Direct Measurements of the Branching Fractions for $D^0 \to K^-e^+\nu_e$ and $D^0 \to \pi^-e^+\nu_e$ and Determinations of the Form Factors $f_{+}^{K}(0)$ and $f^{\pi}_{+}(0)$

The absolute branching fractions for the decays $D^0 \to K^-e ^+\nu_e$ and $D^0 \to \pi^-e^+\nu_e$ are determined using $7584\pm 198 \pm 341$ singly tagged $\bar D^0$ sample from the data collected around 3.773 GeV with the BES-II detector at the BEPC. In the system recoiling against the singly tagged $\bar D^0$ meson, $104.0\pm 10.9$ events for $D^0 \to K^-e ^+\nu_e$ and $9.0 \pm 3.6$ events for $D^0 \to \pi^-e^+\nu_e$ decays are observed. Those yield the absolute branching fractions to be $BF(D^0 \to K^-e^+\nu_e)=(3.82 \pm 0.40\pm 0.27)%$ and $BF(D^0 \to \pi^-e^+\nu_e)=(0.33 \pm 0.13\pm 0.03)%$. The vector form factors are determined to be $|f^K_+(0)| = 0.78 \pm 0.04 \pm 0.03$ and $|f^{\pi}_+(0)| = 0.73 \pm 0.14 \pm 0.06$. The ratio of the two form factors is measured to be $|f^{\pi}_+(0)/f^K_+(0)|= 0.93 \pm 0.19 \pm 0.07$.


I. INTRODUCTION
The semileptonic decays of the charmed mesons are of the weak and strong interactions can be well separated.The decay amplitude is proportional to the product of the Cabibbo-Kobayashi-Maskawa (CKM) matrix element, which parametrizes the mixing between the quark mass eigenstates and the weak eigenstates, and the form factor describing the strong interaction between the final state quarks.The differential decay rate for D 0 → K − (π − )e + ν e process is described by where G F is the Fermi coupling constant, |V cs(d) | is the CKM matrix element and p K(π) is the momentum of the kaon(pion) in the rest frame of the D 0 meson.f (q 2 ) represents the vector form factor of the hadronic weak current depending on the square of the four momentum transfer q = p D − p K(π) .In general theoretical treatment one common form of the form factor is a single pole form and is expressed as where f + (0) is the form factor evaluated at the four momentum transfer q equal to zero, and the pole mass m * is the mass of the lowest-lying Qq ′ meson.
MARK III [1] previously made an absolute measurements of the branching fractions for D 0 → K − e + ν e and D 0 → π − e + ν e by analysing the data taken at the near DD threshold region.In this paper, we report the direct measurements of the branching fractions for the Cabibbo favored decay of D 0 → K − e + ν e (Throughout this paper, charged conjugation is implied.)and the Cabibbo suppressed decay of D 0 → π − e + ν e by analysing the data sample of integrated luminosity of 33 pb −1 collected at and around the center of mass energy of 3.773 GeV with the BES-II detector at the BEPC.Using the measured branching fractions, the well measured |V cs |, |V cd | and the lifetime of the D 0 meson quoted from PDG [4], the vector form factors |f π + (0)| and |f K + (0)| are extracted, and their ratio is determined directly.

II. BES-II DETECTOR
BES-II is a conventional cylindrical magnetic detector that is described in detail in Ref. [2].A 12-layer Vertex Chamber (VC) surrounding the beryllium beam pipe provides input to the event trigger, as well as coordinate information.A forty-layer main drift chamber (MDC) located just outside the VC yields precise measurements of charged particle trajectories with a solid angle coverage of 85% of 4π; it also provides ionization energy loss (dE/dx) measurements which are used for particle identification.Momentum resolution of 1.7% 1 + p 2 (p in GeV/c) and dE/dx resolution of 8.5% for Bhabha scattering electrons are obtained for the data taken at √ s = 3.773 GeV.An array of 48 scintillation counters surrounding the MDC measures the time of flight (TOF) of charged particles with a resolution of about 180 ps for electrons.Outside the TOF, a 12 radiation length, lead-streamer mode, measures the energies of electrons and photons over 80% of the total solid angle with an energy resolution of σ E /E = 0.22/ √ E (E in GeV) and spatial resolutions of σ φ = 7.9 mrad and σ Z = 2.3 cm for electrons.A solenoidal magnet outside the BSC provides a 0.4 T magnetic field in the central tracking region of the detector.Three double-layer muon counters instrument the magnet flux return, and serve to identify muons of momentum greater than 500 MeV/c.They cover 68% of the total solid angle.

III. DATA ANALYSIS
At the center of mass energies around 3.773 GeV, the ψ(3770) resonance is produced in electron-positron (e + e − ) annihilation.The ψ(3770) decays predominately into DD pairs.If one D meson is fully reconstructed, the anti-D meson must exist in the system recoiling against the fully reconstructed D meson (called singly tagged D).Using the singly tagged D 0 sample, the decays of D 0 → K − e + ν e and D 0 → π − e + ν e can be well selected in the recoiling system.Therefore, the absolute branching fractions for these decays can be well measured.

A. Event Selection
The D 0 meson is reconstructed in non-leptonic decay modes of Events which contain at least two reconstructed charged tracks with good helix fits are selected.In order to ensure the well-measured 3-momentum vectors and the reliably charged particle identification, the charged tracks used in the single tag analysis are required to be within |cosθ| <0.85 where θ is the polar angle.All tracks, save those from K 0 S decays, must originate from the interaction region, which require that the closest approach of a charged track in xy plane is less than 2.0 cm and the z position of the charged track is less than 20.0 cm.Pions and kaons are identified by means of TOF and dE/dx measurements.Pion identification requires a consistency with the pion hypothesis at a confidence level (CL π ) greater than 0.1%.In order to reduce misidentification, kaon candidate is required to have a larger confidence level (CL K ) for a kaon hypothesis than that for a pion hypothesis.For electron identification, the combined confidence level (CL e ), calculated for the e hypothesis using the dE/dx, TOF and BSC measurements, is required to be greater than 0.1%, and the ratio CL e /(CL e + CL π + CL K ) is required to be greater than 0.8.The π 0 is reconstructed in the decay of π 0 → γγ.To select good photons from the decay of π 0 , the energy of a photon deposited in the BSC is required to be greater than 0.07 GeV, and the electromagnetic shower is required to start in the first 5 readout layers.In order to reduce the backgrounds the angle between the photon than 22 • and the angle between the direction of the cluster development and the direction of the photon emission to be less than 37 For each event, there may be several different charged track (or charged and neutral track) combinations for each of the four single tag modes.Each combination is subject to center-of-mass energy constraint kinematic fit and is required to have the fit probability P (χ 2 ) greater than 0.1%.If more than one combination are satisfied P (χ 2 ) > 0.1%, the combination with the largest fit probability is retained.For the single tag modes with a neutral kaon and/or neutral pion, one additional constraint kinematic fit for the K 0 S → π + π − and/or π 0 → γγ hypothesis is performed, separately.
The resulting distributions in the fitted invariant masses of Knπ (n = 1, 2, 3) combinations, which are calculated using the fitted momentum vectors from the kinematic fit, are shown in Candidate events D 0 → K − e + ν e and D 0 → π − e + ν e are selected from the surviving tracks in the system recoiling against the tagged D 0 .To select the D 0 → K − e + ν e and D 0 → π − e + ν e events, it is required that there are only two oppositely charged tracks, one of which is identified as an electron and the other as a kaon or pion.The neutrino is undetected, therefore the kinematic quantity U miss ≡ E miss − p miss is used to obtain the information about the missing neutrino, where E miss and p miss are the total energy and the momentum of all missing particles respectively.Fig. 2(a The branching fraction of the Cabibbo favored decay D 0 → K − e + ν e is much larger than that of the Cabibbo suppressed decay D 0 → π − e + ν e .The kaon can be misidentified as pion and therefore the process D 0 → K − e + ν e in the recoil side can be misclassified as D 0 → π − e + ν e .Monte Carlo study shows that this decay process is the main contamination to the selected sample of D 0 → π − e + ν e process.In order to correctly select the events D 0 → π − e + ν e and suppress misidentification from D 0 → K − e + ν e , the quantity U π−as−K is calculated by replacing pion mass with the kaon mass and |U miss | < |U π−as−K | is required.Fig. 2(c) shows the U miss calculated by replacing pion mass with kaon mass for the Monte Carlo events of D 0 → π − e + ν, while Fig. 2(d) shows the distributions of U miss calculated by replacing kaon mass with pion mass for the Monte Carlo events of D 0 → K − e + ν e .The quantity U miss is expected to be closer to zero for the correct particle mass assignment.The decays such as D 0 → K − π 0 e + (µ + )ν e (ν µ ) are suppressed by rejecting the events with extra isolated photons which are not used in the reconstruction of the singly tagged D 0 .The isolated photon should have its energy greater than 0.1 GeV and satisfy photon selection criteria as mentioned earlier.
Fig. 3(a) and Fig. 3(b) show the distributions of the fitted invariant masses of the Knπ combinations for the events in which the D 0 → K − e + ν e and D 0 → π − e + ν e candidate events are observed in the system recoiling against the singly tagged D 0 .In the Fig. 3(a), there are 118 events in the ±3σ signal regions, while there are 10 events in the outside of the signal regions.By assuming that the background distribution is flat, 3.8 ± 1.3 background events are estimated in the signal region.After subtracting the number of background events in the signal region, 114.2 ± 10.9 candidate events are retained.
A similar analysis of the events in Fig. 3(b) gives that there are 11.0 ± 3.6 candidate events after subtracting the number of background events in the signal region.
Events/(0.0025GeV/c 2 ) FIG. 3: Distributions of the fitted invariant masses of Knπ combinations for the events in which (a) the D 0 → K − e + νe and (b) the D 0 → π − e + νe candidate events are observed in the system recoiling against the tagged D 0 .
calculated for the selected events of D 0 → K − e + ν and D 0 → π − e + ν, respectively.Fig. 5 shows distribution of the momentum of the electrons from the selected candidate events of D 0 → K − e + ν e , where the error bars are for the events from the data and the histogram is for the

D. Background Subtraction
There are still some background contaminations in the observed candidate events due to other semileptonic or hadronic decays.These background events must be subtracted from the candidate events.The numbers of back-ple which is 13 times larger than the data.The Monte Carlo events are generated as e + e − → DD and the D and D mesons are set to decay to all possible final states according to the decay modes and branching fractions quoted from PDG [4] except the two decay modes under study.The number of events satisfying the selection criteria is then renormalized to the corresponding data set.Totally 10.2 ± 1.0 and 2.0 ± 0.5 background events are obtained for D 0 → K − e + ν e and D 0 → π − e + ν e , respectively.After subtracting these numbers of background events, 104.0 ± 10.9 and 9.0 ± 3.6 signal events for D 0 → K − e + ν e and D 0 → π − e + ν e decays are retained.

A. Monte Carlo Efficiency
The efficiencies for reconstruction of the semileptonic decay events of D 0 → K − e + ν e and D 0 → π − e + ν e are estimated by Monte Carlo simulation.A detailed Monte Carlo study shows that the efficiencies are ǫ K − e + νe = (35.89± 0.25)% and ǫ π − e + νe = (36.02± 0.25)%, where the errors are statistical.

B. Branching Fractions
The measured branching fractions are obtained by dividing the observed numbers of the semileptonic decay events N (D 0 → K − (π − )e + ν e ) by the number of the singly tagged D 0 meson N D 0 tag and the reconstruction efficiencies ǫ K − e + νe(π − e + νe) , (3) Inserting these numbers into the equation (3), the branching fractions for D 0 → K − e + ν e and D 0 → π − e + ν e decays are obtained to be BF (D 0 → K − e + ν e ) = (3.82± 0.40 ± 0.27)% and BF (D 0 → π − e + ν e ) = (0.33 ± 0.13 ± 0.03)%, where the first errors are statistical and the second systematic.The systematic uncertainties in the measured branching fractions arise from the uncertainties of particle identification (1.1%), tracking efficiency (2.0% per track), photon reconstruction (2.0%), U miss selection (0.6% for D 0 → K − e + ν e , 1.2% for D 0 → π − e + ν e ), the number of the singly tagged D 0 (4.8%), background subtraction (2.3% for D 0 → K − e + ν e , 5.6% for D 0 → π − e + ν e ) and Monte Carlo statistics (0.8%).These uncertainties are added in quadrature to obtained the total systematic errors, which are 7.1% and 8.8% for The decay width [8][9] of the semileptonic decay processes can be derived from the equation (1) by substituting the single pole form of the form factor as given in equation (2) for |f K(π) + (q 2 )| in the equation (1).The relations between the decay widths and the form factors are  I.The ratio of the two form factors can be obtained from the equations (3), ( 4) and ( 5), Inserting the |V cd |, |V cs |, the numbers of the signal events and the efficiencies into the equation ( 6), the value of the ratio is obtained.It is  I.The results are listed in Table II.As a comparison, the values of the CKM matrix elements quoted from PDG [4] are also listed in the Table II.Finally, Table III gives the comparison of the ratio of the CKM matrix elements with that obtained by the MARKIII, in which the ratio of the form factors is taken to be unity.In summary, by analysing the data collected at and around 3.773 GeV with the BES-II detector at the BEPC, the branching fractions for the decay of D 0 → K − e + ν e and D 0 → π − e + ν e have been measured.From a total of 7584 ± 198 ± 341 singly tagged D 0 sample, 104.0 ± 10.9 FIG.1: Distributions of the fitted invariant masses of (a)K + π − , (b) K + π − π + π − , (c) K 0 S π + π − and (d) K + π − π 0 combinations.
) and Fig.2(b)show the U miss distributions for the Monte Carlo D 0 → K − e + ν e and D 0 → π − e + ν e events respectively.The candidate events are required to satisfy the requirement |U miss | < 3σ Umiss , where the σ Umiss is the standard deviation of the U miss distribution.
Γ(D 0 → π − e + ν e ) = 3.01 |V cd | 2 |f π + (0)| 2 × 10 11 s −1 .(5) The form factors |f K + (0)| and |f π + (0)| can be extracted by using the measured values of the branching fractions and the lifetime of the D 0 meson.Inserting the values of |V cs | = 0.996 ± 0.013, |V cd | = 0.224 ± 0.016 and the lifetime τ D 0 = (411.7 ± 2.7) × 10 −15 s into equation (4) and (5), the form factors are obtained to be |f K + (0)| = 0.78 ± 0.04 ± 0.03, |f π + (0)| = 0.73 ± 0.14 ± 0.06, where the first errors are statistical and the second are systematic errors which arise from the systematic uncertainties in the measured values of the branching fractions, the uncertainties in the values of |V cs |, |V cd | and τ D 0 .The values of the form factors are compared with that predicted by various theoretical models and enumerated in Table 0)| = 0.93 ± 0.19 ± 0.07, where the first error is statistical and the second systematic which arises from the systematic uncertainties in the measured values of the branching fractions and the uncertainties in the values of |V cs | and |V cd |.This result is consistent with theoretical predictions, which range from D. CKM Matrix Elements |Vcs| and |V cd | Reversing the argument that presented in the previous section, we obtain the measured values of the CKM matrix elements |V cs | and |V cd | using the predicted form factors as shown in Table ν e and 9.0 ± 3.6 D 0 → π − e + ν e signal events are observed in the system recoiling against the D 0 tags.Those yield the decay branching fractions to be BF (D 0 → K − e + ν e ) = (3.82± 0.40 ± 0.27)% and BF (D 0 → π − e + ν e ) = (0.33 ± 0.13 ± 0.03)%.Using the values of the CKM matrix elements quoted from PDG [4], the form factors |f K + (0)| and |f K + (0)| are determined to be |f K + (0)| = 0.78 ± 0.04 ± 0.03, |f π + (0)| = 0.73 ± 0.14 ± 0.06 and the ratio of the two form factor to be |f π + (0)/f K + (0)| = 0.93 ± 0.19 ± 0.07.In addition, using the form factors predicted by QCDSR and LQCD calculations, the CKM matrix elements |V cs | and |V cd | are also determined.ACKNOWLEDGEMENT The BES collaboration thanks the staff of BEPC for their hard efforts.This work is supported in part by the National Natural Science Foundation of China under contracts Nos.19991480,10225524,10225525, the Chinese Academy of Sciences under contract No. KJ 95T-03, the 100 Talents Program of CAS under Contract Nos.U-11, U-24, U-25, and the Knowledge Innovation Project of CAS under Contract Nos.U-602, U-34(IHEP); by the National Natural Science Foundation of China under Contract No.10175060(USTC),and No.10225522(Tsinghua University).

TABLE I :
Form factor.

TABLE II :
CKM matrix element.

TABLE III :
The ratio of the CKM matrix elements.