Study of the decay $D^0\rightarrow \bar{K}^0\pi^-e^+\nu_e$

We report a study of the decay $D^0 \rightarrow \bar{K}^0\pi^-e^+\nu_{e}$ based on a sample of $2.93~\mathrm{fb}^{-1}$ $e^+e^-$ annihilation data collected at the center-of-mass energy of 3.773~GeV with the BESIII detector at the BEPCII collider. The total branching fraction is determined to be $\mathcal{B}(D^0\rightarrow \bar{K}^0\pi^-e^+\nu_{e})=(1.434\pm0.029({\rm stat.})\pm0.032({\rm syst.}))\%$, which is the most precise to date. According to a detailed analysis of the involved dynamics, we find this decay is dominated with the $K^{*}(892)^-$ contribution and present an improved measurement of its branching fraction to be $\mathcal{B}(D^0\rightarrow K^{*}(892)^-e^+\nu_e)=(2.033\pm0.046({\rm stat.})\pm0.047({\rm syst.}))\%$. We further access their hadronic form-factor ratios for the first time as $r_{V}=V(0)/A_1(0)=1.46\pm0.07({\rm stat.})\pm0.02({\rm syst.})$ and $r_{2}=A_2(0)/A_1(0)=0.67\pm0.06({\rm stat.})\pm0.01({\rm syst.})$. In addition, we observe a significant $\bar{K}^0\pi^-$ $S$-wave component accounting for $(5.51\pm0.97({\rm stat.})\pm0.62({\rm syst.}))\%$ of the total decay rate.


I. INTRODUCTION
The studies on semileptonic (SL) decay modes of charm mesons provide valuable information on the weak and strong interactions in mesons composed of heavy quarks [1]. The semileptonic partial decay width is related to the product of the hadronic form factor describing the strong-interaction in the initial and final hadrons, and the Cabibbo-Kobayashi-Maskawa (CKM) matrix elements |V cs | and |V cd |, which parametrize the mixing between the quark flavors in the weak interaction [2]. The couplings |V cs | and |V cd | are tightly constrained by the unitarity of the CKM matrix. Thus, detailed studies of the dynamics of the SL decays allow measurements of the hadronic form factors, which are important for calibrating the theoretical calculations of the involved strong interaction.
The relative simplicity of theoretical description of the SL decay D →Kπe + ν e [3] makes it a optimal place to study theKπ system, and to further determine the hadronic transition form factors. Measurements ofKπ resonant and nonresonant amplitudes in the decay D + → K − π + e + ν e have been reported by the CLEO [4], BaBar [5] and BESIII [6] collaborations. In these studies a nontrival S-wave component is observed along with the dominant P -wave one. A study of the dynamics in the isospin-symmetric mode D 0 →K 0 π − e + ν e will provide complementary information on theKπ system. Furthermore, the form factors in the D → V e + ν e transition, where V refers to a vector meson, have been measured in decays of D + →K * 0 e + ν e [4][5][6], D → ρe + ν e [7] and D + → ωe + ν e [8], while no form factor in D 0 → K * (892) − e + ν e has been studied yet. Therefore, the study of the dynamics in the decay D 0 → K * (892) − e + ν e provides essentially additional information on the family of D → V e + ν e decays.
In this paper, an improved measurement of the absolute branching fraction (BF) and the first measurement of the form factors of the decay D 0 →K 0 π − e + ν e are reported. These measurements are performed using an e + e − annihilation data sample corresponding to an integrated luminosity of 2.93 fb −1 produced at √ s = 3.773 GeV with the BEPCII collider and collected with the BESIII detector [9].

II. BESIII DETECTOR AND MONTE CARLO SIMULATION
The BESIII detector is a cylindrical detector with a solidangle coverage of 93% of 4π. The detector consists of a Helium-gas based main drift chamber (MDC), a plastic scintillator time-of-flight (TOF) system, a CsI(Tl) electromagnetic calorimeter (EMC), a superconducting solenoid providing a 1.0 T magnetic field and a muon counter. The charged particle momentum resolution is 0.5% at a transverse momentum of 1 GeV/c. The photon energy resolution in EMC is 2.5% in the barrel and 5.0% in the end-caps at energies of 1 GeV. More details about the design and performance of the detector are given in Ref. [9].
A GEANT4-based [10] simulation package, which includes the geometric description of the detector and the detector response, is used to determine signal detection efficiencies and to estimate potential backgrounds. The production of the ψ(3770), initial state radiation production of the ψ(2S) and J/ψ, and the continuum processes e + e − → τ + τ − and e + e − → qq (q = u, d and s) are simulated with the event generator KKMC [11]. The known decay modes are generated by EVTGEN [12] with the branching fractions set to the world-average values from the Particle Data Group [13], while the remaining unknown decay modes are modeled by LUNDCHARM [14]. The generation of simulated signals D 0 →K 0 π − e + ν e incorporates knowledge of the form factors, which are obtained in this work.

III. ANALYSIS
The analysis makes use of both "single-tag" (ST) and "double-tag" (DT) samples of D decays. The single-tag sample is reconstructed in one of the final states listed in Table I, which are called the tag decay modes. Within each ST sample, a subset of events is selected where the other tracks in the event are consistent with the decay D 0 →K 0 π − e + ν e . This subset is referred as the DT sample. For a specific tag mode i, the ST and DT event yields are expressed as where N D 0D0 is the number of D 0D0 pairs, B i ST and B SL are the BFs of theD 0 tag decay mode i and the D 0 SL decay mode, ǫ i ST is the efficiency for finding the tag candidate, and ǫ i DT is the efficiency for simultaneously finding the tagD 0 and the SL decay. The BF for the SL decay is given by where N DT is the total yield of DT events, N ST is the total ST yield, and ǫ SL = ( i N i is the average efficiency of reconstructing the SL decay, weighted by the measured yields of tag modes in data. Selection criteria for photons, charged pions and charged kaons are the same as those used in Ref. [15]. To reconstruct a π 0 candidate in the decay mode π 0 → γγ, the invariant mass of the candidate photon pair must be within (0.115, 0.150) GeV/c 2 . To improve the momentum resolution, a kinematic fit is performed to constrain the γγ invariant mass to the nominal π 0 mass [13]. The χ 2 of this kinematic fit is required to be less than 20. The fitted π 0 momentum is used for reconstruction of theD 0 tag candidates.
The STD 0 decays are identified using the beam constrained mass, where pD0 is the momentum of theD 0 candidate in the rest frame of the initial e + e − system. To improve the purity of the tag decays, the energy difference ∆E = √ s/2 − ED0 for each candidate is required to be within approximately ±3σ ∆E around the fitted ∆E peak, where σ ∆E is the ∆E resolution and ED0 is the reconstructedD 0 energy in the initial e + e − rest frame. The explicit ∆E requirements for the three ST modes are listed in Table I. The distributions of the variable M BC for the three ST modes are shown in Fig. 1. Maximum likelihood fits to the M BC distributions are performed. The signal shape is derived from the convolution of the MC-simulated signal template function with a double-Gaussian function to account for resolution difference between MC simulation and data. An ARGUS function [16] is used to describe the combinatorial background shape. For each tag mode, the ST yield is obtained by integrating the signal function over the D 0 signal region specified in Table I. In addition to the combinatorial background, there are also small wrong-sign (WS) peaking backgrounds in the STD 0 samples, which are from the doubly Cabibbo-suppressed decays ofD 0 → K − π + , K − π + π 0 and K − π + π + π − . TheD 0 → K 0 S K − π + , K 0 S → π + π − decay shares the same final states as the WS background ofD 0 → K − π + π + π − . The sizes of these WS peaking backgrounds are estimated from simulation, and are subtracted from the corresponding ST yields. The background-subtracted ST yields are listed in Table I. The total ST yield summed over all three ST modes is N ST = (2277.2 ± 2.3) × 10 3 , where the uncertainty is statistical only.
Candidates for the SL decay D 0 →K 0 π − e + ν e are selected from the remaining tracks recoiling against the STD 0 2 mesons. TheK 0 meson is reconstructed as a K 0 S . The K 0 S mesons are reconstructed from two oppositely charged tracks 4 and the invariant mass of the K 0 S candidate is required to be within (0.485, 0.510) GeV/c 2 . For each K 0 S candidate, a fit 6 is applied to constrain the two charged tracks to a common Decay vertex, and this K 0 S decay vertex is required to be separated 8 from the interaction point by more than twice the standard deviation of the measured flight distance. A further requirement 10 is that there must only be two other tracks in the event and that they must be of opposite charge. The electron hypoth-12 esis is assigned to the track that has the same charge as that of the kaon on the tag side. For electron particle identifica-14 tion (PID), the specific ionization energy losses measured by the MDC, the time of flight, and the shower properties from 16 the electromagnetic calorimeter (EMC) are used to construct likelihoods for electron, pion and kaon hypotheses (L e , L π 18 and L K ). The electron candidate must satisfy L e > 0.001 and L e /(L e + L π + L K ) > 0.8. Additionally, the EMC en-20 ergy of the electron candidate has to be more than 70% of the track momentum measured in the MDC (E/p > 0.7c).

22
The energy loss due to bremsstrahlung is partially recovered by adding the energy of the EMC showers that are within 24 5 • of the electron direction and not matched to other particles [17]. The pion hyphotesis is assigned to the remaining 26 charged track and must satisfy the same criteria as in Ref. [15]. The background from D 0 →K 0 π + π − decays reconstructed 28 as D 0 →K 0 π − e + ν e is rejected by requiring theK 0 π − e + invariant mass (MK0 π − e + ) to be less than 1.80 GeV/c 2 . The 30 backgrounds associated with fake photons are suppressed by requiring the maximum energy of any unused photon (E γ max ) 32 to be less than 0.25 GeV.
The energy and momentum carried by the neutrino are de-34 noted by E miss and p miss , respectively. They are calculated from the energies and momenta of the tag (ED0 , pD0 ) and the 36 measured SL decay products (E SL = EK0 + E π − + E e + , p SL = pK0 + p π − + p e + ) using the relations E miss = 38 √ s/2 − E SL and p miss = p D 0 − p SL in the initial e + e − rest frame. Here, the momentum p D 0 is given by wherep tag is the momentum direction of the STD 0 and mD0 is the nominalD 0 mass [13]. 42 Information on the undetected neutrino is obtained by using the variable U miss defined by The U miss distribution is expected to peak at zero for signal events.
Figure 2(a) shows the U miss distribution of the accepted candidate events for D 0 →K 0 π − e + ν e in data. To obtain the signal yield, an unbinned maximum likelihood fit of the U miss distribution is performed. In the fit, the signal is described with a shape derived from the simulated signal events convolved with a Gaussian function, where the width of the Gaussian function is determined by the fit. The background is described by using the shape obtained from the MC simulation. The yield of DT D 0 →K 0 π − e + ν e events is determined to be 3131 ± 64(stat.). The backgrounds from the non-D 0 and non-K 0 S decays are estimated by examining the ST candidates in the M BC sideband, defined in the range (1.830, 1.855) GeV/c 2 , and the SL candidates in the K 0 S sidebands, defined in the ranges (0.450, 0.475) GeV/c 2 or (0.525, 0.550) GeV/c 2 in data, respectively. The yield of this type of background is estimated to be 19.4 ± 5.3. After subtracting these background events, we evaluate the number of the signal DT events to be N DT = 3112 ± 64(stat.).
The detection efficiency ε SL is estimated to be (9.53 ± 0.01)%, and the BF of D 0 →K 0 π − e + ν e is determined as B(D 0 →K 0 π − e + ν e ) = (1.434 ± 0.029(stat.))%. Due to the double tag technique, the BF measurement is insensitive to the systematic uncertainty in the ST efficiency. The uncertainties due to the pion and electron tracking efficiencies are estimated to be 0.5% [18] and the uncertainties due to their PID efficiencies are estimated to be 0.5% [18], where the tracking and PID uncertainties are conservatively estimated to account for the possible differences of the momentum spectra in Ref. [18]. The uncertainty due to theK 0 reconstruction is 1.5% [15]. The uncertainty due to the E/p requirement is 0.4% [6]. The uncertainty associated with the E γ max requirement is estimated to be 0.4% by analyzing the DT D 0D0 events where both D mesons decay to hadronic final states. The uncertainty due to the modeling of the signal in simulated events is estimated to be 0.8% by varying the input form factor parameters by ±1σ as determined in this work. The uncertainty associated with the fit of the U miss distribution is estimated to be 0.7% by varying the fitting ranges and the shapes which parametrize the signal and background. The uncertainty associated with the fit of the M BC distributions used to determine N ST is 0.5% and is evaluated by varying the bin size, fit range and background distributions. Further systematic uncertainties are assigned due to the statistical precision of the simulation (0.2%), the background subtraction (0.2%), and the input BF of the decay K 0 S → π + π − (0.1%). The systematic uncertainty contributions are summed in quadrature, and the total systematic uncertainty on the BF measurement is 2.2% of the central value.
The differential decay width of D 0 →K 0 π − e + ν e can be expressed in terms of five kinematic variables: the square of the invariant mass of theK 0 π − system m 2K 0 π − , the square of the invariant mass of the e + ν e system (q 2 ), the angle between theK 0 and the D 0 direction in theK 0 π − rest frame (θK0), the angle between the ν e and the D 0 direction in the e + ν e rest frame (θ e ), and the acoplanarity angle between the two decay planes (χ). Neglecting the mass of e + , the differential decay width of D 0 →K 0 π − e + ν e can be expressed as [19] where X = pK0 π − m D 0 , β = 2p * /mK0 π − , and pK0 π − is the momentum of theK 0 π − system in the rest D 0 system and p * is the momentum ofK 0 in theK 0 π − rest frame. The Fermi coupling constant is denoted by G F . The dependence of the decay density I is given by I = I 1 + I 2 cos2θ e + I 3 sin 2 θ e cos2χ + I 4 sin2θ e cosχ + I 5 sinθ e cosχ + I 6 cosθ e + I 7 sinθ e sinχ + I 8 sin2θ e sinχ + I 9 sin 2 θ e sin2χ, where I 1,...,9 depend on m 2K 0 π − , q 2 and θK0 [19] and can be expressed in terms of three form factors, F 1,2,3 . The form factors can be expanded into partial waves including S-wave (F 10 ), P -wave (F i1 ) and D-wave (F i2 ), to show their explicit dependences on θK0 . Analyses of the decay D + → K + π − e + ν e by using much higher statistics performed by the BaBar [5] and BESIII [6] collaborations do not observe a Dwave component and hence it is not considered in this analysis. Consequently, the form factors can be written as where F 11 , F 21 and F 31 are related to the helicity basis form factors H 0,± (q 2 ) [19,20]. The helicity form factors can in turn be related to the two axial-vector form factors, A 1 (q 2 ) and A 2 (q 2 ), as well as the vector form factor V (q 2 ). The  [13]. The form factor A 1 (q 2 ) is common to all three helicity amplitudes. Therefore, it is natural to define two form factor ratios as r V = V (0)/A 1 (0) and r 2 = A 2 (0)/A 1 (0) at the momentum square q 2 = 0.
The amplitude of the P -wave resonance A(m) is expressed as [5,6] where B(p) = 1 √ 1+R 2 p 2 with R = 3.07 GeV −1 [6] and where p * 0 is the momentum ofK 0 at the pole mass of the resonance m 0 , and . The S-wave related F 10 is described by [5,6] where the term A S (m) corresponds to the mass-dependent S-wave amplitude, and the same expression of A S (m) = r S P (m)e iδS (m) as in Refs. [5,6] is adopted, in which An unbinned five-dimensional maximum likelihood fit to the distributions of mK0 π − , q 2 , cos θ e + , cos θK0 , and χ for the D 0 →K 0 π − e + ν e events within −0.10 < U miss < 0.15 GeV is performed in a similar manner to Ref. [6]. The projected distributions of the fit onto the fitted variables are shown in Figs. 2 (b-f). In this fit, the parameters of r V , r 2 , m 0 , Γ 0 , r S and a S,BG are fixed to 0.08 and −0.81 (GeV/c) −1 due to limited statistics, respectively, based on the analysis of D + → K + π − e + ν e at BESIII [6]. The fit results are summarized in Table II. The goodness of fit is estimated by using the χ 2 /ndof, where ndof denotes the number of degrees of freedom. The χ 2 is calculated from the comparison between the measured and expected number of events in the five-dimensional space of the kinematic variables mK0 π − , q 2 , cos θ e + , cos θK0, and χ which are initially divided into 2, 2, 3, 3, and 3 bins, respectively. The bins are set with different sizes, so that they contain sufficient numbers of signal events for credible χ 2 calculation. Each five-dimensional bin is required to contain at least ten events; otherwise, it is combined with an adjacent bin. The χ 2 value is calculated as where N bin is the number of bins, n data i denotes the measured number of events of the i-th bin, and n fit i denotes the the expected number of events of the ith bin. The ndof is the number of bins minus the number of fit parameters minus 1. The χ 2 /ndof obtained is 96.3/98, which shows a good fit quality. The fit procedure is validated using a large simulated sample of inclusive events, where the pull distribution of each fitted parameter is found to be consistent with a normal distribution.
The fit fraction of each component can be determined by the ratio of the decay intensity of the specific component and that

Variable
Value 1.46 ± 0.07 ± 0.02 r 2 0.67 ± 0.06 ± 0.01 of the total. The fractions of S-wave and P -wave (K * (892) − ) are found to be f S−wave = (5.51 ± 0.97(stat.))% and f K * (892) − = (94.52 ± 0.97(stat.))%, respectively. The systematic uncertainties of the fitted parameters and the fractions of S-wave and K * (892) − components are defined as the difference between the fit results in nominal conditions and those obtained after changing a variable or a condition by an amount which corresponds to an estimate of the uncertainty in the determination of this quantity. The systematic uncertainties due to the E γ max and E/p requirements are estimated by using alternative requirements of E γ max < 0.20 GeV and E/p > 0.75, respectively. The systematic uncertainty because of the background fraction (f ) is estimated by varying its value by ±10%. The systematic uncertainties arising from the requirements placed on the charged pion, the electron and the K 0 S are estimated by varying the pion/electron tracking and PID efficiencies, and K 0 S detection efficiency by ±0.5%, ±0.5% and ±1.5%, respectively. The systematic uncertainty due to neglecting a possible contribution from the D-wave component is estimated by incorporating the D-wave component in Eq. (6). The systematic uncertainties in the fixed parameters of r  Table III and is obtained by adding all contributions in quadrature.