(3872) to the 2³P₁ (3925 MeV). The masses predicted for the states 2¹P₁ (3936 MeV) and 2³P₁ (3925 MeV) are in good agreement with the masses predicted by other models [65, 73, 76, 79, 108, 112, 114]. We assign X(3872) as a candidate for the 2³P₁ state, with well established quantum numbers, although its interpretation as a molecular state [125, 126] was questioned in Ref. [127], while Ref. [128] interpreted it as a virtual state.

Reference [129] predicts X(3872) to be a tetraquarks with a mass difference related to m_u − m_d. Reference [130] described the structures of the X(3872) and X(3915) states as ( c q ) [ c ¯ q ¯ ] tetraquarks with help of the light-front Hamiltonian QCD (LFHQCD) approach.

We have also assigned the charmonium-like states X(3915) and X(4274) to the 2³P₀ (3866 MeV) and 3³P₁(4257 MeV) states respectively. To consider X(3915) as the 2³P₀ state is still problematic, as was also pointed out in Ref. [79, 131] and the references therein. In Ref. [131–133], the authors suggest the X(3915) as the 2³P₀ state faces the following problems. First, scalar mesons should be the open-flavor modes for the dominant decay channels, above the corresponding thresholds. X(3915) should therefore couple in an S-wave and the D D ¯ channel, although this has not been observed in the D D ¯ channel. Second, the mass splitting between the state 1³P₂ and 1³P₀ is 141 MeV, while the mass splitting between the relatively well determined X(3930) as the 2³P₂ state and X(3915) as the 2³P₀ state is 9 MeV, which is too small for the hyperfine splitting.

We observed that new charmonium-like states X(4140) and X(4274) with quantum numbers J^PC = 1⁺⁺ are good candidates for the 3³P₁ state within the screened potential model and linear potential model respectively. However, none of the models can give J^PC = 1⁺⁺ charmonium state masses 4147 MeV and 4273 MeV at the same time, which may indicate the exotic nature of X(4140) and/or X(4274), which was also pointed out in Ref. [79].

The predicted masses for the 1³D₁ (3799 MeV), 1³D₂ (3805 MeV) and 2³D₁ (4145 MeV) states are in accordance with the experimentally observed results [4] as well as in good agreement with other model predictions [65, 73, 75, 76, 79, 112, 114]. The estimated masses of charmonium using our model are overall in agreement (with a few MeV difference) with experimentally observed values. It is found that states with a mass of M < 4.1 GeV are in good agreement with other theoretical estimates.

Table 3 shows the hyperfine splittings for S wave states and fine splittings for some P wave states. For comparison, the experimental data from the PDG [4] and predictions with other theoretical models are also listed. The predicted hyperfine splittings up to the 2S states are in agreement with the world average data [4] and predictions with other theoretical models. The hyperfine splittings for the 3S to 6S states have different values in different theoretical models. By comparing our predicted results with other theoretical models, we observe that the masses of the low-lying nS (n ⩽ 2), nP, and nD (n = 1) charmonium states have less difference, whereas the masses of the higher charmonium states nS(n ⩾ 3), nP, nD (n ⩾ 2) have considerable differences.

The estimated pseudoscalar and vector decay constants, f_P(f_Pcor) and f_V(f_Vcor) respectively, without and with QCD corrections, are shown in Table 1. They are in agreement with the experimental results as well as other theoretical model estimates.

The calculated radiative E1 and M1 dipole transitions widths are shown in Tables 5 and 6. We calculate the E1 transition of Γ[1P → (1S) γ], Γ[2S → (1P) γ], Γ[1D → (1P) γ], Γ[2P → (2S)γ] and Γ[2P → (1D) γ] using the masses predicted by our model. Our calculated E1 transitions for Γ[1P → (1S) γ] and Γ[2S → (1P) γ] are lower than the experimental results as well as other theoretical estimates, whereas for Γ[1D → (1P) γ], Γ[2P → (2S) γ] and Γ[2P → (1D) γ] transition, our results are in agreement with the estimates of other theoretical models. Our predictions for Γ[1³D₁ → (1³P₁) γ] and Γ[1³D₁ → (1³P₀) γ] are almost double that of the PDG average data [4], while our prediction of Γ[1³D₁ → (1³P₂)γ] is in agreement with the PDG average data [4] as well as the values predicted by other models.

We also calculate the M1 transition of the low-lying 1S, 2S and 3S states as well as the 1P states. Our predictions for Γ[1³S₁ → (1¹S₀)γ] and Γ[2³S₁ → (2¹S₀) γ] are in agreement with the PDG average data [4], while our prediction for Γ[2³S₁ → (1¹S₀)γ] is much higher than the PDG average data [4]. Gang Li and Qiang Zhao [134, 135] studied intermediate meson loop contributions to 1³S₁, 2³S₁ → γ 2¹S₀,(γ1¹S₀) apart from the dominant M1 transitions in an effective Lagrangian approach. Their results showed that the IML contributions are relatively small but play a crucial role. Radiative decay widths, including the M1 in the GI model and intermediate hadronic loops, are 1.59 keV for 1³S₁ → γ 2¹S₀ and 0.032(0.86) keV for 2³S₁ → γ 2¹S₀(γ1¹S₀) [134]. Results including the M1 transition amplitude of the GI model and IML transitions are 1.58±0.37 keV for 1³S₁ → γ 2¹S₀ and 0.08±0.03 ( 2.78 − 1.75 + 2.65 ) keV for 2 3 S 1 → γ 2 1 S 0 ( γ 1 1 S 0 ) [135].

Our prediction for Γ[3³S₁ → (3¹S₀)γ] is in agreement with the other theoretical model predictions, while the prediction for Γ[2¹S₀ → (1³S₁)γ] is higher than predictions by other theoretical models. The various models have different estimates for the E1 and M1 transitions, which may be due to the models having different parameters or to treatments in the relativistic corrections. The E1 and M1 transitions in general are strongly model dependent and more studies are required in experiments as well as theory.

We estimate the partial decay widths Γ and Γ^cf (with QCD correction factor) of annihilation processes, using the masses predicted by our potential model and the radial wave function at the origin, for e⁺e⁻, two-photon, three-photon, two-gluon, three-gluon, γgg and q q ¯ + g processes. The results are tabulated in Tables 7–13 and are compared with experimental results from the PDG [4] as well as other theoretically calculated estimates.

Our estimated leptonic decay widths without QCD correction for J/ψ, ψ(2S), ψ(3S) and ψ(4S) are higher than the experimentally observed leptonic decay widths. After QCD correction, the estimated leptonic decay widths are 1.93 keV, 1.24 keV, 0.11 keV and 0.019 keV less than the experimental results for the J/ψ, ψ(2S), ψ(3S) and ψ(4S) states respectively. Also, our estimated leptonic decay width with QCD correction for the n³D₁ state is much lower than the experimental result.

Our estimated two-photon and two-gluon decay widths with QCD correction for the η_c(nS), n³P₀ and n³P₂ states are in accordance with the experimentally observed results as well as with the other theoretical estimates. Our estimated three-photon decay width with QCD correction for J/ψ is lower than the experimentally observed result, while the estimated three-gluon decay widths with QCD correction for the J/ψ and ψ(2S) states are higher than the experimentally observed result as well as other theoretical estimates.

Our estimated γgg decay widths with QCD correction for the J/ψ and ψ(2S) states are in accordance with the experimentally observed results. We have also computed the q q ¯ + g decay width for the n³P₁ states. We observe that the radiative QCD corrections modify the theoretical predictions considerably and bring the estimated result close to the experimental data. We also observe that the estimated values of annihilation decay width by various models show wide variation. Due to the considerable uncertainties which arise from the wave function dependence of the model and possible relativistic as well as QCD radiative corrections, we would like to mention that formulas used for calculation of annihilation decay width should be regarded as estimates of the partial widths rather than precise predictions.

We can conclude that the mass spectra of charmonium, Tables 2 and 4, investigated using a Cornell potential with relativistic correction to the Hamiltonian, are in accordance with the available experimental results as well as those predicted by the other theoretical models. The predicted pseudoscalar (f_Pcor) and the vector (f_Vcor) decay constants with QCD correction using our estimated charmonium masses are in accordance with experimental results as well as those predicted by other theoretical models.

We observe from the Regge trajectories in Figs. 2–5 that the experimental masses of the charmonium states are sitting nicely. In the mass region of the lowest excitations of charmonium, the slope of the trajectories decreases with increasing quark mass. The curvature of the trajectory near the ground state is due to the contribution of the color Coulomb interaction, which increases with mass. Hence, the Regge trajectories of the charmonium are basically nonlinear and exhibit nonlinear behavior in the lower mass region.

From a comparison of our estimated radiative (E1 and M1 dipole) transition widths with other theoretical estimations, we conclude that the various models have very different predictions for the E1 and M1 dipole transitions, which may be due to the different parameters and treatments used in the relativistic corrections in the model. The calculated E1 and M1 dipole transition widths using the masses and parameters estimated by our model are in agreement with other theoretical and experimental predictions. However, in most cases, more precise experimental measurements are required.

We also conclude from the calculated annihilation decay widths using the Van Royen-Weisskopf relation that the inclusion of QCD correction factors is helpful in bringing the estimated results closer to the experimental results. The various models show a wide variation in results for the annihilation decay widths, which may be resolved using the NRQCD (non-relativistic QCD) and pNRQCD (potential non-relativistic QCD) formalisms.