Small violations of CP symmetry are predicted by the standard model6,7 and are a well established phenomenon in weak decays of mesons. However, the mechanisms of the standard model are too specific to yield effects of a size that can explain the observed matter–antimatter asymmetry of the Universe8,9. Therefore, CP tests can be considered a promising area to search for physics beyond the standard model10,11. So far, no CP-violating effects beyond the standard model have been observed in the baryon sector12.

In general, CP symmetry is tested by comparing the decay patterns of a particle to those of its antiparticle. Many CP-symmetry tests in hadron decays rely on strong interactions of the final particles to reveal the signal. This strategy is applied in the determination of the ratio ε′/ε, quantifying the difference between the two-pion decay rates of the two weak eigenstates of neutral kaons. The ε′/ε measurement constitutes the only observation of direct CP violation for light strange hadrons13,14 and provides the most stringent test of contributions beyond the standard model in strange quark systems15. This strategy, however, comes at a price: it is difficult to disentangle, in a model-independent way, the contributions from weak interactions or processes beyond the standard model from those of strong processes. Approaches that do not rely on strong interactions require that the kaon decay into four final-state particles16.

Baryons provide additional information through spin measurements. Known examples involving three-body decays are spin correlations and polarization in nuclear and neutron β decays17. Sequential two-body decays of entangled multi-strange baryon–antibaryon pairs provide another, hitherto unexplored, diagnostic tool to separate the strong and the weak phases.

In this work we explore spin correlations in weak two-body decays of spin-½ baryons. The spin direction of the parent baryon manifests itself in the momentum direction of the daughter particle, enabling straightforward experimental access to the spin properties. Spin-½ baryon decays are described by a parity-conserving (P-wave) and a parity-violating (S-wave) amplitude, quantified in terms of the decay parameters αY, βY and γY (ref. 18). The Y refers to the decaying parent hyperon (for example, Λ or Ξ). These parameters are constrained by the relation \({\alpha }_{Y}^{2}+{\beta }_{Y}^{2}+{\gamma }_{Y}^{2}=1\). By defining the parameter ϕY according to

$${\beta }_{Y}=\sqrt{1-{\alpha }_{Y}^{2}}\,\sin \,{\phi }_{Y},\,{\gamma }_{Y}=\sqrt{1-{\alpha }_{Y}^{2}}\,\cos \,{\phi }_{Y},$$

the decay is completely described by two independent parameters αY and ϕY. In the standard experimental approach3,4,19,20,21, the initial baryon is produced in a well defined spin polarized state, which allows access to the decay parameters through the angular distribution of the final-state particles. For sequentially decaying baryons, for example, the decay of the double-strange Ξ baryon into Λπ, two effects are possible: 1) a polarized Ξ transfers its polarization PΞ to the daughter Λ; 2) a longitudinal component of the daughter Λ polarization is induced by the Ξ decay, even if the Ξ polarization has no component in this direction. In a reference system with the \(\hat{{\bf{z}}}\) axis along the Λ momentum in the Ξ rest frame and the \(\hat{{\bf{y}}}\) axis along \({{\bf{P}}}_{\varXi }\times \hat{{\bf{z}}}\), the Λ polarization vector is given by18

$$\begin{array}{c}{{\bf{P}}}_{\varLambda }\cdot \hat{{\bf{z}}}=\frac{{\alpha }_{\varXi }+{{\bf{P}}}_{\varXi }\cdot \hat{{\bf{z}}}}{1+{\alpha }_{\varXi }{{\bf{P}}}_{\varXi }\cdot \hat{{\bf{z}}}}\,,\\ {{\bf{P}}}_{\varLambda }\times \hat{{\bf{z}}}={{\bf{P}}}_{\varXi }\sqrt{1-{\alpha }_{\varXi }^{2}}\frac{\sin \,{\phi }_{\varXi }\hat{{\bf{x}}}+\,\cos \,{\phi }_{\varXi }\hat{{\bf{y}}}}{1+{\alpha }_{\varXi }{{\bf{P}}}_{\varXi }\cdot \hat{{\bf{z}}}}\,,\end{array}$$

as illustrated in Fig. 1. This means that the longitudinal (\(\hat{{\bf{z}}}\)) component depends on αΞ, and the transversal components are rotated by the angle ϕΞ with respect to the Ξ polarization.

Fig. 1: Illustration of the polarization vectors of Ξ and Λ in relation to the decay parameters α, β and γ of the Ξ → Λπ decay.
figure 1

The Λ polarization \({{\mathscr{P}}}_{{{\Lambda }}^{-}}\) has a component in the longitudinal as well as the transverse direction, where the former (\(\hat{z}\)) is defined by the Λ momentum. The longitudinal component depends on the Λ emission angle and arises from the transferred Ξ polarization \({{\mathscr{P}}}_{{{\Xi }}^{-}}\) combined with the decay parameter α. The remaining Ξ polarization is transferred to the transverse components according to \({\beta }_{\varXi }{{\mathscr{P}}}_{{\varXi }^{-}}\) (\(\hat{x}\)) and \({\gamma }_{\varXi }{{\mathscr{P}}}_{\varXi }\) (\(\hat{y}\)). Quarks: d, down; s, strange; u, up; \(\bar{u}\), antiup.

The decay parameter αΞ appears explicitly in the angular distribution of the direct decay Ξ → Λπ, whereas the sequential decay distribution of the daughter Λ depends on both αΛ and ϕΞ. CP symmetry implies that the baryon decay parameters α and ϕ equal those of the antibaryon \(\bar{\alpha }\) and \(\bar{\phi }\) but with opposite sign. Hence, CP violation can be quantified in terms of the observables

$${A}_{{\rm{CP}}}^{Y}=\frac{{\alpha }_{Y}+{\bar{\alpha }}_{Y}}{{\alpha }_{Y}-{\bar{\alpha }}_{Y}},\,\Delta {\phi }_{{\rm{CP}}}=\frac{{\phi }_{Y}+{\bar{\phi }}_{Y}}{2}.$$

CP violation can only be observed if there is interference between CP-even and CP-odd terms in the decay amplitude. Because the decay amplitude for Ξ → Λπ consists of both a P-wave and an S-wave part, the leading-order contribution to the CP asymmetry, \({A}_{{\rm{CP}}}^{{\Xi }}\), can be written as

$${A}_{{\rm{CP}}}^{{\Xi }}\approx -\,\tan ({\delta }_{{\rm{P}}}-{\delta }_{{\rm{S}}})\tan ({\xi }_{{\rm{P}}}-{\xi }_{{\rm{S}}}),$$

where tan(δP − δS) = β/α denotes the strong-phase difference of the final-state interaction between the Λ and π from the Ξ decay. CP-violating effects would manifest themselves in a nonzero weak-phase difference ξP − ξS (refs. 22,23,24), an observable that is complementary to the kaon decay parameter ε′ (refs. 13,14,25) because the latter only involves an S-wave. The strong-phase difference can be extracted from the ϕΞ parameter, and is found to be small3,26: (−0.037 ± 0.014). Hence, CP-violating signals in \({A}_{{\rm{CP}}}^{{\Xi }}\) are strongly suppressed and difficult to interpret in terms of the weak-phase difference.

An independent CP-symmetry test in Ξ → Λπ is provided by determining the value of ΔϕCP. At leading order, this observable is related directly to the weak-phase difference:

$${({\xi }_{{\rm{P}}}-{\xi }_{{\rm{S}}})}_{{\rm{LO}}}=\frac{\beta +\bar{\beta }}{\alpha -\bar{\alpha }}\approx \frac{\sqrt{1-{\langle \alpha \rangle }^{2}}}{\langle \alpha \rangle }\varDelta {\phi }_{{\rm{CP}}},$$

where \(\langle \alpha \rangle =(\alpha -\bar{\alpha })/2\), and can be measured even if δP = δS. The absence of a strong suppression factor therefore improves the sensitivity to CP-violation effects by an order of magnitude with respect to that of the \({A}_{{\rm{CP}}}^{{\Xi }}\) observable22,23. To measure ΔϕCP using the standard polarimeter technique from refs. 21,28 requires beams of polarized Ξ and \({\bar{{\Xi }}}^{+}\). In such experiments the precision is limited by the magnitude of the polarization and the accuracy of the polarization determination, which in turn is sensitive to asymmetries in the production mechanisms27. In fact, no experiment with a polarized \({\bar{{\Xi }}}^{+}\) has been performed, and the polarization of the Ξ beams were below 5% (ref. 3). Here we present an alternative approach, in which the baryon–antibaryon pair is produced in a spin-entangled CP eigenstate and all decay sequences are analysed simultaneously.

To the best of our knowledge, no direct measurements of any of the asymmetries defined in equation (3) have been performed for the Ξ baryon. The HyperCP experiment28, designed for the purpose of CP tests in baryon decays, used samples of around 107–108 Ξ and \({\bar{{\Xi }}}^{+}\) events to determine the products αΞαΛ and \({\bar{\alpha }}_{{\Xi }}{\bar{\alpha }}_{{\Lambda }}\). From these measurements, the sum \({A}_{{\rm{CP}}}^{{\Lambda }}+{A}_{{\rm{CP}}}^{{\Xi }}\) was estimated to be (0.0 ± 5.5 ± 4.4) × 10−4, where the first uncertainty is statistical and the second systematic. In addition to the aforementioned problem of the smallness of ϕΞ, which limits the sensitivity of \({A}_{{\rm{CP}}}^{{\Xi }}\) to CP violation, an observable defined as the sum of asymmetries comes with other drawbacks: if \({A}_{{\rm{CP}}}^{{\Lambda }}\) and \({A}_{{\rm{CP}}}^{{\Xi }}\) have opposite signs, the sum could be consistent with zero even in the presence of CP-violating effects. A precise interpretation therefore requires an independent measurement of \({A}_{{\rm{CP}}}^{{\Lambda }}\) with matching precision. The most precise result so far is a recent BESIII measurement4 where \({A}_{{\rm{CP}}}^{{\Lambda }}\) was found to be (−6 ± 12 ± 7) × 10−3. Furthermore, ref.4 revealed a 17% disagreement with previous measurements on the αΛ parameter26, a result that rapidly gained some support from a re-analysis of CLAS data5. Although the CLAS result is in better agreement with BESIII than with the Particle Data Group value from 2018 and earlier, there is a discrepancy between the CLAS and BESIII results that needs to be understood. This is particularly important because many physics quantities from various fields depend on the parameter αΛ. Examples include baryon spectroscopy, heavy-ion physics and hyperon-related studies at the Large Hadron Collider29,30,31,32,33,34.

In this work we apply a newly designed method2,35 to study entangled, sequentially decaying baryon–antibaryon pairs in the process \({e}^{+}{e}^{-}\to J/\psi \to {\varXi }^{-}{\bar{\varXi }}^{+}\). This approach enables a direct measurement of all weak decay parameters of the Ξ → Λπ, Λ →  decay, and the corresponding parameters of the \({\bar{\varXi }}^{+}\) . The production and multi-step decays can be described by nine kinematic variables, here expressed as the helicity angles \({\boldsymbol{\xi }}=(\theta ,{\theta }_{\bar{{\Lambda }}},{\varphi }_{\varLambda },{\theta }_{\bar{\varLambda }},{\varphi }_{\bar{\varLambda }},{\theta }_{p},{\varphi }_{p},{\theta }_{\bar{p}},{\varphi }_{\bar{p}})\). The first, θ, is the Ξ scattering angle with respect to the e+ beam in the centre-of-momentum system of the reaction. The angles θΛ and φΛ (\({\theta }_{\bar{{\Lambda }}}\), \({\varphi }_{\bar{{\Lambda }}}\)) are defined by the Λ (\(\bar{{\Lambda }}\)) direction in a reference system denoted \({ {\mathcal R} }_{{\Xi }}\) (\({ {\mathcal R} }_{\bar{{\Xi }}}\)), where Ξ (\({\bar{{\Xi }}}^{+}\)) is at rest and where the \(\hat{z}\) axis points in the direction of the Ξ (\({\bar{{\Xi }}}^{+}\)) in the centre-of-momentum system. The \(\hat{y}\) axis is normal to the production plane. The angles θp and φp (\({\theta }_{\bar{p}}\) and \({\varphi }_{\bar{p}}\)) give the direction of the proton (antiproton) in the Λ (\(\bar{{\Lambda }}\)) rest system, denoted \({ {\mathcal R} }_{{\Lambda }}\) (\({ {\mathcal R} }_{\bar{{\Lambda }}}\)), with the \(\hat{z}\) axis pointing in the direction of the Λ (\(\bar{{\Lambda }}\)) in the \({ {\mathcal R} }_{{\Xi }}\) (\({ {\mathcal R} }_{\bar{{\Xi }}}\)) system and the \(\hat{y}\) axis normal to the plane spanned by the direction of the Ξ (\({\bar{{\Xi }}}^{+}\)) and the direction of the Λ (\(\bar{{\Lambda }}\)). The structure of the nine-dimensional angular distribution is determined by eight global (that is, independent of the Ξ scattering angle) parameters \({\boldsymbol{\omega }}=({\alpha }_{\psi },\varDelta {\Phi },{\alpha }_{{\Xi }},{\phi }_{{\Xi }},{\bar{\alpha }}_{{\Xi }},{\bar{\phi }}_{{\Xi }},{\alpha }_{{\Lambda }},{\bar{\alpha }}_{{\Lambda }})\), and can be written in a modular form as35:

$${\mathscr{W}}({\boldsymbol{\xi }};{\boldsymbol{\omega }})=\mathop{\sum }\limits_{\mu ,\nu =0}^{3}{C}_{\mu \nu }\mathop{\sum }\limits_{\mu {\prime} \nu {\prime} =0}^{3}{a}_{\mu \mu {\prime} }^{{\Xi }}{a}_{\nu \nu {\prime} }^{\bar{{\Xi }}}{a}_{\mu {\prime} 0}^{{\Lambda }}{a}_{\nu {\prime} 0}^{\bar{{\Lambda }}}.$$

Here Cμν(θ; αψ, ΔΦ) is a 4 × 4 spin density matrix, defined in the aforementioned reference systems \({ {\mathcal R} }_{{\Xi }}\) and \({ {\mathcal R} }_{\bar{{\Xi }}}\), describing the spin configuration of the entangled hyperon–antihyperon pair. The parameters αψ and ΔΦ are related to two production amplitudes, where αψ parameterizes the Ξ angular distribution. The ΔΦ is the relative phase between the two production amplitudes (in the so-called helicity representation)36 and governs the polarization Py of the produced Ξ and \({\bar{{\Xi }}}^{+}\) as well as their spin correlations Cij. The matrix elements are related to Py = Py(θ) and Cij = Cij(θ) in the following way:

$${C}_{\mu \nu }=(1+{\alpha }_{\psi }{\cos }^{2}\theta )(\begin{array}{cccc}1 & 0 & {P}_{y} & 0\\ 0 & {C}_{xx} & 0 & {C}_{xz}\\ -{P}_{y} & 0 & {C}_{yy} & 0\\ 0 & -{C}_{xz} & 0 & {C}_{zz}\end{array}).$$

The matrices \({a}_{\mu \nu }^{Y}\) in equation (6) represent the propagation of the spin density matrices in the sequential decays. The elements of these 4 × 4 matrices are parameterized in terms of the weak decay parameters αY and ϕY as well as the helicity angles: \({a}_{\mu \mu {\prime} }^{{\Xi }}({\theta }_{\varLambda },{\varphi }_{\varLambda };\,{\alpha }_{{\Xi }},{\phi }_{{\Xi }})\) in reference system \({ {\mathcal R} }_{{\Xi }}\), \({a}_{\nu \nu {\prime} }^{\bar{{\Xi }}}({\theta }_{\bar{{\Lambda }}},{\varphi }_{\bar{{\Lambda }}};\,{\bar{\alpha }}_{{\Xi }},{\bar{\phi }}_{{\Xi }})\)in system \({ {\mathcal R} }_{\bar{{\Xi }}}\), \({a}_{\mu {\prime} 0}^{{\Lambda }}({\theta }_{p},{\varphi }_{p};{\bar{\alpha }}_{{\Lambda }}\)) in system \({ {\mathcal R} }_{{\Lambda }}\), and \({a}_{\nu {\prime} 0}^{\bar{{\Lambda }}}({\theta }_{\bar{p}},{\varphi }_{\bar{p}};\,{\bar{\alpha }}_{{\Lambda }})\) in system \({ {\mathcal R} }_{\bar{{\Lambda }}}\). The full expressions of Cμν and \({a}_{\mu \nu }^{Y}\) are given in ref. 35.

We have carried out our analysis on a data sample of (1.3106 ± 0.0070) × 109 J/ψ events collected in electron–positron annihilations with the multi-purpose BESIII detector37. The J/ψ resonance decays into the \({\varXi }^{-}{\bar{\varXi }}^{+}\) final state with a branching fraction26 of (9.7 ± 0.8) × 10−4. Our method requires exclusively reconstructed \({\varXi }^{-}{\bar{\varXi }}^{+}\to \varLambda {\pi }^{-}\bar{\varLambda }{\pi }^{+}\to p{\pi }^{-}{\pi }^{-}\bar{p}{\pi }^{+}{\pi }^{+}\) events. The final-state particles are measured in the main drift chamber, where a superconducting solenoid provides a magnetic field allowing momentum determination with an accuracy of 0.5% at 1.0 GeV/c. The Λ (\(\bar{\varLambda }\)) candidates are identified by combining (\(\bar{p}{\pi }^{+}\)) pairs and the Ξ (\({\bar{{\Xi }}}^{+}\)) candidates by subsequently combining Λπ (\(\bar{\varLambda }{\pi }^{+}\)) pairs. Because it was found that the long-lived Ξ and \({\bar{{\Xi }}}^{+}\) can only be reconstructed with sufficient quality if they fulfil |cosθ| < 0.84, only Ξ and \({\bar{{\Xi }}}^{+}\) reconstructed within this range were considered. After applying all selection criteria, 73,244 \({\varXi }^{-}{\bar{\varXi }}^{+}\) event candidates remain in the sample. The number of background events in the signal is estimated to be 199 ± 17. More details of the analysis are given in Methods.

For each event, the complete set of the kinematic variables ξ is calculated from the intermediate and final-state particle momenta. The physical parameters in ω are then determined from ξ by an unbinned maximum log-likelihood fit where the multidimensional reconstruction efficiency is taken into account. The details of the maximum log-likelihood fit procedure and the systematic uncertainties are described in Methods.

The results of the fit, that is, the weak decay parameters Ξ → Λπ and \({\bar{\varXi }}^{+}\to \bar{\varLambda }{\pi }^{+}\), as well as the production-related parameters αψ and ΔΦ, are summarized in Table 1. To illustrate the fit quality, the diagonal spin correlations and the polarization defined in equation (7) are shown in Fig. 2. The upper-left panel of Fig. 2 shows that the Ξ baryon is polarized with respect to the normal of the production plane. The maximum polarization is approximately 30%, as shown in the figure. The data points are determined by independent fits for each cosθ bin, without any assumptions on the cosθ dependence of Cμν. The red curves represent the angular dependence obtained with the parameters αψ and ΔΦ determined from the global maximum log-likelihood fit. The independently determined data points agree well with the globally fitted curves.

Table 1 Summary of results
Fig. 2: Polarization in and spin correlations of the \({e}^{+}{e}^{-}\to {\varXi }^{-}{\bar{\varXi }}^{+}\) reaction.
figure 2

a, Polarization in the \({e}^{+}{e}^{-}\to {\varXi }^{-}{\bar{\varXi }}^{+}\) reaction. bd, Spin correlations of the \({e}^{+}{e}^{-}\to {\varXi }^{-}{\bar{\varXi }}^{+}\) reaction. The coordinate systems \({ {\mathcal R} }_{{\Xi }}\) and \({ {\mathcal R} }_{\bar{{\Xi }}}\) of the Ξ and \({\bar{{\Xi }}}^{+}\), respectively, are described in the text. The data points are determined independently in each bin of the Ξ cosine scattering angle in the e+e centre-of-momentum system. The blue curves represent the expected angular dependence obtained with the production parameters αψ and ΔΦ from the global maximum log-likelihood fit. The error bars indicate the statistical uncertainties.

The extracted values of αΞ, \({\bar{\alpha }}_{{\Xi }}\), ϕΞ, \({\bar{\phi }}_{{\Xi }}\), αΛ and \({\bar{\alpha }}_{{\Lambda }}\) and their correlations allow for three independent CP symmetry tests. The asymmetry \({A}_{{\rm{CP}}}^{{\Xi }}\) is measured for the first time and found to be (6 ± 13 ± 6) × 10−3, where the first uncertainty is statistical and the second systematic. The corresponding standard model (SM) prediction24 is \({A}_{{\rm{CP}},{\rm{SM}}}^{{\Xi }}=(-0.6\pm 1.6)\times {10}^{-5}\).

The result for the decay parameter \({\bar{\phi }}_{{\Xi }}\) is, to our knowledge, the first measurement of its kind for a weakly decaying antibaryon. By combining this parameter with the corresponding ϕΞ measurement, the CP asymmetry \(\varDelta {\phi }_{{\rm{CP}}}^{{\Xi }}\) can be determined, and is found to be (−5 ± 14 ± 3) × 10−3 rad. Because this result is consistent with zero, we can improve our knowledge of the value of ϕΞ by assuming CP symmetry and then calculating the mean value of ϕΞ and \({\bar{\phi }}_{\varXi }\). This procedure yields ϕΞ = 0.016 ± 0.014 ± 0.007 rad, which differs from the HyperCP measurement, ϕΞ,HyperCP = −0.042 ± 0.011 ± 0.011 rad, by 2.6 standard deviations3. It is noteworthy that our method yields a precision in ϕΞ that is similar to that of the HyperCP result, despite the three-orders-of-magnitude larger data sample of the latter measurement. This demonstrates the intrinsically high sensitivity that can be achieved with entangled baryon–antibaryon pairs.

The measurement of ϕΞ, together with the mean value αΞ = −0.373 ± 0.005 ± 0.002, enables a direct determination of the strong-phase difference, which is found to be (δP − δS) = (−4.0 ± 3.3 ± 1.7) × 10−2 rad. This is consistent with the standard model predictions obtained in the framework of heavy-baryon chiral perturbation theory24 of (1.9 ± 4.9) × 10−2 rad but in disagreement with the value (10.2 ± 3.9) × 10−2 rad that one obtains from the HyperCP ϕΞ measurement3 by using αΞ = −0.376 from this work. Because the (δP − δS) value obtained in our analysis is consistent with zero, a calculation of the weak-phase difference from equation (4) is unfeasible. Instead, we apply equation (5), which yields (ξP − ξS) = (1.2 ± 3.4 ± 0.8) × 10−2 rad. This is one of the most precise tests of the CP symmetry for strange baryons and the first direct measurement of the weak-phase difference for any baryon. The corresponding standard model prediction24 is (ξP − ξS)SM = (1.8 ± 1.5) × 10−4 rad.

Sequential Ξ decays also provide an independent measurement of the Λ decay parameters αΛ and \({\bar{\alpha }}_{{\Lambda }}\). Being the lightest baryon with strangeness, Λ appear in the decay chain of many other baryons (Σ0, Ξ0, Ξ, Ω, Λc and so on) decay with appreciable fractions into final states containing Λ. The measurements of spin observables38,39,40 and decay parameters of heavier baryons41,42 therefore implicitly depend on αΛ. Furthermore, because decaying Λ and \(\bar{{\Lambda }}\) beams are used for producing polarized proton and antiproton beams43, all physics from such experiments rely on a correct determination of αΛ. The value of αΛ = 0.757 ± 0.011 ± 0.008 measured in this analysis is in excellent agreement with that obtained from the \(J/\psi \to \varLambda \bar{\varLambda }\) analysis of BESIII4, although it disagrees with the result from the re-analysis of CLAS data5. The precision of our measurement is similar to that of the \(J/\psi \to \varLambda \bar{\varLambda }\) study4, despite being based on a data sample that was six times smaller. The larger sensitivity is primarily explained by the fact that αΛ in equation (6) appears in a product with the polarization, which is much larger in the case of Λ baryons from Ξ decays compared to those directly produced in \(J/\psi \to \varLambda \bar{\varLambda }\). Furthermore, the multi-step process enhances the angular correlations between the baryons and antibaryons to such an extent that αΞ and αΛ can be measured with the same precision even if the \({\varXi }^{-}{\bar{\varXi }}^{+}\) pair is produced unpolarized2.

To summarize, this Article presents a very sensitive test of CP symmetry. This test provides a hunting ground for physics beyond the standard model in strange hadrons that is complementary to ε′/ε measurements in kaon decays44. The contributions to ε and ε′, from hyperon decays on the one hand and kaon decays on the other, are described by different combinations of quark operators. In addition, hyperons provide information on the spin structure of the operators that is not possible to obtain from kaon decays. When applied to future measurements with larger datasets at BESIII45, the upcoming PANDA experiment at FAIR46 and the proposed Super-Charm Tau Factory projects in China and Russia47,48, our method has potential to reach the required precision for CP-violating signals, provided such effects exist.


Monte Carlo simulation

For the selection and optimization of the final event sample, estimation of background sources as well as normalization for the fit method, Monte Carlo simulations have been used. The simulation of the BESIII detector is implemented in the simulation software GEANT450,51. GEANT4 takes into account the propagation of the particles in the magnetic field and particle interactions with the detector material. The simulation output is digitized, converting energy loss to pulse heights and points in space to channels. In this way the Monte Carlo digitized data have the same format as the experimental data. The production of the J/ψ is simulated by the Monte Carlo event generator KKMC52. Particle decays are simulated using the package BesEvtGen53,54,55, where the properties of mass, branching ratios and decay lengths come from the world-averaged values26. We find that although the mass of the Λ in our data agrees with the established value, that of the Ξ is 95 keV/c2 above the central value of the world average26, mΞ,PDG = 1,321.71 ± 0.07 MeV/c2 (PDG, Particle Data Group). Hence, we have adjusted the input mass value in the simulation accordingly55.

The signal channels used for optimization and consistency checks are implemented with the helicity formalism and with parameter values in close proximity to the results presented in Table 1.

Selection criteria

The data were accumulated during two run periods, in 2009 and 2012, where the later set is approximately five times larger than the earlier. For the analysis all charged final-state particles have to be reconstructed. The main drift chamber of the BESIII experimental set-up is used for reconstructing the charged-particle tracks. At least three positively and three negatively charged tracks are required, each track fulfilling the condition that |cosθLAB| < 0.93, where θLAB is the polar angle with respect to the positron beam direction. The momentum distributions of protons and pions from the signal process are well separated and do not overlap, as shown in Extended Data Fig. 1. Therefore a simple momentum criterion suffices for particle identification: ppr > 0.32 GeV/c and pπ < 0.30 GeV/c for protons and pions, respectively. The probability of misidentifying a proton (antiproton) for a π+ (π) is 0.17% (0.18%). Only events with at least one proton, one antiproton, two negatively and two positively charged pions are saved for further analysis. Each Ξ decay chain is reconstructed separately, and is here described for the sequence Ξ → Λπ → π. To find the correct Ξ and Λ particles all proton and π candidates are combined together. The Λ and Ξ particles are reconstructed through vertex fits by first combining the \(p{\pi }_{i}^{-}\) pair to form a Λ and then the \(\varLambda {\pi }_{j}^{-}\) (i ≠ j) pair to form a Ξ. The fits take into account the nonzero flight paths of the hyperons, which can give rise to different production and decay points. All vertex fits must converge and the combination that minimizes ((mpππ − mΞ)2 + (m − mΛ)2)1/2, where mΞ and mΛ are the nominal masses and mpππ (m) is the mass of the candidate Ξ (Λ), is retained for further analysis. The same procedure is performed for the \(\bar{{\Xi }}\) decay chain. For each decay chain the probability that the pions from the Ξ → Λπ and Λ →  decays are wrongly assigned is found to be 0.51% and 0.49% for π+ and π, respectively, which is negligible for the analysis. The \({m}_{{\Lambda }{\pi }^{-}}\) versus \({m}_{\bar{{\Lambda }}{\pi }^{+}}\) scatter plot is shown in Extended Data Fig. 2. A four-constraint kinematic fit requiring energy and momentum conservation (4C) is imposed on the \({e}^{+}{e}^{-}\to J/\psi \to {\varXi }^{-}{\bar{\varXi }}^{+}\) system, and only events where \({\chi }_{4{\rm{C}}}^{2} < 100\) are retained for further analysis. The kinematic fit is effective for removing the background processes \({e}^{+}{e}^{-}\to J/\psi \to \gamma {\eta }_{{\rm{c}}}\to \gamma {\varXi }^{-}{\bar{\varXi }}^{+}\) and \({e}^{+}{e}^{-}\to J/\psi \to \varXi {(1530)}^{-}\bar{\varXi }\to {\pi }^{0}{\varXi }^{-}{\bar{\varXi }}^{+}\) (and its charge conjugate), which have the same charged final-state topology as the signal channel, but contain extra neutral particles.

The invariant masses of the and \(\bar{p}{\pi }^{+}\) pairs are also required to fulfil |m − mΛ,peak| < 11.5 MeV/c2, where mΛ,peak is the peak position of the Λ mass distribution. A similar mass window criterion, optimized to remove the broad resonance \({\varSigma }^{-}(1385){\bar{\varSigma }}^{+}(1385)\)   background contribution, is imposed on the Ξ particle, |mΛπ − mΞ,peak| < 11.0 MeV/c2.

The decay length is defined as the distance between the point of origin and the decay position of the decaying Λ or Ξ particle. If the hyperon momentum points oppositely to the direction from the collision to the decay point, then the decay length becomes negative in the vertex-fit algorithm. These events are removed from the sample.

Differences between experimental data and Monte Carlo simulations are observed for large polar angles. This discrepancy induces a systematic bias on the parameter values. This bias can, however, be reduced to a negligible level by requiring |cosθ| < 0.84. The Ξ scattering angle θ is defined in the main text.

After applying all aforementioned selection criteria, 73,244 \({\varXi }^{-}{\bar{\varXi }}^{+}\) candidates remain in the final sample. This is shown in Extended Data Fig. 3. The number of remaining background events are estimated to be 199 ± 17. The background contribution has a marginal effect on the results at this precision and is therefore neglected.

Definition of the helicity systems

In the \({e}^{+}{e}^{-}\to {\varXi }^{-}{\bar{\varXi }}^{+}\), Ξ → Λπ, Λ → , \({\bar{\varXi }}^{+}\to \bar{\varLambda }{\pi }^{+}\), \(\bar{\varLambda }\to \bar{p}{\pi }^{+}\) process, the ‘master coordinate system’, denoted \( {\mathcal R} \), is defined in the e+e centre-of-momentum system. In this system, we define the unit vector \(\hat{{\bf{z}}}\) in the direction of the positron momentum. The coordinate system \({ {\mathcal R} }_{{\Xi }}\) is then defined in the rest frame of the Ξ baryon, with the z axis along the unit vector \({\hat{{\bf{z}}}}_{{\Xi }}\) defined by the direction of the Ξ momentum in the \( {\mathcal R} \) system. A Cartesian coordinate system with \({\hat{{\bf{x}}}}_{{\Xi }}\) and \({\hat{{\bf{y}}}}_{{\Xi }}\) unit vectors is defined as

$${\hat{{\bf{x}}}}_{{\Xi }}=\frac{\hat{{\bf{z}}}\times {\hat{{\bf{z}}}}_{{\Xi }}}{|\hat{{\bf{z}}}\times {\hat{{\bf{z}}}}_{{\Xi }}|}\times {\hat{{\bf{z}}}}_{{\Xi }},\,{\hat{{\bf{y}}}}_{{\Xi }}=\frac{\hat{{\bf{z}}}\times {\hat{{\bf{z}}}}_{{\Xi }}}{|\hat{{\bf{z}}}\times {\hat{{\bf{z}}}}_{{\Xi }}|}.$$

The helicity system \({ {\mathcal R} }_{\bar{{\Xi }}}\) is defined in the same way in the \({\bar{{\Xi }}}^{+}\) rest frame, and because \({\hat{{\bf{z}}}}_{\bar{{\Xi }}}=-{\hat{{\bf{z}}}}_{\bar{{\Xi }}}\), the axes \({\hat{{\bf{x}}}}_{\bar{{\Xi }}}={\hat{{\bf{x}}}}_{\bar{{\Xi }}}\) and \({\hat{{\bf{y}}}}_{\bar{{\Xi }}}=-{\hat{{\bf{y}}}}_{\bar{{\Xi }}}\). The system \({ {\mathcal R} }_{{\Lambda }}\) is defined in the rest frame of the Λ, with the \({\hat{{\bf{z}}}}_{{\Lambda }}\) pointing in the direction of the Λ momentum in the \({ {\mathcal R} }_{{\Xi }}\) system. A new Cartesian coordinate system is then defined by the unit vectors \({\hat{{\bf{x}}}}_{{\Lambda }}\) and \({\hat{{\bf{y}}}}_{{\Lambda }}\)

$${\hat{{\bf{x}}}}_{{\Lambda }}=\frac{{\hat{{\bf{z}}}}_{\varXi }\times {\hat{{\bf{z}}}}_{{\Lambda }}}{|{\hat{{\bf{z}}}}_{\varXi }\times {\hat{{\bf{z}}}}_{{\Lambda }}|}\times {\hat{{\bf{z}}}}_{{\Lambda }},\,{\hat{{\bf{y}}}}_{{\Lambda }}=\frac{{\hat{{\bf{z}}}}_{\varXi }\times {\hat{{\bf{z}}}}_{{\Lambda }}}{|{\hat{{\bf{z}}}}_{\varXi }\times {\hat{{\bf{z}}}}_{{\Lambda }}|}.$$

In the same way, the system \({ {\mathcal R} }_{\bar{{\Lambda }}}\) can be derived, and hence there is a unique definition of the orientations of the coordinate systems \({ {\mathcal R} }_{{\Xi }}\), \({ {\mathcal R} }_{\bar{{\Xi }}}\), \({ {\mathcal R} }_{{\Lambda }}\) and \({ {\mathcal R} }_{\bar{{\Lambda }}}\) used in the analysis.

The maximum log-likelihood fit procedure

The global fit is performed on the data through the joint angular distribution. For N events the likelihood function is given by

$$\begin{array}{c} {\mathcal L} ({{\boldsymbol{\xi }}}_{1},{{\boldsymbol{\xi }}}_{2},\ldots ,{{\boldsymbol{\xi }}}_{N};{\boldsymbol{\omega }})=\mathop{\prod }\limits_{i=1}^{N}{\mathscr{P}}({{\boldsymbol{\xi }}}_{i};{\boldsymbol{\omega }})\\ =\mathop{\prod }\limits_{i=1}^{N}\frac{{\mathscr{W}}({{\boldsymbol{\xi }}}_{i};{\boldsymbol{\omega }})\varepsilon ({{\boldsymbol{\xi }}}_{i})}{{\mathscr{N}}({\boldsymbol{\omega }})},\end{array}$$

where ε(ξ) is the efficiency, \({\mathscr{W}}({\boldsymbol{\xi }};{\boldsymbol{\omega }})\) is the weight as specified in equation (6), and the normalization factor \({\mathscr{N}}({\boldsymbol{\omega }})=\int {\mathscr{W}}({\boldsymbol{\xi }};{\boldsymbol{\omega }})\varepsilon ({\boldsymbol{\xi }}){\rm{d}}{\boldsymbol{\xi }}\). The normalization factor is approximated as \({\mathscr{N}}({\boldsymbol{\omega }})\approx \frac{1}{M}{\sum }_{j=1}^{M}{\mathscr{W}}({{\boldsymbol{\xi }}}_{j};{\boldsymbol{\omega }})\), using M Monte Carlo events ξj generated uniformly over phase space, propagated through the detector and reconstructed in the same way as data. M is chosen to be much larger than the number of events in data N; our results exploit a simulation sample where M/N ≈ 35. By taking the natural logarithm of the joint probability density, the efficiency function can be separated and removed as it only affects the overall log-likelihood normalization and is not dependent on the parameters in ω. To determine the parameters, the Minuit package from the CERN library is used56. The minimized function is given by S = −ln(\({\mathcal{L}}\)). The operational conditions were slightly different for the 2009 and 2012 datasets, most notably in the nominal value of the magnetic field. For this reason, the likelihoods are constructed separately for the two different run periods.

The results of the simultaneous fit are shown in Table 1. Those results that depend on combinations of decay parameters account for the correlations between the parameters. The correlation coefficients between the decay parameters are given in Extended Data Table 1. Assuming that CP symmetry is conserved we find αΛ = 0.760 ± 0.006 ± 0.003 and αΞ = −0.373 ± 0.005 ± 0.002, where the latter result is in disagreement with the current standard value26 αΞ = −0.401 ± 0.010. The parameter αΞ has previously only been measured indirectly via the product αΞαΛ and the assumed value of αΛ. The current standard value of αΛ = 0.732 ± 0.014 is an average based on the two incompatible results of BESIII and the re-analysed CLAS data4,5, and in disagreement with the value found in this analysis. By contrast, our measured value for the product αΞαΛ = −0.284 ± 0.004 ± 0.002 is compatible with the world average26 αΞαΛ = −0.294 ± 0.005.

Systematic uncertainties

The systematic uncertainties are assigned by performing studies related to the kinematic fit, the Λ and Ξ mass window requirements, the Λ and Ξ decay length selection, and a combined test on the \(\varXi \bar{\varXi }\) fit reconstruction with the p, π main drift chamber track reconstruction efficiency. Searches of systematic effects are tested by varying the criteria above and below the main selection. For each test, i, the parameter values are re-obtained, ωsys,i and the changes evaluated compared to the central values, Δi = |ω − ωsys,i|. Also calculated are the uncorrelated uncertainties \({\sigma }_{{\rm{uc}},i}=\sqrt{|{\sigma }_{\omega }^{2}-{\sigma }_{\omega ,{\rm{sys}},i}^{2}|}\), where σω and σω,sys,i correspond to the fit uncertainties of the main and systematic test results, respectively. If the ratio Δi/σuc,i shows a trending behaviour and larger than two this is attributed to a systematic effect57,58. For each systematic effect the corresponding uncertainty is evaluated. The assigned systematic uncertainties are given in Extended Data Tables 24, where the individual systematic uncertainties are summed in quadrature.

  1. 1.

    Estimator. To test if the method produces systematically biased results, a large Monte Carlo data sample is produced with production and decay distributions corresponding to those of the fit results to the data sample (~10 times the experimental data). The simulated data are divided into subsamples with equal number of events as the experimental sample, and run through the fit procedure. The obtained fit parameters and uncertainties are found to be consistent within one standard deviation of the generated parameter values and hence no bias is detected.

  2. 2.

    Kinematic fit. The systematic differences from the kinematic fit are tested by varying the kinematic fit χ2 value from 40 to 200, with an increment of 20 in each step. Significant effects are seen for the parameters ΔΦ, ϕ and \(\bar{\phi }\) when \({\chi }_{{\rm{4C}}}^{2} > 100\). For \({\chi }_{{\rm{4C}}}^{2} < 100\) systematic deviations occur for αΞ and \({\bar{\alpha }}_{{\Lambda }}\). The difference in track resolution between data and Monte Carlo is the probable cause for these changes in the parameter values. The systematic uncertainty is assigned to be the average difference of the main result to a lower and upper limit, determined to be at \({\chi }_{{\rm{4C}}}^{2}=60\) and 200, respectively.

  3. 3.

    Λ and Ξ mass window selection. Possible systematic effects due to the \(\bar{\varLambda }\varLambda ,\,\bar{\varLambda }\), Ξ and \({\bar{{\Xi }}}^{+}\) mass windows are investigated by varying the selection criteria between 2 and 30 MeV/c2 and 2 and 20 MeV/c2 for the \(\varLambda /\bar{\varLambda }\) and \({\varXi }^{-}/{\bar{\varXi }}^{+}\) candidates, respectively. For the Λ selection systematic deviations are seen for decreasing mass windows. The uncertainty is assigned to be the difference of the nominal result to the result when 95% of the events are included, at |m − mΛ,peak| < 6.9 MeV/c2. For the Ξ and \({\bar{{\Xi }}}^{+}\) mass windows, significant effects are seen for the parameters αJ/ψ, αΛ and ϕΞ. The systematic uncertainties for these parameters are assigned to be the difference of the main result and the results obtained one standard deviation lower than the main selection window, estimated from the mΛπ line shape uncertainty.

  4. 4.

    Λ and Ξ decay length. Possible systematic effects related to the Λ and Ξ lifetimes are studied by varying the decay length selection criteria for the Λ and Ξ candidates. For Ξ no strong trending behaviours are seen, but for Λ a dependence is seen for the asymmetry parameters αΛ and \({\bar{\alpha }}_{{\Lambda }}\), which is accounted for in the final systematic uncertainty.

  5. 5.

    The combined efficiency of \({\varXi }^{-}{\bar{\varXi }}^{+}\) reconstruction and p, π tracking. For the study of systematic effects related to the tracking and the Λ and Ξ reconstruction it is assumed that the combined efficiency for proton, antiproton and π± depends only on the polar angle cosθLAB and the transverse momentum, pT. To study the tracking efficiency the fitted probability density function is modified by allowing for arbitrary efficiency corrections as a function of cosθLAB and pT for each particle type in an iterative procedure. The correction procedure is repeated until the maximum log-likelihood is stable within ln(2) between two successive iterations. The difference between the fit results with and without the tracking correction is assigned as the systematic uncertainty.

  6. 6.

    The cosθ scattering angle. From comparing data to Monte Carlo simulation a discrepancy is seen for charged tracks with polar angles |cosθLAB| > 0.84. The discrepancy is also seen to have a notable effect on some of the decay parameters. The effect can be isolated by removing only the events where |cosθ| > 0.84. Although the observed data–simulation differences are removed by requiring that |cosθ| < 0.84, residual systematic effects are observed for αΞ and αΞαΛ, which are included in the systematic uncertainty.

  7. 7.

    Dataset consistency. When comparing the statistically independent results of the 2009 and 2012 datasets, all parameters are found to agree within two standard deviations. As there is no evidence of systematic bias, no uncertainty is assigned associated with possible dataset differences.