Test of CP Symmetry in Hyperon to Neutron Decays

The quantum entangled J/ψ → Σ + ¯Σ − pairs from (1 . 0087

Charge-parity (CP ) violation is one of Sakharov's three essential conditions for understanding the matterantimatter asymmetry in the universe [1].Despite the established presence of CP violation in the decays of K, B, and D mesons [2][3][4][5][6][7], the standard model (SM) of particle physics, as described by the Kobayashi-Maskawa mechanism, is insufficient in fully explaining the preponderance of matter over antimatter in the universe [8].As a result, it is imperative to continue searching for new sources of CP violation, particularly in the hyperon sector [9].The non-leptonic decays of spin-1/2 hyperons are suitable for CP violation studies.In such decays, the decay asymmetry parameters α, β and γ are defined in terms of the S-wave (parity violating) and P -wave (parity conserving) amplitudes [10]: Only two of these asymmetry parameters are independent.The magnitude of polarization of spin-1/2 hyperons can be inferred in two-body weak decays due to their self-analyzing nature.The polar angle distribution of the daughter nucleons is given by dN/dΩ = 1 4π (1 + αP • p).Here, P is the hyperon polarization vector, and p is the unit vector along the nucleon momentum in the hyperon rest frame.Correspondingly, the decay asymmetry parameter of the anti-hyperon is denoted as ᾱ.Because α and ᾱ are CP -odd, A CP = (α + ᾱ)/(α − ᾱ) can be used to test CP conservation [11,12].A non-zero value of A CP would indicate CP violation.
Theoretically, there are two predictions for CP violation in non-leptonic two-body weak decays of Σ.In the seminal work by Donoghue et al., the CP violation contribution in Σ + → nπ + was predicted to be −1.6×10−4 [13].The most recent study by Tandean and Valencia used heavy baryon Chiral perturbation theory and predicted the CP violation of Σ + → nπ + to be 3.9 × 10 −4 .Although the above two predictions are at the same level, Ref. [13] does not consider the P -wave factorization contribution, which can change the prediction by a factor of ten.To determine the SM CP violation contribution, the experimentally determined asymmetry parameters are used as part of the input.Due to the large experimental uncertainty of the Σ + → nπ + asymmetry parameter α + , the uncertainties in the CP violation estimations of Σ + → nπ + are greater than those of other hyperons, and the predicted CP violation is an order of magnitude greater than those of Σ + → pπ 0 , Σ − → nπ − and Λ → pπ − [14].
Recently it was pointed out that the experimental value of the decay asymmetry α 0 for Σ + → pπ 0 is not consistent with the ∆I = 1/2 rule [15], where ∆I refers to the isospin difference between the initial and final states.Therefore, a precision measurement of the decay asymmetry α + for Σ + → nπ + is needed to determine the contributions of the ∆I = 3/2 and ∆I = 3/2 weak transitions to Σ decays [16].
Experimentally, the decay asymmetry parameters α 0 and its charge-conjugated (c.c.) equivalent ᾱ0 have been well measured [17].For the decay of Σ + → nπ + , there are only two measurements of α + from fixedtarget experiments, performed more than fifty years ago.Although the two existing results 0.069 ± 0.017 [18] and 0.037 ± 0.049 [19] are in agreement with each other, they are relatively imprecise compared with α 0 and also compatible with zero.Furthermore, the corresponding decay parameter ᾱ− of Σ− → nπ − has never been measured before.Precision measurements of the decay parameters of Σ + → nπ + + c.c. would provide a first precision test of CP symmetry in hyperon to neutron decays and supply important experimental input to sharpen CP violation predictions of all non-leptonic two-body weak decays of Σ.However, the relatively small α + and ᾱ− values, the Σ polarization value determination, and the difficulties in neutron and anti-neutron detection all represent a challenge for an accurate experimental measurement.
The BESIII experiment provides a unique environment to study both hyperon production and decay properties in electron-positron annihilation to Σ + Σ− pairs via the intermediate J/ψ resonance, where the above challenges can be well addressed.In this entangled quantum system, the decay parameters of Σ + and Σ− are correlated, allowing for a precise determination of the asymmetry parameters and the CP symmetry.The e + e − → J/ψ → Σ + Σ− process is described by the Ψ electric and magnetic form factors, G Ψ E and G Ψ M [20].These two Ψ form factors are formally equivalent to the Σ electric and magnetic form factors [21][22][23][24][25].They are usually parameterized by two real parameters α J/ψ and ∆Φ, which correspond to the angular decay asymmetry and the relative phase between the two form factors, respectively.The observable ∆Φ is related to the spin-polarization of the produced Σ + Σ− pair.The Σ polarization is perpendicular to the production plane and depends on the opening angle θ Σ + between the Σ + and electron (e − ) beam in the reaction center-of-mass frame, as shown in Fig. 1.The first polarization measurement of J/ψ → Σ + Σ− was reported by the BESIII collaboration with Σ + → pπ 0 and Σ− → pπ 0 based on 1.3 billion J/ψ events [26].The significant polarization provides the prerequisite for α + and ᾱ− measurements.

The production and decay process e
The helicity frame definitions for J/ψ → Σ + Σ− ,Σ + → pπ 0 (nπ + ), Σ− → nπ − (pπ 0 ).In the e + e − center-of-mass system, θΣ is the angle between the Σ + and the electron beam direction.The z Σ + axis is the moving direction of Σ + in the J/ψ rest frame, the y Σ + axis is perpendicular to the plane of Σ + and electron, and the x Σ + axis is defined by the right-handed coordinate system.
are the polar and azimuthal angles of the nucleon and anti-nucleon measured in the rest frames of their respective mother particles.The differential cross section W(ξ) is defined as where T i , (i = 0, 1...6) are angular functions dependent on ξ and described in detail in Ref. [20].According to the above cross-section formula, if the process of J/ψ → Σ + Σ− with Σ + → nπ + , Σ− → nπ − is used with the α and ᾱ parameters close to zero, the cross section will be small, and the determination of the parameters imprecise.Moreover, the simultaneous detection of both the neutron and anti-neutron will be difficult.In addition, the process of J/ψ → Σ − Σ+ with Σ − → nπ − , Σ+ → nπ + with the same final state could contaminate our signal.To overcome these disadvantages, we instead use J/ψ → Σ + Σ− with Σ + → pπ 0 (nπ + ) and Σ− → nπ − (pπ 0 ).Benefiting from the large decay parameters α 0 = −0.998± 0.037 stat ± 0.009 syst and ᾱ0 = 0.990 ± 0.037 stat ± 0.011 syst [26], the measurement accuracy can be improved by 17.4 times compared with the neutron anti-neutron final state.Also, since Σ − cannot decay to pπ 0 , the J/ψ → Σ − Σ+ background is highly suppressed.This Letter is based on a data sample of (1.0087 ± 0.0044) × 10 10 J/ψ events [27] taken with the BESIII detector operating at the BEPCII collider.Details about the design and performance of the BESIII detector are given in Ref. [28].Candidate events for the process J/ψ → Σ + Σ− with subsequent Σ + → pπ 0 (nπ + ) and Σ− → nπ − (pπ 0 ) decays must have two charged tracks with opposite charges and at least two photons.Charged tracks detected in the multi-layer drift chamber (MDC) are required to be within a polar angle (θ) range of | cos θ| < 0.93, where θ is defined with respect to the z axis, the symmetry axis of the MDC.For each track, the distance of the closest approach to the interaction point must be less than 10 cm along the z axis, and less than 2 cm in the transverse plane.
The particle identification (PID) system identifies the two candidate charged tracks as pπ − or pπ + based on the measured energy loss in the MDC and the flight time in the time-of-flight system.Each track is assigned to the particle type corresponding to the hypothesis with the highest confidence level.
Photon candidates are identified using showers in the electromagnetic calorimeter (EMC).The deposited energy of each shower must be more than 25 MeV in the barrel region (| cos θ| < 0.80) and more than 50 MeV in the end-cap region (0.86 < | cos θ| < 0.92).To exclude showers that originate from charged tracks, the opening angle subtended by the EMC shower and the position of the closest charged track at the EMC must be greater than 10 • as measured from the interaction point.To suppress electronic noise and showers unrelated to the event, the difference between the EMC time and the event start time is required to be within [0, 700] ns.
Candidates for π 0 are selected as photon pairs with an invariant mass in the interval of (m π 0 − 60 MeV/c 2 ) < M γγ < (m π 0 + 40 MeV/c 2 ), where m π 0 is the known π 0 mass [17].In addition, a one-constraint (1C) kinematic fit is performed on the selected photon pairs, constraining the invariant mass to the known π 0 mass.The χ 2 1C of the kinematic fit is required to be less than 25.At least one candidate π 0 is required.
To select J/ψ → Σ + Σ− with Σ + → pπ 0 and Σ− → nπ − , the anti-neutron energy deposition in the EMC is required to be at least 0.5 GeV.The second moment, defined as i E i r 2 i / i E i , is required to be greater than 20.Here E i is the energy deposition in the i th crystal and r i is the radial distance of the i th crystal from the cluster center.The opening angle θ γ,n between photon candidates and the n track is required to be greater than 20 • .For this process, a four-constraint (4C) kinematic fit is applied by imposing energy-momentum conservation and an additional π 0 mass constraint, where the direction of the n is measured and the energy is unmeasured.A two-constraint (2C) kinematic fit is applied to the J/ψ → Σ + Σ− process, with Σ + → nπ + and Σ− → pπ 0 .Energy-momentum conservation and an additional π 0 mass constraint are imposed in this fit, with the neutron being treated as a missing particle.The 4C and 2C kinematic fit chi-squares, χ 2 4C and χ 2 2C , are both required to be less than 100.If the number of π 0 candidates in an event is more than one, the combination with the minimum χ 2 4C or χ 2 2C is selected as the final candidate.
To investigate possible background after applying the event selection criteria, an inclusive Monte Carlo (MC) sample of 10 billion J/ψ events has been examined with TopoAna [29].All particle decays are modeled with evtgen [30] using branching fractions either taken from the Particle Data Group (PDG) [17], when available, or otherwise estimated with lundcharm [31].
The main peaking backgrounds are J/ψ → γΣ + Σ− and J/ψ → γη c , η c → Σ + Σ− , which both contribute 0.2% of the signal strength and are negligible.The nonpeaking background mainly includes J/ψ → ∆ + ∆− → pπ 0 nπ − (nπ + pπ 0 ) and J/ψ → pπ 0 nπ − (nπ + pπ 0 ) whose contributions are estimated to be 1.4% and 1.6% with a two-dimensional sideband method.Figure 2 shows the distributions of M nπ − versus M pπ 0 and M nπ + versus M pπ 0 for the two decay modes.The signal regions in the red rectangles are defined as: 1.17 < M pπ 0 (M pπ 0 ) < 1.20 GeV/c 2 and 1.18 < M nπ − (M nπ + ) < 1.20 GeV/c 2 .To estimate the non-peaking background contributions, four sideband regions have been selected, denoted as green rectangles in the plots.Each sideband region has the same area as the signal region and is placed at a distance of about 2σ from the signal boundary, where σ is the invariant mass resolution of Σ + and Σ− .The background events are estimated using f × 4 i=1 B i , where B i is the number of events in the i th sideband region, and the scale factor f is defined as the background ratio between the signal and sideband regions.Using a two-dimensional fit on the distribution of M pπ 0 versus M nπ − or M nπ + versus M pπ 0 , the scale factors are determined to be 0.265±0.001and 0.259 ± 0.001 for these two decay channels, respectively.Here the uncertainties are statistical only.
Following the approach in Ref. [32], the multidimensional fit takes the reconstruction efficiency into account in a model-independent way and background contribution has been considered according to the scale factors f .The numerical fit results are summarized in Table I.The relative phase between the Ψ electric and magnetic form factors is determined to be ∆Φ J/ψ = (−0.2772± 0.0044 stat ± 0.0041 syst ) rad, which implies Σ spin polarization is observed.The moment related to the polarization is defined as Here, m = 40 is the number of bins, n is the total number of events in the data sample, and n k is the number of events in the k th cos θ Σ + bin.The expected angular dependence of the moment is J/ψ α + sin ∆Φ J/ψ cos θ Σ + sin θ Σ + .In Fig. 3, the black points represent data and follow the expectation as shown by the red line.As ∆Φ J/ψ is not zero, it is possible to determine the asymmetry parameters α + and ᾱ− simultaneously.The asymmetry decay parameter α + is measured to be 0.0481 ± 0.0031 stat ± 0.0019 syst , with a precision improved by a factor of 4.7 compared to the previous best measurement [18].The asymmetry decay parameter ᾱ− is determined for the first time as −0.0565 ± 0.0047 stat ± 0.0022 syst .Assuming no CP violation, the average decay asymmetry is calculated to be α + = (α + − ᾱ− )/2 = 0.0506 ± 0.0026 stat ± 0.0019 syst .

Parameter
This work Previous result α J/ψ −0.5156 ± 0.0030 ± 0.0061 −0.508 ± 0.006 ± 0.004 [26] ∆Φ J/ψ (rad) −0.2772 ± 0.0044 ± 0.0041 −0.270 ± 0.012 ± 0.009 [ The systematic uncertainties are listed in Table II, which is divided into two categories.The first category is from the event selection, including the uncertainties for MDC tracking, PID, π 0 and n reconstructions, kinematic fit, background estimations, as well as the Σ + and Σ− mass window requirements.The second category includes the uncertainties associated with the fit procedure.The individual uncertainties are assumed to be uncorrelated and are therefore added in quadrature.The uncertainties due to potential efficiency differences between data and simulation for charged-particle tracking and PID have been investigated with a J/ψ → ppπ + π − control sample, and those due to neutral π 0 and n reconstructions are estimated from J/ψ → Σ + (pπ 0 ) Σ− (pπ 0 ) and J/ψ → pnπ − control samples.Using these control samples, we determine the correction factors and apply them in the MC simulation to obtain the nominal results.The uncertainty of the correction factors is estimated by changing them within 1σ regions.The differences to the nominal results are taken as the MC efficiency correction systematic uncertainties.The systematic uncertainties .The moment M (cos θ Σ + ) for data, that is not corrected for acceptance and reconstruction efficiency, as a function of cos θ Σ + for J/ψ → Σ + Σ− with the two decay channels: Σ + → pπ 0 , Σ− → nπ − and Σ + → nπ + , Σ− → pπ 0 .The black points with error bars are data with background subtracted, the red solid line is the fit result and the blue dashed line represents the distribution without polarization uniformly distributed in phase space.The height of green band shows the absolute difference between the two decay channels with background subtracted.
due to the kinematic fits are examined by comparing the detection efficiencies with and without helix parameter corrections, which are used to reduce the discrepancies between data and MC simulation [34].The differences in detection efficiencies with and without corrections are assigned as the systematic uncertainties.To estimate the systematic uncertainty associated with the signal mass window, the window is changed by 3σ (±5 MeV), where σ is the invariant mass resolution of Σ + and Σ− .The fits are repeated using the new mass window, and the differences of results to the nominal values are regarded as the corresponding systematic uncertainties.The systematic uncertainty caused by the background estimation is studied by varying the length and width of four side-band boxes within ±5 MeV.The largest differences in the parameters are taken as the systematic uncertainties.To validate the reliability of the fit results, a set of 100 pseudo-data samples are simulated and subjected to the same selection criteria.In these samples, the differential cross section is based on the decay parameters listed in Table I.The systematic uncertainties from the fit approach are assumed to be the deviations between the inputs and average outputs.In the nominal fit, the parameters of α 0 and ᾱ0 are fixed at the world average values [17].By changing the parameter within ±1σ (0.006) regions for α 0 and ᾱ0 , the changes between the new and nominal fit results are taken as systematic uncertainties.
In summary, based on a data sample of (1.0087 ± 0.0044) × 10 10 J/ψ events collected at the BESIII detector, the five-dimensional angular analysis of the processes of J/ψ → Σ + Σ− (Σ + → pπ 0 , Σ− → nπ − and Σ + → nπ + , Σ− → pπ 0 ) is performed.The decay parameters α J/ψ and ∆Φ J/ψ are measured to be −0.5156± 0.0030 stat ± 0.0061 syst and (−0.2772 ± 0.0044 stat ± 0.0041 syst ) rad, respectively, which are consistent with the previous measurements but with improved precision [26].The nonzero value of ∆Φ J/ψ in the J/ψ → Σ + Σ− decay, which implies the existence of polarization, is confirmed with two different Σ decay channels, J/ψ → Σ + Σ− → pπ 0 nπ − (nπ + pπ 0 ) and J/ψ → Σ + Σ− → pπ 0 pπ 0 .Therefore, the decay asymmetry parameters α + and ᾱ− are determined simultaneously.The parameters α + and α + /α 0 determined in this work are consistent with the PDG averages but with significantly improved precision, and ᾱ− and ᾱ− /ᾱ 0 are measured for the first time.The average decay asymmetry parameter is 0.0506 ± 0.0026 stat ± 0.0019 syst , which differs from zero by 16σ.In contrast, the previous best measurement only deviates from zero by about 3σ [18].This result is crucial to test the |∆I| = 1/2 rule and study the high order isospin transitions [33].Our precise measurement of the decay asymmetry parameter in the neutron mode is of vital importance to the CP violation prediction [14].This is the first study to test CP symmetry in the hyperon to neutron decay, and the result is consistent with CP -conservation.

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 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 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 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 75270, Pakistan