# Prediction of a missing higher charmonium around 4.26 GeV in \(J/\psi \) family

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## Abstract

Inspired by the similarity between the mass gaps of the \(J/\psi \) and \(\Upsilon \) families, the prediction of a missing higher charmonium with mass 4,263 MeV and very narrow width is made. In addition, the properties of two charmonium-like states, \(X(3940)\) and \(X(4160)\), and charmonium \(\psi (4415)\) are discussed, where our calculation shows that \(X(3940)\) as \(\eta _c(3S)\) is established, while the explanation of \(X(4160)\) to be \(\eta _c(4S)\) is fully excluded and that \(\eta _c(4S)\) is typically a very narrow state. These predictions might be accessible at BESIII, Belle, and BelleII in near future.

### Keywords

Decay Width Narrow Width Invariant Mass Spectrum Belle Collaboration Partial Decay WidthSince the observation of \(J/\psi \) in 1974 [1, 2], the charmonium family has become abundant with more and more such states announced by the experiments [3]. Especially in the past decade, a series of charmonium-like states have been observed, which have further stimulated theorists’ extensive interest in revealing their underlying properties (see a recent review in Ref. [4]), since these novel phenomena reflect non-perturbative behavior of quantum chromodynamics (QCD). Among the studies on these states, it is an important research topic for the whole community of particle physics how to identify the exotic states, whose establishment is, of course, tied with our understanding of the charmonium family.

If this predicted state exists in the \(J/\psi \) family, we must reveal its underlying properties to answer why there does not have any evidence in the present experiment, which will be the main task of this work.

Before studying this missing charmonium, we need to introduce two charmonium-like states \(Y(4260)\) and \(Y(4360)\) with the masses around 4,263 MeV. \(Y(4260)\) was observed by the BaBar Collaboration in the \(J/\psi \pi ^+\pi ^-\) invariant mass spectrum of \(e^+e^-\rightarrow J/\psi \pi ^+\pi ^-\) [7], while \(Y(4360)\) was reported by the Belle Collaboration by studying \(e^+e^-\rightarrow \psi (2S)\pi ^+\pi ^-\) [8]. Thus, both of \(Y(4260)\) and \(Y(4360)\) have \(J^{PC}{=}1^{{-}{-}}\). We will come back later to discuss whether the missing charmonium has the relation to these two charmonium-like states.

In the following, we study the decay behavior of the missing state corresponding to \(\psi (4S)\), which is crucial to explain why this state is still missing at the predicted location and how to search for this in future experiment. In the study we also obtain its full width which is the important information to us. Since the mass of the missing charmonium predicted (in the next discussion we adopt \(\psi (4S)\) to denote this missing charmonium and assume its mass to be 4,263 MeV) is above the thresholds of \(D^{(*)}\bar{D}^{(*)}\) and \(D_s^{(*)}\bar{D}_s^{(*)}\), \(\psi (4S)\) can decay into \(D^{(*)}\bar{D}^{(*)}\) and \(D_s^{(*)}\bar{D}_s^{(*)}\).

In Fig. 2, we give the dependence of the partial decay widths of the predicted \(\psi (4S)\) on the \(R\) value, which covers the \(R\) range discussed in \(\psi (3S)\). Here, \(D\bar{D}\), \(D\bar{D}^*+H.c.\), \(D^*\bar{D}^*\), \(D_s\bar{D}_s\), \(D_s\bar{D}_s^*+H.c.\), \(D_s^*\bar{D}_s^*\) are open for \(\psi (4S)\). A very interesting result of the decay behavior of \(\psi (4S)\) can be found from Fig. 2, i.e., the total decay width of \(\psi (4S)\) is stable over the corresponding \(R\) range adopted, while its partial decay widths strongly depend on the \(R\) value. This phenomenon is due to the node effects. Our result also shows that the node effects are important when discussing the higher charmonium decays because due to these effects we find that the predicted charmonium \(\psi (4S)\) has very narrow width around 6 MeV.

For the higher charmonia above the \(D\bar{D}\) threshold, this phenomenon of \(\psi (4S)\) presented here is unusual since the other higher charmonia, \(\psi (4040)\), \(\psi (4160)\), and \(\psi (4415)\), have widths \(80 \pm 10\), \(103 \pm 8\), and \(62 \pm 20\) MeV, respectively, all of which are large. Even \(\psi (3770)\), which is just 43 MeV above the \(D\bar{D}\) threshold, has the width 27.2 MeV.

If considering the large width difference between the predicted \(\psi (4S)\) and \(Y(4260)/Y(4360)\), it is obvious that there does not exist any correspondence between \(\psi (4S)\) and the observed charmonium-like states \(Y(4260)/Y(4360)\), where the given average values of widths of \(Y(4260)\) and \(Y(4360)\) are \(95 \pm 14\) and \(74 \pm 18\) MeV in particle data group (PDG) [3], respectively. In Refs. [17, 18] the non-resonant explanations to charmonium-like states \(Y(4260)\) and \(Y(4360)\) were proposed, where both of them can be described by the interference effects of the production amplitudes of \(e^+e^-\rightarrow J/\psi (\psi (2S))\pi ^+\pi ^-\) via the intermediate charmonia \(\psi (4160)(\psi (4415))\) and direct \(e^+e^-\) annihilation into \(J/\psi (\psi (2S))\pi ^+\pi ^-\).

As a typical higher charmonium with a very narrow width, the predicted \(\psi (4S)\) is difficult to identify by the analysis of the open-charm decay channels [19, 20, 21, 22] and the \(R\) value scan [23, 24, 25, 26, 27, 28, 29, 30] based on the present experimental data, which can naturally answer why this higher charmonium is still missing in experiment. Thus, we expect future experimental results of the open-charm decays and a more precise study of the \(R\) value scan, especially from BESIII, Belle, and forthcoming BelleII.

We notice a recent analysis of BESIII data in Ref. [31]. BESIII already realized the measurement of the cross sections of \(e^+e^-\rightarrow h_c(1P)\pi ^+\pi ^-\) at center-of-mass energies 3.90–4.42 GeV [32]. Yuan analyzed the data by fitting the line shape with two Breit–Wigner functions, which indicates that there are a narrow structure with mass \(4{,}216 \pm 18\) MeV and width \(39 \pm 22\) MeV and another broad structure with mass \(4{,}293 \pm 9\) MeV and width \(222 \pm 67\) MeV [31], where these two charmonium-like structures have \(J^\mathrm{PC}=1^{{-}{-}}\). Comparing the resonance parameters of this narrow structure in Ref. [31] with our result of the predicted \(\psi (4S)\), one finds their similarity, which means that this narrow charmonium-like structure can be as a good candidate of the predicted \(\psi (4S)\) in this work. As mentioned above, more experimental efforts will be necessary to clarify this point.

In addition, another charmonium-like state \(X(4160)\) was reported by the Belle Collaboration through double charm production, where \(X(4160)\) appears in the \(D^*\bar{D}^*\) invariant mass distribution of \(e^+e^-\rightarrow D^{*+}{D}^{*-}J/\psi \) [34]. In this work, we also discuss whether \(X(4160)\) can be explained as \(\eta _c(4S)\). In Fig. 4, the partial and total decay widths of \(\eta _c(4S)\) are listed. The situation of the decay behavior of \(\eta _c(4S)\) is very similar to that of \(\psi (4S)\) discussed above. Our study shows that the total width of \(\eta _c(4S)\) is also very narrow, which is not consistent with the measured width of \(X(4160)\) that has \(139^{+111}_{-61}\pm 21\) MeV [34]. According to the above analysis, we can fully exclude the \(\eta _c(4S)\) assignment to \(X(4160)\).

Before summarizing our work, we would like to pay attention to \(\psi (4415)\). Here, we discuss the possibility of \(\psi (4415)\) as \(\psi (5S)\).

The experimental information of \(X(3940)\), \(X(4160)\), \(Y(4260)\), \(Y(4360)\), \(\psi (4415)\), and possible \(Y(4216)\)

States | Mass (MeV) | Width (MeV) | Observed channel |
---|---|---|---|

\({3{,}942^{+7}_{-6}\pm 6}\) | \({37^{+26}_{-15}\pm 8}\) | \({e^{+}e^{-}\rightarrow D^{*}\bar{D}J/\psi }\) | |

\(X(4160)\) [34] | \({4{,}156^{+25}_{-20}\pm 15}\) | \({139^{+111}_{-61}\pm 21}\) | \({e^{+}e^{-}\rightarrow D^{{*}{+}}{D}^{{*}{-}}J/\psi }\) |

\(Y(4260)\) [7] | \({4{,}259\pm 8^{+2}_{-6}}\) | \({88\pm 23^{+6}_{-4}}\) | \({e^{+}e^{-}\rightarrow J/\psi \pi ^{+}\pi ^{-}}\) |

\(Y(4360)\) [8] | \({4{,}361\pm 9\pm 9}\) | \({74\pm 15\pm 10}\) | \({e^{+}e^{-}\rightarrow \psi (2S)\pi ^{+}\pi ^{-}}\) |

\(\psi (4415)\) [3] | \({4{,}421\pm 4}\) | \({62\pm 20}\) | See PDG [3] for details |

\({4,216\pm 18}\) | \({39\pm 22}\) | \({e^{+} e^{-}\rightarrow h_{c}\pi ^{+} \pi ^{-}}\) |

In summary, in this work we have predicted a missing charmonium \(\psi (4S)\) with quantum number \(n^{2S+1}L_J=4^3 S_1\), which has mass around 4,263 MeV and very narrow width. This observation has been obtained through the similarity of the mass gaps existing in \(J/\psi \) and \(\Upsilon \) families and further study of its OZI-allowed decay behavior. Comparing this state with the reported higher charmonia \(\psi (4040)\), \(\psi (4160)\), and \(\psi (4415)\), the predicted \(\psi (4S)\) state is the first higher charmonium with such a narrow width, which can explain why there is no evidence in the corresponding analysis of the open-charm decay channels and the \(R\) value scan until now. The relation of \(\psi (4S)\) and the recent evidence of a narrow charmonium-like structure in the \(e^{+}e^{-}\rightarrow h_c(1P)\pi ^{+}\pi ^{-}\) process [31] has also been discussed.

We have also explored two charmonium-like states \(X(3940)\) and \(X(4160)\) under the \(\eta _c(3S)\) and \(\eta _c(4S)\) assignments, respectively. Our study has indicated that \(X(3940)\) can be well explained as \(\eta _c(3S)\) but \(X(4160)\) as \(\eta _c(4S)\) is fully excluded. Here, we have predicted \(\eta _c(4S)\) also has very narrow width similar to the situation of the discussed \(\psi (4S)\). In addition, the properties of \(\psi (4415)\) as \(\psi (5S)\) have been given in this work.

The study presented in this work can enhance our understanding of charmonium family especially for \(J/\psi \) family, which is valuable to reveal the underlying QCD non-perturbative effects. To test our predictions, we expect further experimental progress on higher charmonia, where BESIII, Belle, and BelleII will be good platforms to carry out the search for our predictions.

## Notes

### Acknowledgments

This project is supported by the National Natural Science Foundation of China under Grants No. 11222547, No. 11175073, No. 11035006 and No. 11375240, the Ministry of Education of China (FANEDD under Grant No. 200924, SRFDP under Grant No. 20120211110002, NCET, the Fundamental Research Funds for the Central Universities), the Fok Ying Tung Education Foundation (No. 131006). X.L. would like to expresses his sincere thanks to Professor Takayuki Matsuki for providing the agreeable atmosphere during his stay at Tokyo Kasei University.

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