Study of $e^{+}e^{-}\rightarrow\pi^{0}X(3872)\gamma$ and search for $Z_{c}(4020)^{0}\rightarrow X(3872)\gamma$

The recent discovery of several charmonium-like states has attracted great experimental and theoretical interests pdg . The charmonium-like states are also called $XYZ$ states where $X$ is the isospin-singlet state with $J^{PC}\neq 1^{--}$, $Y$ is the isospin-singlet state with $J^{PC}=1^{--}$, and $Z$ is the isospin-triplet state xyzstates . The masses of these states are above the open-charm thresholds, and due to the unexpected resonance parameters and decay channels, these states can not be described by conventional quark models. Therefore, they are good candidates for exotic states, such as hybrids, tetraquarks, molecules, etc. theory1 ; theory2 ; theory3 .

The first charmonium-like state $X(3872)$, which has been recently renamed as $\chi_{c1}(3872)$ by the Particle Data Group (PDG pdg ), was observed by the Belle experiment in the process $B^{\pm}\rightarrow K^{\pm}X(3872)\rightarrow K^{\pm}\pi^{+}\pi^{-}J/\psi$ x38721 . The $X(3872)$ is a rather narrow state with a mass that is consistent with $D^{0}\bar{D}^{*0}$ threshold. It decays through open-charm, radiative and isospin-violating pion emission decays, and is found to be an isospin singlet with $J^{PC}=1^{++}$ pdg . Among these features, the extremely small mass difference between the $X(3872)$ and $D^{0}\bar{D}^{\ast 0}$ threshold which we will denote as $\delta$, is of particular interest. Taking the values for the $D^{0}$, $D^{*0}$ and $X(3872)$ masses from the PDG pdg , $\delta$ is calculated to be $(-10\pm 180)$ keV/$c^{2}$. Very recently, the LHCb reported a new measurement yielding $\delta=(70\pm 120)$ keV/$c^{2}$ lhcb1 ; lhcb2 . However the improved precision is still insufficient to tell whether the $X(3872)$ mass is above or below the $D^{0}\bar{D}^{\ast 0}$ threshold. Better knowledge of $\delta$ will be an important step towards a deeper understanding of the nature of the $X(3872)$ x3872-a ; x3872-b , and eventually of other related $XYZ$ states. A completely new method to measure the $\delta$ value by measuring the $X(3872)\gamma$ line shape, which is sensitive to the $\delta$ value due to a triangle singularity, is proposed by Ref. guo ; ortega ; guob . Here, the $X(3872)\gamma$ needs to be produced associated with another positive $C$-parity neutral meson, e.g. $e^{+}e^{-}\rightarrow\pi^{0}X(3872)\gamma$. In principle, this method could be applied at the BESIII experiment, based on the sizable data samples taken for $XYZ$ studies. According to the theoretical prediction in Ref. guob , the cross section of the process $e^{+}e^{-}\rightarrow\pi^{0}X(3872)\gamma$ is expected to be small. However, there could be other scenarios where the expected cross section is large.

Recently, the BESIII Collaboration reported an enhancement around 4.2 GeV for the $e^{+}e^{-}\rightarrow\gamma X(3872)$ production cross sections x3872-c , which suggests a connection between $Y$ and $X$ states. BESIII also reported another connection, now between $Y$ and $Z$ states, with the observation of a $Y(4220)$ resonance in the process $e^{+}e^{-}\rightarrow\pi^{0}Z_{c}(3900)^{0}$ x3872-d . Those observations may indicate some common nature among the $XYZ$ states. Therefore, it is important to search for possible connections between $Z$ and $X$ states. Establishing connections among $XYZ$ states may be a clue that can facilitate a better theoretical interpretation of these. One such connection voloshin could be a transition $Z_{c}(4020)^{0}\rightarrow X(3872)\gamma$ in the scenario where the $X(3872)$ is dominantly an $S$-wave $D^{0}\bar{D}^{*0}$ molecule and the $Z_{c}(4020)^{0}$ is an isotopic triplet of near-threshold $S$-wave $D^{*}\bar{D}^{*}$ resonances. Therefore, the search for the transition $Z_{c}(4020)^{0}\rightarrow X(3872)\gamma$ will help to quantitatively study the molecular picture of the $X(3872)$. The $Z_{c}(4020)$ is observed in the $e^{+}e^{-}\rightarrow\pi Z_{c}(4020)$ process, so the study of $e^{+}e^{-}\rightarrow\pi^{0}X(3872)\gamma$ allows one to search for the transition $Z_{c}(4020)^{0}\rightarrow X(3872)\gamma$.

In this paper, we report the search for the reaction $e^{+}e^{-}\rightarrow\pi^{0}X(3872)\gamma$ and $Z_{c}(4020)^{0}\rightarrow X(3872)\gamma$ based on the data of twenty-three energy points recorded with the BESIII detector in the range of $4.178\leq\sqrt{s}\leq 4.600\,\rm{GeV}$. The $X(3872)$ state is reconstructed via $X(3872)\rightarrow\pi^{+}\pi^{-}J/\psi$, $J/\psi\rightarrow\ell^{+}\ell^{-}$ ($\ell=e$ or $\mu$).

The BESIII detector is a magnetic spectrometer besiii located at the Beijing Electron Positron Collider (BEPCII) bepcii . The cylindrical core of the BESIII detector consists of a helium-based multilayer drift chamber (MDC), a plastic scintillator time-of-flight system (TOF), and a CsI(Tl) electromagnetic calorimeter (EMC), which are all enclosed in a superconducting solenoidal magnet, providing a 1.0 T magnetic field. The solenoid is supported by an octagonal flux-return yoke with resistive plate chamber muon identifier modules interleaved with steel. The acceptance of charged particles and photons is 93% over the $4\pi$ solid angle. The charged-particle momentum resolution at $1~{}{\rm GeV}/c$ is $0.5\%$, and the $\textrm{d}E/\textrm{d}x$ resolution is $6\%$ for the electrons from Bhabha scattering. The EMC measures photon energies with a resolution of $2.5\%$ ($5\%$) at $1$ GeV in the barrel (end cap) region. The time resolution of the TOF barrel section is 68 ps, while that of the end cap section is 110 ps. The end cap TOF system was upgraded in 2015 with multi-gap resistive plate chamber technology, providing a time resolution of 60 ps etof . About 70% of the data sample used here was taken after this upgrade.

Simulated data samples produced with the geant4-based geant4 Monte Carlo (MC) package, which includes the geometric description of the BESIII detector and the detector response, are used to determine the detection efficiency and to estimate the background contributions. The simulation includes the beam energy spread and initial-state radiation (ISR) in the $e^{+}e^{-}$ annihilations modeled with the generator kkmc KKMC . The ISR production of vector charmonium(-like) states and the continuum processes are incorporated also in kkmc KKMC . The known decay modes are modeled with evtgen ref:evtgen , using branching fractions summarized and averaged by the PDG pdg , and the remaining unknown decays from the charmonium states are generated with lundcharm ref:lundcharm . Final state radiation from charged final state particles is incorporated with the photos package photos .

Signal MC samples for $e^{+}e^{-}\rightarrow\pi^{0}X(3872)\gamma$ and $e^{+}e^{-}\rightarrow\pi^{0}Z_{c}(4020)^{0}\rightarrow\pi^{0}X(3872)\gamma$ are generated according to phase space at each center-of-mass energy point, assuming that the cross section follows the function fit for the $e^{+}e^{-}\rightarrow\pi^{+}\pi^{-}h_{c}$ line shape in Ref. twobw . The event selection criteria and the detection efficiencies are determined and studied based on signal MC samples of $1\times 10^{5}$ signal events generated for each value of $\sqrt{s}$. Detection efficiencies are determined by the ratio of the reconstructed event yields (after the selection criteria) and the number of generated events. Inclusive MC samples consisting of open charm production processes are employed to investigate potential backgrounds.

For each charged track, the distance of closest approach to the interaction point (IP) is required to be within $10$ cm in the beam direction and within 1 cm in the plane perpendicular to the beam direction. The polar angles ($\theta$) of the tracks with respect to the beam axis (ignoring the small crossing angle), must be within the fiducial volume of the MDC $(|\cos\theta|<0.93)$. Photons are reconstructed from isolated showers in the EMC, which are at least $10^{\circ}$ away from the nearest charged track. The photon energy is required to be at least 25 MeV in the barrel region $(|\cos\theta|<0.80)$ or 50 MeV in the end cap region $(0.86<|\cos\theta|<0.92)$. To suppress electronic noise and energy depositions unrelated to the event, the EMC cluster timing from the reconstructed event start time is further required to satisfy $0\leq t\leq 700$ ns.

Since the reaction $e^{+}e^{-}\rightarrow\pi^{0}X(3872)\gamma$ results in the final states $\gamma\gamma\gamma\pi^{+}\pi^{-}\ell^{+}\ell^{-}$, candidate events are required to have four tracks with zero net charge and at least three photons. Tracks with momenta larger than 1.0 GeV/$c$ are assigned as leptons from the $J/\psi$ decay; otherwise, they are regarded as pions. Leptons from the $J/\psi$ decay with energy deposited in the EMC larger than 1.0 GeV are identified as electrons, while those with less than 0.4 GeV as muons. The $\pi^{0}$ candidates are reconstructed from photon pairs with invariant mass in the range $110<M_{\gamma\gamma}<150\,{\rm MeV}/c^{2}$.

To reduce the background contributions and to improve the mass resolution, a five-constraint (5C) kinematic fit is performed. Four constraints come from the total initial four momentum of the colliding beams; the last one is from constraining the $M_{\gamma\gamma}$ invariant mass to the nominal $\pi^{0}$ value pdg . If there is more than one combination in an event, the one with the smallest $\chi^{2}_{\text{5C}}$ is chosen. Furthermore, the $\chi^{2}_{\text{5C}}$ is required to be less than 60. The $J/\psi$ is reconstructed via $\ell^{+}\ell^{-}$ decays, and the invariant mass of lepton pairs is required to be in the $J/\psi$ mass window $[3.080,3.120]\,{\rm GeV}/c^{2}$.

The systematic uncertainties of $\sigma(e^{+}e^{-}\rightarrow\pi^{0}X(3872)\gamma)\cdot\mathcal{B}(X(3872)\rightarrow\pi^{+}\pi^{-}J/\psi)$ and $\sigma(e^{+}e^{-}\rightarrow\pi^{0}Z_{c}(4020)^{0})\cdot\mathcal{B}(Z_{c}(4020)^{0}\rightarrow X(3872)\gamma)\cdot\mathcal{B}(X(3872)\rightarrow\pi^{+}\pi^{-}J/\psi)$ originate from the luminosity measurement, the tracking efficiency, the photon detection efficiency, the kinematic fit, the $J/\psi$ mass window, the $X(3872)$ mass window, the $Z_{c}(4020)^{0}$ parameters, the line shape, the generator model, the ISR correction, and the input branching fractions.

The integrated luminosity at each point has been measured with a precision of $1.0\%$ using the Bhabha process luminosity .

The uncertainty from the tracking efficiency is $1.0\%$ per track omegachic0 and the uncertainty in photon detection efficiency is $1.0\%$ per photon photon .

The uncertainty due to the kinematic fit requirements is estimated by correcting the helix parameters of charged tracks according to the method described in Ref. helix . The difference between detection efficiencies obtained from MC samples with and without this correction is taken as the uncertainty.

The uncertainty for the $J/\psi$ mass window is estimated using the control sample of $e^{+}e^{-}\rightarrow\gamma_{\rm ISR}\psi(3686),\psi(3686)\rightarrow\pi^{+}\pi^{-}J/\psi$. The difference of the efficiency between data and MC simulation is found to be 1.6$\%$ jpsimasswindow , which is taken as the uncertainty.

The uncertainty from the $X(3872)$ mass window is estimated by changing the window range by $\pm$10%, and the largest efficiency change is taken as the uncertainty.

The uncertainties arising from the $Z_{c}(4020)^{0}$ mass and width are estimated by changing them by one standard deviation values pdg while generating the signal MC. The largest efficiency difference relative to the nominal one is taken as the uncertainty.

The line shape affects the ISR correction factor and the efficiency. No obvious signal was found for our $Z_{c}(4020)^{0}$ search, so we use the line shape from $e^{+}e^{-}\rightarrow\pi^{+}\pi^{-}h_{c}$ in Ref. twobw as the input line shape to get the nominal results. To get the uncertainty introduced by the line shape, we change it to a Breit-Wigner function describing the $\psi(4230)$ or $\psi(4415)$, with the masses and widths fixed to the values from PDG pdg . The largest difference of the final result is taken as a systematic uncertainty.

For the systematic uncertainty from the MC simulation describing the process $e^{+}e^{-}\rightarrow\pi^{0}X(3872)\gamma$, we use the three-body phase space MC simulation to get the nominal efficiency, then change to the $e^{+}e^{-}\rightarrow\pi^{0}Z_{c}(4020)^{0}\rightarrow\pi^{0}X(3872)\gamma$. The difference on the detection efficiency with and without the intermediate resonant state is taken as the uncertainty due to the MC generator model.

The systematic uncertainty from the MC simulation describing the process $e^{+}e^{-}\rightarrow\pi^{0}Z_{c}(4020)^{0}\rightarrow\pi^{0}X(3872)\gamma$ is estimated by varying the distribution of the $\pi^{0}$ polar angle $\theta$. The nominal efficiency is determined assuming a flat distribution in $\cos{\theta}$. A conservative estimate of the systematic uncertainty is obtained using alternative MC samples with angular distributions of $1\pm\cos^{2}{\theta}$. The largest change of efficiency is taken as the uncertainty due to the MC generator model.

The ISR correction factor is obtained from quantum electrodynamics calculations QED ; KKMC . We also analyze MC samples with and without ISR effects considered to get the ISR correction factor, the difference of the two results is taken as the systematic uncertainty on the ISR correction factor.

As uncertainties introduced by the branching fractions of $J/\psi\rightarrow\ell^{+}\ell^{-}$ and $\pi^{0}\rightarrow\gamma\gamma$ we use those quoted by the PDG pdg .

Table 2 summarizes all the systematic uncertainties related to $\sigma(e^{+}e^{-}\rightarrow\pi^{0}X(3872)\gamma)\cdot\mathcal{B}(X(3872)\rightarrow\pi^{+}\pi^{-}J/\psi)$ and $\sigma(e^{+}e^{-}\rightarrow\pi^{0}Z_{c}(4020)^{0})\cdot\mathcal{B}(Z_{c}(4020)^{0}\rightarrow X(3872)\gamma)\cdot\mathcal{B}(X(3872)\rightarrow\pi^{+}\pi^{-}J/\psi)$ for each center-of-mass energy. The total systematic uncertainty for each energy point is calculated as the quadratic sum of the individual uncertainties, assuming them to be uncorrelated.

Using data samples collected at the center-of-mass energies between 4.178 and 4.600 GeV, the processes $e^{+}e^{-}\rightarrow\pi^{0}X(3872)\gamma$ and $Z_{c}(4020)^{0}\rightarrow X(3872)\gamma$ are investigated. In neither of the two processes are significant signals observed. Upper limits at the 90% C.L. on the cross sections multiplied by the branching fractions, $\sigma(e^{+}e^{-}\rightarrow\pi^{0}X(3872)\gamma)\cdot\mathcal{B}(X(3872)\rightarrow\pi^{+}\pi^{-}J/\psi)$ and $\sigma(e^{+}e^{-}\rightarrow\pi^{0}Z_{c}(4020)^{0})\cdot\mathcal{B}(Z_{c}(4020)^{0}\rightarrow X(3872)\gamma)\cdot\mathcal{B}(X(3872)\rightarrow\pi^{+}\pi^{-}J/\psi)$, are reported for each energy point. The average cross sections multiplied by branching fractions are also determined. The measured results of the process $e^{+}e^{-}\rightarrow\pi^{0}X(3872)\gamma$ are not in conflict with the theoretical expectation of about 0.1 fb guob . A three orders of magnitude increase in statistics is needed to test these models.

Using the experimental results on the $\sigma(e^{+}e^{-}\rightarrow\pi^{0}Z_{c}(4020)^{0})\cdot\mathcal{B}(Z_{c}(4020)^{0}\rightarrow(D^{\ast}\bar{D}^{\ast})^{0})$ at $\sqrt{s}=4.226$ and $4.258$ GeV zc40202 , the ratio $\frac{\mathcal{B}(Z_{c}(4020)^{0}\rightarrow X(3872)\gamma)\cdot\mathcal{B}(X(3872)\rightarrow\pi^{+}\pi^{-}J/\psi)}{\mathcal{B}(Z_{c}(4020)^{0}\rightarrow(D^{\ast}\bar{D}^{\ast})^{0})}$ is determined to be less than 0.15% at the 90% C.L. The ratio does not contradict the prediction reported in Ref. voloshin based on the molecular picture. Since no significant $e^{+}e^{-}\rightarrow\pi^{0}X(3872)\gamma$ signals are observed, we cannot study the lineshape as proposed in Ref. guo ; ortega ; this may be achieved at future super tau-charm facilities supertaocharm1 ; supertaocharm2 .

The BESIII collaboration thanks the staff of BEPCII and the IHEP computing center for their strong support. This work is supported in part by National Key Research and Development Program of China under Contracts Nos. 2020YFA0406300, 2020YFA0406400; National Natural Science Foundation of China (NSFC) under Contracts Nos. 11905179, 11625523, 11635010, 11735014, 11822506, 11835012, 11935015, 11935016, 11935018, 11961141012; the Chinese Academy of Sciences (CAS) Large-Scale Scientific Facility Program; Joint Large-Scale Scientific Facility Funds of the NSFC and CAS under Contracts Nos. U1732263, U1832207; CAS Key Research Program of Frontier Sciences under Contracts Nos. QYZDJ-SSW-SLH003, QYZDJ-SSW-SLH040; 100 Talents Program of CAS; INPAC and Shanghai Key Laboratory for Particle Physics and Cosmology; ERC under Contract No. 758462; European Union Horizon 2020 research and innovation programme under Contract No. Marie Sklodowska-Curie grant agreement No 894790; Nanhu Scholars Program for Young Scholars of Xinyang Normal University; German Research Foundation DFG under Contracts Nos. 443159800, Collaborative Research Center CRC 1044, FOR 2359, FOR 2359, GRK 214; Istituto Nazionale di Fisica Nucleare, Italy; Ministry of Development of Turkey under Contract No. DPT2006K-120470; National Science and Technology fund; Olle Engkvist Foundation under Contract No. 200-0605; STFC (United Kingdom); The Knut and Alice Wallenberg Foundation (Sweden) under Contract No. 2016.0157; The Royal Society, UK under Contracts Nos. DH140054, DH160214; The Swedish Research Council; U. S. Department of Energy under Contracts Nos. DE-FG02-05ER41374, DE-SC-0012069.