Evidence for resonant structures in
The cross sections of at center-of-mass energies from 3.90 to 4.42 GeV were measured by the BESIII and the CLEO-c experiments. Resonant structures are evident in the line shape, the fit to the line shape results in a narrow structure at a mass of and a width of MeV, and a possible wide structure of mass and width MeV. Here the errors are combined statistical and systematic errors. This may indicate that the state observed in has fine structure in it.
pacs:14.40.Rt, 14.40.Pq, 13.66.Bc
The observation of the -states in the exclusive production of babary (); belley (); babary_new (); belley_new () and babar_pppsp (); belle_pppsp (); babar_pppsp_new () from the B-factories is a great puzzle in understanding the vector charmonium states epjc-review (). According to the potential models, there are 5 vector states above the well-known 1D state and below around 4.7 GeV/, namely, the 3S, 2D, 4S, 3D, and 5S states epjc-review (). However, experimentally, besides the three well known structures observed in inclusive hadronic cross section, i.e., the , , and pdg (), there are four -states, i.e., the , , , and babary (); belley (); babary_new (); belley_new (); babar_pppsp (); belle_pppsp (); babar_pppsp_new (). This suggests that at least some of these structures are not charmonium states, and thus has arisen various scenarios in interpreting one or more of them epjc-review ().
The BESIII experiment bes3 () running near the open charm threshold supplies further information to understand the properties of these vector states. Amongst these information, the most relevant measurement is the study of zc4020 (). Besides the observation of a charged charmoniumlike state , BESIII reported the cross section measurement of at 13 center-of-mass (CM) energies from 3.900 to 4.420 GeV zc4020 (). The measurements are listed in Table 1. In the studies, the is reconstructed via its electric-dipole (E1) transition with to 16 exclusive hadronic final states: , , , , , , , , , , , , , , , and .
The CLEO-c experiment did a similar analysis, but with significant signal only at CM energy 4.17 GeV cleoc_pipihc (), the result is pb, where the third error is from the uncertainty in .
The cross sections are of the same order of magnitude as those of the measured by BESIII zc3900 () and other experiments babary_new (); belley_new (), but with a different line shape (see Fig. 1). There is a broad structure at high energy with a possible local maximum at around 4.23 GeV. We try to use the BESIII and the CLEO-c measurements to extract the resonant structures in .
As the systematic error () of the BESIII experiment is common for all the data points, we only use the statistical errors in the fits below. The CLEO-c measurement is completely independent from the BESIII experiment, and all the errors added in quadrature ( pb) is taken as the total error and is used in the fits. We use a least method with footnote ()
where is the experimental measurement, and is the cross section value calculated from the model below with the parameters from the fit. Here is the energy corresponds to the th energy point.
As the line shape above 4.42 GeV is unknown, it is not clear whether the large cross section at high energy will decrease or not. We try to fit the data with two different scenarios.
Assuming the cross section follows the three-body phase space and there is a narrow resonance at around 4.2 GeV, we fit the cross sections with the coherent sum of two amplitudes, a constant and a constant width relativistic Breit-Wigner (BW) function, i.e.,
where is the 3-body phase space factor, , is the Breit-Wigner (BW) function for a vector state, with mass , total width , electron partial width , and the branching fraction to , , keep in mind that from the fit we can only extract the product . The constant term and the relative phase, , between the two amplitudes are also free parameters in the fit together with the resonant parameters of the BW function.
The fit indicates the existence of a resonance (called hereafter) with a mass of MeV/ and width of MeV, and the goodness-of-the-fit is , corresponding to a confidence level of 27%. There are two solutions for the which are eV and eV. Here all the errors are from fit only. Fitting the cross sections without the results in a very bad fit, , corresponding to a confidence level of . The statistical significance of the is calculated to be comparing the two s obtained above and taking into account the change of the number-of-degree-of-freedom. Figure 2 (left panel) shows the final fit with the .
Assuming the cross section decreases at high energy, we fit the cross sections with the coherent sum of two constant width relativistic BW functions, i.e.,
where both and take the same form as above but with different resonant parameters.
The fit indicates the existence of the with a mass of MeV/ and width of MeV, as well as a broad resonance, the , with a mass of MeV/ and width of MeV. The goodness-of-the-fit is , corresponding to a confidence level of 97%, an almost perfect fit. There are two solutions for the which are eV and eV. Again, here the errors are from fit only. Fitting the cross sections without the results in a much worse fit, , corresponding to a confidence level of . The statistical significance of the is calculated to be comparing the two s obtained above and taking into account the change of the number-of-degree-of-freedom. Figure 2 (right panel) shows the final fit with the and .
From the two fits showed above, we conclude that very likely there is a narrow structure at around 4.22 GeV/, although we are not sure if there is a broad resonance at 4.29 GeV/. We try to average the results from the fits to give the best estimation of the resonant parameters. For the , we obtain
While for the , we obtain
Here the errors include both statistical and systematic errors. The results from the two solutions and the two fit scenarios are covered by enlarged errors, the common systematic error in the cross section measurement is included in the error of the .
It is noticed that the uncertainties of the resonant parameters of the are large, this is due to two important facts: one is the lack of data at CM energies above 4.42 GeV which may discriminate which of the two above scenarios is correct, the other is the lack of high precision measurements around the peak, especially between 4.23 and 4.26 GeV. The two-fold ambiguity in the fits is a nature consequence of the coherent sum of two amplitudes zhuk (), although high precision data will not resolve the problem, they will reduce the errors in from the above fits. As the fit with a phase space amplitude predicts rapidly increasing cross section at high energy, it is very unlikely to be true, so the results from the fit with two resonances is more likely to be true. More measurements from the BESIII experiments at CM energies above 4.42 GeV and more precise data at around the peak will also be crucial to settle down all these problems.
There are thresholds of zhaoq1 (), zhenghq (); yuancz (), pdg () at the mass region, these make the identification of the nature of this structure very complicated. The fits described in this paper supply only one possibility of interpreting the data. In Ref. zhaoq2 (), the BESIII measurements zc4020 () were described with the presence of one relative S-wave molecular state and a non-resonant background term; while in Ref. voloshin (), the BESIII data zc4020 () were fitted with a model where the and are interpreted as the mixture of two hadroncharmonium states. It is worth to point out that various QCD calculations indicate that the charmonium-hybrid lies in the mass region of these two states ccg_lqcd () and the tend to be in a spin-singlet state. Such a state may couple to a spin-singlet charmonium state such as strongly, this makes the and/or good candidates for the charmonium-hybrid states.
In summary, we fit cross sections measured by BESIII and CLEO-c experiments, evidence for a narrow structure at around 4.22 GeV, as well as a wide one at 4.29 GeV, is observed. More high precision measurements at above 4.42 GeV and around 4.22 GeV are desired to better understand these structures.
This work was supported in part by the Ministry of Science and Technology of China under Contract No. 2009CB825203, and National Natural Science Foundation of China (NSFC) under Contracts Nos. 10825524, 10935008, and 11235011.
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