Investigation of the $\Delta I=1/2$ rule and test of CP symmetry through the measurement of decay asymmetry parameters in $\Xi^{-}$ decays

The non-invariance of fundamental interactions under the combination of charge-conjugation (C) and parity (P) transformations is a necessary condition for baryogenesis Sakharov (1967), a process that dynamically generates the matter-antimatter asymmetry in the universe. Although the Standard Model (SM) accommodates CP violation with the Kobayashi-Maskawa phase Cabibbo (1963); Kobayashi and Maskawa (1973), it can only explain a matter-antimatter asymmetry that is at least ten orders of magnitude smaller than the observed value Barr et al. (1979). Additional sources of CP violation beyond the SM are expected to exist, and the weak hadronic transitions of hyperons are another place to search for such sources of CP violation Salone et al. (2022); Donoghue et al. (1986).

When two or more transition amplitudes interfere with each other, relative weak- and strong-phase contributions exist between them. For $K\to\pi\pi$ Alavi-Harati et al. (1999); Fanti et al. (1999), the CP violating weak phase comes from the interference between $S$-wave isospin $I=0$ ($A_{0}$) and isospin $I=2$ ($A_{2}$) amplitudes, which correspond to $\Delta I=1/2$ and $\Delta I=3/2$ transitions, respectively Cirigliano et al. (2012). The unforeseen large discrepancy between the real parts of the two isospin amplitudes, ${\rm Re}(A_{0})/{\rm Re}(A_{2})=22.45\pm 0.06$, known as the $\Delta I=1/2$ rule Amhis et al. (2022); Manzari et al. (2021), is a long-standing puzzle. Various theoretical models have been proposed to explain this large ratio, but the dual QCD approach Buras et al. (2014) and lattice QCD calculation Abbott et al. (2020) can only partially explain it. A comprehensive understanding of this rule is desirable.

The $\Delta I=1/2$ rule is also applicable in the decays of spin 1/2 hyperons Overseth and Pakvasa (1969); Olsen et al. (1970), which can be described in terms of its decay asymmetry parameters, $\alpha_{\rm Y}$ and $\phi_{\rm Y}$ Lee and Yang (1957). The ratio of decay asymmetry parameters for the two isospin decay modes $\Lambda\to n\pi^{0}$ and $\Lambda\to p\pi^{-}$, $\alpha_{\Lambda 0}/\alpha_{\Lambda-}$, is a sensitive probe to determine the contribution of $\Delta I=3/2$ transitions. In their absence, the ratio $\alpha_{\Lambda 0}/\alpha_{\Lambda-}$ is predicted to be unity Olsen et al. (1970). A recent BESIII result suggests that this might not be the case Ablikim et al. (2019). Further studies of the isospin amplitude in hyperon decays is required to rigorously test the $\Delta I=1/2$ rule.

Moreover, contrary to kaon decays, CP-violation in hyperon decays could arise from the interference between parity-conserving ($P$-wave) and parity-violating ($S$-wave) amplitudes with a CP-odd weak phase. The decay asymmetry parameters of hyperon are CP-odd and assuming CP conservation $\alpha_{\rm Y}=-\bar{\alpha}_{\rm Y}$ and $\phi_{\rm Y}=-\bar{\phi}_{\rm Y}$, where $\bar{\alpha}_{\rm Y}$ and $\bar{\phi}_{\rm Y}$ are decay asymmetry parameters for antihyperon $\bar{Y}$ Donoghue et al. (1986). CP symmetry, which is broken in the presence of non-negligible weak phase contributions, is gauged by the CP-observables $A_{\rm CP}$ and $\Delta\Phi_{\rm CP}$ Ablikim et al. (2022a):

$$A_{\rm CP}^{\rm Y}=\frac{\alpha_{\rm Y}+\bar{\alpha}_{\rm Y}}{\alpha_{\rm Y}-\bar{\alpha}_{\rm Y}}=-\tan\left(\delta_{P}-\delta_{S}\right)\tan\left(\xi_{P}-\xi_{S}\right),$$

(1)

$$\Delta\phi_{\rm CP}^{\rm Y}=\frac{\phi_{\rm Y}+\bar{\phi}_{\rm Y}}{2}=\frac{\langle\alpha\rangle}{\sqrt{1-\langle\alpha\rangle^{2}}}\cos\langle\phi\rangle\tan\left(\xi_{P}-\xi_{S}\right),$$

(2)

where $\langle\alpha\rangle=(\alpha_{{\rm Y}}-\bar{\alpha}_{{\rm Y}})/2$, $\langle\phi\rangle=(\phi_{{\rm Y}}-\bar{\phi}_{{\rm Y}})/2$, $\delta_{P}-\delta_{S}$ denotes the strong phase difference of the final-state interaction, and $\xi_{P}-\xi_{S}$ denotes the weak phase difference. Experimentally, the weak phase difference has been directly determined to be $(1.2\pm 3.4\pm 0.8)\times 10^{-2}$ rad Ablikim et al. (2022a) for the decay $\Xi^{-}\to\Lambda\pi^{-}$ using entangled $\Xi^{-}$ and $\bar{\Xi}^{+}$ produced at BESIII.

In this Letter, the process $J/\psi\to\Xi^{-}\bar{\Xi}^{+}\to\Lambda(p\pi^{-})\pi^{-}\bar{\Lambda}(\bar{n}\pi^{0})\pi^{+}$ is studied with $(10087\pm 44)\times 10^{6}$ $J/\psi$ events Ablikim et al. (2022b) collected by the BESIII detector. Benefiting from the transversely-polarized hyperons and the spin correlation between hyperon and anti-hyperons, various decay properties of $\Xi^{-}$ and $\Lambda$ are determined by an extended formalism that completely describes the angular distributions of the production and decay processes Perotti et al. (2019).

The design and performance of the BESIII detector are described in Refs. Ablikim et al. (2010); Huang et al. (2022). The corresponding simulation, analysis framework, and software are presented in Refs. Ping (2008); Deng et al. (2006); Li et al. (2006). Simulated Monte Carlo (MC) samples are produced with Geant4-based Agostinelli et al. (2003) MC software, which models the experimental conditions including the electron-positron collision, the decays of the particles, and the response of the detector. A sample of simulated events of generic $J/\psi$ decays, corresponding to the luminosity of data, is used to study the potential background reactions. To eliminate experimenter bias, the central values were blinded by using the hidden answer technique Klein and Roodman (2005) until all selections, fits, and uncertainty evaluations were finalized. Simulated signal and background samples are used to verify the analysis approaches and to study the systematic effects. Unless otherwise indicated, the charge-conjugate channel is implied throughout the text.

Four charged tracks are required in the multilayer drift chamber (MDC) within the range $\lvert\cos\theta\rvert<0.93$, where $\theta$ is the polar angle with respect to the $z$-axis, which is the symmetry axis of the MDC. Due to the non-overlapping momentum ranges of the proton and pions, a positively charged track with momentum greater than $0.32$ GeV/$c$ is assigned to be a proton, while a positively and two negatively charged tracks with momentum less than $0.30$ GeV/$c$ are assigned to be pions. The probability of misidentifying a proton for a $\pi^{+}$ is negligible. The sequential decay $\Xi^{-}\to\Lambda\pi^{-}\to p\pi^{-}\pi^{-}$ is reconstructed by a vertex fit Xu et al. (2010); Ablikim et al. (2022a), which takes into account the flight paths of the hyperons. The combination with the smallest $(M_{p\pi^{-}\pi^{-}}-m_{\Xi^{-}})^{2}+(M_{p\pi^{-}}-m_{\Lambda})^{2}$ is retained, where $M_{p\pi^{-}\pi^{-}(p\pi^{-})}$ denotes the invariant mass of $p\pi^{-}\pi^{-}$ ($p\pi^{-}$) and $m_{\Xi^{-}(\Lambda)}$ refers to the nominal mass of $\Xi^{-}$ ($\Lambda$) Workman and Others (2022). The probability of a $\pi^{-}$ from the $\Lambda$ and $\Xi^{-}$ decays being wrongly assigned is found to be $0.1\%$, which is negligible. The candidate events are required to satisfy $\lvert M_{p\pi^{-}}-m_{\Lambda}\rvert<11$ MeV/$c^{2}$ and $\lvert M_{p\pi^{-}\pi^{-}}-m_{\Xi^{-}}\rvert<11$ MeV/$c^{2}$. The decay lengths of the $\Xi^{-}$ and $\Lambda$ are calculated in the vertex fit and required to be positive. To improve the resolution and minimize the discrepancy between data and MC simulation, the polar angle $\theta_{\Xi^{-}}$ of the reconstructed $\Xi^{-}$ in the $e^{+}e^{-}$ center-of-mass frame is required to satisfy $\lvert\cos\theta_{\Xi^{-}}\rvert<0.84$.

At least two photon candidates in the electromagnetic calorimeter (EMC) are required. A photon candidate should have energy greater than $25$ MeV in the barrel region ($\lvert\cos\theta\rvert<0.8$) or $50$ MeV in the end-cap region ($0.86<\lvert\cos\theta\rvert<0.92$). For antiprotons, which may annihilate in the detector, photon candidates must be separated from charged tracks with an opening angle greater than $20^{\circ}$, while for other tracks the angle must be greater than $10^{\circ}$. To suppress electronic noise and showers unrelated to the event, photon candidates are required to have the EMC time difference from the event start time within $[0,700]$ ns. To veto the showers from antineutron interactions in the EMC, the photon candidates should be separated from the direction of the $\Xi^{-}\pi^{+}$ recoiling system with an opening angle greater than $15^{\circ}$. A boosted decision tree (BDT) classifier Therhaag (2010) is constructed based on the shower shape variables to discriminate a signal photon from a noise shower. The shower shape variables include the deposited energy, number of hits, second and Zernike moments, and deposition shape clu . The signal efficiency of the BDT is $90\%$, and $55\%$ of the background is rejected. The $\pi^{0}$ candidates are reconstructed from a pair of photons by constraining their invariant mass to the $\pi^{0}$ nominal mass, and the corresponding $\chi^{2}_{1C}$ is required to be less than 25. Due to combinatorial effects, it is possible to have more than one unique $\pi^{0}$ candidate in a single event.

A kinematic fit under the hypothesis of $J/\psi\to\Xi^{-}\pi^{+}\bar{n}\gamma\gamma$ is performed imposing energy-momentum conservation and constraining the invariant masses of $\gamma\gamma$ and $\gamma\gamma\bar{n}$ to the nominal masses of $\pi^{0}$ and $\bar{\Lambda}$, respectively. The kinematics of the $\Xi^{-}$ are obtained from the above vertex fit. The antineutron is treated as a missing particle with unknown mass. The fit is performed for each $\pi^{0}$ candidate. If there is more than one $\pi^{0}$ candidate, the candidate with the smallest $\chi^{2}$ is retained, and $\chi^{2}<200$ is required. The invariant mass of $\bar{n}\gamma\gamma\pi^{+}$ is required to satisfy $\lvert M_{\bar{n}\gamma\gamma\pi^{+}}-m_{\bar{\Xi}^{+}}\rvert<11$ MeV/$c^{2}$. Since all other final state particles are detected, the kinematic fit allows for the reconstruction of the four momentum of antineutron. The signal is identified by the antineutron’s missing mass, as shown in Fig. 1 with a prominent signal peak in the antineutron vicinity and a low level background.

Detailed studies are performed with MC simulation and data in the $\Xi^{-}$ and $\bar{\Xi}^{+}$ sideband regions to evaluate the potential backgrounds. The dominant background, referred to as combinatorial background, is from signal events with misreconstructed $\pi^{0}$ candidates, which does not peak in the antineutron missing mass distribution. The remaining background sources are classified into two categories Zhou et al. (2021): resonant background that contains $\Xi^{-}\bar{\Xi}^{+}$ intermediate states, such as, $J/\psi\to\gamma\eta_{c}\to\gamma\Xi^{-}\bar{\Xi}^{+}\to\gamma\Lambda(\to p\pi^{-})\pi^{-}\bar{\Lambda}(\bar{n}\pi^{0})\pi^{+}$ and $J/\psi\to\Xi^{-}\bar{\Xi}^{+}\to\Lambda(\to p\pi^{-})\pi^{-}\bar{\Lambda}(\bar{p}\pi^{+})\pi^{+}$; non-resonant background without $\Xi^{-}\bar{\Xi}^{+}$ intermediate states. The decay processes of resonant backgrounds are well understood, and the corresponding contributions are evaluated by MC simulation, which are generated according to the helicity amplitudes and weighted according to the branching fractions Workman and Others (2022). MC simulation shows that the distributions of $M_{p\pi^{-}\pi^{-}}$ and $M_{\bar{n}\gamma\gamma\pi^{+}}$ of non-resonant background are almost flat. Therefore, the corresponding contribution can be evaluated from the $\Xi^{-}$ and $\bar{\Xi}^{+}$ sideband regions.

Signal yields are obtained from an unbinned maximum likelihood fit of the missing mass distribution. In the fit shown in Fig. 1, the signal is described by an MC-simulated shape convolved by a Gaussian function accounting for the resolution difference between data and MC simulation. The combinatorial background is described by the signal MC sample, and it is parameterized by a product of an ARGUS function Albrecht et al. (1990) and a cubic function. Fixing both the magnitude and shape, the resonant and non-resonant backgrounds are described with the MC simulation and data events in the sideband region, respectively. The normalization of the background and the definition of the sideband region are shown in Sec. 2 of the Supplemental Material sup . The fit yields $143973\pm 414$ signal events and a purity of $91.2\%$ in the mass range $[0.925,0.955]$ GeV/$c^{2}$. The same procedure is performed for the charge-conjugate process and results in $123208\pm 382$ signal events and a purity of $91.0\%$.

The joint angular amplitude of the full decay chain can be written in a modular form as

$$\mathcal{W}(\xi;\omega)=\sum_{\mu,\nu=0}^{3}C_{\mu\nu}\sum_{\mu^{\prime},\nu^{\prime}=0}^{3}a_{\mu\mu^{\prime}}^{\Xi}a_{\nu\nu^{\prime}}^{\Xi}a_{\mu^{\prime}0}^{\Lambda}a_{\nu^{\prime}0}^{\bar{\Lambda}}.$$

(3)

Here $C_{\mu\nu}$ is a $4\times 4$ real-valued spin density matrix describing the spin configuration of the entangled $\Xi^{-}\bar{\Xi}^{+}$ pair, $a^{{\rm Y}}_{\mu\nu}$ is also a $4\times 4$ real-valued matrix representing the propagation of the spin density matrix in the decays of a spin 1/2 hyperon into a spin 1/2 baryon and a pseudoscalar, ${\rm Y}\to{\rm B}\pi$. Therefore, the distribution of the nine helicity angles $\xi=(\theta_{\Xi},\theta_{\Lambda},\phi_{\Lambda},\theta_{\bar{\Lambda}},\phi_{\bar{\Lambda}},\theta_{p},\phi_{p},\theta_{\bar{n}},\phi_{\bar{n}})$ is determined by eight global parameters $\omega=(\alpha_{J/\psi},\Delta\Phi_{J/\psi},\alpha_{\Xi},\phi_{\Xi},\bar{\alpha}_{\Xi},\bar{\phi}_{\Xi},\alpha_{\Lambda-},\bar{\alpha}_{\Lambda 0})$. In this analysis, ${\rm Y}\to{\rm B}\pi$ stands for $\Xi^{-}\to\Lambda\pi^{-}$, $\Lambda\to p\pi^{-}$ and $\bar{\Lambda}\to\bar{n}\pi^{0}$. The distribution of the helicity angle $\theta_{p}$ in the $\Lambda$ rest frame is written as

$$\frac{\mathrm{d}N}{\mathrm{d}\cos\theta_{p}}\propto 1+\alpha_{\Lambda-}\alpha_{\Xi}\cos\theta_{p}$$

(4)

by integrating over the remaining eight helicity angles. The formalism of the full angular distribution and the definition of the reference system are discussed in detail in Ref. Ablikim et al. (2022a).

A simultaneous fit on the joint angular distribution is carried out with the production parameters, $\alpha_{J/\psi}$ and $\Delta\Phi_{J/\psi}$, and decay asymmetry parameters of $\Xi^{-}$ shared between the two charge-conjugate channels. For each channel, the probability distribution function of the eight unknown parameters $\omega$ can be defined in terms of the helicity angles $\xi$

$$\mathcal{P}(\xi;\omega)=\mathcal{W}(\xi;\omega)\varepsilon(\xi)/\mathcal{N}(\omega),$$

(5)

where the normalization factor $\mathcal{N}(\omega)$ is calculated with $\mathcal{N}(\omega)=\frac{1}{M}\sum_{j=1}^{M}\frac{\mathcal{W}(\xi_{j};\omega)}{\mathcal{W}(\xi_{j};\omega_{\rm gen})}$ by a signal MC sample generated with parameters $\omega_{\rm gen}$. The sum runs over all events in the generated sample $M$, which is chosen to be thirty times the yield obtained in data after the full selection. The log-likelihood function for $N$ observed events is

$$\small\mathcal{S}=-\mathcal{G}\left(\sum_{i=1}^{N}\ln\mathcal{P}(\xi_{i};\omega)-\sum_{j}{\sc w_{j}}\sum_{i}^{N_{j}^{\rm bkg}}\ln\mathcal{P}(\xi_{i};\omega)\right),$$

(6)

where the second term in brackets with $j$ from one to three represents the three different sources of background remaining in the final event sample. Their contributions are evaluated with the corresponding MC samples or data events in the sideband region and their associated weight factors $w_{j}$. The global factor, $\mathcal{G}=(N-\sum_{j}N_{j}^{\rm bkg}\times\sc{w_{j}})/(N+\sum_{j}N_{j}^{\rm bkg}\times\sc{w_{j}}^{2})$ , corrects for the statistical uncertainties in the weighted likelihood fit Langenbruch (2022).

The $\mathcal{S}$ function is minimized using Minuit2 Hatlo et al. (2005) to determine the production and decay asymmetry parameters $\omega$. The results from the fit, as shown in Table 1, are consistent with previous measurements, but with improved precision. In particular, $\alpha_{\Lambda 0}$ is almost the same in magnitude and opposite in sign as $\bar{\alpha}_{\Lambda 0}$, and its precision is improved by a factor of four over previous measurements.

If CP is conserved, the product of the decay asymmetry parameters $\alpha_{\Lambda-}\alpha_{\Xi}$ and $\alpha_{\Lambda+}\bar{\alpha}_{\Xi}$ should be equal to each other, and the ratios of helicity angular distributions for different nucleons in the final states, $R(\cos\theta_{p},\cos\theta_{\bar{p}})=(1+\alpha_{\Lambda-}\alpha_{\Xi}\cos\theta_{p})/(1+\alpha_{\Lambda+}\bar{\alpha}_{\Xi}\cos\theta_{\bar{p}})$ and $R(\cos\theta_{n},\cos\theta_{\bar{n}})=(1+\alpha_{\Lambda 0}\alpha_{\Xi}\cos\theta_{n})/(1+\bar{\alpha}_{\Lambda 0}\bar{\alpha}_{\Xi}\cos\theta_{\bar{n}})$, are flat and equal to unity. In a similar way, if there is no $\Delta I=3/2$ transition in $\Lambda$ decay, $\alpha_{\Lambda-}$ should be equal to $\alpha_{\Lambda 0}$ and the ratios, $R(\cos\theta_{n},\cos\theta_{p})=(1+\alpha_{\Lambda 0}\alpha_{\Xi}\cos\theta_{n})/(1+\alpha_{\Lambda-}\alpha_{\Xi}\cos\theta_{p})$ and $R(\cos\theta_{\bar{n}},\cos\theta_{\bar{p}})=(1+\bar{\alpha}_{\Lambda 0}\bar{\alpha}_{\Xi}\cos\theta_{\bar{n}})/(1+\alpha_{\Lambda+}\bar{\alpha}_{\Xi}\cos\theta_{\bar{p}})$, are also flat and equal to unity. The accuracy of the CP symmetry and the $\Delta I=1/2$ rule tests can be improved by using the isospin average for $R_{1}$, $R_{1}=(1+\alpha_{\Lambda}\alpha_{\Xi}\cos\theta)/(1+\bar{\alpha}_{\Lambda}\bar{\alpha}_{\Xi}\cos\theta)$, where $\cos\theta$ stands for the helicity angle of nucleon, $\alpha_{\Lambda}$ is defined as $(2\alpha_{\Lambda-}+\alpha_{\Lambda 0})/3$, and the average of the decay symmetry parameters of hyperon and antihyperon for $R_{2}$, $R_{2}=(1+\langle\alpha_{\Lambda 0}\rangle\langle\alpha_{\Xi}\rangle\cos\theta)/(1+\langle\alpha_{\Lambda-}\rangle\langle\alpha_{\Xi}\rangle\cos\theta)$.

To illustrate the tests of CP symmetry and the $\Delta I=1/2$ rule, four ratios of the helicity angular distributions for different nucleons in the final states are shown in Fig. 2 by dots with error bars. $R_{1}$ and $R_{2}$ with parameters from Table 1 are also presented in Fig. 2. The ratios obtained by fitting the events in bins of $\cos\theta$ are in good agreement with the global curves obtained for $R_{1}$ and $R_{2}$. The nearly flat distribution of $R_{1}$ is consistent with CP conservation. The sloping distribution of $R_{2}$ indicates the existence of the contribution of $\Delta I=3/2$ transition in $\Lambda$ decay.

The systematic uncertainties are split into different categories: reconstruction and event selection of the signal candidates, the uncertainties related to the background contributions, and the effects which arise from the final fit procedure. The uncertainty of the $\pi^{0}$ reconstruction is investigated by studying the decay $J/\psi\to\Sigma^{+}(p\pi^{0})\pi^{-}\bar{\Lambda}(\bar{p}\pi^{+})+c.c.$ as it has a similar final state topology and decay length as the signal. The systematic uncertainty from $\pi^{\pm}$ reconstruction is investigated by using a control sample of $J/\psi\to\Xi^{-}\bar{\Xi}^{+}\to\Lambda(p\pi^{-})\pi^{-}\bar{\Lambda}(\bar{p}\pi^{+})\pi^{+}+c.c.$. The systematic uncertainties related to the selection criteria (the decay points and invariant masses of $\Lambda$ and $\Xi^{-}$, the polar angle of $\Xi^{-}$, the missing mass and the $\chi^{2}$ of the kinematic fit) are studied by varying their required values around the nominal ones and repeating the fit. The uncertainty due to the combinatorial background is determined by both smearing the parameters of model and varying its yield from the fit to the missing mass distribution by $\pm 1\sigma$. The uncertainties associated with the resonant backgrounds, which are propagated from the uncertainties in branching fractions, number of $J/\psi$ events and MC sample statistics, are also evaluated by varying the background yield by $\pm 1\sigma$. In the case of non-resonant background, the fit is repeated without this background component, and the deviation from the nominal fit is taken as the systematic uncertainty. To estimate the systematic uncertainty of the fit procedure, 1000 sets of toy MC samples are generated with the parameters from Table 1. Each set is fitted to obtain the distribution of the output parameters. The average values of the difference between the input and output parameters and statistical errors of the average differences are regarded as systematic uncertainties. More details can be found in the Supplemental Material sup .

In summary, the decay asymmetry parameters listed in Table 1 are simultaneously determined from the process $J/\psi\to\Xi^{-}\bar{\Xi}^{+}\to\Lambda(p\pi^{-})\pi^{-}\bar{\Lambda}(\bar{n}\pi^{0})\pi^{+}$ and its charge-conjugate channel with $(10087\pm 44)\times 10^{6}$ $J/\psi$ events collected by the BESIII detector. Using Eq. 1 and Eq. 2, the CP observables $A_{\rm CP}^{\Xi}$ and $\Delta\phi_{\rm CP}^{\Xi}$ for $\Xi^{-}$ decay, as well as $A_{\rm CP}^{-}=(\alpha_{\Lambda-}+\alpha_{\Lambda+})/(\alpha_{\Lambda-}-\alpha_{\Lambda+})$ and $A_{\rm CP}^{0}=(\alpha_{\Lambda 0}+\bar{\alpha}_{\Lambda 0})/(\alpha_{\Lambda 0}-\bar{\alpha}_{\Lambda 0})$ of the charged and neutral $\Lambda$ decays, are obtained from the corresponding decay asymmetry parameters and correlations. $A_{\rm CP}^{\Xi}$ and $\Delta\phi_{\rm CP}^{\Xi}$ are measured with world-leading precision, and $A_{{\it CP}}^{0}$ is measured for the first time. The correlations $\rho(\alpha_{\Lambda-},\alpha_{\Lambda+})$ and $\rho(\alpha_{\Lambda 0},\bar{\alpha}_{\Lambda 0})$ measured from two charge-conjugate channels are negligible. The precise CP symmetry test of the $\Lambda$ decay is conducted with its isospin averages, $A_{\rm CP}^{\Lambda}=(2A_{{\it CP}}^{-}+A_{{\it CP}}^{0})/3$, which improves the sensitivity of the CP symmetry test by 20% compared to the individual tests for each isospin decay mode. The strong phase and weak phase differences of $\Xi^{-}\to\Lambda\pi^{-}$, derived from Eq. 1 and Eq. 2, are both consistent with previous BESIII results Ablikim et al. (2022a). The strong phase difference is also in agreement with the HyperCP measurement Holmstrom et al. (2004). The CP symmetry is conserved in the decay of $\Xi^{-}$ and $\Lambda$ within the current precision. The theoretical predictions within the Standard Model Tandean and Valencia (2003); He et al. (2022) are $0.5\times 10^{-5}\leqslant(A_{\rm CP}^{\Xi})_{\mathrm{SM}}\leqslant 6\times 10^{-5}$, $-3.8\times 10^{-4}\leqslant(\xi_{P}-\xi_{S})_{\mathrm{SM}}\leqslant-0.3\times 10^{-4}$ and $-3\times 10^{-5}\leqslant(A_{CP}^{\Lambda})_{\mathrm{SM}}\leqslant 3\times 10^{-5}$.

The ratios of $\alpha_{\Lambda 0}/\alpha_{\Lambda-}$ and $\bar{\alpha}_{\Lambda 0}/\alpha_{\Lambda+}$ deviate from unity by more than $5$ standard deviations, which signifies the existence of the $\Delta I=3/2$ transition in both $\Lambda$ and $\bar{\Lambda}$ decays for the first time. Using the averages of the ratio $\langle\alpha_{\Lambda 0}\rangle/\langle\alpha_{\Lambda-}\rangle=0.870\pm 0.012^{+0.011}_{-0.010}$ with combinations of the decay rates $\Gamma(\Lambda\to p\pi^{-})$, $\Gamma(\Lambda\to n\pi^{0})$ Workman and Others (2022) and the $N$-$\pi$ scattering phase shift Hoferichter et al. (2016), the ratio of $\Delta I=3/2$ to $\Delta I=1/2$ transitions in $S$-wave is determined to be $S_{1}/S_{3}=28.4\pm 1.3^{+1.1}_{-1.0}\pm 3.9$, while in $P$-wave $P_{1}/P_{3}=-13.0\pm 1.4^{+1.1}_{-1.2}\pm 0.7$ according to Ref. Salone et al. (2022), where the first uncertainties are statistical, the second systematic and the third from the input parameters. The ratio in $S$-wave is consistent with ${\rm Re}(A_{0})/{\rm Re}(A_{2})$ in $K\to\pi\pi$ within the uncertainty, while the ratio in $P$-wave is measured for the first time and found different from that in $S$-wave. This measurement provides a constraint for lattice QCD Abbott et al. (2020) and dual QCD Buras et al. (2014) approach to understand the $\Delta I=1/2$ rule.

The authors thank Professor X.G. He and Professor X. Feng for helpful discussions. The BESIII Collaboration thanks the staff of BEPCII and the IHEP computing center and the supercomputing center of the University of Science and Technology of China (USTC) for their strong support. This work is supported in part by National Key R&D Program of China under Contracts Nos. 2020YFA0406300, 2020YFA0406400; National Natural Science Foundation of China (NSFC) under Contracts Nos. 11635010, 11735014, 11835012, 11935015, 11935016, 11935018, 11961141012, 12022510, 12025502, 12035009, 12035013, 12061131003, 12192260, 12192261, 12192262, 12192263, 12192264, 12192265, 12221005, 12225509, 12235017, 12122509, 12105276, 11625523; 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, U2032111, U1732263, U1832103; 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; 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. 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 of Mongolia; National Science Research and Innovation Fund (NSRF) via the Program Management Unit for Human Resources & Institutional Development, Research and Innovation of Thailand under Contract No. B16F640076; Polish National Science Centre under Contract No. 2019/35/O/ST2/02907; The Swedish Research Council; U. S. Department of Energy under Contract No. DE-FG02-05ER41374; Olle Engkvist Foundation under Contract No. 200-0605 and Lundström-Åman Foundation.