How higher charmonia shape the puzzling data of the e + e − → η J /ψ cross section

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I. INTRODUCTION
With the continuous accumulation of data, an increasing number of charmoniumlike XYZ states have been documented in various experimental endeavors, such as Belle, BESIII, LHCb, and CMS.This surge in reported states has invigorated the field of hadron spectroscopy, giving it a remarkable level of activity and importance (see review articles [1][2][3][4][5][6][7][8][9] for more details).This research landscape has led to significant advances in our construction of hadron spectroscopy.In particular, it has advanced our understanding of the intricate nonperturbative dynamics associated with the strong interactions, which is a central frontier of contemporary physics research.
As collected by the Particle Data Group [20], the family of charmonium encompasses two well-established entities, namely ψ(4160) and ψ(4415), in addition to ψ(4040).However, upon meticulous examination of the experimental data reported by BESIII, a conspicuous absence is observed within the cross section distribution of e + e − → ηJ/ψ, pertaining specifically to the aforementioned charmonium states, ψ(4160) and ψ(4415).Remarkably, the established charmonium ψ(4160) possesses sizable decay fractions into the ηJ/ψ channel according to the theoretical calculation [21].Moreover, if carefully checking the BESIII's data, we may find an asymmetric line shape, associated with the resonance structure denoted as Y(4230).Confronted by this puzzling phenomenon existing in the cross section data of e + e − → ηJ/ψ [10], it is necessary to come up with a convincing solution to this problem.We should mention the distinct characterized energy level structure of higher charmonia within the framework of the unquenched picture, a contribution credited to the diligent efforts of the Lanzhou group [22][23][24].In the newly constructed J/ψ family, there are six vector charmonium states in the energy range of 4−4.5 GeV, i.e., ψ(4040), ψ(4160), ψ(4220), ψ(4380), ψ(4415), and ψ(4500) [22].In this work, we reveal how these higher charmonia shape the puzzling data of the e + e − → ηJ/ψ cross section.II.We then illustrate the calculation details of these charmonium decays into ηJ/ψ via the hadronic loop mechanism in Sec.II A. Based on these results, we perform a combined fit to show how these higher charmonia shape the puzzling data of the e + e − → ηJ/ψ cross section in Sec.II B. The paper ends with a short summary in Sec.III.

II. AN ANALYSIS TO THE CROSS SECTION OF
e + e − → ηJ/ψ The continued accumulation of the experimental data in the range of 4.0−4.5 GeV has deepened our understanding of the spectrum of higher vector charmonium above 4.0 GeV.Recently, the BESIII collaboration performed a precise measurement of the e + e − → ηJ/ψ cross section over a center-of-mass energy range of √ s = 3.808−4.951GeV, and obtained the charmonium state ψ(4040) and the other two charmoniumlike states named Y(4230) and Y(4360) from a Breit-Wigner fit, but the extracted resonance parameters of the observed Y(4230) state differ from other measurements of the opencharmed and hidden-charmed decay channels [11][12][13][14][15][16][17][18][19], so we can conclude that the ψ(4230) resonance parameters can be influenced by the contribution of the nearby ψ(4160) state.The line shape of the observed Y(4230) looks asymmetric and irregular, and could not be formed by the contribution of just one resonance.It seems that the structure around 4.2 GeV from the measured cross section data of e + e − → ηJ/ψ may have some substructures, which is also endorsed in a recent theoretical study conducting a global coupled-channel analysis for the BESIII's data [25].
The contribution of genuine intermediate charmonium to e + e − → ηJ/ψ can be described by a phase space corrected Breit-Wigner function, i.e., , where m ψ , Γ ψ , and Γ e + e − ψ are the mass, total width and the dielectron width of the intermediate charmonium, respectively.Φ 2→2 is the phase space and s donates the center-of-mass energy.The only remaining unknown term BR(ψ → ηJ/ψ) is the branching ratio of the associated charmonium decay into ηJ/ψ, and we will present the details of the calculation for the ψ → ηJ/ψ decay next.In Ref. [22], an S -D mixing scheme was proposed to construct the energy level structure of vector charmonia in the range of 4.0−4.5 GeV, as shown below, where θ denotes the mixing angle.In this scheme, ψ(4220) was assigned to a 4S -3D mixing state while its partner ψ(4380) was predicted.The wellestablished ψ(4415) was assigned into a 5S -4D mixing state while its partner ψ(4500) was predicted and was found to exist in the BESIII measurement of e + e − → K + K − J/ψ [23].The assignment of ψ(4040) and ψ(4160) with a 3S -2D mixing scheme has also been proposed in Ref. [23] to match the experimental dielectron width of ψ(4160) [30].The mixing angles are listed in Table I.Next, we will utilize the hadronic loop mechanism to calculate the branching ratios of the hidden-charm decays of ψ → ηJ/ψ for the above charmonia.
The hadronic loop mechanism has been widely used to study the hidden-flavor decays of heavy quarkonia above the open-flavor thresholds  and the calculated branching ratios are usually comparable to the experimental measurements.In the framework of the hadronic loop mechanism, the higher charmonium ψ ′ /ψ ′′ within an S -D mixture first decay into a pair of charmed mesons D ( * ) D( * ) , and reach the ηJ/ψ final states by exchanging a D or D * meson as shown in Fig. 1.The general expression for the amplitude mediated by the charmed meson loop is where V i (i = 1, 2, 3) are interaction vertices, and P i (i = 1, 2, E) denote the corresponding propagators of intermediate charmed mesons.The form factor F (q 2 , m 2 E ) is introduced to compensate for the off-shell effect of the exchanged D ( * ) meson and to depict the structure effect of the interaction vertices.In our calculation, the monopole form factor is taken as where m E and q are the mass and four momentum of the exchanged intermediate meson, respectively.The cutoff Λ can be parametrized as Λ = m E + αΛ QCD , with Λ QCD = 220 MeV [32][33][34], α is usually of order of 1 and depends on the specific processes [53].
The effective Lagrangian approach is used to give the concrete expressions for the decay amplitudes defined in Eq. ( 3).The Lagrangians of the concrete interactions involved in Fig. 1 are listed below [49], With the above preparations, we can write down the amplitudes of ψ(4040) and ψ(4160) decays into ηJ/ψ within the 3S -2D mixing scheme defined in Eq. ( 2), as shown in Fig. 1, The concrete expressions of M (i) ψ(4040) and M (i) ψ(4160) are given in Appendix A, the amplitudes of the others are similar to the group of ψ(4040) and ψ(4160) with different coupling constants and mixing angles.Finally, the branching ratio of the charmonium decay into ηJ/ψ can be obtained by where m ψ and Γ ψ are the mass and total width of the initial charmonium listed in Table I, and ⃗ p 1 is the three-momentum of the η meson in the rest frame of the initial state.The coefficient 1/3 and the sum spin come from the averaging over the Below is a brief description of the method for determining the relevant coupling constants, as they appear in the concrete amplitudes outlined in Appendix A.
B. A combined fit to the cross section data of e + e − → ηJ/ψ With the input of the branching ratios BR(ψ → ηJ/ψ) predicted by the hadronic loop mechanism as shown in Fig. 2, we can now obtain the contribution of each vector charmonium to the cross section of e + e − → ηJ/ψ from Eq. ( 1), and apply a combined fit to the experimental data.
Since the branching ratio of ψ(4500) → ηJ/ψ is at least 2 orders of magnitude smaller than the others, and there is indeed no signal of structures around 4.5 GeV in the measured data of e + e − → ηJ/ψ, we believe that the contribution of ψ(4500) to the cross section of e + e − → ηJ/ψ can be safely dropped.In the following, we will use the left five theoretically constructed charmonia, i.e., ψ(4040), ψ(4160), ψ(4220), ψ(4380), and ψ(4415), as well as their theoretical branching ratios of decays into ηJ/ψ, dependent on the α parameter, to perform a combined fit to the cross section data of e + e − → ηJ/ψ.FIG.3: Our fit to the higher vector charmonium contribution in the cross section distribution of e + e − → ηJ/ψ within scheme I. Here, the data points are from BESIII measurement [10], the five dashed lines represent the contributions of the higher charmonium states, the blue line represents the background, and the red line with a band represents the total contribution and uncertainties.

BESIII @2023
The total amplitude of e + e − → ηJ/ψ can be written as with an exponentially parametrized background formulated as where ϕ i is the phase angle between the resonance amplitude M ψ i (s) and the background M 0 (s), g and a are two free parameters.The total cross section can be represented by Here, we employ two fitting schemes.In the first scheme, we assume a same α parameter for all charmonium states.The best fitting result in scheme I, represented by a red curve, is shown in Fig. 3, with χ 2 /d.o.f.= 1.49.The relevant fitting parameters in scheme I are listed in Table IV.
A direct observation in Fig. 3 shows that the line shape around 4.0 GeV is not well reproduced in scheme I.This is mainly due to the scarcity of cross section data points around 4.0 GeV, which puzzles us in determining the contribution of the resonance state ψ(4040).Our analysis suggests two resonance contributions from ψ(4160) and ψ(4220) around 4.2 GeV.With the same α parameter, the branching ratio of ψ( 4415) is significantly larger than that of ψ(4380) (see Fig. 2), thus the peak around 4.4 GeV is dominated by the contribution of ψ(4415), which affects the quality of the fit and is one of the largest contributions to the χ 2 /d.o.f.. Therefore, it should be a focus of future BESIII and Belle II experiments to have a more clear understanding around 4.4 GeV in this reaction process.In particular, we found that the interference effect plays a crucial role in broadening the distribution of the resonance signal associated with Y(4230), explaining its large width of 80.4 ± 4.4 ± 1.4 MeV in the experimental fit and the line shape puzzle around 4.2 GeV in the e + e − → ηJ/ψ cross section.
In scheme II, we consider a range of α parameters [1,5] for the five charmonium states without extra limit.The three solutions A, B, and C in scheme II, represented by the red curve, are shown in Fig. 4 with χ 2 /d.o.f.around 0.93, which is obviously improved compared to that of scheme I.The relevant fitting parameters in scheme II are listed in Table IV.
Similar to scheme I, our analysis in scheme II shows the significant role of the interference effects in broadening the distribution of the resonance signal associated with ψ(4160) and ψ(4220), which naturally explains the asymmetric line shape associated with the resonance structure denoted as Y(4230).Of the three solutions, ψ(4040) gives different contributions to the cross section, and we hope that the experiment will complement the data points around in this part and clarify the contribution of ψ(4040).Around 4.4 GeV, each solution in scheme II shows an unremarkable resonance signal of ψ( 4415) in e + e − → ηJ/ψ.Further investigation in future BESIII and Belle II experiments is indeed necessary to solve the broad width puzzle associated with the resonance structure denoted as Y(4360).FIG.4: Our fit to the cross section of the e + e − → ηJ/ψ process between E cm = 3.808 to 4.600 GeV by scheme II.Here, the five dashed lines represent the contributions of the higher charmonium states, the blue line represents the background, and the red line with a band represents the total contribution and uncertainties.The insets show the branching ratios for each state with the central values and uncertainties in the fit as shown in Table IV and are represented by the corresponding α values in the hadronic loop mechanism [in Eq. ( 4)] as shown in Fig. 2, where the central values and the errors are represented by the solid black lines and the colored bands, respectively.

III. SUMMARY
Recently, the BESIII collaboration measured the cross section of e + e − → ηJ/ψ from 3.808 to 4.951 GeV, and reported three structures, the charmonium ψ(4040) and two other charmoniumlike states, named Y(4230) and Y(4360) [10].It is curious that the measured width of Y(4230) is so large and has a large difference from the other open-charm and hiddencharm measurements, and the more subtle details of the line shape around 4.2 GeV with asymmetry suggest that the structure of the observed Y(4230) may not be formed by a single resonance.
In order to understand the puzzles arising in the measured data, the energy level structures of the charmonia from theoretical inputs are crucial.In this work, we focus on the six charmonia constructed by an unquenched potential model in the range of 4.0−4.5 GeV, i.e., ψ(4040), ψ(4160), ψ(4220), ψ(4380), ψ(4415), and ψ(4500).Using the hadronic loop mechanism, we are able to quantitatively calculate the branching ratios of the higher charmonium state decays into ηJ/ψ.The results indicate that the contribution of ψ(4500) to the cross section of e + e − → ηJ/ψ is too small and is ignored in our later fitting analysis, which is in agreement with the measured data, while others have sizable contributions.We then perform a combined fit with the left five charmonia to the newly measured cross section data of e + e − → ηJ/ψ, the corresponding branching ratios of these charmonia decaying into ηJ/ψ are constrained within a reasonable range, suggested by the hadronic loop mechanism.We have made two attempts with different cutoff parameter constraint schemes.Both fitting results show that the newly measured cross section of e + e − → ηJ/ψ can be reproduced by the five charmonia from the theoretical inputs, the puzzling large width and asymmetric line shape of the 4.2 GeV structure can be naturally explained by the contributions of the neighboring charmonia ψ(4160) and ψ(4220).Moreover, the introduction of ψ(4380) predicted by the theory in the combined fit is compatible with the experimental data, which together with ψ(4415) can reproduce the third broad enhancement structure around 4.4 GeV reported by BESIII.More importantly, the results support the characterized energy level construction of higher vector char-monia in the range of 4.0−4.5 GeV.
With the increasing accumulation of experimental data on the varieties of final states in this particular charmonium energy range, as well as the continuing attention of experimentalists and theorists, our understanding of the inner nature of higher charmonium states is becoming more mature and profound.

TABLE IV :
The fitting parameters in scheme I, and the solutions A, B and C in scheme II.The g and a are parameters for the background, and BR i and ϕ i (rad) are decay branching ratios into J/ψη and phases for five ψ states [successively ψ(4040), ψ(4160), ψ(4220), ψ(4380) and ψ(4415)], respectively.
polarizations of the initial state and summing up the polarizations of the final state.