First Measurement of the Decay Asymmetry in the pure W -boson-exchange Decay

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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 4.4 fb −1 of e + e − annihilation data collected at the center-of-mass energies between 4.60 and 4.70 GeV with the BESIII detector at the BEPCII collider, the pure W -boson-exchange decay Λ + c → Ξ 0 K + is studied with a full angular analysis.The corresponding decay asymmetry is measured for the first time to be α Ξ 0 K + = 0.01 ± 0.16(stat.)± 0.03(syst.).This result reflects the non-interference effect between the Sand P -wave amplitudes.The phase shift between Sand P -wave amplitudes has two solutions, which are δp − δs = −1.55 ± 0.25(stat.)± 0.05(syst.)rad or 1.59 ± 0.25(stat.)± 0.05(syst.)rad.
Investigations of charmed baryon decay dynamics are essential to explore the weak and strong interactions in the Standard Model (SM) of particle physics.The ground state of the singly-charmed baryons Λ + c was discovered in 1979 [1].Many studies have since been made of the properties of charmed baryons, such as the decay branching fractions (BFs) and decay asymmetries.But experimental results of the decay asymmetries, which are sensitive to the different amplitudes in the decay dynamics, were only a few.Since 2014, there has been some progress on the weak hadronic decays of Λ + c , Ξ +,0 c , and Ω 0 c , both experimentally and theoretically [2][3][4].This provides crucial information about the properties of all the singly-charmed baryons and the searches for doubly-charmed baryons (Ξ cc and Ω cc ) [5].Nonetheless, the understanding of the decay dynamics of charmed baryons is still limited, due to the lack of precision experimental measurements and the difficulties in the theoretical treatment of strong interaction effects.
Compared to heavy meson decays, charmed baryon decays have a significant dependence on nonfactorizable contributions from W -boson-exchange diagrams.However, these contributions cannot currently be calculated using theoretical approaches.Additionally, no experimental measurements exist for the decay asymmetries of W -boson-exchange hadronic decays.An example of such a process is the decay Λ + c → Ξ 0 K + , which can only be produced via a W -boson-exchange process as depicted in Fig. 1.Experimental measurements of the asymmetry parameters of the decay Λ + c → Ξ 0 K + can aid understanding of the internal dynamics and can also explore charge-parity (CP ) violation in baryons [6].
Table I lists the theoretical calculations of the BF and asymmetry parameters of Λ + c → Ξ 0 K + , as well as the experimental measurements of the BF.Various predictions of the BFs based on the covariant confined quark model (CCQM), the pole model, and current algebra (CA) in the 1990s are all smaller than the experimental results [2].This is explained as a strong cancellation in the S-and P -wave amplitudes, corresponding to the L = 0 and L = 1 orbital angular momenta of the Ξ 0 -K + system, respectively.Moreover, the decay asymmetry parameter was predicted to be zero in these models owing to the vanishing Swave amplitude [7][8][9][10][11].This long-standing puzzle has recently experienced a renewed interest in the theoretical community [6,12,13], especially after the report of new BF measured by BESIII in 2018 [14].To reproduce the relatively large experimental branching fraction of this decay, Ref. [6] adopted a variant of the CA approach and obtained a larger B(Λ + c → Ξ 0 K + ) ≃ 0.71%.This modification introduces a large positive decay asymmetry of 0.90, which is quite close to the calculations based on SU(3) symmetry [12,13].However, regarding the significant enhancement of α Ξ 0 K + from 0 to about 0.9, the authors of Ref. [15] pointed out that the particular construction of the S-wave amplitude in Ref. [6] is not well-justified.So experimental measurement of the asymmetry parameter of the decay Λ + c → Ξ 0 K + will be crucial to test these calculations and confirm the vanishing S-wave contribution [15].
In the SM, the amplitude for a spin-1/2 baryon decaying into a spin-1/2 baryon and a spin-0 meson can be written as M = iū f (A − Bγ 5 )u i , where A and B are constants, u i and ūf are spinors describing the initial and final baryons [8].For the decay Λ + c → Ξ 0 K + , the decay asymmetry is defined by ; here E Ξ 0 and ⃗ p Ξ 0 are the energy and momentum of the Ξ 0 in the Λ + c rest frame [2].The effect of the S-and P -wave phase shift difference, δ p − δ s , is not well accounted for in the theoretical calculations of decay asymmetries.It can be extracted from experiments combined with the BF cited from the Particle Data Group (PDG) [2] and provides an important experimental parameter for the theoretical prediction of CP violation [16].
A detailed description of the selection criteria for charged tracks, showers, π 0 , and Λ candidates is provided in Ref. [28].The only difference is the χ 2 of vertex fit, which constrains the daughter tracks pπ − from Λ decays to a common originating vertex, is imposed to be less than 20 in order to better suppress the background.The Ξ 0 candidates are formed by Λπ 0 combinations and the invariant mass M Λπ 0 must be within the mass region (1.30, 1.33) GeV/c 2 .The mass region is selected to be about three times the resolution.

Two kinematic variables, the energy difference ∆E ≡ E Λ +
c − E beam and the beam-constrained mass Here, E Λ + c and ⃗ p Λ + c are the reconstructed energy and momentum of the Λ + c candidates calculated in the e + e − rest frame, and E beam is the average energy of the e + and e − beams.All candidates are required to satisfy |∆E| < 0.05 GeV and 2.25 GeV/c 2 < M BC < E beam .If more than one candidate satisfies the above requirements, the one with minimal |∆E| is kept.After applying these conditions, the M BC distribution in data collected at 4.60 GeV is shown in the Fig. 2. In the fit to this data, the correctly and mis-reconstructed signal shapes are modeled with the MC-simulated shape convolved with a Gaussian function representing the resolution difference between data and MC simulation, and the background shape is described by an ARGUS function [29].Finally, 378 ± 21 signal events are obtained by combining the six energy points.The decay asymmetry parameters are determined by analyzing the multi-dimensional angular distributions, where the full cascade-decay chain is considered.The joint angular formula is obtained using the helicity basis [30].Figure 3 illustrates the definitions of the helicity angles for the three-level cascade decay Λ + c → Ξ 0 K + , Ξ 0 → Λπ 0 , and Λ → pπ − following the process of e + e − → γ * → Λ + c Λ− c .In the helicity frame of the e + e − → Λ + c Λ− c , θ 0 is the polar angle of the Λ + c with respect to the e + beam axis in the e + e − CM system.For the Λ + c → Ξ 0 K + decay, ϕ 1 is the angle between the e + Λ + c and Ξ 0 K + planes, and θ 1 is the polar angle of the Ξ 0 with respect to the direction of Λ− c evaluated in Λ + c 's rest frame.For the Ξ 0 → Λπ 0 decay, ϕ 2 is the angle between the Ξ 0 K + and Λπ 0 planes, and θ 2 is the polar angle of the Λ with respect to the direction of K + evaluated in Ξ 0 's rest frame.For the helicity angles describing the Λ → pπ − decay, ϕ 3 is the angle between the Λπ 0 and pπ − planes, and θ 3 is the polar angle of the proton with respect to the direction of π 0 evaluated in Λ's rest frame.In Ref. [30], ∆ 0 is defined as the phase shift between two individual helicity amplitudes, H λ1,λ2 , for the Λ + c production process γ * (λ 0 ) → Λ + c (λ 1 ) Λ− c (λ 2 ) with γ * 's helicity λ 0 = ±1, and total helicities |λ 1 − λ 2 | = 0 and 1, respectively.In the case where one-photon exchange dominates the production process, ∆ 0 is also the phase between the electric and magnetic form factors of Λ + c [33,34], and α 0 is the angular distribution parameter of Λ + c defined by the helicity amplitude . Similarly, the Λ + c → Ξ 0 K + decay is described by two parameters, α Ξ 0 K + and ∆ Ξ 0 K + , where the latter one is the phase shift between the two helicity amplitudes.The Lee-Yang parameters [30,35] can be obtained with the relations In this analysis, the common free parameters (α Ξ 0 K + and ∆ Ξ 0 K + ) describing the angular distributions for the six data sets are determined by a simultaneous unbinned maximum likelihood fit.The likelihood function is constructed from the joint probability density function (PDF) by Here, f s ( ⃗ ξ i ) is the PDF of the signal process, N data is the number of events in the data and i is the event index.The signal PDF f s ( ⃗ ξ i ) is formulated as where ⃗ ξ i denotes the kinematic angular observables (θ 0,1,2,3 and ϕ 1,2,3 ) and ⃗ η denotes the free parameters (α Ξ 0 K + and ∆ Ξ 0 K + ) to be determined.M ( ⃗ ξ i ; ⃗ η) is the total amplitude [30] of all decay chain and ϵ( ⃗ ξ i ) is the detection efficiency parameterized in terms of the kinematic variables ⃗ ξ i .The background contribution to the joint likelihood is subtracted according to the calculated likelihoods for the combinatorial background based on the inclusive MC simulation and for the misreconstructed signal events based on the signal MC simulation.The integration of the normalization factor is calculated with a large phase space MC sample as where N gen is the total number of the simulated phase space MC events, N MC is the number of the phase space MC events surviving all selection criteria and k MC is the event index.
The systematic uncertainties arise mainly from the reconstruction of final state particles, which is studied with J/ψ → K 0 s K ± π ∓ for kaon, Λ + c → ΛX for Λ, ψ(3686) → J/ψ π 0 π 0 and e + e − → ωπ 0 for π 0 .The systematic uncertainties for ∆E requirement and M BC signal regions are estimated by smearing the phase space MC samples using resolution parameters, and for the background subtraction by taking into account the background shape and size.The uncertainties from the quoted values of α 0 , ∆ 0 , α Λπ 0 , αΛ π 0 , ∆ Λπ 0 , ∆Λ π 0 , α pπ − and α pπ + are estimated by Gaussian sampling considering their uncertainties and refit the angular distribution, and by taking the values in one time uncertainty of Gaussian fit as the uncertainties of this part.A further source of uncertainty is the fit bias, which is the difference before and after the correction from a pull distribution check.Systematic uncertainties from all sources are combined in quadrature to calculate the total systematic uncertainties.All details of systematic uncertainties can be found in Ref. [30].
FIG. 5.The comparison between this work and theoretical predictions, where the branching fraction is taken from PDG (2022) [2].The 1σ, 2σ and 3σ contours correspond to 68.2%, 95.4% and 99.7% conference level, respectively.The blue symbols are theoretical predictions and the red star is result from this work.The definitions of the superscripts a and b can be found in Table I and the theory acronyms are explained in the text.

FIG. 2 .
FIG.2.Distribution of the MBC fitting result at 4.60 GeV, and the corresponding signal yield is 70 ± 8. Black points with error bars are data; the blue shaded region indicates the combinatorial background events and pink shaded region is the mis-reconstructed signal events.

1 FIG. 4 .
FIG. 4. Projections of the best fit onto various variables.Black points with error bars are data; red solid lines are phase space MC re-weighted by angular distribution formula, and represent the fitting result; the blue shaded region denotes the combinatorial background events and the pink shaded region is the mis-reconstructed signal events.

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 MOE 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 (Dated:January 23, 2024)

TABLE I .
Theoretical calculations and experimental measurements of the BF, α Ξ 0 K + , |A|, |B| and δp − δs of Λ + c → Ξ 0 K + .GF is the Fermi constant.The superscript a denotes a model with SU(3) symmetry, while model b includes SU(3) symmetrybreaking effects.The PDG Fit BF also includes a CLEO result on B