---
title: 'Supplemental Material for Study of the ${e^+e^-}\to{\pi^0}{\pi^0}{J/\psi}$ process and neutral ${Z_{c}(3900)^0}$'
---

Table \[tab:crs\] summarizes the measured Born cross sections as a function of center-of-mass energy and the various input parameters that are used in the calculation of the cross section.  Table \[tab:fit2R\_pars\] lists the parameter values of the lineshape fit of the Born cross sections of ${e^+e^-}\to{\pi^0}{\pi^0}{J/\psi}$.

   $\sqrt{s}({\rm ~GeV})$   ${\mathcal{L}}({\rm {~pb^{-1}})}$   ${\varepsilon}^{\rm PWA}(\%)$   ${\varepsilon}^{\rm PHSP}(\%)$   $N^{\rm est}_{\gamma{\psi^\prime}}$   $N^{\rm est}_{\eta{J/\psi}}$   $N_{{\pi^0}{\pi^0}{J/\psi}}^{\rm obs}$   1+$\delta^{r}$   1+$\delta^{v}$    $\sigma^{\rm Born}$ (pb)    ${{\cal R}}(\frac{{\pi^0}{\pi^0}{J/\psi}}{\pi^{+}\pi^{-}{J/\psi}})$
  ------------------------ ----------------------------------- ------------------------------- -------------------------------- ------------------------------------- ------------------------------ ---------------------------------------- ---------------- ---------------- ---------------------------- ---------------------------------------------------------------------
           3.808                      $50.5\pm0.5$                           ...                            20.44                                ...                               ...                             $11.5\pm4.0$                    0.815            1.056         $11.11\pm 3.87 \pm 0.78$                           $0.70\pm0.28\pm0.17$ 
           3.896                      $52.6\pm0.5$                           ...                            19.89                                ...                               ...                             $6.5\pm3.3$                     0.855            1.049         $5.95 \pm 3.02 \pm 0.42$                           $0.36\pm0.23\pm0.10$
           4.008                      $482.0\pm4.8$                          ...                            19.64                               11.3                               1.4                            $88.6\pm11.2$                    0.904            1.044         $8.51 \pm 1.14 \pm 0.60$                           $0.56\pm0.08\pm0.07$
           4.086                      $52.9\pm0.4$                           ...                            19.60                                ...                               ...                             $10.6\pm3.6$                    0.933            1.052         $8.96 \pm 3.04 \pm 0.63$                           $0.63\pm0.22\pm0.14$
           4.178                     $3194.5\pm31.9$                        20.68                           20.09                               25.4                               8.6                            $365.7\pm24.6$                   0.900            1.054         $5.00 \pm 0.34 \pm 0.35$   
           4.189                      $43.3\pm0.3$                           ...                            20.97                                ...                               ...                             $11.2\pm3.8$                    0.861            1.056         $11.65\pm 3.91 \pm 0.82$                           $0.79\pm0.27\pm0.21$
           4.189                      $524.6\pm0.4$                         22.91                           20.78                                4.4                               2.4                            $89.3\pm11.4$                    0.859            1.056         $7.03 \pm 0.90 \pm 0.49$                           $0.48\pm0.06\pm0.12$
           4.200                      $526.0\pm0.4$                         22.01                           21.54                                4.6                               2.6                            $122.6\pm12.5$                   0.810            1.056         $10.62\pm 1.08 \pm 0.74$   
           4.208                      $55.0\pm0.4$                           ...                            22.44                                ...                               ...                             $26.9\pm5.7$                    0.768            1.057         $23.10\pm 4.86 \pm 1.62$                           $0.46\pm0.10\pm0.06$
           4.210                      $518.0\pm0.4$                         21.94                           22.16                                4.5                               2.9                            $235.9\pm16.7$                   0.762            1.057         $22.12\pm 1.57 \pm 1.55$                           $0.44\pm0.03\pm0.05$
           4.217                      $54.6\pm0.4$                           ...                            22.89                                ...                               ...                             $28.6\pm5.8$                    0.741            1.057         $25.07\pm 5.08 \pm 1.76$                           $0.44\pm0.09\pm0.06$
           4.219                      $514.6\pm0.4$                         22.95                           22.60                                4.3                               3.0                            $351.0\pm20.1$                   0.739            1.056         $32.66\pm 1.87 \pm 2.29$                           $0.57\pm0.03\pm0.07$
           4.226                     $1100.9\pm7.0$                         22.75                           23.27                                8.8                               5.2                            $890.4\pm32.1$                   0.744            1.056         $38.84\pm 1.40 \pm 2.72$                           $0.48\pm0.02\pm0.04$
           4.236                      $530.3\pm0.5$                         21.80                           23.22                                4.4                               2.4                            $452.9\pm22.6$                   0.779            1.056         $40.88\pm 2.04 \pm 2.86$   
           4.242                      $55.9\pm0.4$                           ...                            23.26                                ...                               ...                             $49.1\pm7.6$                    0.805            1.056         $38.15\pm 5.87 \pm 2.67$                           $0.48\pm0.07\pm0.06$
           4.244                      $538.1\pm0.5$                         23.34                           23.04                                3.9                               1.9                            $435.3\pm22.0$                   0.815            1.056         $34.59\pm 1.75 \pm 2.42$   
           4.258                      $828.4\pm5.5$                         22.91                           23.15                                5.9                               1.3                            $585.1\pm25.9$                   0.857            1.054         $29.29\pm 1.30 \pm 2.05$                           $0.52\pm0.02\pm0.05$
           4.267                      $531.1\pm0.6$                         24.64                           22.68                                3.6                               0.8                            $356.0\pm20.6$                   0.869            1.053         $25.51\pm 1.48 \pm 1.79$   
           4.278                      $175.7\pm0.6$                         22.98                           22.44                                1.1                               0.2                            $110.0\pm11.2$                   0.877            1.053         $25.32\pm 2.58 \pm 1.77$   
           4.308                      $45.1\pm0.3$                           ...                            22.35                                ...                               ...                             $23.3\pm5.4$                    0.886            1.052         $21.32\pm 4.91 \pm 1.49$                           $0.43\pm0.10\pm0.06$
           4.358                      $543.9\pm3.6$                         21.24                           20.38                                2.9                               0.4                            $196.4\pm15.9$                   1.062            1.051         $13.07\pm 1.06 \pm 0.92$                           $0.54\pm0.04\pm0.06$
           4.387                      $55.6\pm0.4$                           ...                            18.52                                ...                               ...                             $4.2\pm2.6$                     1.177            1.051         $2.83 \pm 1.77 \pm 0.20$                           $0.15\pm0.09\pm0.03$
           4.416                     $1090.7\pm6.9$                         17.59                           17.18                                4.7                               1.1                            $182.8\pm16.2$                   1.238            1.052         $6.27 \pm 0.56 \pm 0.44$                           $0.55\pm0.05\pm0.06$
           4.467                      $111.1\pm0.7$                          ...                            16.28                                ...                               ...                             $17.2\pm4.9$                    1.268            1.055         $6.11 \pm 1.72 \pm 0.43$                           $0.48\pm0.14\pm0.09$
           4.527                      $112.1\pm0.7$                          ...                            16.32                                ...                               ...                             $14.1\pm5.0$                    1.265            1.055         $4.96 \pm 1.74 \pm 0.35$                           $0.49\pm0.17\pm0.10$
           4.575                      $48.9\pm0.1$                           ...                            16.52                                ...                               ...                             $7.2\pm3.1$                     1.259            1.055         $5.72 \pm 2.47 \pm 0.40$                           $0.45\pm0.19\pm0.12$
           4.600                      $586.9\pm3.9$                          ...                            16.76                                1.8                               0.1                             $32.6\pm7.8$                    1.252            1.055         $2.15 \pm 0.52 \pm 0.15$                           $0.36\pm0.09\pm0.05$

\[tab:crs\]

                               Parameters                                 Solution I       Solution II       Solution III       Solution IV  
  -------------------------------------------------------------------- ---------------- ----------------- ------------------ -----------------
                                $p_{0}$                                                                                      
                                $p_{1}$                                                                                      
                     $M({R_1})~({\rm ~MeV/c^{2}})$                                                                           
                 $\Gamma_{\rm tot}(R_1))~({\rm ~MeV})$                                                                       
   $\Gamma_{\rm ee}{{\cal B}}(R_1\to{\pi^0}{\pi^0}{J/\psi})~(\rm eV)$   0.99$\pm$0.17     4.13$\pm$0.28     1.38$\pm$0.30      5.72$\pm$0.57
                         $\phi(R_1)~(\rm rad)$                           16.8$\pm$5.5    $-73.6\pm$ 3.6      63.0$\pm$8.4      $-28.2\pm$2.0
                      $M(R_2)~({\rm ~MeV/c^{2}})$                                                                            
                  $\Gamma_{\rm tot}(R_2)~({\rm ~MeV})$                                                                       
   $\Gamma_{\rm ee}{{\cal B}}(R_2\to{\pi^0}{\pi^0}{J/\psi})~(\rm eV)$   0.31$\pm$0.15     0.41$\pm$0.20     6.44$\pm$0.41      8.51$\pm$0.44
                         $\phi(R_2)~(\rm rad)$                          133.5$\pm$12.3     99.2$\pm$1.6     $-96.0\pm$5.2     $-131.3\pm$2.8

\[tab:fit2R\_pars\]

The systematic uncertainty for the cross section measurements includes those associated with the luminosity, detection efficiency, radiative correction, fitting procedure, and the branching fractions (BFs) of intermediate states.  The uncertainty of the integrated luminosity is 1% from the analysis of large-angle Bhabha scattering events [@2015Lumi].  The uncertainties of detection efficiency include those of tracking, photon detection, $e/\mu$ identification, ${\pi^0}$ reconstruction, kinematic fit, and MC modeling.  The uncertainty of the tracking efficiency for leptons is $1.0\%$ per track [@systrack].  The uncertainty for the photon detection is 1.0% per photon [@sysphoton].  The uncertainty of the $e/\mu$ identification is 0.2% per track estimated with a control sample ${e^+e^-}\to{\pi^0}{\pi^0}{J/\psi}$ at $\sqrt{s}=3.686~{\rm ~GeV}$. The uncertainty for ${\pi^0}$ reconstruction, 0.3%, is the efficiency difference by smearing the $M_{\gamma\gamma}$ distribution of signal MC sample to match that of data.  The uncertainty due to the kinematic fit is 2.6%, estimated with a clean control sample ${e^+e^-}\to\gamma{\psi^\prime},~{\psi^\prime}\to{\pi^0}{\pi^0}{J/\psi}$ at $\sqrt{s}=4.178{\rm ~GeV}$. The uncertainty associated with the MC modeling is estimated by two approaches.  For high-statistics c.m.  energy points for which a PWA is performed, we generate 100 sets of signal MC samples at each c.m.  energy point by altering the input parameters with an additional residual generated randomly with a multi-variable Gaussian function according to the PWA results, and the resultant standard deviation of detection efficiencies is taken as the systematic uncertainty.  We also take differences between the nominal efficiencies and efficiencies estimated by PHSP MC samples at the c.m.  energy points with large statistics as the uncertainty for those with low statistics.  The uncertainty of the radiative correction includes those from the radiative correction factor and vacuum polarization factor, which are 0.3% obtained by altering the input cross section lineshape taken from Ref.  [@LiuZQ2017] to the measurements in this analysis, and 0.5% taken from a QED calculation [@vacuum], respectively.  The uncertainty of the fit procedure, 1.2%, including uncertainties of the background shape and the fit range, is estimated by altering the first oder polynomial function to the second and by varying the fit range by 100 ${\rm ~MeV/c^{2}}$. The (largest) change of signal yield is taken as the uncertainty.  The uncertainty of peaking backgrounds is negligible by considering the uncertainties of their cross sections.  The uncertainties on the BFs of ${J/\psi}\to{\ell^+\ell^-}$ and ${\pi^0}\to\gamma\gamma$ are quoted from Ref.  [@2019PDG].  All above uncertainties are summarized in Table \[tab:sys\_crs\], assuming all sources are independent, the total uncertainty is found to be 6.1% by adding all individual values in quadrature. 

  Sources                             Uncertainties ($\%$)
  ---------------------------------- ----------------------
  Tracking for $e^{\pm}/\mu^{\pm}$            2.0
  $e/\mu$ identification                      0.4
  Photon efficiency                           4.0
  ${\pi^0}$ mass window                       0.3
  Kinematic fit                               2.6
  Fit to $M_{{\ell^+\ell^-}}$                 1.2
  Radiative correction                        0.3
  Vacuum polarization                         0.5
  MC model                                    2.7
  Luminosities                                1.0
  ${{\cal B}}_{\rm{inter}}$                   0.6
  Total                                       6.1

  : Summary of the systematic uncertainties of ${e^+e^-}\to{\pi^0}{\pi^0}{J/\psi}$ cross-section measurements. 

\[tab:sys\_crs\]

The resonant parameters of the $Y(4220)$ are obtained by fitting the measured cross sections, and the corresponding uncertainties are from the measured cross sections, the c.m.  energy measurement and spread, and the fit procedure.  The uncertainties from the measured cross section are considered by including their systematic uncertainties in the $\chi^2$ fit.  The c.m.  energy uncertainty is 0.8 ${\rm ~MeV}$, obtained from a measurement of di-muon events [@2016Ecms], and its value propagates to the mass of the $Y(4220)$. The uncertainties from the c.m.  energy spread are obtained by convolving the resonant PDF with a Gaussian function, of which the width, 1.6 ${\rm ~MeV}$, is the c.m.  energy spread obtained with the Beam Energy Measurement System [@enespread].  The uncertainties associated with the fit procedure include those of the PDF model, the PHSP factor, and the resonant parameters of the $Y(4320)$. The uncertainties of the PDF model are estimated by replacing the exponential function with a broad resonance $Y(4008)$ [@LiuZQ2017] and by changing the constant full width $\Gamma_{\rm tot}$ to a phase-space dependent full width $\Gamma_{\rm tot}\frac{\Phi(\sqrt{s})}{\Phi(M)}$. The uncertainty of the three-body PHSP factor, due to the existence of intermediate states, is estimated by considering the PHSP of cascade two-body decays of ${e^+e^-}\to R{J/\psi}$ with $R\to{\pi^0}{\pi^0}$ ($R=\sigma,~f_0(980), ~f_0(1370)$) and ${e^+e^-}\to{\pi^0}{Z_{c}(3900)^0}$ with ${Z_{c}(3900)^0}\to{\pi^0}{J/\psi}$. The uncertainties associated with the resonant parameters of the $Y(4320)$ are obtained by changing their values by 1$\sigma$ of their uncertainty.  In the above scenarios, the alternative fits are performed, and the resultant differences with respect to the nominal values are considered as the uncertainties.  All above systematic uncertainties are summarized in Table \[tab:sys4220\], assuming all of them are independent, the total uncertainties, 2.3$~{\rm ~MeV/c^{2}}$ for the mass and 2.1$~{\rm ~MeV}$ for the width of the $Y(4220)$, are the quadratic sum of the individual values. 

  --------------------------- -------------------------------- ---------------------------------
                                                               
                                   M $({\rm ~MeV/c^{2}})$           $\Gamma$ $({\rm ~MeV})$    
  Cross section measurement                 0.2                               0.3
  c.m.  energies                             0.8                               ...
  c.m.  energy spread                        0.01                              0.4
  Fit procedure                             2.2                               2.0
  Total                                     2.3                               2.1
  --------------------------- -------------------------------- ---------------------------------

  : Summary of the systematic uncertainties of resonant parameters of the $Y(4220)$.

\[tab:sys4220\]

Table \[tab:crszc\] summarizes the Born cross sections of ${e^+e^-}\to{\pi^0}{Z_{c}(3900)^0}\to{\pi^0}{\pi^0}{J/\psi}$ extracted from PWA results.  Table \[tab:pwasys\] summarizes the systematic uncertainties of the ${Z_{c}(3900)^0}$ resonant parameters and the corresponding cross sections, while Table \[tab:zclineshapesys\] summarizes the systematic uncertainties of resonant parameters observed in the ${e^+e^-}\to{\pi^0}{Z_{c}(3900)^0}\to{\pi^0}{\pi^0}{J/\psi}$ cross-section lineshape. 

   $\sqrt{s}~({\rm ~GeV})$        $\sigma_{{e^+e^-}\to{\pi^0}{Z}_{c}^{0}} ~(\rm pb)~~~~~ $
  ------------------------- ---------------------------------------------------------------
            4.178                               $1.24\pm0.29 \pm 0.87$
            4.189                               $1.62\pm0.57 \pm 1.13$
            4.200                               $4.65\pm1.06 \pm 3.26$
            4.210                               $2.75\pm0.94 \pm 1.93$
            4.219                               $6.66\pm1.50 \pm 4.68$
            4.226                               $6.14\pm1.09 \pm 4.31$
            4.236                               $7.87\pm1.80 \pm 2.59$
            4.244                               $5.03\pm1.21 \pm 1.38$
            4.258                               $2.34\pm0.82 \pm 0.74$
            4.267                               $1.82\pm0.75 \pm 1.28$
            4.278                               $4.78\pm2.78 \pm 3.36$

  : Summary of the Born cross section of ${e^+e^-}\to{\pi^0}Z_{c}^{0}\to{\pi^0}{\pi^0}{J/\psi}$. The first uncertainties are statistical and the second systematic. 

\[tab:crszc\]

The systematic uncertainties for the ${Z_{c}(3900)^0}$ resonant parameters and the corresponding cross sections include those associated with the amplitude modeling and background treatment in the PWA, as well as the detection efficiency difference between data and the MC simulation.  The uncertainties associated with the parameterizations of intermediate states are studied individually by describing $\sigma$ with the PKU ansatz [@E791sigma] or the Zou-Bugg approach [@E791sigma], varying $f_0(980)$ couplings by 1$\sigma$ of uncertainties [@f0980flatte], describing $f_0(1370)$ with a mass-dependent width BW function, and parameterizing ${Z_{c}(3900)^0}$ with a Flatt$\acute{e}$-like formula as described for ${e^+e^-}\to{\pi^+\pi^-}{J/\psi}$ [@2017PingRG].  The uncertainty related to the barrier radius $r$ is estimated by varying $r$ in the range 1.0$\sim$5.0 ${\rm ~GeV}^{-1}$. The uncertainty due to extra components is studied by including the process ${e^+e^-}\to f_2(1270){J/\psi}$, which is the most significant (3.2$\sigma$) amplitude not included in the nominal fit.  The uncertainty related to the background treatment is studied by varying the $M_{{\ell^+\ell^-}}$-sideband region.  We perform the alternative PWA for the above scenarios individually, and the resultant (largest) changes with respect to the nominal values are regarded as the uncertainties.  The uncertainties related with the detection efficiency include those for tracking of leptons and photons, ${\pi^0}$ reconstruction, kinematic fit and ISR correction factor, as well as the additional MUC hits requirement.  The uncertainties associated with the former few sources discussed in the measurement of the ${e^+e^-}\to{\pi^0}{\pi^0}{J/\psi}$ cross sections, are 5.2%, which directly propagate to the measured ${Z_{c}(3900)^0}$ cross section, while not affecting the ${Z_{c}(3900)^0}$ resonant parameters.  To estimate the uncertainties from the MUC hits requirement, we perform alternative PWA fits by assigning a correction factor for the efficiency to ${J/\psi}\to\mu^+\mu^-$ MC events according to the $\cos\theta$ distribution of $\mu$, and the resultant changes are regarded as uncertainties.  The detection resolution, about 8.8 ${\rm ~MeV/c^{2}}$ in the $M_{{\pi^0}{J/\psi}}$ distribution, is not considered in the PWA.  To estimate its effect, we generate 300 sets of MC samples according to the nominal PWA results with the same statistics of data, and repeat the PWA.  The resultant $|\mu_{pull}|+\sigma_{pull}$ are regarded as the corresponding uncertainties conservatively, where $\mu_{pull}$ and $\sigma_{pull}$ are the mean values and standard deviations of the differences between the input and fitted ${Z_{c}(3900)^0}$ values of 300 pseudo-experiments.  Assuming all of the individuals are uncorrelated, the total uncertainties are the quadratic sum of individual sources. 

  ----------------------------------- ------ ------ ----------- ----------- ----------- -----------
                                                                                        
                                                       4.226       4.236       4.244       4.258  
  ${Z_{c}(3900)^0}$ parametrization    ...    ...      65.7        16.5        11.1        12.4
  $\sigma$ parametrization             0.03   4.9      11.7        25.3         8.6         4.2
  $f_0(980)$ Coupling constant         0.01   0.6       1.8         1.3         1.2         1.6
  $f_0(1370)$ parametrization          0.01   2.7       7.5         3.2         5.6         5.6
  $f_2(1270)$ amplitude                0.05   2.9      15.1         5.0        18.1        25.8
  Barrier radius                       0.01   13.4     11.7         2.9         8.0         3.4
  Background estimation                0.01   1.3       3.8        10.2         9.8         5.1
  Event selection                      0.01   0.2       5.2         5.2         5.2         5.2
  Detection Resolution                 0.06   11.4      6.0         4.1         6.4         9.3
  Total                                0.08   18.8     70.4        33.3        28.0        32.0
  ----------------------------------- ------ ------ ----------- ----------- ----------- -----------

  : Summary of the systematic uncertainties of ${Z_{c}(3900)^0}$ parameters and the cross section of ${e^+e^-}\to{\pi^0}{Z_{c}(3900)^0}\to{\pi^0}{\pi^0}{J/\psi}$ in percentage ($\%$). 

\[tab:pwasys\]

  --------------------------- --------------------------- ------------------------
                                                          
                               Mass (${\rm ~MeV/c^{2}}$)   $\Gamma$(${\rm ~MeV}$)
  cross section measurement               0.6                       13.6
  c.m.  energies                           0.8                       ...
  c.m.  energy spread                     0.01                       0.4
  Fit procedure                           4.7                       9.2
  Total                                   4.9                       16.4
  --------------------------- --------------------------- ------------------------

  : Summary of the systematic uncertainties of structure parameters observed in the $\sigma({e^+e^-}\to{\pi^0}{Z_{c}(3900)^0}\to{\pi^0}{\pi^0}{J/\psi})$ lineshape. 

\[tab:zclineshapesys\]

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