Precise Measurements of Decay Parameters and $CP$ Asymmetry with Entangled $\Lambda-\bar{\Lambda}$ Pairs

Based on 10 billion $J/\psi$ events collected at the BESIII experiment, a search for $CP$ violation in $\Lambda$ decay is performed in the difference between $CP$-odd decay parameters $\alpha_{-}$ for $\Lambda \rightarrow p\pi^-$ and $\alpha_{+}$ for $\bar\Lambda \rightarrow \bar{p}\pi^+$ by using the process $e^+e^- \to J/\psi \rightarrow \Lambda \bar\Lambda $. With a five-dimensional fit to the full angular distributions of the daughter baryon, the most precise values for the decay parameters are determined to be $\alpha_{-} = 0.7519 \pm 0.0036 \pm 0.0024$ and $\alpha_{+} = -0.7559 \pm 0.0036 \pm 0.0030$, respectively. The $\Lambda$ and $\bar{\Lambda}$ averaged value of the decay parameter is extracted to be $\alpha_{\rm{avg}} = 0.7542 \pm 0.0010 \pm 0.0024$ with unprecedented accuracy. The $CP$ asymmetry $A_{CP}=(\alpha_{-}+\alpha_{+})/(\alpha_{-}-\alpha_{+})$ is determined to be $-0.0025 \pm 0.0046 \pm 0.0012$, which is one of the most precise measurements in the baryon sector. The reported results for the decay parameter will play an important role in the studies of the polarizations and $CP$ violations for the strange, charmed and beauty baryons.

a Also at the Moscow Institute of Physics and Technology, Moscow 141700, Russia b Also at the Novosibirsk State University, Novosibirsk, 630090, Russia c Also at the NRC "Kurchatov Institute", PNPI, 188300, Gatchina, Russia Based on 10 billion J/ψ events collected at the BESIII experiment, a search for CP violation in Λ decay is performed in the difference between CP -odd decay parameters α− for Λ → pπ − and α+ for Λ → pπ + by using the process e + e − → J/ψ → Λ Λ.With a five-dimensional fit to the full angular distributions of the daughter baryon, the most precise values for the decay parameters are determined to be α− = 0.7519 ± 0.0036 ± 0.0024 and α+ = −0.7559± 0.0036 ± 0.0030, respectively.The Λ and Λ averaged value of the decay parameter is extracted to be αavg = 0.7542±0.0010±0.0024with unprecedented accuracy.The CP asymmetry ACP = (α− + α+)/(α− − α+) is determined to be −0.0025± 0.0046 ± 0.0012, which is one of the most precise measurements in the baryon sector.The reported results for the decay parameter will play an important role in the studies of the polarizations and CP violations for the strange, charmed and beauty baryons.
Charge-parity (CP ) violation is a subject of continuing interest.To date, CP violation has been discovered in the K [1], B [2,3], and D meson [4] systems, but it has never been observed in the decays of any baryon.Hence it is vital to search for additional sources of CP violation.Moreover, the Standard Model(SM) is difficult to explain the phenomenon of matter-antimatter asymmetry of the universe [5][6][7].Thus, the CP test is an ideal and sensitive method to search for the physics beyond SM [8,9].
The promising CP -violating signature in spin- 1  2 nonleptonic hyperon decays is the difference between hyperon and antihyperon decay distributions in their parity-violating two-body weak decays [10].In such decays the angular distribution of the daughter baryon is proportional to (1 + α Y P Y • pd ), where α Y is the hyperon decay parameter, and P Y and pd are the hyperon polarization and the unit vector in the direction of the daughter baryon momentum, respectively, both in the hyperon rest frame.The CP asymmetry is defined as The parameters α Y and α Ȳ are CP odd so that a nonzero A CP indicates CP violation.In the SM, the Cabibbo-Kobayashi-Maskawa mechanism predicts a tiny A CP value of ∼ 10 −4 [11].Therefore, the hyperon decay is sensitive to the sources of CP asymmetry from physics beyond the SM [12,13].A precise measurement of the Λ( Λ) decay parameters is important for studies of spin polarization [14][15][16][17] and decay parameters [18][19][20][21][22][23][24] of many other baryons(Σ 0 , Ξ 0 , Ξ − , Ω − , Λ c , Λ b etc.) decays into final states involving Λ.
A total sample of 10 billion J/ψ events has been collected by the BESIII experiment, about 3.2 million quantum-entangled Λ-Λ pairs are expected to be fully reconstructed in the decay J/ψ → Λ Λ with Λ( Λ) → pπ − (pπ + ) [25].Hence, in this Letter, we present the most precise measurements of Λ decay parameters and CP asymmetry with a five-dimensional fit to the full angular distributions of the daughter baryon.
Two real parameters, the J/ψ → Λ Λ angular distribution parameter α J/ψ and the helicity phase difference ∆Φ, describe the angular distribution and polarization of the produced Λ and Λ [26].If the phase difference ∆Φ is nonvanishing, the polarization of the Λ and Λ will be oriented perpendicular to the production plane.For the decay Λ → pπ − , the angular distribution of the proton is , where α − is Λ decay parameter, P Λ is the polarization vector of the Λ, n is the unit vector of the proton momentum in the Λ rest frame.The definition of the decay parameter α + for Λ → pπ + follows an analogous convention [27].
For the cascade decay J/ψ → Λ Λ with Λ( Λ) → pπ − (pπ + ), the angular distribution of each event is uniquely characterized by the kinematic variable ξ = (θ Λ , θ p , φ p , θ p, φ p), where θ p , φ p and θ p, φ p are the polar and azimuth angles of the proton and antiproton in their mother particles rest frames.The components of these vectors are expressed using a right-handed coordinate system (x, ŷ, ẑ) shown in Fig. 1.The ẑ axis is taken along the Λ momentum p Λ = −pΛ ≡ p in the e + e − center-of-mass system (CMS).The ŷ axis is perpendicular to the production plane and oriented along the vector k × p, where k e − = −k e + ≡ k is the electron beam momentum.The scattering angle of the Λ is given by cos θ Λ = p • k.
The differential distribution function can be expressed as (1) where the angular functions F i (ξ) (i = 0, 1, ...6) are described in detail in Ref. [26].The terms proportional to α − α + in Eq. ( 1) represent the contribution from Λ-FIG. 1. Definition of the right-hand coordinate system used to describe the J/ψ → Λ Λ process.In the e + e − center-ofmass system, the Λ is emitted along the ẑ axis direction.ŷ axis is perpendicular to the plane of Λ and e − .The hyperons are polarized along the ŷ direction.
Λ spin correlations, and the terms proportional to α − and α + separately represent the contribution from the hyperon transverse polarization P y , defined as: The analysis presented here is based on the aforementioned sample of 10 billion J/ψ events [28] collected at the BESIII detector [29,30].A Monte Carlo (MC) simulation of J/ψ samples is used to determine the detector efficiency, optimize the event selection, and estimate the background.The simulation is performed by the GEANT4-based [31] BESIII Object Oriented Simulation Tool project [32], which includes the geometric description of the BESIII detector and the detector response.The MC event generators KKMC [33], BesEvtGen [34], and Lundcharm [35,36] are used to describe J/ψ production together with known and unknown decay modes.For the signal process, J/ψ → Λ Λ, the parameters of the angular distribution are obtained from previous measurements [12].For the dominant background channel J/ψ → γη c (η c → Λ Λ), the decay J/ψ → γη c is generated with an angular distribution of 1+cos 2 θ γ [37], where θ γ is the angle between the photon and positron beam direction in the CMS, and another background channel J/ψ → γΛ Λ is described by the phase space model.
The Λ and Λ baryons are reconstructed from their dominant hadronic decay mode, Λ( Λ) → pπ − (pπ + ).Charged tracks detected in the main drift chamber must satisfy |cos(θ)| < 0.93 , where θ is the angle between the charged track and the positron beam direction.Events with at least four charged tracks are retained.Tracks with momentum larger than 0.5 GeV/c are considered as proton candidates, otherwise as pion candidates.There are no further particle identification requirements.Vertex fits are performed by looping over all combinations with oppositely charged proton and pion candidates, constraining them to a common vertex.The pairs with vertex fit χ 2 lower than 200 and decay length larger than 0 are regarded as Λ − Λ candidates.A four-momentum constrained kinematic fit is applied to the ppπ + π − hypothesis, and events with a minimum χ 2 lower than 60 are selected as J/ψ candidates.
An inclusive MC sample of 10 billion J/ψ events is used for studying potential backgrounds.After applying the same selection criteria as for the data, the main backgrounds are divided into two types according to the shapes of m pπ − and m pπ + : (1)BKGI, nonpeaking backgrounds, including J/ψ → pπ − pπ + , ∆ ++ pπ − , ∆++ pπ + , ∆ ++ ∆++ ; (2)BKGII, peaking backgrounds, including J/ψ → γΛ Λ, γη c (η c → Λ Λ).The number of nonpeaking backgrounds is estimated by the two-dimensional sideband regions of the m pπ − versus m pπ + distribution from the data sample which is shown in Fig. 2. The signal region is defined as m pπ − / pπ + ∈ [1.111, 1.121] GeV/c 2 , and the lower and higher sideband regions are defined as m pπ − / pπ + ∈ [1.098, 1.107] GeV/c 2 and m pπ − / pπ + ∈ [1.125, 1.134] GeV/c 2 , respectively.The yields of various peaking background sources are estimated by individual exclusive MC samples, then normalized to the data sample according to their branching fractions [27].The final data sample contains 3231781 events including the estimated background yield of 3801 ± 63 events.The sample has a high purity of 99.9%.Based on the joint angular distribution, a maximum likelihood fit with four free parameters (α J/ψ , ∆Φ, α − , and α + ) is performed.The joint likelihood function is defined as: where P(ξ i ; α J/ψ , ∆Φ, α − , α + ) is the probability density function of ξ i , the kinematic variable of event i, and W(ξ i ; α J/ψ , ∆Φ, α − , α + ) is given by Eq. ( 1).The detection efficiency is denoted by (ξ i ).The normalization factor C −1 = 1 NMC NMC j=1 W(ξ j ; α J/ψ , ∆Φ, α − , α + ) (ξ j ) is estimated with the N MC events generated with the phase space model, applying the same event selection criteria as for the data.To improve the accuracy of the normalization factor, we generate a MC sample about 100 times larger than the selected experimental data.We use the ROOFIT package [38] to determine the fit parameters from the minimization of the function: where L data and L BKGI are the likelihood function of events in the signal region and sideband regions, respectively.The L BKGII is the likelihood function of background events obtained by exclusive MC samples.The likelihood function of background event is the same as the data.The results of the maximum likelihood fit of data are given in Table I, with the CP asymmetry given by A CP = (α − +α + )/(α − −α + ), and the average value of the Λ and Λ decay parameters α avg = (α − − α + )/2.The correlation coefficient between α − and α + is ρ(α − , α + ) = 0.850.[12] for comparison.The first uncertainty is statistical, the second one is systematic.

Par. This work
Previous results [12] α J/ψ 0.4748 ± 0.0022 ± 0.0031 0.461 ± 0.006 ± 0.007 ∆Φ 0.7521 ± 0.0042 ± 0.0066 0.740 ± 0.010 ± 0.009 α− 0.7519 ± 0.0036 ± 0.0024 0.750 ± 0.009 ± 0.004 α+ −0.7559 ± 0.0036 ± 0.0030 −0.758 ± 0.010 ± 0.007 ACP −0.0025 ± 0.0046 ± 0.0012 0.006 ± 0.012 ± 0.007 αavg 0.7542 ± 0.0010 ± 0.0024 - The moment which related to the polarization, is used to compare the consistency between the data and the fit results.Hereby, N is the total number of events in the data set, and m = 100 is the number of bins in cos(θ Λ ) for calculating the moment.N k denotes the number of events in the k th cos(θ Λ ) bin.The expected angular dependence of the moment for the acceptance-corrected data reads A significant transverse polarization of Λ and Λ can be seen in Fig. 3, in which the points with error bars are the data, and the solid line is obtained from signal MC sample generated by Eq. ( 1), where the input parameters are taken from fit results.The data are consistent with the fit results.The systematic uncertainties in this analysis can be divided into two categories: (A) the uncertainties from event selection, including background estimation, tracking, the Λ/ Λ vertex fit and kinematic fit; (B) the uncertainty associated with the fit procedure.The uncertainty from background is estimated by varying the input background numbers by 1 standard deviation.The differences on the fitted parameters are taken as the systematic uncertainty.For the tracking and Λ − Λ vertex fit and kinematic fit, a correction to the MC efficiency is made.We use control samples to get the efficiencies of the data and the MC simulation in tracking, Λ − Λ vertex fit, and kinematic fit, and use the data and MC difference to calibrate the MC sample.The uncertainty due to the charged particle tracking efficiency has been investigated with a J/ψ → pπ − pπ + control sample.The systematic uncertainties due to the Λ and Λ vertex reconstruction and kinematic fits are estimated by a control sample J/ψ → Λ Λ → pπ − pπ + .In order to reduce the impact of statistical fluctuations, the fit with a corrected MC sample is performed 100 times by varying the correction factor randomly within 1 standard deviation.The differences between the mean value of the fit results with corrections and the nominal fit are taken as the systematic uncertainties.The MC simulation is used to estimate the uncertainty of the fit method.The sum of the differences between the input and output values and their uncertainty are regarded as systematic uncertainties.The absolute systematic uncertainties for various sources are summarized in Table II.The total systematic uncertainty of each parameter is obtained by summing the individual contributions in quadrature.In summary, by analyzing 10 billion J/ψ events, we report the most precise measurements of the decay parameters of Λ( Λ), with results given in Table I.The results are consistent with those in the previous analysis [12], however, with significantly improved accuracy.The measured CP asymmetry provides a hunting ground for physics beyond the standard model [39].A clear transverse polarization is observed for the Λ and Λ as shown in Fig. 3.The phase between helicity flip and helicity conserving transitions is determined to be ∆Φ = 0.7521 ± 0.0042 ± 0.0066, where the first uncertainty is statistical and the second one is systematic.The large value of the phase makes it possible to simultaneously determine the decay parameters of Λ → pπ − and Λ → pπ + to be α − = 0.7519 ± 0.0036 ± 0.0024 and α + = 0.7559 ± 0.0036 ± 0.0030, which represents the most precise measurements to date.Owing to the large correlation coefficient of the two decay parameters ρ(α − , α + ) = 0.850, the Λ and Λ averaged value is determined to be α avg = 0.7542 ± 0.0010 ± 0.0024 for the first time, which are the most precise measurements in the baryon sector.Being the lightest baryon with strangeness, the measurements of polarizations, decay parameters, and CP asymmetries of heavier baryons, therefore, implicitly depend on α Λ [20][21][22][23][24].
Results of the Λ decay parameter from different experiments are shown in Fig. 4. The α − value obtained in this work agrees with the previous BESIII measurements [12] and the BESIII result extracted from the J/ψ → Ξ − Ξ+ decay [40], but deviates from the CLAS result by 3.5σ.In addition, we obtain the value of CP violation for the Λ decay A CP = (α − + α + )/(α − − α + ) = −0.0025±0.0046±0.0012,which is compatible with zero, thereby, indicating a non-CP -violation scenario.The next generation of charm factories [41,42] will greatly improve the accuracy of the CP -violating measurements, and shed light on the mechanism of CP violation in the baryon sector.

FIG. 2 .
FIG.2.Distribution of the invariant mass spectra of pπ + versus the invariant mass spectra of pπ − from the data.The signal and sideband regions are denoted by red and green boxes, respectively.The z axis is in logarithm style.

FIG. 3 .
FIG. 3.Distribution of moment µ(cos θΛ) versus cos θΛ.The points with error bars are the data, the red histogram is the signal MC sample with input parameters fixed to fit results.The blue histogram shows the result from phase space(PHSP) MC sample.The distribution of χ = (µ data − µMC)/σ(µ data ) is shown at the bottom, where µ data and µMC are the moments of data and signal MC sample.The σ(µ data ) is the statistical uncertainty of µ data .

FIG. 4 .
FIG. 4.Results of the Λ decay parameter from different experiments.The green band represents the PDG 2018 value, and the pink band represents the PDG 2022 value.

d
Also at Goethe University Frankfurt, 60323 Frankfurt am Main, Germany e Also at Key Laboratory for Particle Physics, Astrophysics and Cosmology, Ministry of Education; Shanghai Key Laboratory for Particle Physics and Cosmology; Institute of Nuclear and Particle Physics, Shanghai 200240, People's Republic of China Also at School of Physics and Electronics, Hunan University, Changsha 410082, China i Also at Guangdong Provincial Key Laboratory of Nuclear Science, Institute of Quantum Matter, South China Normal University, Guangzhou 510006, China j Also at Frontiers Science Center for Rare Isotopes, Lanzhou University, Lanzhou 730000, People's Republic of China f Also at Key Laboratory of Nuclear Physics and Ion-beam Application (MOE) and Institute of Modern Physics, Fudan University, Shanghai 200443, People's Republic of China g Also at State Key Laboratory of Nuclear Physics and Technology, Peking University, Beijing 100871, People's Republic of China h k Also at Lanzhou Center for Theoretical Physics, Lanzhou University, Lanzhou 730000, People's Republic of China l Also at the Department of Mathematical Sciences, IBA, Karachi , Pakistan (Dated: September 26, 2022)