On leptonic width of $X(4260)$

New measurements on cross sections in $e^+e^-\to J/\psi\pi^+\pi^-$, $h_c\pi^+\pi^-$, $D^0D^{*-}\pi^++c.c.$, $\psi(2S)\pi^+\pi^-$, $\omega\chi_{c0}$ and $J/\psi\eta$ channels have been carried out by BESIII, Belle and BABAR collaborations, and also in the $D_s^*\bar D_s^*$ channel. We perform extensive numerical analyses by combining all these data available, together with those in $D\bar D^*$ and $D^*\bar D^*$ channels. Though the latter show no evident peak around $\sqrt{s}=4.230$ GeV, the missing $X(4260)$ is explained as that it is concealed by the interference effects of the well established charmonia $\psi(4040)$, $\psi(4160)$ and $\psi(4415)$. Our analyses reveal that the leptonic decay width of $X(4260)$ ranges from $O(10^2)$ eV to $O(1)$ keV, and hence may be explained in the conventional quark model picture. That is, the $X(4260)$ may well be interpreted as a mixture of $4^3S_1$ and $3^3D_1$ 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 quark model predictions [6]. The only such 1 −− states expected up to 4.4 GeV are 1S, 2S, 1D, 3S, 2D and 4S, and they seem to be well established [7] -the situation is depicted in Figure 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 generally 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 Figure 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/ψππ channel would become a very important, if not the dominant one.   Hence the leptonic width Γ e + e − of X(4260) would become very small, making it even harder to be understood as a conventional 1 −−c c state. For example, in a previous publication, we have 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 Γ e + e − is found to be 25eV. However the nearby 4 3 S 1 state is expected to have a leptonic width 1 keV, whereas for a pure 3 3 D 1 state Γ e + e − 44 eV [19].
Many new experimental results have appeared since the work of Ref. [27], measured by BESIII, Belle and BABAR collaborations, such as e + e − → J/ψπ + π − [29][30][31], h c π + π − [32], D 0 D * − π + + c.c. [33], ψ(2S)π + π − [34][35][36], ωχ c0 [37,38], J/ψη [39,40] and D * sD * s [41]. Hence the analyses of Ref. [27] urgently need to be upgraded. Among all it is worthwhile mentioning the D * sD * s data near the X(4260) region [41], which indicates a strong enhancement of events above the D * sD * 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 * sD * s continuum (maybe a small mixing with the 3 3 D 1 state as well). If the D * sD * s data are excluded from the fit, however, the final result on Γ e + e − can still be 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 section 1 is the introduction. A detailed description of the numerical fit program will be given in section 2 . In section 3, combined fits to the hidden charm and open charm decay channels are performed, with two scenarios: one includes the D * sD * s cross section data and the other does not. We leave physical discussions and conclusions in section 4.

Theoretical Discussions
To begin with, the X(4260) propagator is written in the following form: where Γ tot (s) is the total momentum dependent width comprising of all partial ones: 2 Considering the isospin symmetry, it is noticed that Γ J/ψππ (s) As for the J/ψπ + π − , h c π + π − , D 0 D * − π + and ψ(2S)π + π − channels, the three body partial decay width takes the standard form [4], where p ij = p i + p j , m 2 ij = p 2 ij , |p * 1 |, Ω * 1 are the momentum and angle of particle 1 in the rest frame of the system of particle 1 and 2, respectively; Ω 3 is the angle of particle 3 in the rest frame of particle M ; and |p * 1 |, |p 3 | are defined as In addition, the two body decay widths take the following simple forms: In above k ωχc0 , k J/ψη , k D * sD * s , k D + D − , k D + D * − and k D * + D * − are three momentums of ωχ c0 , J/ψη, D * sD * s , D + D − , D + D * − and D * + D * − in X(4260) rest frame, respectively. Further, except for the channels just discussed, there could be other channels with lower thresholds, for these channels we use a constant width Γ 0 to describe the overall effects.
Because the quantum number of X(4260) is J P C = 1 −− , the interaction between X(4260) and photon can be written as where X µν = ∂ µ V ν − ∂ ν V µ and V µ represents the X(4260) field, F µν = ∂ µ A ν − ∂ ν A µ describes the photon field. So the decay width of X(4260) → e + e − reads Γ e + e − = 4α 3 In h c π + π − , D 0 D * − π + , ψ(2S)π + π − , ωχ c0 , J/ψη, D * sD * s , D + D * − 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, Γ ee follows Eq. (8) and Γ f takes the form of Eq. (3) and Eq. (6). The term parameterized as a resonance propagator with a mass M i and width Γ 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 section 3.

Numerical Analyses and Discussions
In section 3.1, we will perform comprehensive fits to relevant data available in the vicinity of X(4260), which include the J/ψπ + π − [29][30][31] [37,38], J/ψη [39,40], D * sD * s [41], 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 * sD * s cross section data is also performed in section 3.2. Different results are carefully compared and discussed, and we believe that a clearer understanding on the nature of X(4260) emerges.

The fit with
For the J/ψπ + π − channel, the experimental data sets come from: BESIII [29], √ s ∈   (14) in Ref. [27] to describe the e + e − → γ * → X(4260) → J/ψπ + π − process and the propagator 1 D X (s) 4 of that equation is rewritten as the following form: where m th is the threshold of J/ψπ + π − . M 4415 and Γ 4415 is introduced to represent the mass and width of ψ(4415), respectively. See Figure 3 and Table 1 for fit results.
where i = 3, 4, 5 represents the D 0 D * − π + , h c π + π − and ψ(2S)π + π − channels, respectively. See Figures 4(a), 4(b), 4(c) and Table 2 for fit quality and results. s cross section data: (a) Fit to the cross section of J/ψπ + π − . The dots(blue) and squares(orange) are from BABAR [30] and Belle [31], respectively. (b) Fit to the cross section of J/ψπ + π − measured by BESIII [29]. The squares(orange) are from the XY Z data sample at BESIII [29] and the dots(blue) are from the R-scan data sample by BESIII [29]. The solid red curves show the best fit. s cross section data: (a) Fit to D 0 D * − π + data by BESIII [33]. (b) Fit to h c π + π − data by BESIII [32]. The squares(orange) are from the XY Z data sample at BESIII [32] and the dots(blue) are from the R-scan data sample by BESIII [32]. (c) Fit to ψ(2S)π + π − data. The orange dots come from BESIII [34], the green squares are from Belle [35] and the blue rhombuses come from BABAR [36]. The solid red curves show the best fit, and the dashed black ones represent X(4260) components.  Table 2: Fit parameters in D 0 D * − π + , h c π + π − and ψ(2S)π + π − channels with c i (i = 3, 4, 5) in unit of GeV 2 . s cross section data scheme. The blue dots and the orange squares come from BESIII experiments in Ref. [37] and Ref. [38], respectively. We do not fit the two data points (orange squres) on the left side since they are below the threshold.  [40] and the orange squares indicate the BESIII data [39].

The ωχ c0 process
The ωχ c0 data comes from Ref. [ and the fit results are presented as Figure 5. simultaneously. On account of the influence of ψ(4160), which was taken into account in the fit by Belle in Ref. [40], the cross section is written as the following:

The J/ψη process
Besides, M 4160 and Γ 4160 are introduced to represent the mass and width of ψ(4160).
See Figure 7 and Table 4 for fit results.
in which M 4040 and Γ 4040 are used to describe the mass and width of ψ(4040). Additionally, 18 data points released by Belle [25] from √ s ∈ [4.11, 4.45] are adopted in the D * + D * − process and the cross section is written as    Table 6: Fit parameters in D * + D * − channel with c i (i = 10, 11, 12) in GeV 2 .

Further discussions on the fit program with D *
sD * s process 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 and the widths are fit parameters in this research. The parameters related to backgrounds in each process are listed above, and the widths of ψ(4040), ψ(4160) and ψ(4415) are listed in Table 7, 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 7 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 [44] to be g D + D − : g D + D * − +c.c. : g D * + D * − = 1 : 4 : 7, which is used in our fit. Therefore, there is only one parameter in need to describe the coupling coefficient in these three channels. The goodness of the fit is χ 2 /d.o.f. = 293.099/(357 − 42) = 0.930. The value of g 0 corresponds to the leptonic decay width Γ e + e − = 1.302 keV, which gives a strong support for X(4260) to be a 4 3 S 1 vector charmonium [19].

The fit without D * sD * s cross section data
Since the D * sD * s cross section data from BESIII [41]are preliminary, the program without fitting the D * sD * 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 * sD * 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 * sD * s cross section data or not. However, the leptonic decay width Γ e + e − turns out to be distinct from that in the fit with D * sD * s cross section data, as will be seen below.

The J/ψπ + π − process
The fit formula for the J/ψπ + π − process is also used as the form in the section 3.1. The fit results are displayed in Figure 9 and Table 8.  Figure 9: The results of the fit without D * sD * s cross section data: (a) Fit to the cross section of J/ψπ + π − . The dots(blue) and squares(orange) are from BABAR [30] and Belle [31], respectively. (b) Fit to the cross section of J/ψπ + π − measured by BESIII [29]. The squares(orange) are from the XY Z data sample at BESIII [29] and the dots(blue) are from the R-scan data sample by BESIII [29]. The solid red curves show the best fit.  Table 8: Fit parameters in J/ψπ + π − channel.

The ωχ c0 process
The fit formula can be parameterized as the Eq. (12) and the fit result is presented as Figure 11.

The J/ψη process
The fit equation used in the J/ψη process is denoted as Eq. (13), too. So the fit results are displayed in Figure. 12 and Table. 10. s cross section data scheme. The blue dots and the orange squares come from BESIII experiments in Ref. [37] and Ref. [38], respectively. We do not fit the two data points (orange squres) on the left side since they are below the threshold.   keV, which may imply that X(4260) is a mixture of 4 3 S 1 and 3 3 D 1 cc charmonium state [19].  3.2.6 Further discussions on the D + D * − and D * + D * − processes Now following the strategy of section 3.1.7 to test the stability of outputs against the variation of backgrounds, we add a complex coherent background in D * + D * − channel. The fit is plotted in Figure 14. The fit quality is χ 2 /d.o.f. = 277.047/(352 − 42) = 0.894. It is found, however, unlike the result of section 3.1.7, the fit is not quite stable here. The difference is clearly seen when comparing Figure 13 and Figure 14: the destructive interference between different resonances are done in rather different manner. The leptonic width behaves quite differently, with a value of Γ e + e − = 0.157 keV, comparing with the result of section 3.2.5, Γ e + e − = 0.466 keV.

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 sec- For the fit excluding both D + D * − , D * + D * − and D * sD * s cross sections, the parameter g 0 is 4.584 MeV with Γ e + e − = 0.48 keV, which is still twice as large as the value estimated in Ref. [27]. However, we believe that 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. If taking the D * sD * s data into account, the leptonic width will easily exceed 1 keV. Even though there is a variation on the value of Γ e + e − , in any case, however, the X(4260) persists in a pole structure on Riemann sheets like an "elementary" particle, i.e., a confining state, according to the pole counting criteria. 5

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 channels -reveals that the X(4260) can have a sizable leptonic width up to O(10 2 )eV. Further the fit including the D * sD * s data can raise the value up to 1 keV. In Ref. [19], which use 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 Γ e + e − 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 * sD * s continuum. To further determine the accurate portion of these mixing is still an open question awaiting more fine studies both theoretically and experimentally.

Acknowledgement
We are grateful to illuminating discussion with Kuang-ta Chao, Ce Meng and Chang-Zheng Yuan at early stage of this work. This work is supported in part by National Nature Science Foundations of China (NSFC) under Contract Nos. 10925522, 11021092.