Combined fit to the e+e-→π+π-J/ψ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$e^+e^-\rightarrow \pi ^{+}\pi ^{-}J/\psi $$\end{document} and e+e-→π+π-ψ(3686)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$e^+e^-\rightarrow \pi ^{+}\pi ^{-}\psi (3686)$$\end{document} line shape

We analyze the cross section of e+e-→π+π-J/ψ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$e^+e^-\rightarrow \pi ^{+}\pi ^{-}J/\psi $$\end{document} measured by Belle, BABAR and BESIII experiments. The parameters of the two resonances Y(4220) and Y(4360) are consistent with that in e+e-→π+π-ψ(3686)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$e^+e^-\rightarrow \pi ^{+}\pi ^{-}\psi (3686)$$\end{document}. A combined fit is performed to the two cross sections assuming the two resonances Y(4220) and Y(4360) have the same parameters. The parameters of Y(4220), Y(4360) and Y(4660) are determined to be MY(4220)=(4223.3±1.6±2.5)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M_{Y(4220)}=(4223.3\pm 1.6\pm 2.5)$$\end{document} MeV/c2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{2}$$\end{document}, ΓY(4220)=(54.2±2.6±1.0)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma _{Y(4220)}=(54.2\pm 2.6\pm 1.0)$$\end{document} MeV; MY(4360)=(4386.4±2.1±6.4)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M_{Y(4360)}=(4386.4\pm 2.1\pm 6.4)$$\end{document} MeV/c2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{2}$$\end{document}, ΓY(4360)=(96.0±6.7±2.7)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma _{Y(4360)}=(96.0\pm 6.7\pm 2.7)$$\end{document} MeV; MY(4660)=(4646.4±9.7±4.8)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M_{Y(4660)}=(4646.4\pm 9.7\pm 4.8)$$\end{document} MeV/c2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{2}$$\end{document}, ΓY(4660)=(103.5±15.6±4.0)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma _{Y(4660)}=(103.5\pm 15.6\pm 4.0)$$\end{document} MeV, where the first uncertainties are statistical and the second systematic. The ratios B(Y(4220)→π+π-ψ(3686))B(Y(4220)→π+π-J/ψ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{\mathcal {B}(Y(4220) \rightarrow \pi ^{+}\pi ^{-}\psi (3686))}{\mathcal {B}(Y(4220) \rightarrow \pi ^{+}\pi ^{-}J/\psi )}$$\end{document} and B(Y(4360)→π+π-ψ(3686))B(Y(4360)→π+π-J/ψ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{\mathcal {B}(Y(4360) \rightarrow \pi ^{+}\pi ^{-}\psi (3686))}{\mathcal {B}(Y(4360) \rightarrow \pi ^{+}\pi ^{-}J/\psi )}$$\end{document} are also obtained, which may help in understanding the nature of these structures.

Another famous Y -state Y (4360) was firstly observed by BABAR [6] in π + π − ψ(3686) mass distribution, then confirmed by Belle [7] experiment. Belle also observed another structure, Y (4660), in the π + π − ψ(3686) [7]. Recently, using the results for e + e − → π + π − ψ(3686) from Belle [15], BABAR [16] and BESIII [17] experiments, the authors of Ref. [18] observed the fine structure for Y (4360) in e + e − → π + π − ψ(3686), inferring that also the Y (4360) is a combination of two resonances, the lower one's mass is around 4.22 GeV/c 2 and the higher is around 4.38 GeV/c 2 . There are many theoretical models to explain these Y -states, such as hybrid charmonium, tetraquark state and the molecular state [19]. However, there is still no definite conclusion. On the other hand, there are still missing excited charmonium states predicted by the potential models, which we have not observed. So it is important to confirm which Ystates are charmonium and which Y -states are exotic states. In this paper, we revisit the e + e − → π + π − J/ψ and e + e − → π + π − ψ(3686) line shape, and we perform a combined fit to the two cross sections e + e − → π + π − J/ψ and e + e − → π + π − ψ(3686). These results will be useful in understanding the nature of charmonium(-like) states in this energy region.
Assuming that π + π − J/ψ comes from three resonances [9], we use a least χ 2 method to fit the cross section with a coherent sum of three mass dependent relativistic Breit-Wigner (BW) width functions, that is, where PS( √ s) is the three-body phase space factor, φ 1 and , is the BW function for a vector state, with mass M, total width , electron partial width ee , and the branching fraction to π + π − J/ψ, B(π + π − J/ψ). The relative phases φ 1 , φ 2 and the parameters M, , ee B(π + π − J/ψ) of the three BW functions are allowed to float in the fit. Figure 2 shows the fit results. Four solutions are found with the same fit quality. From the fit results, the three resonances (called Y 1 , Y (4220) and Y (4360)) as for mass and width are 6) MeV, and the goodness of the fit is χ 2 /nd f = 164.6/190, corresponding to a confidence level of 91%. The obtained parameters of the cross section of e + e − → π + π − J/ψ are listed in Table 1.
From the fit results, we find that the parameters of the two resonances Y (4220) and Y (4360) are consistent with that in e + e − → π + π − ψ(3686) [18] within 3σ . ψ(3686) is the radial excited state of J/ψ, and it is expected that they have the same source. So a combined fit is performed to the two cross sections assuming the two resonances Y (4220) and Y (4360) have the same parameters. The fit functions are where BW 2 , BW 3 , and BW 4 represent the resonances Y (4220), Y (4360) and Y (4660), respectively, and the φ j are the relative phases. Because a different decay mode has a different final state interaction, it will lead to the relative phases between BW 2 and BW 3 are not the same for the two decay modes. The cross section of e + e − → π + π − ψ(3686) is consistent with zero below 4.1 GeV, so the contribution from Y 1 can be neglected for e + e − → π + π − ψ(3686). And we also do not observe the resonance Y (4660) in e + e − → π + π − J/ψ from the Belle, BABAR and BESIII experiments, so the contribution from Y (4660) in e + e − → π + π − J/ψ can also be neglected. Thus we do not include Y (4660) (Y 1 ) in the function σ J/ψ ( √ s) (σ ψ(3686) ( √ s)). We fit to the e + e − → π + π − J/ψ and e + e − → π + π − ψ(3686) line shape simultaneously. Figure 3 shows the fit results. Four solutions are found with the same fit quality for π + π − J/ψ and π + π − ψ(3686), respectively.   Fig. 3 The results of the combined fit to e + e − → π + π − J/ψ and e + e − → π + π − ψ(3686). Four solutions are shown in plots (a), (b), (c), (d) for e + e − → π + π − J/ψ and (e), (f), (g), (h) for e + e − → π + π − ψ(3686). The solid red curves show the best fits, and the dashed green ones are individual components and the goodness of the fit is χ 2 /nd f = 284.1/305, corresponding to a confidence level of 80%. The obtained parameters of the cross sections of e + e − → π + π − J/ψ and e + e − → π + π − ψ(3686) are listed in Table 2.
The systematic uncertainties on the resonant parameters for resonances Y (4220)/Y (4360)/Y (4660) in the combined fit to the cross sections of e + e − → π + π − J/ψ and e + e − → π + π − ψ(3686) are mainly from the uncertainty of the center-of-mass energy determination, energy spread, parametrization of the BW function, background shape and the cross section measurements.
Since the uncertainty of the beam energy is about 0.8 MeV at BESIII, the uncertainty of the resonant parameters caused by the beam energy is estimated by varing √ s within 0.8 MeV for BESIII data. The energy spread of BEPC-II per beam is about 1.3 MeV, so the energy spread of √ s is about 1.9 MeV. To estimate the uncertainty caused by the energy spread for BESIII data, the BW function convoluted with a Gaussian function which the resolution is 1.9 MeV is used to fit the data, and the uncertainties are estimated by comparing the fit results with the nominal results. To estimate the uncertainty from the parametrization of BW, the width in the denominator of the BW function is set to be of constant width, the BW function will change to √ 12π ee B(π + π − J/ψ) The difference of the fit results and the nominal results are taken as the uncertainty from the parametrization of the BW function. To model the e + e − → π + π − J/ψ cross section near 4 GeV, we use the BW formula (Y 1 ) in the normal fit. An exponential function as used in Ref. [9] also can describe the cross section near 4 GeV. So we cannot get the enhancement around √ s = 4 GeV is from a true resonance or the background. Here, we take the enhancement around √ s = 4 GeV as the background, such that the exponential function is taken instead of using the Y 1 resonance in the function σ J/ψ ( √ s). Because there is no Y 1 in e + e − → π + π − ψ(3686), there is not any change for the function σ ψ(3686) ( √ s) in the combined fit. The difference of the fit results in the two methods are taken as the uncertainty from background shape. The uncertainty of the cross section will affect the resonant function in the fit, we vary the cross section within the systematic Table 2 The fitted parameters from the combined fit to the cross sections of e + e − → π + π − J/ψ and e + e − → π + π − ψ(3686). The errors are statistical. For Y (4220)/Y (4360)/Y (4660) M and , the first errors are statistical, the second are systematic 0.03) − (0.55 ± 0.18 ± 0.19) and B(Y (4360)→π + π − ψ(3686)) B(Y (4360)→π + π − J/ψ) = (0.81 ± 0.12 ± 0.13) − (42 ± 15 ± 15). These results may help in understanding the nature of these structures. With different assumptions for Y (4220)/Y (4360)/Y (4660) and (Y (4220)/Y (4360)/Y (4660) → e + e − ), we also can get the branching fractions B(Y (4220)/Y (4360) → π + π − J/ψ) and B(Y (4220)/Y (4360)/Y (4660) → π + π − ψ(3686)). It is worth mentioning that now BESIII is collecting more data above 4 GeV; these data would help to better understand the nature of charomonium(-like) states in the future.