Measurements of the branching fractions of the semileptonic decays $\Xi_{c}^{0} \to \Xi^{-} \ell^{+} \nu_{\ell}$ and the asymmetry parameter of $\Xi_{c}^{0} \to \Xi^{-} \pi^{+}$

Using data samples of 89.5 and 711 fb$^{-1}$ recorded at energies of $\sqrt{s}=10.52$ and $10.58$ GeV, respectively, with the Belle detector at the KEKB $e^+e^-$ collider, we report measurements of branching fractions of semileptonic decays $\Xi_{c}^{0} \to \Xi^{-} \ell^{+} \nu_{\ell}$ ($\ell=e$ or $\mu$) and the $CP$-asymmetry parameter of $\Xi_{c}^{0} \to \Xi^{-} \pi^{+}$ decay. The branching fractions are measured to be ${\cal B}(\Xi_{c}^{0} \to \Xi^{-} e^{+} \nu_{e})=(1.31 \pm 0.04 \pm 0.07 \pm 0.38)\%$ and ${\cal B}(\Xi_{c}^{0} \to \Xi^{-} \mu^{+} \nu_{\mu})=(1.27 \pm 0.06 \pm 0.10 \pm 0.37)\%$, and the decay parameter $\alpha_{\Xi\pi}$ is measured to be $0.63 \pm 0.03 \pm 0.01$ with much improved precision compared to the current world average. The corresponding ratio ${\cal B}(\Xi_{c}^{0} \to \Xi^{-} e^{+} \nu_{e})/{\cal B}(\Xi_{c}^{0} \to \Xi^{-} \mu^{+} \nu_{\mu})$ is $1.03 \pm 0.05\pm 0.07$, which is consistent with the expectation of lepton flavor universality. The first measured asymmetry parameter ${\cal A}_{CP} = (\alpha_{\Xi^{-}\pi^{+}} + \alpha_{\bar{\Xi}^{+}\pi^{-}})/(\alpha_{\Xi^{-}\pi^{+}} - \alpha_{\bar{\Xi}^{+}\pi^{-}}) = 0.024 \pm 0.052 \pm 0.014$ is found to be consistent with zero. The first and the second uncertainties above are statistical and systematic, respectively, while the third ones arise due to the uncertainty of the $\Xi_{c}^{0} \to \Xi^{-} \pi^+$ branching fraction.

Charmed baryons play an important role in studies of strong and weak interactions, especially via investigations of their semileptonic decays [1][2][3][4][5][6][7][8] and charge-parity violation (CPV) [9,10].Such decay amplitudes are the product of a well-understood leptonic current for the lepton system and a more complicated hadronic current for the quark transition.For semileptonic decays of SU (3) anti-triplets, Λ + c and Ξ +,0 c , thanks to the spin-zero light diquark constituents, a simpler and more powerful theoretical calculation of form factors, hadronic structures, and nonperturbative aspects of strong interactions can be performed in a relatively simple version of quantum chromodynamics (QCD) [1].
Though the Standard Model accommodates CPV which is one of the conditions needed to explain our matter-dominated universe [31], the magnitude of this effect as predicted by the KM mechanism is not sufficient [32].CPV has been established in many meson decays [33][34][35][36][37][38][39][40][41], but CPV has never been observed in any baryon system.Studies of CP -violating processes in the charm baryon sector are very scarce [13,14,[42][43][44].Since there should be CPV sources other than those currently known, it is imperative to search for those also in the charm baryon sector, and several phenomenology studies about CPV in charmed baryon decays have been conducted [45][46][47][48].
LFU is tested using these measured results.Charm baryons are produced in processes such as e + e − → cc → Ξ 0 c + anything.Ξ − is reconstructed via the Λπ − mode, and Λ decays into pπ − .The decay parameters of α + and α − and the CP -asymmetry parameter A CP are first measured for Ξ 0 c ( Ξ0 c ) → Ξπ.To optimize the signal selection criteria and calculate the signal reconstruction efficiency, we use Monte Carlo (MC) simulated events.The e + e − → cc process is simulated with pythia [61], while the signal events of Ξ 0 c semileptonic decays are generated using form factors from Lattice QCD calculation [62], and Ξ 0 c → Ξ − π + decays are generated with EvtGen [63].The MC events are processed with a detector simulation based on geant3 [64].Simulated Υ(4S) → B B events with B = B + or B 0 , and e + e − → q q events with q = u, d, s, c at √ s = 10.52 GeV and 10.58 GeV, are used as background samples in which the signals are removed, which are called generic simulated samples.
For leptons and pions which are direct childs of Ξ 0 c , the impact parameters perpendicular to and along the e + beam direction with respect to the interaction point are required to be less than 0.5 cm and 4 cm, respectively, and transverse momentum is restricted to be higher than 0.1 GeV/c.For charged tracks, information from different detector subsystems is combined to form the likelihood L i for species (i), where i = e, µ, π, K, or p [65].A track not from Λ with a likelihood ratio L π /(L K + L π )>0.6 is identified as a pion.With this selection, the pion identification efficiency is about 94%, while 5% of the kaons are misidentified as pions.A track with a likelihood ratio L e /(L e + L non−e )>0.9 is identified as an electron [66].The γ conversions are removed by examining all combinations of an e ± track with other oppositely-charged tracks in the event that are identified as e ∓ , and requiring e + e − invariant mass larger than 0.2 GeV/c 2 .Tracks with L µ /(L µ + L K + L π )>0.9 are considered as muon candidates [67].Furthermore, the muon tracks required to hit at least five layers of the K 0 L and muon subdetector, and not to be identified as kaons with L K /(L K +L π ) < 0.4 to suppress backgrounds due to misidentification.With the above selections, the efficiencies of electron and muon identification are 96% and 75%, respectively, with pion fake rates less than 2%.

Candidate Λ baryons are reconstructed in the decay
), where σ denotes the mass resolution.Here and throughout the text, M i represents a measured invariant mass and m i denotes the nominal mass of the particle i [30].The proton track from Λ decay is required to satisfy L p /(L π + L p )>0.2 and L p /(L K + L p )>0.2 with an efficiency of 95%.We define the Ξ − signal region as |M Λπ − − m Ξ − | < 6.5 MeV/c 2 (∼ 3σ), and Ξ − mass sidebands as 1.294 GeV/c 2 < M Λπ − < 1.307 GeV/c 2 and 1.337 GeV/c 2 < M Λπ − < 1.350 GeV/c 2 .To suppress combinational background, we require the flight directions of Λ and Ξ − candidates, which are reconstructed from their fitted production and decay vertices, to be within five degrees of their momentum directions.
We also require the scaled momentum p * Ξ − X /p * max >0.45 (X = e + , µ + or π + ), where p * Ξ − X is the momentum of the Ξ − X system in the center-of-mass frame and (E beam is the beam energy).This requirement removes all Ξ 0 c → Ξ − π + decays with Ξ 0 c produced in B decays from the √ s = 10.58GeV sample.For Ξ 0 c → Ξ − ℓ + ν ℓ , the cosine of the opening angle between Ξ − and ℓ + is further required to be larger than 0.25.
After the above selections, the obtained Ξ − e + , Ξ − µ + , and Ξ − π + mass spectra from data in p * Ξ − X /p * max regions of (0.45, 0.55), (0.55, 0.65), (0.65, 0.75), and (0.75, 1) are shown in Fig. 1.The Ξ 0 c signals are extracted from maximum-likelihood fits to these invariant mass spectra.For Ξ 0 c semileptonic decays, the signal shapes are taken directly from MC simulation.The background shapes from wrongly constructed Ξ candidates can be described by the M Ξ − ℓ + distributions of Ξ − mass sidebands.The backgrounds from Ξ c → Ξπℓ + ν ℓ are taken from MC simulations of those processes.The backgrounds from e + e − → q q due to mis-selected ℓ + can be represented by the M Ξ − ℓ + distributions of Ξ − ℓ − events with their normalized Ξ − mass sidebands subtracted.The other backgrounds are from e + e − → B B with Ξ − from one B and ℓ + from another B, whose shapes are taken from generic simulated samples.Background from Ω 0 c → Ξ − ℓ + ν ℓ decays is assumed to be negligible since it is a c → d process and should be suppressed strongly.In fitting the Ξ − µ + mass spectrum, an additional background of simulated Ξ 0,+ c → Ξ − π + +hadrons events from generic simulated samples is added.In the fit above, the shapes of all fit components are fixed while their yields are floated.In fitting the Ξ − π + mass spectrum, the Ξ 0 c signal shape is parameterized with a double-Gaussian function with same mean value and all other parameters floated, while the background shape is represented with a 1st-order polynomial.Figure 1  The Ξ 0 c semileptonic decay branching fractions are calculated using where are the fitted signal yield and detection efficiency, respectively, in each is the efficiency of the p * Ξ − X /p * max >0.45 requirement for each channel and is 0.783, 0.574, and 0.588 for Ξ 0 c → Ξ − π + , Ξ − e + ν e , and Ξ − µ + ν µ , respectively.
Using the results listed in Table I, we obtain    Table I: List of the fitted signal yields and the corresponding detection efficiencies in each ) of data at √ s = 10.52 GeV and 10.58 GeV.The last column gives the ratios of branching fractions  In the following, Ξ 0 c → Ξ − π + and Ξ0 c → Ξ+ π − decays are treated separately to extract decay parameters of α + and α − , and To obtain the θ Ξ distribution, we divided the 2D plane of p * Ξπ /p * max versus cos θ Ξ into 4 × 5 bins with the bin edges for p * Ξπ /p * max and cos θ Ξ set as (0.45, 0.55, 0.65, 0.75, 1.0) and (−1.0, −0.6, −0.2, 0.2, 0.6, 1.0), respectively.The detection efficiency in each 2D bin is calculated individually.The number of Ξ 0 c ( Ξ0 c ) signal events in each 2D bin is obtained by fitting the corresponding M Ξπ distribution with the method used in the branching fraction measurements.The number of signal events in each cos θ Ξ bin is the sum of the efficiency-corrected signal yields in corresponding p * Ξπ /p * max bins.The fitting method was checked using special simulated samples with a range of values of A CP .The final efficiencycorrected cos θ Ξ distributions for (a) Ξ 0 c → Ξ − π + and (b) Ξ0 c → Ξ+ π − decays are shown in Fig. 2. Using Eq. ( 1) with α Ξ − = −0.376± 0.008 and αΞ+ = 0.371 ± 0.007 [50], the fits yield α + = −0.64 ± 0.05 and α − = 0.61 ± 0.05, resulting in A CP = 0.024±0.052.Here, the uncertainties are statistical only.There are several sources of systematic uncertainties contributing to the branching fraction measurements.Using the D * + → D 0 π + , D 0 → K − π + , Λ → pπ, and J/ψ → ℓℓ control samples, the particle identification uncertainties (σ PID ) are 0.51% − 0.55% per pion, 0.55% − 0.93% per electron, and 0.44% − 0.84% per muon, depending on the p * Ξ − X /p * max region.The systematic uncertainties associated with tracking efficiency and Ξ − selection cancel in the branching fraction ratio measurements.We estimate the systematic uncertainties associated with the fitting procedures (σ fit ) for Ξ 0 c → Ξ − ℓ + ν ℓ and Ξ 0 c → Ξ − π + separately.For Ξ 0 c → Ξ − ℓ + ν ℓ decays, we change the bin width of the M Ξ − ℓ + spectra by ±5 MeV/c 2 , change the Ξ − mass sidebands from two times that of the signal region to three times that of the signal region, add the background component from Ξ c → Ξπ + π − ℓ + ν ℓ with its shape taken from MC simulation and yields floated, and take the difference of the fitted signal yields as σ fit for each p * Ξ − ℓ + /p * max bin (2.30% − 4.54% for the electron mode and 2.34% − 5.10% for the muon mode).For Ξ 0 c → Ξ − π + , we estimate σ fit by changing the range of the fit and the order of the background polynomial, and take the differences of the fitted signal yields as systematic uncertainties (1.03% − 1.46% depending on the p * Ξ − π + /p * max region).By using the control sample Ξ 0 c → Ξ − π + , the maximum difference in selection efficiency of the requirement p * Ξ − π − /p * max >0.45 between weighted MC simulation based on p * Ξ − X /p * max distribution from data and different signal MC simulations with different fragmentation functions in PYTHIA generator [61] is 3.0%, which is taken as the systematic uncertainty (σ ε pop ).For semileptonic decays, the uncertainties of the form factors in Ref. [62] introduce a 3.1% (3.6%) uncertainty in the electron (muon) mode (σ FF ).The change of the branching fraction measured with the sub-datasets with p * Ξ − X /p * max >0.75 that removes all background from B decay is taken as the uncertainty associated with modeling of the B-decay background (σ B B ) which is 2.5% (6.3%) for electron (muon) mode.The systematic uncertainties σ PID (σ fit ) are added linearly (in quadrature) weighted by and then summed with σ ε pop , σ FF , and σ B B in quadrature to yield the total systematic uncertainty (σ B ) for each Ξ 0 c decay mode, which yields 4.6%, 7.6%, and 3.1% for the electron, muon, and pion mode, respectively.The final systematic uncertainty on the branching fraction is the sum of the corresponding two σ B s in quadrature, which yields 5.6% for B(Ξ 0 c → Ξ − e + ν e ), and 8.2% for B(Ξ 0 c → Ξ − µ + ν µ ).The uncertainty of 28.9% on B(Ξ 0 c → Ξ − π + ) [29] is treated as an independent systematic uncertainty.The total systematic uncertainty for B(Ξ The sources of systematic uncertainties in α ± include fitting procedures (σ α ± fit ) and uncertainties on α Ξ ± values (σ α ± α Ξ ∓ ).σ α ± fit are estimated to be 0.2% with a toy MC method whose simulated distributions of α ± are found to be unbiased.The uncertainties on α Ξ ± values are σ α + α Ξ − = 2.1% and σ α − α Ξ+ = 1.9% [50], which are the leading systematic uncertainties.The final systematic uncertainties of α ± are σ Finally, the systematic uncertainties for α + , α − , and A CP are estimated to be 0.01, 0.01, and 0.014, respectively.
Y. B. Li acknowledges the support from China Postdoctoral Science Foundation (2020TQ0079).We thank the KEKB group for excellent operation of the accelerator; the KEK cryogenics group for efficient solenoid operations; and the KEK computer group, the NII, and PNNL/EMSL for valuable computing and shows the fitted results in each p * Ξ − X /p * max bin labelled at the bottom for (a) Ξ 0 c → Ξ − e + ν e , (b) Ξ 0 c → Ξ − µ + ν µ , and (c) Ξ 0 c → Ξ − π + .The fitted result in each p * Ξ − X /p * max bin together with the corresponding detection efficiency are listed in TableI.The background sources and fit methods are validated with generic simulated samples.

Figure 1 :
Figure 1: The fits to the M Ξ − e + , M Ξ − µ + , and M Ξ − π + distributions of the selected (a) Ξ 0 c → Ξ − e + νe, (b) Ξ 0 c → Ξ − µ + νµ, and (c) Ξ 0 c → Ξ − π + candidates in each p * Ξ − X /p * max bin listed at the bottom.The points with error bars represent the data from √ s = 10.52 GeV and 10.58 GeV, the solid blue lines are the best fits, and the violet dashed lines are the fitted total backgrounds.The other components of the fits are indicated in the legends.

8 Figure 2 :
Figure 2: The maximum likelihood fits to the efficiencycorrected cos θΞ distributions of data to extract (a) α Ξ − π + and (b) αΞ+ π − for Ξ 0 c → Ξ − π + and Ξ0 c → Ξ+ π − decays.The points with error bars represent data from the combined samples at √ s = 10.52 GeV and 10.58 GeV, and the red solid lines are the best fits.
p * max range.Quoted uncertainties are statistical only.