First simultaneous measurement of $\Xi^{0}$ and $\bar{\Xi}^{0}$ asymmetry parameters in $\psi(3686)$ decay

The universe began with the Big Bang, where it is commonly assumed that matter and antimatter were created in equal amounts. However, at present, only traces of antimatter can be seen. $CP$ violation (CPV) is one of the necessary conditions to possibly explain this asymmetry CPsource . The existence of CPV in the decays of $K^{0}$, $B^{0}$, and $D^{0}$ mesons KCP ; BCP1 ; BCP2 ; DCP , as well as in neutrino oscillations $\nu_{l}$ nuCP , are firmly established. However, these CPV effects are too small to explain the large matter-antimatter asymmetry in the universe.

Recently, a technique to test CPV in the hyperon sector has been developed by simultaneously analyzing the spin polarization and the asymmetry parameters of the entangled hyperon-antihyperon pairs produced in the decays of the $J/\psi$, $\psi(3686)$, and $\psi(3770)$ mesons at the BESIII experiment xghe . For cascade hyperon decays, the angular distribution of the daughter hyperon is proportional to $(1+\alpha_{H}\boldsymbol{P}_{H}\cdot\hat{\boldsymbol{n}})$, where $\alpha_{H}$ is the hyperon decay parameter, $\boldsymbol{P}_{H}$ and $\hat{\boldsymbol{n}}$ are the hyperon polarization and the unit vector in the direction of the daughter hyperon momentum, respectively, both in the hyperon rest frame. The $CP$ asymmetry is defined as $A_{CP}=(\alpha_{H}+\alpha_{\bar{H}})/(\alpha_{H}-\alpha_{\bar{H}})$, where the parameters $\alpha_{H}$ and $\alpha_{\bar{H}}$ are $CP$ odd, and a nonzero $A_{CP}$ indicates CPV. In the Standard Model (SM), a tiny $A_{CP}$ value of $\sim 10^{-4}$ 10m4 is predicted in the hyperon sector. Therefore, a test of CPV in hyperon decays is sensitive to possible sources of CPV from physics beyond the SM theory ; Liu:2023xhg . At present, BESIII has performed CPV tests in the decays of $\Lambda$ Lambda ; 3773Lambda ; zhangjianyu , $\Sigma^{+}$ Sigmap , and $\Xi^{-}$ Patrik ; zhangjingxu hyperons, where for $\Xi^{-}$ hyperons, the most precise asymmetry parameter measurement was reported in $J/\psi$ decay. BESIII has also performed the first determination of the weak phase of the $\Xi^{-}$ hyperon using entangled $\Xi^{-}\bar{\Xi}^{+}$ pairs Patrik . However, CPV in $\Xi^{0}$ hyperon decays has not so far been searched for, the asymmetry parameter $\alpha_{\Xi^{0}}$ in the decay $\Xi^{0}\to\Lambda\pi^{0}$ has not been measured directly, only the product $\alpha_{\Lambda}\cdot\alpha_{\Xi^{0}}$ has been reported product1 ; product2 , and the weak decay phase $\phi_{\Xi^{0}}$ was measured with large uncertainty HBC1 ; HBC2 ; HBC3 .

The process of $e^{+}e^{-}\to\psi(3686)\to\Xi^{0}\bar{\Xi}^{0}\to\pi^{0}\pi^{0}\Lambda\bar{\Lambda}$ with $\Lambda\to p\pi^{-}$ can be fully described by the vector $\boldsymbol{\xi}=(\theta_{\Xi},\theta_{\Lambda},\varphi_{\Lambda},\theta_{\bar{\Lambda}},\varphi_{\bar{\Lambda}},\theta_{p},\varphi_{p},\theta_{\bar{p}},\varphi_{\bar{p}})$, where the coordinate systems and angles are shown in Fig. LABEL:xyz with the same convention as Refs. Patrik and zhangjingxu . The cascade $\Xi$ and $\Lambda$ polarization vector, $\boldsymbol{P}_{\Xi}$ and $\boldsymbol{P}_{\Lambda}$, are related as $\boldsymbol{P}_{\Lambda}=\alpha_{\Xi^{0}}\hat{\boldsymbol{z}}_{\Xi^{0}}+\beta_{\Xi^{0}}(\boldsymbol{P}_{\Xi^{0}}\times\hat{\boldsymbol{z}}_{\Xi^{0}})+\gamma_{\Xi^{0}}[\hat{\boldsymbol{z}}_{\Xi^{0}}\times(\boldsymbol{P}_{\Xi^{0}}\times\hat{\boldsymbol{z}}_{\Xi^{0}})]$, where $\alpha_{\Xi^{0}}$, $\beta_{\Xi^{0}}$, and $\gamma_{\Xi^{0}}$ are defined in Ref. LeeYang and $\hat{\boldsymbol{z}}_{\Xi^{0}}$ is the unit vector in the direction of the $\Xi$ momentum. The joint angular distribution function is described by formula

In this paper, we present the first simultaneous measurement of the $\Xi^{0}$ asymmetry parameters using entangled $\Xi^{0}$-$\bar{\Xi}^{0}$ pairs from $(448.1\pm 2.9)\times 10^{6}$ $\psi(3686)$ decays psip collected with the BESIII detector BESIII . In addition, a study of the transverse polarization in $\psi(3686)\to\Xi^{0}\bar{\Xi}^{0}$ and a test of CPV in $\Xi^{0}$ hyperon decays are performed.

Candidate $\psi(3686)\to\Xi^{0}\bar{\Xi}^{0}$ events are selected by fully reconstructing the subsequent decays $\Xi^{0}\to\pi^{0}\Lambda$, $\Lambda\to p\pi^{-}$ and $\pi^{0}\to\gamma\gamma$ (as well as the charge conjugate final states for $\bar{\Xi}$ and $\bar{\Lambda}$ decays). Potential background contributions are studied with an inclusive Monte Carlo (MC) simulation sample of $\psi(3686)$ decays TopoAna , and an exclusive simulation of the signal process with $5\times 10^{6}$ events is generated with a phase space model for normalization. The production of the $\psi(3686)$ resonance for both MC samples is simulated with the kkmc generator kkmc1 ; kkmc2 , and the subsequent decays are processed by evtgen evtgen1 ; evtgen2 . Additionally for the inclusive MC sample, the branching fractions of cascade decays are fixed according to the Particle Data Group (PDG) PDG2022 . All the remaining unmeasured decay modes are generated with lundcharm lundcharm1 ; lundcharm2 . The response of the BESIII detector is modeled with MC simulations using a framework based on geant4 geant41 ; geant42 .

Candidate events are required to contain at least four charged particles (two positive and two negative) and at least four photons. Charged particles are reconstructed as tracks within the multilayer drift chamber (MDC). Only tracks fully contained in the acceptance region of the MDC, $|\cos\theta|<0.93$ (with $\theta$ defined with respect to the $z$-axis, which is the symmetry axis of the MDC), are kept for the analysis. Because of the momentum separation in the two body decay, the momenta of (anti-)proton and charged pion candidates are required to be greater and less than 0.5 GeV/$c$, respectively.

$\Lambda$($\bar{\Lambda}$) candidates are reconstructed as $p\pi^{-}$ ($\bar{p}\pi^{+}$) pairs that satisfy a vertex fit. The four-track combination that minimizes $\sqrt{(M_{p\pi^{-}}-m_{\Lambda})^{2}+(M_{\bar{p}\pi^{+}}-m_{\Lambda})^{2}}$ is selected, where $M_{p\pi^{-}}(M_{\bar{p}\pi^{+}})$ is the invariant mass of the $p\pi^{-}(\bar{p}\pi^{+})$ pair and $m_{\Lambda}$ is the $\Lambda$ mass from the PDG PDG2022 . To further suppress non-$\Lambda$ background, the $\Lambda$ decay length is required to be greater than zero, where negative decay lengths are caused by the detector resolution and background contributions.

Photon candidates are reconstructed from isolated showers in the electromagnetic calorimeter (EMC). The energy deposited in the nearby time of flight (TOF) counter is included to improve the reconstruction efficiency and energy resolution. The shower energies are required to be greater than 25 MeV in the EMC barrel region ($|\cos\theta|<0.8$), or greater than 50 MeV in the EMC end-cap region ($0.86<|\cos\theta|<0.92$). In order to reject electronic noise and energy deposits unrelated to the event start time, the EMC shower time, measured with respect to the collision signal, is required to satisfy $0<t<700$ ns.

To further suppress background from soft $\pi^{0}$s and radiated photon events and to improve the mass resolution, a six-constraint (6C) kinematic fit is applied to all possible $\gamma\gamma\gamma\gamma\Lambda\bar{\Lambda}$ combinations by imposing energy-momentum conservation and constraining the masses of the two pairs of photons from the $\pi^{0}$ mesons to the $\pi^{0}$ mass. The $\Xi^{0}$ and $\bar{\Xi}^{0}$ candidates are then reconstructed as the $\pi^{0}\Lambda$ and $\pi^{0}\bar{\Lambda}$ combinations that minimize the discriminant $\delta=\sqrt{(M_{\pi^{0}\Lambda}-m_{\Xi^{0}})^{2}+(M_{\pi^{0}\bar{\Lambda}}-m_{\Xi^{0}})^{2}}$ from all $\pi^{0}\Lambda(\pi^{0}\bar{\Lambda})$ combinations, where $M_{\pi^{0}\Lambda}(M_{\pi^{0}\bar{\Lambda}})$ is the invariant mass of the $\pi^{0}\Lambda(\pi^{0}\bar{\Lambda})$ system and $m_{\Xi^{0}}$ is the $\Xi^{0}$ mass from the PDG PDG2022 . Finally, background contributions from the $\psi(3686)\to\pi^{0}\pi^{0}J/\psi$ process are rejected by requiring the recoil mass of $\pi^{0}\pi^{0}$ combinations to be at least $20$ MeV/$c^{2}$ away from the nominal $J/\psi$ mass PDG2022 . Figure LABEL:box shows the distribution of $M_{\pi^{0}\Lambda}$ versus $M_{\pi^{0}\bar{\Lambda}}$ for candidate events selected in data. A clear signal around the $\Xi^{0}(\bar{\Xi}^{0})$ mass is observed. Signal events are required to simultaneously satisfy $|M_{\pi^{0}\Lambda}-m_{\Xi^{0}}|<15$ MeV/$c^{2}$ and $|M_{\pi^{0}\bar{\Lambda}}-m_{\Xi^{0}}|<15$ MeV/$c^{2}$ (region marked as $\rm{S}$ in Fig. LABEL:box). Most of the background contributions arise from $\psi(3686)$ decays that do not contain a $\Xi^{0}\bar{\Xi}^{0}$ pair, such as $\psi(3686)\to\pi^{0}\pi^{0}\Lambda\bar{\Lambda}$. The background yield is evaluated by the mean of the three sideband regions ${\rm B}_{i}$ (with $i=1,2,3$) depicted in Fig. LABEL:box. The sideband regions have the same size as the signal region, but are centered on the following values of $(M_{\pi^{0}\Lambda},M_{\pi^{0}\bar{\Lambda}})=(1.27,1.27)$, $(1.36,1.27)$, and $(1.27,1.36)$ GeV/$c^{2}$. The sideband region $\rm{B}_{4}$ is not suitable for background evaluation as it is close to the $\psi(3686)\to\Sigma(1385)^{0}\bar{\Sigma}(1385)^{0}$ region and would lead to an overestimation of the signal contamination. In the signal region $N=1934$ events are counted with an expected background contribution of $23\pm 5$ events, resulting in a $1.2$% contamination level. The signal contamination can, therefore, be considered as negligible in the following analysis.

To determine the set of $\boldsymbol{\Omega}$ parameters, an unbinned maximum likelihood fit (MLL fit) is performed, where the decay parameters $\alpha_{\Lambda}$ and $\alpha_{\bar{\Lambda}}$ are fixed to 0.754 Lambda assuming $CP$ conservation in $\Lambda$ and $\bar{\Lambda}$ decays. In the fit, the likelihood function $\mathscr{L}$ is given by

Systematic uncertainties arise from the difference of detection efficiencies between data and simulations (tracking, $\pi^{0}$ and $\Xi^{0}(\bar{\Xi}^{0})$ reconstruction, 6C kinematic fit, $\pi^{0}\pi^{0}J/\psi$ background veto) as well as from the sideband technique, background from the continuum process $e^{+}e^{-}\to\Xi^{0}\bar{\Xi}^{0}$, the uncertainties of the $\Lambda\to p\pi^{-}$ decay parameters and the MLL fit method as listed in Table 2. Correction factors for tracking and $\pi^{0}$ reconstruction efficiency differences between data and simulations are evaluated on a control sample of $\psi(3686)\to\Xi^{0}\bar{\Xi}^{0}$ events, where one of the hyperon is fully reconstructed and one of the charged particles ($p$, $\bar{p}$, $\pi^{+}$ and $\pi^{-}$) or the $\pi^{0}$ from the second is not considered. The uncertainties arising from the correction procedure are evaluated by mean of 100 variation of the correction factors, following a Gaussian distribution with the nominal value as mean and the statistical uncertainty as width. The difference between the nominal results of the decay parameters and the mean values of those obtained through the variations is regarded as systematic uncertainty. The correction to the $\Xi^{0}$reconstruction efficiency differences is evaluated as in Ref. preXi0 ; BESIII:2016ssr ; BESIII:2019dve ; BESIII:2019cuv ; BESIII:2021aer ; BESIII:2020ktn ; BESIII:2021gca ; BESIII:2021ccp ; BESIII:2022mfx , and the same procedure is applied to estimate the systematic uncertainty. The 6C kinematic fit is sensitive to differences in the momentum resolution of the charged tracks between data and simulations. Corrections to the helix parameters of charged tracks are evaluated and applied in the measurement, and the difference between the spin polarization parameters obtained with and without the corrections is considered as the systematic uncertainty. The systematic uncertainty from vetoing $\psi(3686)\to\pi^{0}\pi^{0}J/\psi$ background is estimated by varying the range of the mass window requirement by 5 MeV/$c^{2}$. The largest difference is taken as the uncertainty. The uncertainties related to background contributions (sideband evaluation and continuum process) are evaluated by introducing a background term to the MLL function $S\to S^{\prime}=-\ln\mathscr{L}+\ln\mathscr{L}_{\rm{BKG}}$. The difference in the fit results is taken as the uncertainty. The possible bias introduced by the $\Lambda\to p\pi^{-}$ decay parameters are estimated by changing $a_{\Lambda}$ and $a_{\bar{\Lambda}}$ values reported in Ref. Lambda by $\pm 1\sigma$. The largest variation with respect to the central value is considered as the systematic uncertainty. To validate the fit procedures, an input-output check based on 300 pseudo-experiments is performed with the helicity amplitude formula Eq. (1). The polarization and the asymmetry decay parameters measured in this analysis are used as input in the formula. The number of events in each generated MC sample is 5000, and the check is performed independently 300 times. The difference between the input and output Gaussian fit values is taken as the systematic uncertainty caused by the fit method. Assuming all sources to be independent, the total systematic uncertainties in the measurement of $\alpha_{\bar{\Xi}^{0}}$, $\Delta\Phi$ and the decay asymmetry parameters via analyzing for $\psi(3686)\to\Xi^{0}\bar{\Xi}^{0}$ are determined as the sum in quadrature of the mentioned sources.

In summary, based on a data sample of $(448.1\pm 2.9)\times 10^{6}$ $\psi(3686)$ events collected with the BESIII detector, the $\Xi^{0}$ asymmetry parameters are measured with high precision using entangled quantum $\Xi^{0}$-$\bar{\Xi}^{0}$ pairs. The numerical results are summarized in Table 1. The polarization signal related with Eq. (3) is shown in Fig. 3. The value of $\alpha_{\psi(3686)}$ is measured to be $0.665\pm 0.086\pm 0.081$, which is consistent with the previous BESIII measurement preXi0 , and $\Delta\Phi=-0.050\pm 0.150\pm 0.020$ rad is measured for the first time and is significantly different from the one for $\Xi^{-}$ reported in Refs. Patrik ; zhangjingxu . The relative phase that is approximately zero implies an insignificant transverse polarization, which differs from the polarization observed in $\Lambda$ decays from $J/\psi$ decays, $\Sigma^{+}$ decays from both $J/\psi$ and $\psi(3686)$ decays, and $\Xi^{-}$ decays from both $J/\psi$ and $\psi(3686)$ decays Lambda ; Sigmap ; Patrik ; zhangjingxu . The asymmetry parameters $\alpha_{\Xi}$ and $\alpha_{\bar{\Xi}}$ are determined simultaneously for the first time. Previously only the product of $\alpha_{\Lambda}\cdot\alpha_{\Xi^{0}}$ product1 ; product2 was reported. The parameter $\phi_{\Xi^{0}}$ is measured more precisely compared with the value reported by the HBC group almost half a century ago HBC1 ; HBC2 ; HBC3 . In addition, the $\Xi^{0}$ hyperon $CP$ asymmetry parameters $A_{CP}^{\Xi}$, $\Delta\phi_{CP}^{\Xi}$, as summarized in Table 1, indicate no CPV effect at the current level of accuracy. It is expected that the test of CPV will reach sensitivities comparable to the SM prediction when a large data sample will be available at BESIII newdata , the upcoming PANDA experiment at FAIR panda , and the proposed Super Tau-Charm Factory projects in China and Russia superTC ; STCF .

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 R&D Program of China under Contracts Nos. 2020YFA0406400, 2020YFA0406300; National Natural Science Foundation of China (NSFC) under Contracts Nos. 12075107, 12247101, 11635010, 11735014, 11835012, 11905236, 11935015, 11935016, 11935018, 11961141012, 12022510, 12025502, 12035009, 12035013, 12047501, 12061131003, 12192260, 12192261, 12192262, 12192263, 12192264, 12192265, 12225509; the Chinese Academy of Sciences (CAS) Large-Scale Scientific Facility Program; the CAS Center for Excellence in Particle Physics (CCEPP); Joint Large-Scale Scientific Facility Funds of the NSFC and CAS under Contract No. U1832207; CAS Key Research Program of Frontier Sciences under Contracts Nos. QYZDJ-SSW-SLH003, QYZDJ-SSW-SLH040; 100 Talents Program of CAS; The Institute of Nuclear and Particle Physics (INPAC) and Shanghai Key Laboratory for Particle Physics and Cosmology; ERC under Contract No. 758462; European Union’s Horizon 2020 research and innovation programme under Marie Sklodowska-Curie grant agreement under Contract No. 894790; German Research Foundation DFG under Contracts Nos. 443159800, 455635585, Collaborative Research Center CRC 1044, FOR5327, GRK 2149; Istituto Nazionale di Fisica Nucleare, Italy; Ministry of Development of Turkey under Contract No. DPT2006K-120470; National Research Foundation of Korea under Contract No. NRF-2022R1A2C1092335; National Science and Technology fund; National Science Research and Innovation Fund (NSRF) via the Program Management Unit for Human Resources & Institutional Development, Research and Innovation under Contract No. B16F640076; Polish National Science Centre under Contract No. 2019/35/O/ST2/02907; Suranaree University of Technology (SUT), Thailand Science Research and Innovation (TSRI), and National Science Research and Innovation Fund (NSRF) under Contract No. 160355; The Royal Society, UK under Contract No. DH160214; The Swedish Research Council; U. S. Department of Energy under Contract No. DE-FG02-05ER41374.