Charmonium resonances in the 3.9 GeV/$c^2$ energy region and the $X(3915)/X(3930)$ puzzle

An interesting controversy has emerged challenging the widely accepted nature of the $X(3915)$ and the $X(3930)$ resonances, which had initially been assigned to the $\chi_{c0}(2P)$ and $\chi_{c2}(2P)$ $c\bar c$ states, respectively. To unveil their inner structure, the properties of the $J^{PC}\!\!=\!0^{++}$ and $J^{PC}\!\!=\!2^{++}$ charmonium states in the energy region of these resonances are analyzed in the framework of a constituent quark model. Together with the bare $q\bar q$ states, threshold effects due to the opening of nearby meson-meson channels are included in a coupled-channels scheme calculation. We find that the structure of both states is dominantly molecular with a probability of bare $q\bar q$ states lower than $45\%$. Our results favor the hypothesis that $X(3915)$ and $X(3930)$ resonances arise as different decay mechanisms of the same $J^{PC}\!\!=\!2^{++}$ state. Moreover we found an explanation for the recently discovered $M=3860$ MeV$/c^2$ as a $J^{PC}\!\!=\!0^{++}$ $2P$ state and rediscovery the lost $Y(3940)$ as an additional state in the $J^{PC}\!\!=\!0^{++}$ family.

distribution of the X(3930) in the γγ center of mass measured by Belle follows the one expected for a J = 2 state. Hence, the X(3930) was rapidly assigned to the χ c2 (2P ) charmonium state and incorporated to the PDG [5], despite most of the quark models predict a mass higher than the experimental one. For instance, the widely used Godfrey-Isgur relativistic quark model [6] finds the aforementioned state at M=3979 MeV/c 2 .
The situation is worse in the case of the X(3940) resonance. It has not been seen in the DD channel which rules out the J P C = 0 ++ assignment. The dominance of theDD * decay mode suggests that the X(3940) is the cc(2 3 P 1 ) state with J P C = 1 ++ , but these quantum numbers coincide with the ones of the X(3872). In addition, a decay to ωJ/ψ was not observed indicating that the X(3940) and the Y (3940) are not the same state. The history of the Y (3940) is more complicated. In 2008, three years after its discovery, the Babar Collaboration claimed the confirmation of the Y (3940) in the B → J/ψωK decay, but with a mass somewhat smaller (3914 MeV/c 2 ) [7]. In 2010, the Belle Collaboration reported a resonance-like enhancement in the γγ → ωJ/ψ process [8], at M = 3915 ± 3 ± 2 MeV/c 2 and Γ = 17±10±3 MeV with possible quantum numbers J P C = 0 ++ and J P C = 2 ++ . Finally, the BaBar Collaboration confirmed the existence of the X(3915) and its spin-parity analysis clearly prefers the assignment J P C = 0 ++ [9]. These authors pointed out that these values are consistent with those of the Y (3940) and both signals are renamed as X(3915). Then the state was eventually labeled as the χ c0 (2P ) state by the PDG [5]. This assignment was also supported by the χ c0 (2P ) mass value, 3916 MeV, predicted by the Godfrey-Isgur relativistic quark model [6].
However problems do not end here. The J P C = 0 ++ assignment was challenged by Guo and Meissner [10] and also by Olsen [11] mainly for three reasons: • The partial width for the X(3915) → ωJ/ψ is too large for an OZI-suppressed decay.
• There is not signal for the X(3915) → DD decay, which is expected to be the dominant decay mechanism.
Beyond the discussion above, very recent studies have altered the previous situation. On the one hand, from the theoretical side, Z.-Y. Zhou et al. [12] revealed that BaBar Collaboration's conclusion on the X(3915) quantum numbers is largely based on the assumption that the dominant amplitude for a J P = 2 + state has helicity-2, which originally comes from quark models [13]. Abandoning this assumption the reanalysis of the data made by Zhou et al. concluded that the assignment J P = 2 + for the X(3915) is more consistent with the data, showing a sizable helicity-0 contribution in both γγ → DD and γγ → ωJ/ψ amplitudes. This large helicity contribution implies that the X(3915) state might not be a pure qq state. As a consequence of this analysis, PDG relabeled the resonance back to X(3915), with the extra clarification: "was χ c0 (3915)".
On the other hand, from the experimental side, a novel charmonium-like state dubbed X(3860), decaying to DD, has been reported by the Belle Collaboration [14], having a mass of 3862 +26 +40 −32 −13 MeV/c 2 and a width of 201 +154 +88 −67 −82 MeV. The J P C = 0 ++ option is favored over the 2 ++ hypothesis, but its quantum numbers are not definitively determined. This state coincides with the suggestion of Ref [10]. These authors, contrary to Belle and BaBar analysis, assume that all the cross section of the γγ → DD process is due to resonant structures. Therefore, the broad bump below the narrow peak of the χ c2 (2P ) can be identified with the authentic χ c0 (2P ), with a mass and width of 3837.6 ± 11.5 MeV/c 2 and 221 ± 19 MeV, respectively. It is worth emphasizing that the previous mass coincides with the predictions of some dynamical coupled-channel models [15][16][17].
To analyze these resonances it is necessary to take into account that, in the energy region around 3.9 GeV/c 2 , a significant number of open-charm channels are opened. There are convincing arguments [17,18] that open-charm thresholds play an important role in this energy region of the charmonium spectrum, being the charmonium-like resonances better described as states with a significant non-qq component. Thus, the X(3872) resonance together with the X(3940) have been explained as two J P C = 1 ++ states, being the X(3872) basically a DD * + h.c. molecule with a small amount of 2 3 P 1 cc state, while the X(3940) is a mixture with more than 60% of cc structure [19]. These compositenesses are essential to reproduce their properties. Taking into account that the X(3915), the X(3930) and the Y (3940) resonances belong to the same energy region it is reasonable to assume that the nature of these states are determined by the interplay between two and four quark channels.
In view of these arguments, this work explores the possible non-qq components of the X(3915), the X(3930) and the Y (3940) as suggested by Zhou et al. [12]. For that purpose we perform a coupled-channels calculation in the framework of the constituent quark model (CQM) proposed in Ref. [20]. This model has been extensively used to describe the hadron phenomenology both in the light [21] and the heavy quark sectors [22].
The basis of the aforestated CQM is the emergence of the light-quark constituent mass as a consequence of the dynamical chiral symmetry breaking in QCD at some momentum scale. Regardless of the breaking mechanism, the simplest Lagrangian which describes this situation must contain Goldstone-boson fields to compensate the mass term. In the heavy quark sector chiral symmetry is explicitly broken and Goldstone-boson exchanges do not appear.
However, it constrains the model parameters through the light-meson phenomenology [23] and provides a natural way to incorporate the pion exchange interaction in the molecular dynamics.
The potential coming from the Goldstone-boson fields is supplemented by a screened linear confinement potential and the one-gluon exchange interaction. A scale dependent quark-gluon coupling constant α s [20] allows a consistent description of light, strange and heavy mesons (see Refs. [24,25] for review).
To find the quark-antiquark bound states we solve the Schrödinger equation, following Ref. [26], we employ Gaussian trial functions with ranges in geometric progression. This enables the optimization of ranges employing a small number of free parameters. Moreover, the geometric progression is dense at short distances, so that the description of the dynamics mediated by short range potentials is properly treated. Additionally, the fast damping Gaussian tail generated by this method can represent a problem for describing the long range. Fortunately, this issue can be easily overcome by choosing the maximal range much larger than the hadronic size.
In order to explore the J P C = 0 ++ and 2 ++ charmonium sectors we employ the coupled-channels formalism described in Ref. [17]. We assume that the hadronic state is where |ψ α are cc eigenstates of the two body Hamiltonian, φ M are qq eigenstates describing the A and B mesons, |φ A φ B β is the two meson state with β quantum numbers coupled to total J P C quantum numbers and χ β (P ) is the relative wave function between the two mesons in the molecule.
In the framework of the CQM, we can derive the meson-meson potential from the qq interaction using the Resonating Group Method (RGM). For this work, the possible interactions include a direct potential, which connects open-charm meson channels, and an exchange one, which describes the coupling between open-charm meson channels and J/ψω, done by simple quark rearrangement driven by the qq interaction (see Ref. [19] for more details).
In this formalism, two-and four-quark configurations are coupled using the same transition mechanism that, within our approach, allows us to compute open-flavor meson strong decays, namely the 3 P 0 model [27,28]. This model assumes that the transition operator is where µ (ν =μ) are the quark (antiquark) quantum numbers and γ ′ = 2 5/2 π 1/2 γ with γ = g 2m is a dimensionless constant that gives the strength of the qq pair creation from the vacuum. From this operator we define the transition potential h βα (P ) within the 3 P 0 model as [29] The usual version of the 3 P 0 model gives vertices that are too hard, specially when working at high momenta. Following the suggestion of Ref. [30], we use a momentum dependent form factor to truncate the vertex as where Λ = 0.84 GeV is the value used herein [31].
Using the latter coupling mechanism, the coupledchannels system can be expressed as a Schrödinger-type equation, where χ β (P ) is the meson-meson relative wave function for channel β and H β ′ β is the RGM Hamiltonian for the two-meson states obtained from the qq interaction. The effective potential V eff β ′ β encodes the coupling with the cc bare spectrum, and can be written as where M α are the masses of the bare cc mesons. This potential has two general effects. On the one hand, it adds additional attraction or repulsion to the qq interaction provided by the RGM potentials via the exchange of intermediate cc bare states between the two interacting mesons, which can generate new states, as it is the case for the X(3872) [19]. On the other hand, the bare charmonium spectrum is renormalized by the presence of nearby meson-meson channels.
Alternatively, Eq. (7) can be solved by means of the T matrix [17], solution of the Lippmann-Schwinger equation, which is more convenient for such states above thresholds. Resonances will appear as poles of the T matrix, namely as zeros of the inverse propagator of the mixed state, defined as withĒ the pole position and G α ′ α the complete massshift of the coupled-channels state, written as where φ αβ are the 3 P 0 verteces dressed by the RGM meson-meson interaction [32]. This equivalent formalism leads to a more appropriate definition of branching ratios and partial widths, following Ref. [33]. The detailed derivation has been described in Ref. [17], so here we will only summarize the most relevant aspects. The coupled-channels S matrix for an arbitrary number of cc states can be expressed as where k is the on-shell momentum of the two meson state and S β ′ β bg (E) is the non-resonant term. Then, in the neighborhood of the poleĒ, the S matrix can be approximated as where So, assuming that we can write Z α ′ α (Ē) = λ Z 1/2 where we can identify the decay vertex From there, the partial width of a two meson decayΓ β can be written aŝ whereĒ = M r − i Γr 2 , k β is the on-shell momentum for the meson-meson β channel and E i is the on-shell total energy of mesons i = {1, 2}.
The previous equation (16) does not, in general, satisfy that the sum of the partial widths must be equal to the total width. This issue can be easily solved by defining the branching ratios as [33] B so the physical partial widths are Γ f = B f Γ r . We have performed two calculations for the quantum numbers J P C = 0 ++ and J P C = 2 ++ . The first one includes, for the J P C = 0 ++ charmonium sector, the naive 2 3 P 0 cc state together with the following channels (their corresponding threshold energies are indicated in parenthesis): DD (3734 MeV/c 2 ), ωJ/ψ (3880 MeV/c 2 ), D sDs (3937 MeV/c 2 ) and D * D * (4017 MeV/c 2 ). For the J P C = 2 ++ case we add to the former channels the DD * + h.c. (3877 MeV/c 2 ) one, which in this case will be coupled to the bare 2 3 P 2 cc state. These thresholds have been considered because of their closeness to the masses of the naive 2 3 P J (J=0,2) states predicted by the quark model. Moreover, the D * D * threshold, though located at higher energies compared to the other channels, must be included because it is the only one contributing with an S−wave in the J P C = 0 ++ sector and can have a major impact on the dynamics of the system. Its inclusion for the J P C = 2 ++ case is needed to compare both sectors.
Using the original parameters of Ref. [19] (which will be denoted as model A) we obtain the masses and widths shown in Table I.
We find two states with J P C = 0 ++ and only one with J P C = 2 ++ because the interaction in the mesonmeson channel for the latter sector is not strong enough to generate a second resonance. The mass and width of the J P C = 2 ++ state is compatible with those of the X(3930), whereas the mass of the first J P C = 0 ++ state is more similar to the new X(3860) resonance than the one of the X(3915). However, our width is smaller than the experimental one. Such small value is connected with the position of the node in the 2 3 P 0 bare wave function, which affects the 3 P 0 transition amplitudes and, hence, causes a higher sensitivity of the width to small changes in the wave function structure or, alternatively, the mass of the X(3860) resonance. A recent analysis of the decay width of the X(3860) has been performed by Ref. [34], using a simple harmonic oscillator (SHO) approximation for the meson wave function. The X(3860) width shows a strong dependence with the oscillator parameter, finding agreement with the experimental data with a resonable value. In our case, all the parameters are fixed by the strong decays of light and heavy quark mesons [35] and the qq dynamics and, thus, a similar fine-tuning cannot be done.
The mass of the second J P C = 0 ++ state allows us to assign it to the Y (3940) resonance. However, as in the former case, its width is far from the experimental value. This disagreement in the width of both states suggests a new, that may be more interesting, assignment. One can identify the second 0 ++ state with the X(3860), as the width of the state (229.8 MeV in Table I) matches with the experimental data, whereas, considering that the measured mass even reaches more than 3900 MeV, the discrepancy of the experimental mass value with the theoretical one is within the range of the uncertainties of the model. Additionally, the extra state with a width of 6.7 MeV is too narrow and can hardly be observed in the experiment of Ref. [14]. With the assignment of the X(3860) to the broader 0 ++ resonance, we do not find any candidate to the Y (3940) signal, which would be in agreement with BaBar suggestion that this resonance is the same as the X(3915) [7].
Certainly, all the states show a sizable no-qq structure and therefore cannot be assigned to pure qq states. This fact overrides the concern about the hyperfine splitting because the masses of the qq states are renormalized by the coupling with the different meson-meson channels.
To explore the robustness of the results, taken into account the uncertainties of the model parameters, we have performed a second calculation (named model B) where we have slightly changed the bare mass of the 2 3 P J cc pairs (0.25%) and used the coupling of the 3 P 0 model from Ref. [35], which represent a change from γ = 0.226 to γ = 0.286 for the charmonium sector. The results of the new calculation are shown in Table II.
Interestingly, this new parametrization leads to prac- tically the same results for the first J P C = 0 ++ state and the same compositeness for the J P C = 2 ++ , although now the mass is more similar to the X(3915) resonance. The mass of the second J P C = 0 ++ state is slightly increased, although such modification is of the order of the experimental error of the Y (3940) resonance.
In view of these results, we can proceed and calculate for the J P C = 2 ++ state the product of the two-photon decay width and the branching fraction to ωJ/ψ and DD channels, assuming the X(3915) and X(3930) are the same J P C = 2 ++ resonance. The results are quoted in Table III where we also include the decay to the DD * channel.
Our model predicts a value for the branching fraction of the 2 ++ state to DD some standard deviations below the experimental one. This value is obtained from the decay to the I = 0 DD channel as incorporated in the coupled-channels calculation. However, it does not include possible contributions from higher open-charm channels decaying to DD pairs, such as the decay of D * to Dγ or Dπ in the DD * channel. As shown in Table III, our calculated value for the branching fraction to DD * channel is higher than the one for DD, so it is reasonable to assume that part of the DD * pairs decaying to DDγ and DDπ are, in fact, measured as DD pairs, increasing our theoretical branching fraction for the DD channel. Under this assumption, the disagreement between our value and the experimental branching fraction can be easily explained if just one third of the DD * decays are measured as DD pairs.
As indicated by Table III, the results for both model A and B are very similar and not far from the experimental data. Then, both models describe the experimental branchings providing that the X(3915)/X(3930) resonances are J P C = 2 ++ . This conclusion agrees with Ref. [38].
Assuming the assignment of the broader resonance to the Y (3940), we can estimate the product branching function B(B → KY (3940)) × B(Y (3940) → ωJ/ψ). Following Olsen [11], we can assume that, due to the significant χ c0 (2P ) component, the B(B → KY (3940)) should be less than or equal to B(B → Kχ c0 (1P )). This assumption is based on the fact that the width of P-wave mesons is proportional to the derivative of the qq radial wave function at the origin, which decreases with increasing radial excitation. Moreover, the available phase space is smaller. With this assumption we obtain B(B → KY (3940)) × B(Y (3940) → ωJ/ψ) ≤ 3.3 × 10 −5 for the model A and B(B → KY (3940)) × B(Y (3940) → ωJ/ψ) ≤ 2.9 × 10 −5 for the model B, which in both cases is of the same order of magnitude as the experimental result, (7.1 ± 1.3 ± 3.1) × 10 −5 [2].
In summary, within a coupled-channels calculation we have obtained two J P C = 0 ++ and one J P C = 2 ++ resonances in the energy region of 3.9 GeV/c 2 . Using the parametrization of Ref [17] we obtain two possible description of the charmonium-like states experimentally measured in this region. On the one hand, the X(3860) is identified with the second J P C = 0 ++ state, with the right width but slightly higher mass, and the J P C = 2 ++ state with the X(3915)/X(3930). On the second hand, the two J P C = 0 ++ states are identify with the X(3860) and the Y (3940), maintaining the assignment for the other resonances. Including the results of Ref. [17] for the J P C = 1 ++ charmonium sector, where two resonances, the X(3872) and the X(3940), are described, the present work completes the picture of the P-wave charmonia around 3.9 GeV/c 2 . All these states are mixtures of χ cJ (2P ) charmonium states and meson-meson channels. Therefore neither can be identified with pure cc states, which explains their deviations from the naive quark model predictions. Among other characteristics, this compositeness is able to explain the properties of the X(3872) [17].
Within the uncertainties of our model, the mass and width of the J P C = 2 ++ state can be identified either with the X(3930) or with the X(3915), suggesting that the two resonances X(3915) and X(3930) are in fact the same J P C = 2 ++ as claimed by Z.-Y. Zhou et al. [12]. We may identify the new X(3860) resonance with a J P C = 0 ++ as suggested in Ref [10]. Finally, in the second scenario we find a resonance which reproduces the experimental data of the Y (3940) as a J P C = 0 ++ , which may encourage new experimental searches for this state. In any case, further theoretical and experimental work is necessary to fully unveil the nature of these cc resonances in this energy region.