On leptonic width of X(4260)

New measurements on cross sections in e+e-→J/ψπ+π-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$e^+e^-\rightarrow J/\psi \pi ^+\pi ^-$$\end{document}, hcπ+π-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h_c\pi ^+\pi ^-$$\end{document}, D0D∗-π++c.c.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D^0D^{*-}\pi ^++c.c.$$\end{document}, ψ(2S)π+π-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\psi (2S)\pi ^+\pi ^-$$\end{document}, ωχc0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega \chi _{c0}$$\end{document} and J/ψη\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$J/\psi \eta $$\end{document} channels have been carried out by BESIII, Belle and BABAR experiments, as well as in the Ds∗+Ds∗-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D_s^{*+}D_s^{*-}$$\end{document} channel. We perform extensive numerical analyses by combining all these data available, together with those in D+D∗-+c.c.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D^+D^{*-}+c.c.$$\end{document} and D∗+D∗-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D^{*+}D^{*-}$$\end{document} channels. Though the latter show no evident peak around s=4.230\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sqrt{s}=4.230$$\end{document} GeV, the missing X(4260) is explained as that it is concealed by the interference effects of the well established charmonia ψ(4040)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\psi (4040)$$\end{document}, ψ(4160)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\psi (4160)$$\end{document} and ψ(4415)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\psi (4415)$$\end{document}. Our analyses reveal that the leptonic decay width of X(4260) ranges from O(102)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {O}(10^2)$$\end{document} eV to O(1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {O}(1)$$\end{document} keV, and hence it is probably explained in the conventional quark model picture. That is, the X(4260) may well be interpreted as a mixture of 43S1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$4^3S_1$$\end{document} and 33D1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$3^3D_1$$\end{document} (23D1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2^3D_1$$\end{document}) states.

The property of X (4260) becomes a very interesting topic since its discovery, because it is generally thought that there are not enough unassigned vector states in charmonium spectrum (taking into account the recently reported X (4360), X (4630)/X (4660) states), according to the naive a e-mail: caoqf@pku.edu.cn b e-mail: qihongrong@tsinghua.edu.cn (corresponding author) quark model predictions [6]. The only such 1 −− states expected up to 4.4 GeV are generally 1S, 2S, 1D, 3S, 2D and 4S, and they seem to be well established [7] -the situation is depicted in Fig. 1.
It is noticed that above DD threshold the number of 1 −− states given by quark model prediction is inconsistent with that given by experiments. It is considered that the discrepancy between the naive quark model prediction and the observed spectrum is ascribed, at least partially, to the existence of many open charm thresholds, since the latter will distort the spectrum (see related discussions in, for example, Refs. [8,9]). The situation is depicted in Fig. 2.
The open charm channels such as DD, DD * , D * D * do not seem to be found in the final states of X (4260) decays [24][25][26]. If this is indeed the case, then it would make X (4260) even more mysterious, since the J/ψπ π channel would become a very important, if not the dominant one. Hence the leptonic width e + e − of X (4260) would become very small, making it even harder to be understood as a conventional 1 −−c c state, since the nearby 4 3 S 1 state is expected to have a leptonic width at the order ∼ 10 3 eV [19].
In a previous publication, we have also studied the X (4260) issue and suggested that there could be a sizable ωχ c0 coupling [27], later confirmed by experimental researches [28]. At the same time, a very small e + e − was found ∼ 25eV. However, many new experimental results have appeared since then, measured by BESIII, Belle and BABAR experiments, such as e + e − → J/ψπ + π − [3,5,29], , D 0 D * − π + +c.c. [31], ψ(2S)π + π − [32][33][34], ωχ c0 [35,36], J/ψη [37,38] and D * + s D * − s [39]. Hence the analyses of Ref. [27] urgently need to be upgraded. Among all it is worthwhile mentioning the D * + s D * − s data near the X (4260) region [39], which indicates a strong enhancement of events above the D * + s D * − s threshold. If this is true, our analyses show that it decisively changes our previous understandings on X (4260) resonance: It could probably be described by the conventional 4 3 S 1 state heavily renormalized by the D * + s D * − s continuum (maybe a small mixing with the 3 3 D 1 state as well). If the D * + s D * − s data are excluded from the fit, however, the final result on e + e − can still be at least O(10 2 ) eV, i.e., much larger comparing with that of Ref. [27], owing to other new data available as mentioned above. As a consequence, the X (4260) resonance may still be considered as a mixture of 3 3 D 1 and 4 3 S 1 states, i.e., a conventionalcc resonance.
The paper is organized as follows: this Sect. 1 is the introduction. A detailed description of hadronic cross sections of e + e − annihilation will be given in Sect. 2. In Sect. 3, combined fits to the hidden charm and open charm decay channels are performed, with two scenarios: one includes the D * + s D * − s cross section data and the other does not. We leave physical discussions and conclusions in Sect. 4.
respectively. Further, except for the channels just discussed, there could be other channels with lower thresholds, for these channels we use a constant width 0 to describe the overall effects.
Because the quantum number of X (4260) is J PC = 1 −− , the interaction between X (4260) and photon can be written as where In and D * + D * − channels, using narrow width approximation, the cross section formulae take the form where k is the 3-momentum of incoming electron in c.m. frame, ee follows Eq. (8) and f takes the form of Eqs. (3) and (6). The term parameterized as a resonance propagator with a mass M i and width i and the complex constantc play the role of a background in each decay channel here, as will be declared in details in the forthcoming Sect. 3.

Numerical analyses and discussions
In Sect. 3.1, we will perform comprehensive fits to relevant data available in the vicinity of X (4260), which include [32][33][34], ωχ c0 [35,36], J/ψη [37,38], D * + s D * − s [39], together with the previous D + D * − and D * + D * − data in Ref. [25]. To be cautious, for the reason as already mentioned previously, the fit without D * + s D * − s cross section data is also performed in Sect. 3.2. Different results are carefully compared and discussed, and we believe that a clearer understanding on the nature of X (4260) emerges.  (14) in Ref. [27] to describe the e + e − → γ * → X (4260) → J/ψπ + π − process and the propagator 1 D X (s) 5 of that equation is rewritten as the following form: where m th is the threshold of J/ψπ + π − , M 4415 and 4415 are introduced to represent the mass and width of ψ(4415), respectively; φ 11 and φ 12 are interference phases; c 11 , c 12 , and c 13 are free constants (see Fig. 3a and b for fit results).

The ωχ c0 process
The ωχ c0 data comes from Ref. [ 5 Here we use s = q 2 . and the fit results are illustrated as Fig. 3f.  [38] are adopted simultaneously. On account of the influence of ψ(4160), which was taken into account in the fit by Belle in Ref. [38], the cross section is written as the following:

The J/ψη process
Besides, M 4160 and 4160 are introduced to represent the mass and width of ψ(4160), respectively.

The D * + s D * − s process
In D * + s D * − s channel we take the data from Ref. [39], √ s ∈ [4.23, 4.36] GeV, 5 data points. Then the cross section reads See Fig. 3h for fit results. For the D + D * − channel, we take the Belle data [25], √ s ∈ [3.93, 4.37] GeV, with 23 data points, as shown in Fig. 3i. Two additional Breit-Wigner resonances and a complex constant are imposed in the fit to simulate the contribution of interference backgrounds:  Fig. 3 The results of the fit with D * + s D * − s cross section data. The solid curves in all subgrpaphs are the projections from the best fit: a and b fit to the cross section of J/ψπ + π − ; c fit to D 0 D * − π + data, where the dashed black one represents X (4260) components; d fit to h c π + π − data, where the dashed black one indicates X (4260) components; e fit to ψ(2S)π + π − data, where the dashed black one describes X (4260) components; f fit to ωχ c0 cross section data, and we do not fit the two data points (orange squres) on the left side since they are below the threshold; g fit to J/ψ η data; h fit to D * +   the cross section is written as The fit projection is presented in Fig. 3j.

The fit results
We have attempted to fit the experimental data with three well established charmonia, ψ(4040), ψ(4160) and ψ(4415), together with other coherent background contributions in the above decay channels. Since X (4260) is our only interest here, the mass of ψ(4040), ψ(4160) and ψ(4415) is fixed whereas the widths are left free in this research. The parameters related to backgrounds in each process are mentioned above, and the widths of ψ(4040), ψ(4160) and ψ(4415) are listed in Table 1, which are found in reasonable agreement with the widths given by Particle Data Group [4]. The coupling coefficients between X (4260) and different final states are presented in the Table 1 as well. Especially, with heavy quark spin symmetry considered, the relationship between g D + D − , g D + D * − +c.c. and g D * + D * − can be calculated [43] to be g D + D − : g D + D * − +c.c. : g D * + D * − = 1 : 4 : 7, which is uti-lized in our fit. Therefore, there is only one parameter in need of describing the coupling coefficient in these three channels. e + e − = 1.314 ± 0.066 keV, which gives a strong support for X (4260) to be a 4 3 S 1 vector charmonium [19].
The above conclusions are rather stable against variations of background parameterizations. For example, the complexconstant coherent background can be employed in Eq. (16) for the D * + D * − channel, and the expression is Since the D * + s D * − s cross section data from BESIII [39] are preliminary, the program without fitting the D * + s D * − s has also been carried out. In this subsection, the total width of the X (4260) propagator is also Eq. (2), which includes the D * + s D * − s decay width, even though the data are not fitted. It is noticed that the branching ratios of each decay channel remain similar whether the program includes fitting the D * + s D * − s cross section data or not. Likewise, the fit formula for the J/ψπ + π − , D 0 D * − π + , h c π + π − , ψ(2S)π + π − , ωχ c0 , J/ψη, D + D * − and D * + D * − processes are used as the forms in the Sect. 3.1 respectively. The fit results are displayed in Fig. 4.
In addition, another solution to the fit without D * + s D * − s cross section data is found, with the fit quality to be 1.038±0.157 keV, which is different from the previous solution but is similar to the fit with D * + s D * − s cross section data. We find that, unfortunately, the destructive interference in the open charm channels is rather unstable, which makes it impossible to distinguish the physical one between the two solutions with the similar fit quality, and we can not figure out another solution better than the two. So only the range of the leptonic width rather than a definitive value may be trustworthy. Even so, the leptonic width of over hundreds eV still favor X (4260) as a conventional charmonium.
Owing to the instability brought by the open charm channels, then we add a complex coherent background in D * + D * − channel following the strategy of Sect. 3.1.7 to test the stability of outputs against the variation of backgrounds. The fit is plotted in Fig. 5. The fit quality is

Summary and discussions on numerical fits
To compare with the fits discussed above and to further test the stability of the whole fit program, here we also test the fit without including the D + D * − and D * + D * − cross  Ref. [27]. However, we believe this scenario does not have much chance to be physically correct, since there is no reason a priori to exclude the couplings between X (4260) and these states. We may conclude, in the most conservative situation, one still get a leptonic width well above 10 2 eV, which is compatible with the upper limit 580 eV [44] obtained by reanalyzing BESII R-scan data [45,46]. If taking the D * + s D * − s data into account, the leptonic width will easily exceed 1 keV.
The pole location of X (4260) propagator with D * + s D * − s coupling is also searched for in order to achieve a better understanding on the nature of X (4260). Since DD, DD * and D * D * channels play a vital role and the threshold of D * + s D * − s channel is close to the location of X (4260), the complicated multi-sheets structure of the complex plane is simplified as 4 sheets defined in Table 3. It is noticed that there are two poles on sheet II and III as shown in Table 4. The pole width is a bit smaller than the line shape width excluding the coupling to D * + s D * − s . We think this is well understood and be a typical situation in p waves, when the pole lies below the second threshold. Since one partial width is ∝ k 3 where k is the 2nd channel momentum, below the second threshold the k 2 factor provides an additional minus sign. However, it should be emphasized that the appearance of two poles, is not a manifestation of the "elementariness" of

Conclusions
Studies on X (4260) resonance play an important role in deepening our understandings on exotic particles and strong interactions. Ref. [27] pointed out that X (4260) is a confining state with a very small leptonic decay width which is hard to be understood by a simple quark model calculation. Thanks to the new experimental data available, a correct understanding gradually emerges, as we believe: a combined fit with the "old" D + D * − and D * + D * − data -even though there is no apparent X (4260) peak showing up in these channelsreveals that the X (4260) can have a sizable leptonic width up to at least O(10 2 )eV. Further the fit including the D * + s D * − s data can raise the value up to 1 keV. It is worth mentioning that Ref. [51] gives the muonic width to be from 1.09 to 1.53 keV in the range from 4212.8 to 4219.4 MeV, which provides a strong support to our results. 7 In Ref. [19], which uses a screening potential instead of a linear confining potential to calculate the spectrum, it is estimated that a 4 3 S 1 state has a leptonic width ∼ 1 keV, whereas a 3 3 D 1 state has a leptonic width ∼ 50 eV. Hence the smaller e + e − (∼ 300 eV) obtained in this paper may be provided by a 3 3 D 1 state (maybe a small portion of 2 3 D 1 state as well) mixed with certain portion of 4 3 S 1 state, and the larger value estimated in this paper may corresponds to a 4 3 S 1 state, and is probably largely renormalized by the D * + s D * − s continuum. To further determine the accurate portion of these mixing is still an open question awaiting more fine studies both theoretically and experimentally. are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecomm ons.org/licenses/by/4.0/. Funded by SCOAP 3 .