Measurement of the absolute branching fraction of D⁺ → K̅⁰ e⁺ν_e via K̅⁰ → π⁰π^0*

With a mass of 3.773 GeV just above the open charm threshold, the ψ(3770) resonance decays predominately into D⁰D̅⁰ or D⁺D⁻ meson pairs. In each event, if a D⁻ meson can be fully reconstructed via its decay into hadrons (in the following called the single tag (ST) D⁻), there must be a recoiling D⁺ meson. Using a double tag technique which was first employed by the MARKIII Collaboration [22], we can measure the absolute branching fraction of the D⁺ → K̅⁰e⁺ν_e decay. Throughout the paper, charge conjugation is implied.

The ST D⁻ mesons are reconstructed using six hadronic decay modes: K⁺π⁻π⁻, , K⁺π⁻π⁻π⁰, , and K⁺K⁻π⁻. The daughter particles and π⁰ are reconstructed via and π⁰ → γγ, respectively.

All charged tracks are required to be reconstructed within the good MDC acceptance |cosθ| < 0.93, where θ is the polar angle of the track with respect to the positron beam direction. All tracks except those from decays are required to originate from the interaction region defined as V_xy < 1.0 cm and |V_z| < 10.0 cm. Here, V_xy and |V_z| are the distances of closest approach to the Interaction Point (IP) of the reconstructed track in the plane transverse to and along the beam direction, respectively. For PID of charged particles [23], we combine the dE/dx and TOF information to calculate Confidence Levels for the pion and kaon hypotheses (CL_π and CL_K). A charged track is taken as kaon (pion) if it has CL_K > CL_π (CL_π > CL_K).

The charged tracks from decays are required to satisfy |V_z| < 20.0 cm. The two oppositely charged tracks, which are assumed as π⁺π⁻ without PID, are constrained to originate from a common vertex. A π⁺π⁻ combination is considered as a candidate if its invariant mass lies in the mass window , where is the nominal mass [24]. The π⁺π⁻ combinations with L/σ_L > 2 are retained, where σ_L is the uncertainty of the reconstructed decay length L.

Photon candidates are selected by using the EMC information. The shower time is required to be within 700 ns of the event start time, which is the interval of the trigger start time to the real collision time [25]. The shower energy is required to be greater than 25 (50) MeV in the barrel (endcap) region. The opening angle between the candidate shower and the closest charged track is required to be greater than 10°. A γγ combination is considered as a π⁰ candidate if its invariant mass falls in (0.115, 0.150) GeV/c². To obtain better mass resolution for the D⁻ candidates, the γγ invariant mass is constrained to the π⁰ nominal mass [24] via a kinematic fit.

To suppress combinatorial backgrounds, we define the variable ΔE = E_mKnπ − E_beam, which is the difference between the measured energy of the mKnπ (m = 1, 2; n = 1, 2, 3) combination (E_mKnπ) and the beam energy (E_beam). For each ST mode, if there is more than one mKnπ combination satisfying the above selection criteria, only the one with the minimum |ΔE| is kept. The ΔE is required to be within (−25,+25) MeV for the K⁺π⁻π⁻, , and K⁺K⁻π⁻ combinations, and be within (−55,+40) MeV for the K⁺π⁻π⁻π⁰ and combinations.

To measure the yield of ST D⁻ mesons, we perform maximum likelihood fits to the spectra of the beam energy constrained masses of the accepted mKnπ combinations, as shown in Fig. 1. Here, is the measured momentum of the mKnπ combination. In the fits, the D⁻ signal is modeled by the MC simulated M_BC distribution convolved with a double Gaussian function, and the combinatorial background is described by an ARGUS function [26]. The parameters of the double Gaussian function and the ARGUS function are float. The candidates in the ST D⁻ signal region defined as (1.863,1.877) GeV/c² are kept for further analysis. Single-tag reconstruction efficiencies ∊_ST are estimated by analyzing the inclusive MC sample. The ST yields N_ST and the ST efficiencies are summarized in Table 1. The total ST yield is , where the uncertainty is the quadratic sum of the uncertainties from all the M_BC fits.

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Table 1. Summary of the ST yields , the ST and DT efficiencies ( and ), and the reconstruction efficiencies of . The efficiencies do not include the branching fractions for (used in the reconstruction of ST D⁻ mesons), K̅⁰ → π⁰π⁰ and π⁰ → γγ. The uncertainties are statistical only. The index i represents the ith ST mode.

ST mode i
D− → K+π−π−	782669±990	50.61±0.06	13.39±0.07	26.45±0.14
	91345±320	50.41±0.17	13.81±0.22	27.40±0.44
D− → K+π−π−π0	251008±1135	26.74±0.09	6.23±0.06	23.29±0.25
	215364±1238	27.29±0.07	6.88±0.07	25.21±0.28
	113054±889	28.31±0.12	6.74±0.10	23.79±0.37
D− → K+K−π−	69034±460	40.83±0.24	10.54±0.20	25.81±0.50

In the system recoiling against the ST D⁻ mesons, the D⁺ → K̅⁰e⁺ν_e candidates, called the double tag (DT) events, are selected via . It is required that there be at least four good photons and only one good charged track that have not been used in the ST selection. The good charged track, photons and π⁰ mesons are selected using the same criteria as those used in the ST selection. If there are multiple π⁰π⁰ combinations satisfying these selection criteria, only the combination with the minimum value of is retained, where the and are the chi-squares of the mass constrained fits on π⁰ → γγ. A π⁰π⁰ combination is considered as a K̅⁰ candidate if its invariant mass falls in (0.45, 0.51) GeV/c². For electron PID, we combine the dE/dx, TOF and EMC information to calculate Confidence Levels for the electron, pion and kaon hypotheses (CL_e, CL_π and CL_K), respectively. The electron candidate is required to have CL_e > 0.001 and CL_e/(CL_e + CL_π + CL_K) > 0.8, and to have a charge opposite to the ST D⁻ meson. To partially recover the effects of FSR and bremsstrahlung, the four-momenta of photon(s) within 5° of the initial electron direction are added to the electron four-momentum measured by the MDC. To suppress the backgrounds associated with fake photon(s), we require that the maximum energy of any of the extra photons, which have not been used in the DT selection, be less than 300 MeV.

In order to obtain the information of the missing neutrino, we define the kinematic quantity

where E_miss and are the total energy and momentum of the missing particle in the event, respectively. E_miss is calculated by

where E_K̅⁰ and E_e⁺ are the energies carried by K̅⁰ and e⁺, respectively. is calculated by

where , and are the momenta of D⁺, K̅⁰ and e⁺, respectively. To obtain better U_miss resolution, is constrained by

where is the momentum direction of the ST D⁻ meson and m_D⁺ is the D⁺ nominal mass [24].

To determine the number of DT events, we perform a maximum likelihood fit to the U_miss distribution of the accepted DT candidates, as shown in Fig. 2. In the fit, the DT signal and the combinatorial background are modeled by the MC simulated U_miss shapes, respectively. From the fit, we obtain the DT yield in data as

where the uncertainty is from U_miss fit.

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The efficiency of reconstructing the DT events, called the DT efficiency _DT, is determined by analyzing the signal MC events. The DT efficiencies obtained from MC simulations are corrected by the differences of π⁰ reconstruction efficiencies between data and MC simulations for the signal side. Dividing _DT by _ST, we obtain the reconstruction efficiency for D⁺ → K̅⁰e⁺ν_e in each ST mode, _D⁺ → K̅⁰e⁺ν_e, as summarized in Table 1. Weighting them by the ST yields observed in data, we obtain the averaged reconstruction efficiency of D⁺ → K̅⁰e⁺ν_e

which does not include the branching fractions of K̅⁰ → π⁰π⁰ and π⁰ → γγ.

The branching fraction of D⁺ → K̅⁰e⁺ν_e is determined by

where N_DT is the DT yield, is the total ST yield, is the averaged reconstruction efficiency of D⁺ → K̅⁰e⁺ν_e, ℬ(K̅⁰ → π⁰π⁰) and ℬ(π⁰ → γγ) are the branching fractions of K̅⁰ → π⁰π⁰ and π⁰ → γγ [24], respectively. Here, we assume that constitutes half the decays of the neutral kaons.

Inserting the numbers of N_DT, , , ℬ(K̅⁰ → π⁰π⁰) and ℬ(π⁰ → γγ) in Eq. (7), we obtain

where the uncertainty is statistical only.

In the measurement of the branching fraction, the systematic uncertainty arises from the uncertainties in the fits to the M_BC spectra of the ST candidates, the ΔE, M_BC and K̅⁰(π⁰π⁰) mass requirements, the π⁰ reconstruction, the e⁺ tracking, the e⁺ PID, the requirement, the U_miss fit, the selection method, the MC statistics, the quoted branching fractions and the MC generator.

The uncertainty in the fits to the M_BC spectra of the ST candidates is estimated to be 0.5% by observing the relative change of the ST yields of data and MC when varying the fit range, the combinatorial background shape or the endpoint of the ARGUS function. To estimate the uncertainties in the ΔE, M_BC and K̅⁰(π⁰π⁰) mass requirements, we examine the change in branching fractions when enlarging the ΔE selection window by 5 or 10 MeV; varying the M_BC selection window by ±1 MeV/c² and using alternative K̅⁰(π⁰π⁰) mass windows (0.460, 0.505), (0.470, 0.500), (0.480, 0.500) GeV/c², respectively. The maximum changes in the branching fractions, 0.3%, 0.2%, and 0.9%, are assigned as the systematic uncertainties. The π⁰ reconstruction efficiency is examined by analyzing the DT hadronic decays D⁰ → K⁻π⁺ and K⁻π⁺π⁺π⁻ versus D̅⁰ → K⁺π⁻π⁰ and . The difference of the π⁰ reconstruction efficiencies between data and MC is found to be (−1.0±1.0)% per π⁰. The systematic uncertainty in π⁰ reconstruction is taken to be 1.0% for each π⁰ after correcting the MC efficiency of D⁺ → K̅⁰e⁺ν_e to data. The data-MC differences of the e⁺ tracking and PID efficiencies are estimated by analyzing e⁺e⁻ → γe⁺e⁻events. To consider different kinematic distributions of e⁺, the data-MC differences are re-weighted by the momentum and cosθ distributions of e⁺ in the D⁺ → K̅⁰e⁺ν_e decays. The re-weighted data-MC difference 0.5% is quoted as the systematic uncertainties of the e⁺ tracking and PID efficiencies. The uncertainty in the requirement is estimated to be 0.1% by analyzing the DT hadronic DD̅ decays. The uncertainty in the U_miss fit is assigned to be 0.5%, which is obtained by comparing with the nominal value of the branching fraction measured with an alternative signal shape obtained with different requirements on the MC-truth matched signal shape, an alternative background shape after changing the relative ratios of the dominant backgrounds (doubling each of the simulated backgrounds for D⁰D̅⁰, D⁺D⁻ and qq̅ continuum processes), and alternative fit range (± 50 MeV). The difference of 0.3% in the π⁰π⁰ acceptance efficiencies of the minimum requirement between data and MC, which is estimated by the DT hadronic decays D⁰ → K⁻π⁺π⁰ versus D̅⁰ → K⁺π⁻π⁰, is assigned as a systematic uncertainty due to the selection method. In this analysis, the mesons from the signal side are formed with photon candidates reconstructed under the assumption that they originate at the IP. We examine the DT efficiencies of the signal MC events in which the lifetimes of meson from the signal side are set to the nominal value and 0, respectively. The difference of these two DT efficiencies, which is less than 0.2%, is taken as the systematic uncertainty of the reconstruction. The uncertainties in the MC statistics and the ℬ(K̅⁰ → π⁰π⁰) are 0.5% and 0.2% [24], respectively. In our previous work, the uncertainty in the signal MC generator is estimated to be 0.1%, which is obtained by comparing the DT efficiencies before and after re-weighting the q²(=(p_D − p_K)²) distribution of the signal MC events of D⁰ → K⁻e⁺ν_e to the distribution found in data [11], where the p_D and p_K are the four-momenta of the D and K mesons. The systematic uncertainties are summarized in Table 2. Adding all uncertainties in quadrature, we obtain the total systematic uncertainty to be 2.5%.

Table 2. Relative systematic uncertainties (in %) in the measurement of ℬ(D⁺ → K̅⁰e⁺ν_e).

source	uncertainty
MBC fit	0.5
ΔE requirement	0.3
MBC∈(1.863,1.877) GeV/c2	0.2
Mπ0π0∈(0.45, 0.51) GeV/c2	0.9
π0 reconstruction	2.0
tracking for e+	0.5
PID for e+	0.5
	0.1
Umiss fit	0.5
selection method	0.3
reconstruction	0.2
MC statistics	0.5
ℬ(K̅0 → π0π0)	0.2
MC generator	0.1
total	2.5

The analysis procedure is examined by an input and output check using an inclusive MC sample equivalent to a luminosity of 3.26 fb⁻¹. Using the same selection criteria as those used in data analysis, we obtain the ST yield, the DT yield and the weighted reconstruction efficiency of D⁺ → K̅⁰e⁺ν_e to be 1683631±1768, 5802±85 and (26.07±0.11)%, where no efficiency correction has been performed. Based on these numbers, we determine the branching fraction ℬ(D⁺ → K̅⁰e⁺ν_e) = (8.82±0.13)%, where the uncertainty is statistical only. The measured branching fraction is in excellent agreement with the input value of 8.83%.

To validate the reliability of the MC simulation, we examine the cosθ and momentum distributions of K̅⁰ and e⁺ of the D⁺ → K̅⁰e⁺ν_e candidates, as shown in Fig. 3. We can see that the consistency between simulation and data is very good.

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Based on the analysis of 2.93 fb⁻¹ data collected at with the BESIII detector, we measure the absolute branching fraction ℬ(D⁺ → K̅⁰e⁺ν_e) = (8.59 ± 0.14 ± 0.21)%, using . Figure 4 presents a comparison of ℬ(D⁺ → K̅⁰e⁺ν_e) measured in this work with the results obtained by other experiments. Our result is well consistent with the other measurements within uncertainties and has a precision comparable to the PDG value [24]. Our measurement will be helpful to improve the precision of the world average value of ℬ(D⁺ → K̅⁰e⁺ν_e).

Combining the PDG values for ℬ(D⁰ → K⁻e⁺ν_e), ℬ(D⁺ → K̅⁰e⁺ν_e) [24], and the lifetimes of D⁰ and D⁺ mesons (τ_D⁰ and τ_D⁺) [24] with the value of ℬ(D⁺ → K̅⁰e⁺ν_e) measured in this work, we determine

where is the uncertainty averaged branching fraction based on the PDG value and the one measured in this work. Combining with the branching fraction measured in this work, the precision of the test of the isospin symmetry is improved.

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