Determination of spin and parity of the $Z_c(3900)$

TL;DR

This study determines the spin and parity of the $Z_c(3900)$ state as $J^P=1^+$ with high significance, measures its pole mass and width, and investigates related production processes using BESIII data.

Contribution

The paper provides the first definitive assignment of the $Z_c(3900)$ quantum numbers and precise measurements of its pole parameters, advancing understanding of its nature.

Findings

$J^P=1^+$ quantum numbers confirmed with >7σ significance

Pole mass measured as approximately 3881 MeV/c^2

Pole width measured as approximately 52 MeV

Abstract

The spin and parity of the $Z_c(3900)^\pm$ state are determined to be $J^P=1^+$ with a statistical significance larger than $7\sigma$ over other quantum numbers in a partial wave analysis of the process $e^+e^-\to \pi^+\pi^-J/\psi$. We use a data sample of 1.92 fb$^{-1}$ accumulated at $\sqrt{s}=4.23$ and 4.26 GeV with the BESIII experiment. When parameterizing the $Z_c(3900)^\pm$ with a Flatte-like formula, we determine its pole mass $M_\textrm{pole}=(3881.2\pm4.2_\textrm{stat}\pm52.7_\textrm{syst})\textrm{MeV}/c^2$ and pole width $\Gamma_\textrm{pole}=(51.8\pm4.6_\textrm{stat}\pm36.0_\textrm{syst})\textrm{MeV}$. We also measure cross sections for the process $e^+e^-\to Z_c(3900)^+\pi^-+c.c.\to J/\psi\pi^+\pi^-$ and determine an upper limit at the 90\% confidence level for the process $e^+e^-\to Z_c(4020)^+\pi^-+c.c.\to J/\psi\pi^+\pi^-$.

Tables

Table 1. Table 1: Significance of the spin parity 1 + superscript 1 1^{+} over other quantum numbers for Z c ± superscript subscript 𝑍 𝑐 plus-or-minus Z_{c}^{\pm} . The significance is obtained for given change in ndf, Δ ( ndf \Delta({\rm ndf} ). In each case, Δ ​ ( ndf ) = 2 × 4 + 5 Δ ndf 2 4 5 \Delta({\rm ndf)}=2\times 4+5 , where 2 × 4 2 4 2\times 4 ndf account for the coupling strength for e + ​ e − → Z c ± ​ π ∓ → superscript 𝑒 superscript 𝑒 superscript subscript 𝑍 𝑐 plus-or-minus superscript 𝜋 minus-or-plus e^{+}e^{-}\to Z_{c}^{\pm}\pi^{\mp} at the two data sets, and the additional five ndf are the contribution of the common degrees of freedom for the Z c subscript 𝑍 𝑐 Z_{c} resonant parameters and the coupling strength for Z c ± → J / ψ ​ π ± → superscript subscript 𝑍 𝑐 plus-or-minus 𝐽 𝜓 superscript 𝜋 plus-or-minus Z_{c}^{\pm}\to J/\psi\pi^{\pm} .

Hypothesis	$\Delta \left(-2\ln L\right)$	$\Delta \left(\mathrm{ndf}\right)$	Significance
$1^{+}$ over $0^{-}$	94.0	13	7.6$\sigma$
$1^{+}$ over $1^{-}$	158.3	13	$10.8\sigma$
$1^{+}$ over $2^{-}$	151.9	13	$10.5\sigma$
$1^{+}$ over $2^{+}$	96.0	13	$7.7\sigma$

Table 2. Table 2: Summary of systematic uncertainties on the Z c ​ ( J P = 1 + ) subscript 𝑍 𝑐 superscript 𝐽 𝑃 superscript 1 Z_{c}~{}(J^{P}=1^{+}) mass, parameters g 1 ′ superscript subscript 𝑔 1 ′ g_{1}^{\prime} and g 2 ′ superscript subscript 𝑔 2 ′ g_{2}^{\prime} , and the signal yields at 4.23 GeV ( N Z c I superscript subscript 𝑁 subscript 𝑍 𝑐 I N_{Z_{c}}^{\textrm{I}} ) and 4.26 GeV ( N Z c II superscript subscript 𝑁 subscript 𝑍 𝑐 II N_{Z_{c}}^{\textrm{II}} ). The uncertainties shown for the Z c subscript 𝑍 𝑐 Z_{c} mass, parameter g 1 ′ superscript subscript 𝑔 1 ′ g_{1}^{\prime} and the ratio g 2 ′ / g 1 ′ superscript subscript 𝑔 2 ′ superscript subscript 𝑔 1 ′ g_{2}^{\prime}/g_{1}^{\prime} are absolute values, while the uncertainties for N Z c I superscript subscript 𝑁 subscript 𝑍 𝑐 I N_{Z_{c}}^{\textrm{I}} and N Z c II superscript subscript 𝑁 subscript 𝑍 𝑐 II N_{Z_{c}}^{\textrm{II}} are relative ones.

Sources	$Z_c$ Mass (MeV/$c^2$)	$g_1^{\prime }\times {10}^3$ (GeV2)	$g_2^{\prime }/g_1^{\prime }$	$N_{Z_c}^{\text{I}}$ (%)	$N_{Z_c}^{\text{II}}$ (%)
Event selection	1.8	…	…	4.8	4.8
$\sigma$ lineshape	19.5	12.0	0.3	2.5	31.0
$Z_c$ parametrization	3.9	…	…	15.5	7.9
Backgrounds	13.9	8.0	0.1	1.9	9.3
$f_0\left(980\right),g_1,g_2/g_1$	17.5	14.0	0.6	2.4	24.6
$f_0\left(1370\right)$	16.7	11.0	0.4	11.5	14.0
Barrier radius	7.9	2.0	1.7	0.5	12.9
$Z_c$ mass resolution	1.0	2.0	…	0.4	0.5
Nonresonance	14.3	9.0	0.0	0.1	18.0
Total	38.0	24.8	1.9	20.3	49.2

Equations

$$F_{\lambda ,\nu }=\sum _{lS}{g}_{lS}\sqrt{\frac{2l+1}{2J+1}}⟨l0S\delta |J\delta ⟩⟨s\lambda \sigma -\nu |S\delta ⟩r^l\frac{B_l(r)}{B_l(r_0)},$$

$$BW(s,M,g_1^{\prime },g_2^{\prime })=\frac{1}{s-M^2+i[g_1^{\prime }{\rho }_1(s)+g_2^{\prime }{\rho }_2(s)]},$$

Full text

A charged charmoniumlike state, $Z_{c}^{\pm}$ ($Z_{c}$ denotes $Z_{c}(3900)$ throughout this Letter except when its mass is explicitly mentioned), was observed by the BESIII bes3zc and Belle bell collaborations in the process $e^{+}e^{-}\to\pi^{+}\pi^{-}J/\psi$ and confirmed using CLEO-c’s data cleoc . As there are at least four quarks in the structure, many theoretical interpretations of the nature and the decay dynamics of the $Z_{c}$ have been put forward theoryzc1 ; theoryzc2 ; theoryzc3 ; theoryzc4 ; theoryzc5 ; theoryzc6 .

A similar charged structure, the $Z_{c}(3885)^{\pm}$, was observed in the process $e^{+}e^{-}\to(D\bar{D}^{*})^{\pm}\pi^{\mp}$ xuxp , with spin parity ($J^{P}$) assignment of $1^{+}$ favored over the $1^{-}$ and $0^{-}$ hypotheses. However, its mass and width are $2\sigma$ and $1\sigma$, respectively, below those of the $Z_{c}^{\pm}$ observed in $e^{+}e^{-}\to\pi^{+}\pi^{-}J/\psi$. Are the $Z_{c}(3885)^{\pm}$ and the $Z_{c}^{\pm}$ the same state and do they have the same spin and parity? This is one of the most important piece of information desired in many theoretical analyses theoryzc3 ; braaten . Finally, the $Z_{c}(4020)$ was observed for the first time in the processes $e^{+}e^{-}\to\pi^{+}\pi^{-}h_{c}$ pipihc and $e^{+}e^{-}\to(D^{*}\bar{D}^{*})^{\pm}\pi^{\mp}$ ddpi , but it has not been searched for in the $\pi^{+}\pi^{-}J/\psi$ final state yet.

In this Letter, we report on the determination of spin and parity of the $Z_{c}$ and a search for the $Z_{c}(4020)^{\pm}$ in the process $e^{+}e^{-}\to\pi^{+}\pi^{-}J/\psi$. The results are based on a partial wave analysis (PWA) of the $e^{+}e^{-}\to\pi^{+}\pi^{-}J/\psi$ events accumulated with the BESIII detector bes3 . The data sample includes 1092 $\rm pb^{-1}$ $e^{+}e^{-}$ collision data at a center-of-mass (c.m.) energy $\sqrt{s}=4.23$ GeV, and 827 $\rm{pb}^{-1}$ data at $\sqrt{s}=4.26$ GeV lumi . The precise c.m. energies are measured with the di-muon process ecms .

The $e^{+}e^{-}\to\pi^{+}\pi^{-}J/\psi$ candidate events are selected with the same selection criteria as described in Ref. bes3zc ; eventslc with $J/\psi$ reconstructed from lepton pairs ($\ell^{+}\ell^{-}=\mu^{+}\mu^{-},~{}e^{+}e^{-}$). The numbers of selected candidate events are 4154 at $\sqrt{s}=4.23$ GeV and 2447 at $\sqrt{s}=4.26$ GeV; the event samples are estimated to contain 365 and 272 background events, respectively, at these two points, using the $J/\psi$ mass sidebands as has been done in Ref. bes3zc .

Amplitudes of the PWA are constructed with the helicity-covariant method chung ; the process $e^{+}e^{-}\to\pi^{+}\pi^{-}J/\psi$ is assumed to proceed via the $Z_{c}$ resonance, i.e., $e^{+}e^{-}\to Z_{c}^{\pm}\pi^{\mp}$, $Z_{c}^{\pm}\to J/\psi\pi^{\pm}$, and via the non-$Z_{c}$ decay $e^{+}e^{-}\to RJ/\psi$, $R\to\pi^{+}\pi^{-}$. All processes are added coherently to obtain the total amplitude chenh . For a particle decaying to the two-body final state, i.e., $A(J,m)\to B(s,\lambda)C(\sigma,\nu)$, where spin and helicity are indicated in the parentheses, its helicity amplitude $F_{\lambda,\nu}$ is related to the covariant amplitude via chung ; vf

[TABLE]

where $\delta=\lambda-\nu$, and $g_{lS}$ is the coupling constant in the $l$-$S$ coupling scheme, the angular brackets denote Clebsch-Gordan coefficients, $r$ is the magnitude of the momentum difference between the two final state particles, $r_{0}$ corresponds to the momentum difference at the nominal mass of the resonance, and $B_{l}$ is a barrier factor barrierform . The nonresonant process, $e^{+}e^{-}\to\pi^{+}\pi^{-}J/\psi$, is parameterized with an amplitude based on the QCD multipole expansion contactterm .

The relative magnitudes and phases of the complex coupling constants $g_{lS}$ are determined by an unbinned maximum likelihood fit to data. The minimization is performed using the package minuit minuit , and the backgrounds are subtracted from the likelihood as in Ref. zhuc .

In the nominal fit, we assume the $Z_{c}$ to have $J^{P}=1^{+}$, and its lineshape is described with a Flatté-like formula taking into account the fact that the $Z_{c}^{\pm}$ decays are dominated by the final states $(D\bar{D}^{*})^{\pm}$ xuxp and $J/\psi\pi^{\pm}$ bes3zc , i.e.,

[TABLE]

where the subscripts in $g^{\prime}_{i}~{}(i=1,2)$ represent the $Z_{c}^{\pm}\to\pi^{\pm}J/\psi$ and $(D\bar{D}^{*})^{\pm}$ decays, respectively; $\rho_{i}(s)=2k_{i}/\sqrt{s}$ is a kinematic factor with $k_{i}$ being the magnitude of the three-vector momentum of the final state particle ($J/\psi$ or $D$) in the $Z_{c}$ rest frame; and $g^{\prime}_{1}$ and $g^{\prime}_{2}$ are the coupling strengths of $Z_{c}^{\pm}\to\pi^{\pm}J/\psi$ and $Z_{c}^{\pm}\to(D\bar{D}^{*})^{\pm}$, respectively, which will be determined by the fit to data.

To describe the $\pi^{+}\pi^{-}$ mass spectrum, four resonances, $\sigma$, $f_{0}(980)$, $f_{2}(1270)$ and $f_{0}(1370)$, are introduced. $f_{0}(980)$ is described with a Flatté formula bes2gi , and the others are described with relativistic Breit-Wigner (BW) functions. The width of the wide resonance $\sigma$ is parameterized with $\Gamma_{\sigma}(s)=\sqrt{1-{4m^{2}_{\pi}\over s}}\Gamma$ berman ; besiib , and the masses and widths for the $f_{2}(1270)$ and $f_{0}(1370)$ are taken from the Particle Data Group (PDG) pdg . The statistical significance for each resonance is determined by examining the probability of the change in log likelihood $(\log L)$ values between including and excluding this resonance in the fits, and the probability is calculated under the $\chi^{2}$ distribution hypothesis taking the change of the number of degrees of freedom $\Delta({\rm ndf})$ into account. With this procedure, the statistical significance of each of these states and the nonresonant process is estimated to be larger than 5$\sigma$. All of them are therefore included in the nominal fit, which includes the $e^{+}e^{-}\to\sigma J/\psi$, $f_{0}J/\psi$, $f_{0}(1370)J/\psi$, $f_{2}(1270)J/\psi$, $Z_{c}^{\pm}\pi^{\mp}$ and nonresonant processes.

A simultaneous fit is performed to the two data sets. The coupling constants are set as free parameters and are allowed to be different at the two energy points except for the common ones describing $Z_{c}$ decays. The oppositely charged $Z_{c}$ states are regarded as isospin partners; they share a common mass and coupling parameters $g^{\prime}_{1}$ and $g^{\prime}_{2}$. Figure 1 shows projections of the fit results at $\sqrt{s}=4.23$ and 4.26 GeV. The mass of $Z_{c}^{\pm}$ is measured to be $M_{Z_{c}}=(3901.5\pm 2.7_{\textrm{stat}})$ MeV/$c^{2}$ and the coupling parameters $g^{\prime}_{1}=(0.075\pm 0.006_{\textrm{stat}})$ GeV2 and $g^{\prime}_{2}/g^{\prime}_{1}=27.1\pm 2.0_{\textrm{stat}}$. This measurement is consistent with the previous result $g^{\prime}_{2}/g^{\prime}_{1}=27.1\pm 13.1$ estimated based on the measured decay width ratio $\Gamma(Z_{c}^{\pm}\to(D\bar{D}^{*})^{\pm})/\Gamma(Z_{c}^{\pm}\to J/\psi\pi^{\pm})=6.2\pm 2.9$ xuxp . If the $Z_{c}^{\pm}$ is parameterized as a constant width BW function, the simultaneous fit gives a mass of $(3897.6\pm 1.2_{\textrm{stat}})\textrm{~{}MeV}/c^{2}$ and a width of $(43.5\pm 1.5_{\textrm{stat}})\textrm{~{}MeV}$, but the value of $-\ln L$ increases by 22 with $\Delta(\text{ndf})=1$. The BW parametrization is thus disfavored with a significance of 6.6$\sigma$.

Figure 2 shows the polar angle ($\theta_{Z_{c}^{\pm}}$) distribution of $Z_{c}^{\pm}$ in the process $e^{+}e^{-}\to Z_{c}^{+}\pi^{-}+c.c.$ and the helicity angle $(\theta_{J/\psi})$ distribution in the decay $Z_{c}^{\pm}\to\pi^{\pm}J/\psi$ for the combined data within the $Z_{c}$ mass region $m_{J/\psi\pi^{\pm}}\in(3.86,3.92)$ GeV/$c^{2}$, where $\theta_{J/\psi}$ is the angle between the momentum of $J/\psi$ in the $Z_{c}$ rest frame and the $Z_{c}$ momentum in the $e^{+}e^{-}$ rest frame. The fit results, using different assumptions for the $Z_{c}$ spin and parity, are drawn with a global normalization factor. The distribution indicates that data favors a spin and parity assignment of $1^{+}$ for the $Z_{c}^{\pm}$. The significance of the $Z_{c}^{\pm}(1^{+})$ hypothesis is further examined using the hypothesis test cl , in which the alternative hypothesis is our nominal fit with an additional $Z_{c}^{\pm}(J^{P}\neq 1^{+})$ state. Possible $J^{P}$ assignments, other than $1^{+}$, are $0^{-}$, $1^{-}$, $2^{-}$, and $2^{+}$. The changes $-2\Delta\ln L$ when the $Z_{c}(1^{+})\pi^{\mp}$ amplitude is removed from the alternative hypothesis are listed in Table 1. Using the associated change in the ndf when the $Z_{c}^{\pm}(1^{+})$ is excluded, we determine the significance of the $1^{+}$ hypothesis over the alternative $J^{P}$ possibilities to be larger than 7$\sigma$.

The fit results shown in Fig. 1 indicate that process is dominated by the $\pi\pi~{}S-$wave resonances, i.e. the $\sigma$, $f_{0}(980)$ and $f_{0}(1370)$. The fraction of all $\pi^{+}\pi^{-}$ $S$-wave components including the interference between them is measured to be $(61.7\pm 2.1_{\textrm{stat}})$% of the total $\pi^{+}\pi^{-}J/\psi$ events at $\sqrt{s}=4.23$ GeV and $(71.4\pm 4.1_{\textrm{stat}})\%$ at $\sqrt{s}=4.26$ GeV. The signal yields $N_{Z_{c}^{\pm}}$ of $Z_{c}^{\pm}$ are calculated by scaling its partial signal ratio with the total number of signal events. They are measured to be $N_{Z_{c}^{\pm}}=952.3\pm 39.3_{\textrm{stat}}$ at $\sqrt{s}=4.23$ GeV and $343.3\pm 23.3_{\textrm{stat}}$ at $\sqrt{s}=4.26$ GeV. Here, the errors are statistical only, and they are estimated using the covariance matrix from the fits.

To measure amplitudes associated with the polarization of $Z_{c}^{\pm}$ in $e^{+}e^{-}\to Z_{c}^{\pm}\pi^{\mp}$ and that of $J/\psi$ in $Z_{c}^{\pm}\to J/\psi\pi^{\pm}$ decays in the nominal fit, the ratios of helicity amplitudes with different polarizations as defined in Eq. (1) are calculated to be $|F^{Z_{c}}_{1,0}|^{2}/|F^{Z_{c}}_{0,0}|^{2}=0.22\pm 0.05_{\textrm{stat}}$ at 4.23 GeV, and $0.21\pm 0.11_{\textrm{stat}}$ at 4.26 GeV for $e^{+}e^{-}\to Z_{c}^{\pm}\pi^{\mp}$, and $|F^{\psi}_{1,0}|^{2}/|F^{\psi}_{0,0}|^{2}=0.45\pm 0.15_{\textrm{stat}}$ for $Z_{c}^{\pm}\to J/\psi\pi^{\pm}$, at both energy points. Here $F^{Z_{c}/\psi}_{1,0}$ and $F^{Z_{c}/\psi}_{0,0}$ correspond to transverse and longitudinal polarization amplitudes in the decay, respectively. The results show that the $Z_{c}$ polarization is dominated by the longitudinal component.

The Born cross section for $Z_{c}$ production is measured with the relation $\sigma=N_{Z_{c}^{\pm}}/(\mathcal{L}(1+\delta)\epsilon\mathcal{B})$, where $N_{Z_{c}^{\pm}}$ is the signal yield for the process $e^{+}e^{-}\to Z_{c}^{+}\pi^{-}+c.c.\to\pi^{+}\pi^{-}J/\psi$, $\mathcal{L}$ is the integrated luminosity, and $\epsilon$ is the detection efficiency obtained from a MC simulation which is generated using the amplitude parameters determined in the PWA. The radiative correction factor $(1+\delta)$ is determined to be 0.818 bes3zc . The Born cross section is measured to be $(22.0\pm 1.0_{\textrm{stat}})$ pb at $\sqrt{s}=4.23$ GeV and $(11.0\pm 1.2_{\textrm{stat}})$ pb at $\sqrt{s}=4.26$ GeV.

Using these two data sets, we also search for the process $e^{+}e^{-}\to Z_{c}(4020)^{+}\pi^{-}+c.c.\to\pi^{+}\pi^{-}J/\psi$, with the $Z_{c}(4020)^{\pm}$ assumed to be a $1^{+}$ state. In the PWA, its mass is taken from Ref. pipihc , and its width is taken as the observed value, which includes the detector resolution. The statistical significance for $Z_{c}(4020)^{\pm}\to J/\psi\pi^{\pm}$ is found to be 3$\sigma$ in the combined data. The Born cross sections are measured to be $(0.2\pm 0.1_{\textrm{stat}})$ pb at $4.23$ GeV and $(0.8\pm 0.4_{\textrm{stat}})$ pb at $s=4.26$ GeV, and the corresponding upper limits at the 90% confidence level are estimated to be $0.9$ pb and $1.4$ pb, respectively.

Systematic errors associated with the event selection, including the luminosity measurement, tracking efficiency of charged tracks, kinematic fit, initial state radiation (ISR) correction factor and the branching fraction of $Br(J/\psi\to\ell^{+}\ell^{-})$, have been estimated to be 4.8% for the cross section measurement and 1.8 MeV for the $Z_{c}$ mass in the previous analysis bes3zc .

Uncertainties associated with the amplitude analysis come from the $\sigma$ and $Z_{c}$ parametrizations, the background estimation, the parameters in the $f_{0}(980)$ Flatté formula, the barrier radius in the barrier factor, the mass resolution and the component of non-resonant amplitude.

The systematic uncertainty due to the $\sigma$ lineshape is estimated by comparing the nominal fit with two other parameterizations, the PKU ansatz pku and the Zou-Bugg approach zb . The differences in the $Z_{c}$ signal yields and mass measurement are taken as the errors, which are 2.5% (31.0%) for the signal yields at 4.23 (4.26) GeV and 19.5 MeV for the $Z_{c}$ mass.

The uncertainty due to the $f_{0}(980)$ lineshape is estimated by varying the couplings by 1$\sigma$ as determined in the decays $J/\psi\to\phi\pi^{+}\pi^{-}$ and $\phi K^{+}K^{-}$ bes2gi . Uncertainties associated with the $f_{0}(1370)$ are estimated by varying the mass and width by one standard deviation around the world average values pdg .

The uncertainty due to the $Z_{c}$ parametrization is estimated by using a constant-width relativistic BW function. The simultaneous fit gives the $Z_{c}$ mass of $(3897.6\pm 1.2_{\textrm{stat}})$ MeV/$c^{2}$ and the width of $(43.5\pm 1.5_{\textrm{stat}})$ MeV. The difference in the $Z_{c}$ signal yields is 15.5% (7.9%) for the data taken at 4.23 (4.26) GeV.

The uncertainty due to the background level is estimated by changing the number of background events by $1\sigma$ around the nominal value, that is, $\pm 25$ around 637 events.

The barrier radius is usually taken in the range $r_{0}\in(0.25,~{}0.76)$ fm, with 0.6 fm being used in the nominal fit. Uncertainties at both ends are checked. For a conservative estimation, the radius $r_{0}=0.76$ fm, which results in the larger difference, is used to estimate the uncertainty.

The uncertainty due to the mass resolution in the $J/\psi\pi$ invariant mass is estimated with an unfolded $Z_{c}$ width. A truth width is unfolded from the observed $Z_{c}$ width using a relation determined by the MC simulation, and its difference from the unfolded width, $\delta\Gamma/\Gamma=\delta g_{1}^{\prime}/g_{1}^{\prime}$, is taken as the systematic uncertainty for the coupling constant $g_{1}^{\prime}$. The uncertainties in the signal yields and the $Z_{c}$ mass are determined with the truth coupling constant.

The nonresonant process is described with a formula derived from the QCD multipole expansion contactterm . It includes the $S$- and $D$-wave components. The uncertainty associated with this amplitude is estimated by removing the insignificant $D$-wave component and using the $S$-wave component only.

Table 2 summarizes the systematic uncertainties. Assuming all of these sources are independent, the total systematic uncertainties are 38.0 MeV for the measurement of the $Z_{c}$ mass, and 20.3% (49.2%) for the measurement of $Z_{c}$ cross sections at $\sqrt{s}=4.23$ (4.26) GeV.

In summary, with 1.92 fb*-1* data taken at $\sqrt{s}=4.23$ and 4.26 GeV, the $Z_{c}^{\pm}$ state is studied with an amplitude fit to the $e^{+}e^{-}\to\pi^{+}\pi^{-}J/\psi$ samples, and its spin and parity have been determined to be $1^{+}$ with a statistical significance larger than 7$\sigma$ over other quantum numbers. The mass is measured to be $M_{Z_{c}}=(3901.5\pm 2.7_{\textrm{stat}}\pm 38.0_{\textrm{syst}})$ MeV$/c^{2}$ in the parametrization of a Flatté-like formula with parameters $g^{\prime}_{1}=0.075\pm 0.006_{\textrm{stat}}\pm 0.025_{\textrm{syst}}$ GeV2, and $g^{\prime}_{2}/g^{\prime}_{1}=27.1\pm 2.0_{\textrm{stat}}\pm 1.9_{\textrm{syst}}$, which corresponds to the $Z_{c}$ pole mass $M_{\text{pole}}=(3881.2\pm 4.2_{\textrm{stat}}\pm 52.7_{\text{syst}})\text{~{}MeV}/c^{2}$ and pole width $\Gamma_{\textrm{pole}}=(51.8\pm 4.6_{\textrm{stat}}\pm 36.0_{\textrm{syst}})\textrm{~{}MeV}$, where $M_{\textrm{pole}}-i\Gamma_{\textrm{pole}}/2$ is the solution for which the denominator of Flatté-like formula is zero. The pole mass is consistent with the previous measurement xuxp . The Born cross sections for the process $e^{+}e^{-}\to\pi^{+}Z_{c}^{-}+c.c.$ are measured to be $(21.8\pm 1.0_{\textrm{stat}}\pm 4.4_{\textrm{syst}})$ pb at $\sqrt{s}=4.23$ GeV and $(11.0\pm 1.2_{\textrm{stat}}\pm 5.4_{\textrm{syst}})$ pb at $\sqrt{s}=4.26$ GeV. The contributions from $Z_{c}(4020)^{\pm}$ are also searched for, but no significant signals are observed, and an upper limit for the $e^{+}e^{-}\to\pi^{+}Z_{c}(4020)^{-}+c.c.$ process is determined to be 0.9 (1.4) pb at $\sqrt{s}=4.23~{}(4.26)$ GeV.

The BESIII collaboration thanks the staff of BEPCII and the computing center for their strong support. This work is supported in part by the Ministry of Science and Technology of China under Contract No. 2009CB825200; Joint Funds of the National Natural Science Foundation of China under Contracts Nos. U1332201; National Natural Science Foundation of China (NSFC) under Contracts Nos. 11175188, 11375205, 11235011, 11375221, 11565006, 10825524; German Research Foundation DFG under Contract No. Collaborative Research Center CRC-1044, 627240; Istituto Nazionale di Fisica Nucleare, Italy; Ministry of Development of Turkey under Contract No. DPT2006K-120470; U.S. Department of Energy under Contracts Nos. DE-SC-0012069, DE-SC-0010504, DE-SC-0010118, DE-FG02-05ER41374; U.S. National Science Foundation; University of Groningen (RuG) under Contracts No. 530-4CDP03, and the Helmholtzzentrum fuer Schwerionenforschung GmbH (GSI), Darmstadt; WCU Program of National Research Foundation of Korea under Contract No. R32-2008-000-10155-0.