Measurement of the $K_S \to \pi e \nu$ branching fraction with the KLOE experiment

The branching fraction for the decay $K_S \to \pi e \nu$ has been measured with a sample of 300 million $K_S$ mesons produced in $\phi \to K_L K_S$ decays recorded by the KLOE experiment at the DA$\Phi$NE $e^+e^-$ collider. Signal decays are selected by a boosted decision tree built with kinematic variables and time-of-flight measurements. Data control samples of $K_L \to \pi e \nu$ decays are used to evaluate signal selection efficiencies. A fit to the reconstructed electron mass distribution finds 49647$\pm$316 signal events. Normalising to the $K_S \to \pi^+\pi^-$ decay events the result for the branching fraction is $\mathcal{B}(K_S \to \pi e \nu) = (7.211 \pm 0.046_{\rm stat} \pm 0.052_{\rm syst}) \times10^{-4}$. The combination with our previous measurement gives $\mathcal{B}(K_S \to \pi e \nu) = (7.153 \pm 0.037_{\rm stat} \pm 0.043_{\rm syst}) \times10^{-4}$. From this value we derive $f_+(0)|V_{us}| = 0.2170 \pm 0.009$.

The beam energy, the energy spread, the φ transverse momentum and the position of the interaction point are measured with high accuracy using Bhabha scattering events [12].
The K S (K L ) mesons are identified (tagged ) with high efficiency and purity by the observation of a K L (K S ) in the opposite hemisphere. This tagging procedure allows the selection efficiency for K S → πeν to be evaluated with good accuracy using a sample of the abundant decay K L → πeν tagged by the detection of K S → π + π − decays. The branching fraction B(K S → πeν) is obtained from the ratio R and the value of B(K S → π + π − ) measured by KLOE [13].

The KLOE detector
The detector consists of a large-volume cylindrical drift chamber, surrounded by a leadscintillating fibers finely-segmented calorimeter. A superconducting coil around the calorimeter provides a 0.52 T axial magnetic field. The beam pipe at the interaction region is spherical in shape with 10 cm radius, made of a 0.5 mm thick beryllium-aluminium alloy. Low-beta quadrupoles are located at ±50 cm from the interaction region. Two small lead-scintillating-tile calorimeters [14] are wrapped around the quadrupoles.
The drift chamber (DC) [15], 4 m in diameter and 3.3 m long, has 12582 drift cells arranged in 58 concentric rings with alternated stereo angles and is filled with a low-density gas mixture of 90% helium-10% isobutane. The chamber shell is made of carbon fiberepoxy composite with an internal wall of 1.1 mm thickness at 25 cm radius. The spatial resolution is σ xy = 0.15 mm and σ z = 2 mm in the transverse and longitudinal projection, respectively. The momentum resolution for tracks with polar angle 45 • < θ < 135 • is σ p T /p T = 0.4%. Vertices formed by two tracks are reconstructed with a spatial resolution of about 3 mm.
The calorimeter (EMC) [16] is divided into a barrel and two endcaps and covers 98% of the solid angle. The readout granularity is 4.4×4.4 cm 2 , for a total of 2440 cells arranged in five layers. Each cell is read out at both ends by photomultipliers. The energy deposits are obtained from signal amplitudes, the arrival times of particles and their position along the fibres are determined from the signals at the two ends. Cells close in space and time are grouped into energy clusters. The cluster energy E is the sum of the cell energies, the cluster time and position are energy-weighted averages. Energy and time resolutions are σ E /E = 0.057/ E (GeV) and σ t = 54 ps/ E (GeV) ⊕ 100 ps, respectively. The cluster spatial resolution is σ = 1.4 cm/ E (GeV) along the fibres and σ ⊥ = 1.3 cm in the orthogonal direction.
The level-1 trigger [17] uses both the calorimeter and the drift chamber information; the calorimeter trigger requires two energy deposits with E > 50 MeV in the barrel and E > 150 MeV in the endcaps; the drift chamber trigger is based on the number and topology of hit drift cells. A higher-level cosmic-ray veto rejects events with at least two energy deposits above 30 MeV in the outermost calorimeter layer. The trigger time is determined by the first particle reaching the calorimeter and is synchronised with the DAΦNE r.f. signal. The time interval between bunch crossings is smaller than the time spread of the signals produced by the particles, thus the event T 0 related to the bunch crossing originating the event is determined after event reconstruction and all the times related to that event are shifted accordingly. Data for reconstruction are selected by an online filter [18] to reject beam backgrounds. The filter also streams the events into different output files for analysis according to their properties and topology. A fraction of 5% of the events are recorded without applying the filter to control inefficiencies in the event streaming.
The KLOE Monte Carlo (MC) simulation package, GEANFI [18], has been used to produce an event sample equivalent to the data. Energy deposits in EMC and DC hits from beam background events triggered at random are overlaid onto the simulated events which are then processed with the same reconstruction algorithms as the data.
3 The measurement of Γ( where N πeν and N ππ are the numbers of selected K S → πeν and K S → π + π − events, πeν and ππ are the respective selection efficiencies, and R = ( ππ / πeν ) com is the ratio of common efficiencies for the trigger, on-line filter, event classification and preselection that can be different for the two decays. The number of signal events, N πeν in Eq. (3.1), is the sum of the two charge-conjugated decays to π − e + ν and π + e −ν . These are separated in a parallel analysis of the same dataset based on the same selection criteria presented in this section, optimised for measuring the charge asymmetry Γ(π − e + ν)−Γ(π + e −ν ) Γ(π − e + ν)+Γ(π + e −ν ) [19].

Data sample and event preselection
Neutral kaons from φ-meson decays are emitted in two opposite hemispheres with λ S = 5.9 mm and λ L = 3.4 m mean decay path for K S and K L respectively. About 50% of K L mesons reach the calorimeter before decaying and the K L velocity in the φ-meson reference system is β * = 0.22. K S mesons are tagged by K L interactions in the calorimeter, K Lcrash in the following, with a clear signature of a delayed cluster not associated to tracks. To select K L -crash and then tag K S mesons, the requirements are: • one cluster not associated to tracks (neutral cluster) and with energy E clu > 100 MeV, the centroid of the neutral cluster defining the K L direction with an angular resolution of ∼1 • ; • 15 • < θ clu < 165 • for the polar angle of the neutral cluster, to suppress small-angle beam backgrounds; • 0.17 < β * < 0.28 for the velocity in the φ reference system of the K L candidate; β * is obtained from the velocity in the laboratory system, β = r clu /ct clu , with t clu being the cluster time and r clu the distance from the nominal interaction point, the φmeson momentum and the angle between the φ-meson momentum and the K L -crash direction.
The K S momentum p K S = p φ − p K L is determined with an accuracy of 2 MeV, assigning the neutral kaon mass. K S → πeν and K S → π + π − candidates are preselected requiring two tracks of opposite curvature forming a vertex inside the cylinder defined by After preselection, the data sample contains about 300 million events and its composition evaluated by simulation is shown in Table 1. The large majority of events are K S → π + π − decays, together with a large contribution from φ → K + K − events where one kaon produces a track and the kaon itself or its decay products generate a fake K L -crash while the other kaon decays early into π ± π 0 .
18 397 0.006 φ → π + π − π 0 24 153 0.008 others 4 852 0.002 The β * distribution is shown in Figure 1, for data and simulated events. Two peaks are visible, the first is associated to events triggered by photons or electrons, and the second to events triggered by charged pions. The trigger is synchronised with the bunch crossing and the time difference between an electron (or photon) and a pion (or muon) arriving at the calorimeter corresponds to about one bunch-crossing shift.

Signal selection and normalisation sample
Signal selection is performed in two steps based on uncorrelated information: 1) the event kinematics using only DC tracking variables, and 2) the time-of-flight measured with the EMC.
Time assignment to tracks requires track-to-cluster association (TCA): for each track connected to the vertex a cluster with E clu > 20 MeV and 15 • < θ clu < 165 • is required whose centroid is within 30 cm of the track extrapolation inside the calorimeter. Track-tocluster association is required for both tracks in the event.
A multivariate analysis is performed with a boosted decision tree (BDT) classifier built with the following five variables with good discriminating power against background: p 1 , p 2 : the tracks momenta; α 1,2 : the angle at the vertex between the two momenta in the K S reference system; α LS : the angle between the momentum sum, p sum = p 1 + p 2 , and the K L -crash direction; ∆p : the difference between | p sum | and the absolute value | p K S | of the K S momentum; m ππ : the invariant mass reconstructed from p 1 and p 2 , in the hypothesis of charged-pion mass. Figure 2 shows the distributions of the variables for data and simulated signal and background events. Two selection cuts are applied to avoid regions far away from the signal where MC does not reproduce well the data: p < 320 MeV for both tracks and ∆p < 190 MeV.
Training of BDT classifier is done with MC samples: 5,000 K S → πeν events and 50,000 background events. Samples of the same size are used for the test. After training and test the classification is run on both MC and data samples. Figure 3 shows the BDT classifier output for data and simulated signal and background events. To suppress the large background contribution from K S → π + π − and φ → K + K − events, a cut is applied on the classifier output: BDT > 0.15.
(3.4) Figure 2. Distributions of the variables used in the multivariate analysis for data and simulated events after preselection. From top left: track momenta (p 1 , p 2 ), angle between the two tracks in the K S reference system (α 1,2 ), angle beween K L and K S directions (α SL ), two-track invariant mass in the hypothesis of charged pions (m ππ ), ∆p = | p sum | − | p K S |.
Track pairs in the selected events are eπ for the signal and are Kπ, ππ, µπ for the main backgrounds. A selection based on time-of-flight measurements is performed to identify eπ pairs. For each track associated to a cluster, the difference between the time-of-flight measured by the calorimeter and the flight time measured along the particle trajectory is computed, where t clu,i is the time associated to track i, L i is the length of the track, and the velocity β i = p i / p 2 i + m 2 i is function of the mass hypothesis for the particle with track i. The times t clu,i are referred to the trigger and the same T 0 value is assigned to both clusters. To reduce the uncertainty from the determination of T 0 the difference is used to determine the mass assignment. The ππ hypothesis is tested first. Figure 4 shows the δt ππ = δt 1,π − δt 2,π distribution. A fair agreement is observed between data and simulation, with K S → πeν and K S → πµν distributions well separated and large part of the K + K − background isolated in the tails of the distribution. However the signal is hidden under a large K S → π + π − background, therefore a cut 2.5 ns < |δt ππ | < 10 ns (3.6) is applied. Then, the πe hypothesis is tested by assigning the pion and electron mass to where the label as track-1 and track-2 is chosen at random. Figure 5 shows the twodimensional (δt πe , δt eπ ) distribution for data and MC where signal events populate either band around δt = 0. The mass assignment is based on the comparison of two hypotheses: if |δt 1,π − δt 2,e | < |δt 1,e − δt 2,π | track-1 is assigned to the pion and track-2 to the electron, otherwise the other solution is taken; the corresponding time difference, δt e , is the value defined by min[|δt πe |, |δt eπ |]. A cut is applied on this variable |δt e | < 1 ns. (3.7) The number of events selected by the time-of-flight requirements is 57577 and the composition as predicted by simulation is listed in Table 2. The background comprises K S → π + π − , φ → K + K − and K S → πµν, the other contributions being small.
The mass of the charged secondary identified as the electron is evaluated as with p 2 miss = ( p K S − p π − p e ) 2 , E K S and p K S being the energy and momentum reconstructed using the tagging K L , and p π , p e , the momenta of the pion and electron tracks, respectively.
A fit to the m 2 e distribution with the MC shapes of three components, K S → πeν, K S → π + π − and the sum of all other backgrounds, allows the number of signal events to   Figure 6 shows the m 2 e distribution for data and simulated events before the fit, and the comparison of the fit output with the data. The fit result is reported in Table 3. The number of signal events is N πeν = 49647 ± 316 with χ 2 /ndf = 76/96.  The K S → π + π − normalisation sample is selected requiring K L -crash, two opposite curvature tracks, the vertex as in Eq. (3.2) and 140 < p < 280 MeV for both tracks (Figure 2(a)). A total of N ππ = (282.314 ± 0.017) × 10 6 events are selected with an efficiency of 97.4% and a purity of 99.9% as determined by simulation.

Determination of efficiencies
The signal efficiency for a given selection is determined with a K L → πeν control sample (CS) and evaluated as where CS is the efficiency of the control sample, and MC πeν , MC CS are the efficiencies obtained from simulation for the signal and the control sample, respectively. Extensively studied with the KLOE detector [20], K L → πeν decays are kinematically identical to the signal, the only difference being the much longer decay path. Tagging is done with K S → π + π − decays selected requiring two opposite curvature tracks and the vertex defined in Eq. (3.2) with the additional requirement |m ππ − m K 0 | < 15 MeV to increase the purity, ensuring the angular and momentum resolutions are similar to the K L -crash tagging for the signal. The radial distance of the K L vertex is required to be smaller than 5 cm, to match the signal selection, but greater than 1 cm to minimise the ambiguity in identifying K L and K S vertices. Weighting the K L vertex position to emulate the K S vertex position has negligible effect on the result.
The control sample composition is K L → πeν (B = 0.405), K L → πµν (B = 0.270) and K L → π + π − π 0 (B = 0.125) decays, while most of K L → π 0 π 0 π 0 decays are rejected requiring two tracks and the vertex. The distribution of the m 2 miss missing mass, with respect to the two tracks connected to the K L vertex and in the charged-pion mass hypothesis, shows a narrow isolated peak at the π 0 mass. K L → π + π − π 0 decays are efficiently rejected with the m 2 miss < 15000 MeV 2 cut. Two control samples are selected, based on the two-step analysis strategy using largely uncorrelated variables and presented in Section 3.2: the first CS kinBDT applying a cut on the TOF variables to evaluate the efficiency of the selection based on the kinematic variables and the BDT classifier, the second CS TCATOF applying a cut on kinematic variables to evaluate TCA and TOF selection efficiencies.
The CS kinBDT control sample is selected applying a cut on the two-dimensional (δt πe , δt eπ ) distribution, rejecting most of the K L → πµν events. The sample contains 0.44×10 6 events with a 97% purity as determined from simulation. The Monte Carlo BDT distributions for the signal and control sample are compared in Figure 7 The CS TCATOF control sample is selected applying a cut on the (m ππ , m 2 miss ) distribution. The sample contains 1.3 × 10 6 events with a 95% purity as determined from simulation. In the K S → πeν analysis, the T 0 is determined by the first cluster in time, associated with one of the tracks of the K S decay. Then, for the control sample the first cluster in time is required to be associated with the K L decay, in order not to bias TOF variables. Figure 7   For the K S → π + π − normalisation sample, the efficiency of the momentum selection 140 < p < 280 MeV is determined using preselected data. The cut on the vertex transverse position in Eq. (3.2) is varied in 1 cm steps from ρ max vtx = 1 cm to ρ max vtx = 4 cm, based on the observation that ρ vtx and the tracks momenta are the least correlated variables, the correlation coefficient being 13%. Using Eq. (3.8) and extrapolating to ρ max vtx = 5 cm, the efficiency is data ππ = (96.569 ± 0.004)%. Alternatively, the efficiency is evaluated using the K S → π + π − data sample (with ρ max vtx = 5 cm and MC ππ = MC pres ), the efficiency is data ππ = (96.657 ± 0.002)%. The second value, free from bias of variables correlation, is used for the efficiency and the difference between the two values is taken as systematic uncertainty. The number of K S → π + π − events corrected for the efficiency is N ππ / ππ = (292.08±0.27)×10 6 . The ratio R in Eq. (3.1) includes several effects depending on the event global properties: trigger, on-line filter, event classification, T 0 determination, K L -crash and K S identification. In Table 5 the various contributions to R evaluated with simulation are listed with statistical uncertainties only, the resulting value is R = 1.1882 ± 0.0017. Systematic uncertainties are detailed in Section 4. Table 5. Ratios of MC efficiencies common to the K S → πeν and K S → π + π − selections with statistical uncertainties. The error on R is calculated as the quadratic sum of the errors of the single ratios.

Systematic uncertainties
The signal count is affected by three main systematic uncertainties: BDT selection, TOF selection, and the m 2 e fit. The distributions of the BDT classifier output for the data and simulated signal and control sample events are shown in Figures 3 and 7. The resolution of the BDT variable predicted by simulation comparing the reconstructed events with those at generation level is σ BDT = 0.005. The analysis is repeated varying the BDT cut in the range 0.135-0.17. The ratio of the number of signal events determined with the m 2 e fit and the efficiency evaluated with Eq. (3.8) is found to be stable and the half-width of the band defined by the maximum and minimum values, ±0.27%, is taken as relative systematic uncertainty.
The number of reconstructed clusters can be different for the signal (K L -crash,πeν) and control sample (ππ,πeν), thus the TCA efficiency calculation is repeated by weighting the events of the control sample by the number of track-associated clusters. The difference, less than 0.1%, is taken as relative systematic uncertainty for the TCA efficiency.
The main source of uncertainty in the TOF selection is the lower cut on |δt ππ | in Eq. (3.6) because the signal and background distributions in Figure 4 are steep and with opposite slopes. The resolution is the combination of the time resolution of the calorimeter, the tracking resolution of the drift chamber and the track-to-cluster association and is determined by the width of the δt e distribution.
The comparison of the δt e distributions for the signal and the K L → πeν control sample is shown in Figure 8, they are fitted with a Gaussian and a 2 nd degree polynomial, obtaining σ = 0.44 ± 0.02 ns in both cases. The analysis is repeated varying the |δt ππ | lower cut in the range 2.0-3.0 ns, the half-width of the band gives a relative systematic uncertainty of ±0.28%. With the same procedure the cut on |δt e | in Eq. (3.7) is varied in the range 0.8-1.2 ns and the half-width of the band, ±0.12%, is taken as relative systematic uncertainty.
Possible effects in the evaluation of the TCA and TOF efficiencies due to a detector response different for the π + e −ν and π − e + ν final states are negligible. The fit to the m 2 e distribution in Figure 6 is repeated varying the range and the bin size. The fit is also done using two separate components for K S → πµν and φ → K + K − , the χ 2 is good but the statistical error is slightly increased. Half of the difference between maximum and minimum result of the different fits, ±0.15%, is taken as relative systematic uncertainty. The systematic uncertainties are listed in Table 6. The dependence of R on systematic effects has been studied in previous analyses for different K S decays selected with the K L -crash tagging method: K S → π + π − and K S → π 0 π 0 [13], and K S → πeν [19]. The systematic uncertainties are evaluated by a comparison of data with simulation, the difference from one of the ratio Data MC is taken as systematic uncertainty.
Trigger -Two triggers are used for recording the events, the calorimeter trigger and the drift chamber trigger. The validation of the MC relative efficiency is derived from the comparison of the single-trigger and coincidence rates with the data. The data over MC ratio is 0.999 with negligible error.
On-line filter -The on-line filter rejects events triggered by beam background, detector noise, and events surviving the cosmic-ray veto. A fraction of non-filtered events prescaled by a factor of 20 allows to validate the MC efficiency of the filter. The data over MC ratio does not deviate from one by more than 0.1%.
Event classification -The event classification produces different streams for the analyses. The K L K S stream used in this analysis selects events based on the properties of K S and K L decays. In more than 99% of the cases the events are selected based on the K S decay topology and the K L -crash signature and differences between MC and data are accounted for in the systematic uncertainties described below for the K L -crash and K S vertex reconstruction.
T 0 -The trigger time is synchronised with the r.f. signal and the event T 0 is redefined after event reconstruction. The systematic uncertainty is evaluated analysing the data and MC distributions of T 0 for the decays with the most different timing properties: K S → π + π − and K S → π 0 π 0 [13]. The data over MC ratio does not deviate from one by more than 0.1%.
K L -crash and β * selection. -The systematic uncertainty is evaluated comparing data and simulated events tagged by K S → π + π − and K S → π 0 π 0 decays which have different timing and topology characteristics. The data over MC ratio is 1.001 with negligible error.
K S vertex reconstruction -The systematic uncertainty of the requirement of two tracks forming a vertex in the cylinder defined by Eq. (3.2) is evaluated for signal and normalisation using a control sample of φ → π + π 0 π − events selected requiring one track with minimum distance of approach to the beamline in the cylinder and a well-reconstructed π 0 . Energy-momentum conservation determines the momentum of the second track. The momentum distribution of tracks in the control sample covers a range wider than both signal and normalisation samples. The efficiency for reconstructing the second track and the vertex is computed for data and simulation and the ratio r(p L , p T ) = Data MC is parameterised as function of the longitudinal and transverse momentum p L and p T . The ratios relative to the signal and normalisation events, r πeν and r π + π − , are obtained as convolution of r(p L , p T ) with the respective momentum distribution after preselection. The ratio r π + π − rπeν deviates from one by 0.45% with an uncertainty of 0.2% due to the knowledge of the parameters of the r(p L , p T ) function.
The R total systematic uncertainty is estimated by combining the differences from one of the data over MC ratios and amounts to 0.48%. Including the systematic uncertainties the factors in Eq. (3.1) are: π + π − = (96.657 ± 0.088)%, πeν = (19.38 ± 0.10)%, and R = 1.1882 ± 0.0059. The previous result from KLOE based on an independent data sample corresponding to an integrated luminosity of 0.41 fb −1 is R = (1.019 ± 0.011 stat ± 0.007 syst ) × 10 −3 [10]. Correlations exist between the two measurements in the determination of efficiencies for the event preselection and time-of-flight analysis, correlations in the determination of R and the fit being negligible. The correlation coefficient is 12%. The combination of the two measurements gives R = Γ(K S → πeν) Γ(K S → π + π − ) = (1.0338 ± 0.0054 stat ± 0.0064 syst ) × 10 −3 .
The value of |V us | is related to the K S semileptonic branching fraction by the equation where I K is the phase-space integral, which depends on measured semileptonic form factors, S EW is the short-distance electro-weak correction, δ K EM is the mode-dependent longdistance radiative correction, and f + (0) is the form factor at zero momentum transfer for the ν system. Using the values S EW = 1.0232 ± 0.0003 [21], I e K = 0.15470 ± 0.00015 and δ Ke EM = (1.16 ± 0.03) 10 −2 from Ref. [5], and the world average values for the K S mass and lifetime [22] we derive f + (0)|V us | = 0.2170 ± 0.0009.

Conclusion
A measurement of the ratio R = Γ(K S → πeν)/Γ(K S → π + π − ) is presented based on data collected with the KLOE experiment at the DAΦNE φ-factory corresponding to an integrated luminosity of 1.63 fb −1 . The φ → K L K S decays are exploited to select samples of pure and quasi-monochromatic K S mesons and data control samples of K L → πeν decays. The K S decays are tagged by the detection of a K L interaction in the detector. The K S → πeν events are selected by a boosted decision tree built with kinematic variables and by measurements of time-of-flight. The efficiencies for detecting the K S → πeν decays are derived from K L → πeν data control samples. A fit to the m 2 distribution of the identified electron track finds 49647 ± 316 signal events. Normalising to K S → π + π − decay events recorded in the same dataset, the result is R = (1.0421±0.0066 stat ±0.0075 syst )×10 −3 . The combination with our previous measurement gives R = (1.0338 ± 0.0054 stat ± 0.0064 syst ) × 10 −3 . From this value we derive the branching fraction B(K S → πeν) = (7.153±0.037 stat ± 0.044 syst ) × 10 −4 and the value of |V us | times the form factor at zero momentum transfer, f + (0)|V us | = 0.2170 ± 0.009.