Quark structure of the $\chi_{\rm c}(3P)$ and $X(4274)$ resonances and their strong and radiative decays

We calculate the masses of $\chi_{\rm c}(3P)$ states with threshold corrections in a coupled-channel model. The model was recently applied to the description of the properties of $\chi_{\rm c}(2P)$ and $\chi_{\rm b}(3P)$ multiplets [Phys.\ Lett.\ B {\bf 789}, 550 (2019)]. We also compute the open-charm strong decay widths of the $\chi_{\rm c}(3P)$ states and their radiative transitions. According to our predictions, the $\chi_{\rm c}(3P)$ states should be dominated by the charmonium core plus small meson-meson components. The $X(4274)$ is interpreted as a $c \bar c$ $\chi_{\rm c1}(3P)$ state. More informations on the other members of the $\chi_{\rm c}(3P)$ multiplet, as well as a more rigorous analysis of the $X(4274)$'s decay modes, are needed to provide further indications on the quark structure of the previous resonance.

In a previous paper [49], we discussed a novel coupledchannel model approach to the spectroscopy and structure of heavy quarkonium-like mesons based on the Unquenched Quark Model (UQM) formalism [36,41,42,[45][46][47][48][50][51][52]. In the UQM, quarkonium-like exotics are interpreted as the superposition of a heavy quarkonium core plus meson-meson molecular-type components. In the approach of Ref. [49], the UQM formalism was used to compute the self-energy corrections to the bare masses of χ c (2P ) and χ b (3P ) states due to virtual particle effects. However, differently from previous UQM calculations, see e.g. Refs. [37,41,42,45], we did not perform a global fit to the whole heavy quarkonium spectrum. We applied the formalism to a single heavy quarkonium multiplet at a time. Moreover, we introduced a "renormalization" prescription for the UQM results. Thanks to this, we could suggest a solution to some long-standing problems of UQM calculations. They include: I) the lack of convergence of UQM results; II) the fact that the selfenergy corrections to a state |A , both when |A is close to meson-meson decay thresholds and when |A is far away from them, are of the same order of magnitude, which is unphysical.
According to our coupled-channel model results, threshold effects should be small to medium-sized in the χ c (3P ) multiplet. Our 3 P 0 model prediction for the open-charm strong decay width of X(4274) is compatible with the experimental data within the experimental error. Therefore, it is reasonable to treat the X(4274) as a charmonium state. However, due to the total lack of experimental data on the other members of the multiplet, we cannot exclude the presence of small meson-meson components in the X(4274) wave function. Our results for the radiative transitions of the X(4274) and χ c (3P )s will be an important check and may help to assess the quark structure of the previous resonances.

A. 3 P0 pair-creation model
In the 3 P 0 pair-creation model, the open-flavor strong decay of a hadron A into hadrons B and C takes place in the rest frame of A. The decay proceeds via the creation of an additional qq pair with J P C = 0 ++ quantum numbers from QCD vacuum [53][54][55] (see Fig. 1) and the width is computed as [53,54,56] where l is the relative angular momentum between B and C and J represents the total angular momentum of B and C. The coefficient is the phase-space factor for the decay; it depends on the relative momentum q 0 between B and C, the energies of the two decay products, E B,C (q 0 ), and the mass of the decaying meson, M A . We assume harmonic oscillator wave functions for the hadrons A, B and C, depending on a single oscillator parameter α ho ; see [41, Table II] and [60, Table II]. The values of the oscillator parameter, α ho , and of the other pair-creation model parameters, r q and γ 0 , were fitted to the open-charm strong decays of higher charmonia [41]. Picture from Ref. [51]. APS copyright.
B. Threshold mass-shifts in a coupled-channel model We briefly summarize the main features of the coupledchannel model of Ref. [49]. There, higher Fock components, |BC , due to virtual particle effects are superimposed on the QQ bare meson wave functions, |A , of heavy quarkonium states. One has [36,41,42,[50][51][52]: The sum is extended over a complete set of meson-meson intermediate states |BC , with energies E B,C (q) = M 2 B,C + q 2 ; M A is the physical mass of the meson A; q is the on-shell momentum between B and C, ℓ is the relative orbital angular momentum between them, and J is the total angular momentum, with J = J B + J C + ℓ. Finally, the amplitudes BCq ℓJ| T † |A are computed within the 3 P 0 pair-creation model [53][54][55][56][57][58][59][60]. See also Sec. II A.
In the coupled-channel approach of Ref. [49], one can study a single multiplet at a time, like χ c (2P ) or χ b (3P ). The physical masses of the meson multiplet members are given by In the previous equation, is a self-energy correction, E A is the bare mass of the meson A, and ∆ th is a free parameter. Contrary to our previous UQM studies [41,42] The introduction of ∆ th in Eq. (4) represents our "renormalization" or "subtraction" prescription for the threshold corrections in the UQM.

C. Radiative Transitions in the QM and UQM formalisms
Radiative transitions of higher charmonia are of considerable interest, since they can shed light on their internal structure and provide one of the few pathways between different cc multiplets. Particularly, for those states which cannot be directly produced at e + e − colliders (such as P -wave charmonia), the radiative transitions serve as an elegant probe to explore such systems. In the quark model, the electric dipole (E1) transitions can be expressed as [65][66][67] Here, e c = 2 3 is the c-quark charge, α the fine structure constant, E γ denotes the energy of the emitted photon, γ is the total energy of the final meson. The spatial matrix elements involve the initial and final meson radial wave functions and are obtained numerically; for further details, we refer to [47,48]. From Eq. (7), we know that the value of the decay width depends on the details of the wave functions, which are highly model dependent. The angular matrix elements C AB are given by where S A,B , L A,B and J A,B are the spin, orbital angular momentum and total angular momentum of the initial/final charmonia, respectively. In the UQM formalism, the wave function of a heavy quarkonium state consists of both a QQ valence configuration and meson-meson higher Fock components, which are the result of the creation of light qq pairs from the vacuum; see Eq. (3). Therefore, the heavy quarkonium bare meson wave function has to be properly renormalized [51,Eq. (9)].
In our specific case, the radial wave functions R A,B of Eq. (7) have to be multiplied by the factors P cc (A, B) ≤ 1, which are the probabilities of finding the wave functions of the A and B states in their valence components. Given this, in the UQM formalism the width of Eq. (6) becomes [48]: III. RESULTS

A. Open-charm strong decays of χc(3P ) states
In this section, we calculate the open-charm strong decay widths of χ c (3P ) states within the 3 P 0 pair-creation model. The main features of the model are briefly described in Sec. II A. When available, we extract the masses of both the initial-and final-state mesons from the PDG [1]; otherwise, we use the relativized QM predictions of Refs. [57,64]. Our theoretical results are given  Table IV, second column), except for the value of the χc1(3P ) [or X(4274)] mass, which is extracted from the PDG [1]. The values of the charmed and charmed-strange meson masses are taken from the PDG [1], the mixing angle between D1(1P1) and D1(1P ′ 1 ) states is taken from [60, Table III].
in Table I and can be compared to the 3 P 0 pair-creation model results of [57, Table XI].
It is worth noting that: I) our predictions are of the same order of magnitude as those of [57, Table XI]. The discrepancies are in the order of 10 − 20%, except for the h c (3P ), where they are larger. These differences between our results and those of Ref. [57] arise partly because of different choices of the 3 P 0 model parameters, and partly because of the values of the masses of the decaying mesons given as inputs in the calculations. In particular, in our case we use for the decaying meson masses either the experimental values [1] or relativized QM predictions [64]. On the contrary, in Ref. [57] the authors extracted the masses from a non-relativistic potential model fit to the charmonium spectrum. Moreover, the use of different masses for the cc decaying mesons determines the opening of decay channels, like χ c2 (3P ) → DD * 2 (2460), which were below threshold in [57, Table XI]. Finally, as a check we have also computed the decay widths of χ c (3P )s by using the same input masses and model parameters as Ref. [57] and we have obtained the same results as Ref. [57]; II) according to our results, the χ c (3P )s are characterized by relatively large open-charm widths, which are of the order of 40 − 60 MeV. If our predictions were confirmed by the experiments, we may argue that χ c (3P ) mesons should be charmonium-like states, with their wave functions being dominated by a cc core; III) of particular interest are our results for the χ c1 (3P ) state. Specifically, our theoretical prediction for the total open-charm width of the χ c1 (3P ), i.e. 43.6 MeV, is compatible with the total experimental width of the χ c1 (4274) [1], namely 49 ± 12 MeV, under the hypothesis that the open-charm contribution to the total width of the χ c1 (4274) is the dominant one. As discussed in the previous point, this suggests that the wave function of the χ c1 (4274) should be dominated by the charmonium component.
Here, we discuss our UQM results for the E1 radiative transitions of χ c (3P ) states. Our predictions, denoted as Γ E1,UQM and computed by means of Eq. (9), are given in Table II; see also Tables V and VI. The QM widths of Eq. (6), Γ E1,QM , are computed by using Cornell potential model [45,47,65] wave functions for both the parent and daughter charmonium states. Our results for the Γ E1,QM widths coincide with those reported in Ref. [57]; therefore, they are not shown in the present paper.
The UQM predictions, denoted as Γ E1,UQM , are calculated by renormalizing the A and B meson wave functions according to the valence probabilities P cc (A) and P cc (B). For simplicity, due to the large amount of A and B states taken into account (1S, 2S, 1P , 1D, and so on), the calculation of the probabilities P cc (A, B) of Table  II is not performed in the UQM-based coupled-channel formalism of Secs. II B and III C, but rather in the standard UQM formalism [45,51], with the model parameter values, α ho = 0.5 GeV and γ 0 = 0.4, extracted from Ref. [57], and considering only 1S1S open-charm intermediate states. Moreover, when the initial state is above a D ( * )D( * ) or D ( * ) sD ( * ) s threshold, we ignore the contribution of this channel in wave function renormalization, even though the mass shift caused by the previous channel is not zero.
Finally, it is worth noting that: I) the radiative decay widths of χ c (3P ) states span a wide interval, from O(300 MeV) to O(1 MeV), in the case of 3P → 3S + γ and 3P → 1D + γ transitions, respectively. In particular, the 3P → 3S + γ decay widths are quite large; thus, they might be observed in the next few years; II) our UQM results for χ c (3P ) states are roughly of the same order of magnitude as the QM ones [57]. The difference between them is of the order of 5 − 10% in the case of the 3P → 3S + γ transitions and 30 − 40% for 3P → 2S + γ decays. This is a confirmation of our statement that loop effects can play a relatively important role in determining the properties of χ c (3P )s, though their importance is far from being conclusive. In this respect, it is interesting to estimate the importance of loop effects in the case of other charmonium radiative transitions. See Appendix A, Tables V and VI, and the QM results of Ref. [57]. For example, consider the χ c2 (2P ) → ψ 3 (1 3 D 1 ) + γ decay, where the ratio between the QM [57] and UQM widths is almost a factor of 2.5; III) finally, we also show  that the E1 transition widths of χ c (3P )s into J/ψ + γ are one order of magnitude suppressed with respect to those into ψ(3S) + γ. A similar pattern was previously observed in the χ b (3P ) case [48].
In conclusion, these results may provide solid references to search for the other members of the χ c (3P ) multiplet by analyzing the χ c (3P ) → ψ(2S, 3S) + γ radiative transitions. Recently, the CMS Collaboration was able to distinguish for the first time between two candidates of the bottomonium 3P multiplet, χ b1 (3P ) and χ b2 (3P ), through their Υ(nS) + γ (n = 1, 2, 3) decays [68]. We expect the charmonium 3P multiplet to be easily searched by means of the same strategy.   Table  II]. The first rows show the partial contributions to Σ(MA) from channels BC, such as DD * 0 (2300), DD1(2420), and so on. The last rows provide the total results, obtained by summing the previous partial contributions. The contributions of those channels denoted by -are suppressed by selection rules.

C. Threshold mass shifts within the χc(3P ) multiplet
We calculate the relative threshold mass shifts between the χ c (3P ) multiplet members due to a complete set of 1S1P meson-meson loops, like DD * 0 (2300), DD 1 (2420), and so on. 1 As shown in Ref. [41], charmonium loops, like η c χ c0 (1P ), are negligible because of the suppression mechanism of [51, Eq. (12)]. Therefore, these loops are not taken into account in the calculation of the selfenergy corrections of χ c (3P ) states. Following Ref. [49], the values of the bare meson masses, E A , are extracted from the relativized QM predictions of [57, Table I, sixth column] and [64]. We have: E hc(3P ) = 4318 MeV, E χc0(3P ) = 4292 MeV, E χc1(3P ) = 4317 MeV and E χc2(3P ) = 4337 MeV. The values of the physical masses, M A , of the χ c (3P ) states should be extracted from the data [1]. However, except for the mass of the χ c1 (4274), 4274 +8 −6 MeV, there are no experimental results for the masses of the remaining and still unobserved χ c (3P ) states, namely the h c (3P ), χ c0 (3P ) and χ c2 (3P ). Therefore, for the physical masses of the previous unobserved states we use the same values as the bare ones [57,64]. Moreover, for simplicity, we do not consider mixing effects between 1 1 P 1 and 1 3 P 1 charmed and charmed-strange mesons in the self-energy calculation of this section. Therefore, for the wave functions of the previous states we make the assumptions: The self-energy corrections are computed according to the UQM formalism of Sec. II B and Refs. [41,42]. Our results are reported in Table III.
Compared to our previous results for χ c (2P )s and χ b (3P )s [49], the present results for χ c (3P ) states are more model-dependent. The reason is the lack of experimental data for three of the four multiplet members. Finally, our results for the "renormalized" threshold corrections, Σ(M A ) − ∆ th , and the calculated physical masses, M th A , of χ c (3P ) states are reported in Table IV.  It is worth noting that: I) the threshold corrections of Table IV are larger than those of χ b (3P )s, but smaller than those of χ c (2P ) states; see [49, Table 1]. In light of this, we expect the χ c (3P ) states to be dominated by the cc core component; II) at present, the only decay mode of the X(4274) which has been observed experimentally is that into J/ψφ. This may be compatible with the interpretation of the X(4274) as a multiquark state with non-zero hidden-charm hidden-strange components. However, as discussed in Sec. III A, several properties of the X(4274) (e.g. its total decay width) are compatible with those of a χ c1 (3P ) state.
In conclusion, the present results indicate that the X(4274)'s wave function should be dominated by the χ c1 (3P ) component. More informations on the other members of the χ c (3P ) multiplet, as well as a more rig-orous analysis of the X(4274)'s decay modes, are needed to provide further indications on the quark structure of the previous resonance.
IV. X(4274): OTHER INTERPRETATIONS As pointed out in Ref. [69], molecular states cannot account for the 1 + nature of the X(4274). A possible interpretation of the X(4274) is that of a sscc compact tetraquark state. The spectrum of strange and nonstrange hidden-charm compact tetraquark states was computed in Ref. [22] within a relativized diquarkantidiquark model. There, the authors could provide tetraquark assignments to 13 suspected XY Z exotics, including the Z c (3900), X(4500) and X(4700); however, they could not accommodate the X(4274) within the tetraquark picture. A similar investigation on sscc compact tetraquarks was conducted within the relativized quark model [20]. There, the authors discussed possible assignments to the X(4140), X(4500) and X(4700), but they could not accommodate the X(4274) within a sscc compact tetraquark description [20]. In Ref. [70], the authors made use of QCD sum rules to study the properties of the X(4140) and X(4274). They interpreted the X(4140) as a 1 ++ diquark-antidiquark compact tetraquark in the3 c 3 c color configuration, while the X(4274) was described as a diquark-antidiquark bound state with a 6 c6c color wave function. Finally, in Ref. [71] it was suggested that the X(4140), X(4274), X(4500) and X(4700) could be accommodated within two tetraquark multiplets, with the X(4274) characterized by 0 ++ or 2 ++ quantum numbers.
In Ref. [72], the authors investigated possible assignments for the four J/ψφ structures, reported by LHCb, CMS, D0 and BaBar [73][74][75][76], in a coupled channel scheme by using a nonrelativistic constituent quark model [77]. In particular, they showed that the X(4274), X(4500) and X(4700) can be described as conventional 3 3 P 1 , 4 3 P 0 , and 5 3 P 0 charmonium states, respectively. The same interpretation for the X(4274) was proposed in Ref. [78]. In a study of heavy quarkonium hybrids based on the strong coupling regime of pNRQCD [79], the authors found out that the X(4274) is compatible with a χ c1 (3P ) state, which may be affected by the D * + s D * − s threshold.
In Ref. [80], an interpretation of the X(4274) as a Pwave D sDs0 (2317) molecular state in a quasi-potential Bethe-Salpeter equation approach was proposed. If the previous state is a hadronic molecule, an S-wave D sDs0 (2317) bound state below the J/ψφ threshold should also exist. Finally, in Ref. [81] the authors suggested to assign the X(4274) to a ψ(2S)φ S-wave hadrocharmonium configuration.
The present coupled-channel model was previously used to study the properties and quark structure of the χ c (2P ) and χ b (3P ) multiplets [49]. There, a prescription to "renormalize" the UQM results for the selfenergy/threshold corrections made it possible to distinguish between quarkonia, the χ b (3P ), and quarkoniumlike states with significant meson-meson components in their wave functions, the χ c (2P )s.
According to our new results, the X(4274) can be described as a χ c1 (3P ) state. The other members of the χ c (3P ) multiplet can be interpreted as 3P charmonium cores plus small to medium-sized open-charm mesonmeson components.
A comparison between theoretical results for the radiative transitions of χ c (3P )s (including ours and, for example, those from Ref. [57]) and the forthcoming experimental data may provide exploratory pathways to search for still unobserved 3P charmonia. Hence, we suggest the experimentalists to focus on the study of the χ c (3P ) → ψ(nS) + γ decay modes, and especially on the ψ(2S, 3S) + γ transitions.
In conclusion, we hope that this study might be helpful to fulfill a better understanding of higher P -wave charmonia. More precise conclusions regarding the quark structure of the χ c (3P ) states will necessarily require more experimental informations on the properties of the still unobserved h c (3P ), χ c0 (3P ) and χ c2 (3P ).   In Table V, we enlist our UQM results for the E1 radiative transition widths of higher-lying charmonia, including 2S, 3S, 1P and 2P resonances. The widths are calculated as explained in Sec. III B. In Table VI, we compare our UQM predictions to the available experi-mental data [1]. Our results can also be compared to the QM predictions of Ref. [57].
It is worth noting that our predictions are in good accordance with the existing experimental results [1]. This is a further indication of the importance of the radiative transitions in the study of the properties of both the well-established and still unobserved heavy quarkonium resonances.