Measurement of the charge asymmetry for the $K_S \rightarrow \pi e \nu$ decay and test of CPT symmetry with the KLOE detector

Using 1.63 fb$^{-1}$ of integrated luminosity collected by the KLOE experiment about $7\times 10^4$ $K_S \rightarrow \pi^{\pm}e^{\mp}\nu$ decays have been reconstructed. The measured value of the charge asymmetry for this decay is $A_S = (-4.9 \pm 5.7_{stat} \pm 2.6_{syst}) \times 10^{-3}$, which is almost twice more precise than the previous KLOE result. The combination of these two measurements gives $A_S = (-3.8 \pm 5.0_{stat} \pm 2.6_{syst}) \times 10^{-3}$ and, together with the asymmetry of the $K_L$ semileptonic decay, provides significant tests of the CPT symmetry. The obtained results are in agreement with CPT invariance.


Introduction
Semileptonic decays have been of fundamental importance in establishing several properties of the neutral kaon system, and of the Standard Model in general, including the ∆S = ∆Q rule [1], CP violation [2], and the unitarity of the quark mixing matrix [3,4].
The asymmetries which can be constructed from the decay rates into the two CP conjugated semileptonic final states, π − e + ν and π + e −ν , constitute a powerful probe in the study of discrete symmetries [5]. In particular, the charge asymmetries for the physical states K S and K L defined as: are sensitive to CP violation effects. At first order in small parameters [6]: with Re ( K ) and Re (δ K ) implying T -and CP T -violation in the K 0 − K 0 mixing, respectively, Re(y) and Re(x − ) implying CP T violation in ∆S = ∆Q and ∆S = ∆Q decay amplitudes, respectively 1 , and all parameters implying CP violation. If CP T symmetry holds then the two asymmetries are expected to be identical A S = A L = 2 Re ( K ) 3×10 −3 each accounting for the CP impurity in the mixing in the corresponding physical state.
The CP T theorem ensures exact CP T invariance for quantum field theories -like the Standard Model -formulated on flat space-time and assuming Lorentz invariance, locality, and hermiticity [7]. CP T violation effects might arise in a quantum gravity scenario [8,9] and their observation would constitute an unambiguous signal of processes beyond the Standard Model.
In this context the measurement of the difference A S − A L = 4 (Re δ K + Re x − ) is of particular importance as a test of the CP T symmetry. This observable is well constrained and can provide a test based on the direct comparison of a transition probability with its CP T conjugated transition -realised with entangled neutral kaon pairs -which constitutes one of the most precise, robust and model independent tests of the CP T symmetry [10].
The sum A S + A L = 4 (Re K − Re y) can be used to extract the CP T -violating parameter Re(y) once the measured value of Re( K ) is provided as input.
The two combinations A S ± A L (dominated by the uncertainty on A S ) constitute also a fundamental ingredient for improving the semileptonic decay contribution to the CP T test obtained imposing the unitarity relationship, originally derived by Bell and Steinberger [11], and yielding the most stringent limits on Im(δ) and the mass difference [12,13]. 1 More explicitly y and x− are described in terms of the decay amplitudes A± = A(K 0 → e ± π ∓ ν(ν)) andĀ± = A(K 0 → e ± π ∓ ν(ν)) as: At present, the most precise measurement of A L has been performed by the KTeV collaboration: A L = (3.322 ± 0.058 stat ± 0.047 syst ) × 10 −3 [14]. The measurement of its counterpart, A S , requires a very pure K S beam which can only be realised exploiting the entangled neutral kaons pairs produced at a φ-factory [15]. The first measurement of A S has been performed by the KLOE collaboration using 410 pb −1 of integrated luminosity collected at DAΦNE [16], the φ-factory of the INFN laboratories of Frascati: A S = (1.5 ± 9.6 stat ± 2.9 syst ) × 10 −3 [17], with an accuracy dominated by the statistical uncertainty. The new measurement reported here is based on a four times larger data sample, corresponding to an integrated luminosity of 1.63 fb −1 collected in [2004][2005]. The combination of the two results has a precision approaching the level of the CP violation effects expected for K S under the assumption of CP T invariance. New limits on Re(y) and Re(x − ) have been also derived.

The KLOE detector
The KLOE detector operates at the DAΦNE electron-positron collider. The energy of the two colliding beams is set to the mass of the φ meson which decays predominantly into a pair of charged or neutral kaons. Since the beams cross at an angle of 2 × 12.5 mrad the φ-meson is produced with a small momentum of p φ ≈ 13 MeV.
The KLOE detector consists of two main components: the cylindrical drift chamber and the electromagnetic calorimeter, both surrounding the beam pipe and immersed in a 0.52 T axial magnetic field. The drift chamber (DC) is a 3.3 m long cylinder with internal and external radii of 25 cm and 2 m, respectively. The chamber structure is made of carbonfiber epoxy composite and the gas mixture used is 90% helium, 10% isobutane. These features maximize transparency to photons and reduce charged particle multiple scattering and K L → K S regeneration. About 40% of produced K L mesons decay inside the DC volume, while most of the surviving K L 's interact and are detected in the electromagnetic calorimeter. Around 12500 sense wires stretched between the DC endplates allow to obtain a track spatial resolution of ∼2 mm along the axis and better than 200 µm in the transverse plane. The accuracy on the decay vertex determination is ∼1 mm and the resolution of the particle transverse momentum is 0.4 % [18]. The electromagnetic calorimeter (EMC) made by lead and scintillating fibers is divided into a barrel and two end-caps, has a readout granularity of ∼ (4.4 × 4.4) cm 2 , for a total of 2440 cells arranged in five layers covering 98% of the solid angle. It has energy and time resolution of σ(E)/E = 5.7%/ E[GeV], σ t = 54 ps/ E[GeV] ⊕ 140 ps for photons and electrons [19].
The data acquisition is enabled by a two-level trigger system [20]. The first level trigger is a fast trigger with a minimal delay which starts the acquisition at the front-end electronics. It requires two local energy deposits above threshold (50 MeV on the barrel, 150 MeV on the end-caps). The trigger time is determined by the first particle reaching the calorimeter and is synchronized with the DAΦNE RF signal.
The second level trigger uses information from both the drift chamber and the electromagnetic calorimeter. The trigger decision can be vetoed if the event is recognised as Bhabha scattering or cosmic ray event. For control purposes these events are accepted and saved as dedicated downscaled samples.
The time interval between bunch crossings (T bunch = 2.715 ns) is smaller than the time spread of the registered signals originating from K L K S events that can reach 30-40 ns. The offline reconstruction procedure therefore has to determine the true bunch crossing time T 0 for each event and correct all times related to that event accordingly. In the reconstruction algorithm the T 0 is determined by using the EMC information. In the studied channel, since the K S decay time is smaller than the K L interaction time in the calorimeter, the T 0 time has to be corrected in the offline analysis.
The data sample used for this analysis has been processed and filtered with the KLOE standard reconstruction software and the event classification procedure. The simulated data samples are based on the Monte Carlo (MC) GEANFI program [21].

Measurement of K S → πeν charge asymmetry
The charge asymmetry for the short-lived kaon is given by: where N + and N − are the numbers of observed K S → π − e + ν and K S → π + e −ν decays, respectively, while + and − are the corresponding efficiencies. Negative and positive charged pions interact differently in the detector material, therefore the efficiency is separately estimated for π − e + ν and π + e −ν final charge states.

K S tagging
The interaction of a K L meson in the calorimeter (crash) tags the presence of a K S meson. K L candidates must deposit an energy E clu (crash) > 100 MeV in the calorimeter in the polar angle range 40 • < θ < 140 • and not associated with a track from the DC. Since the kaon velocity in the φ meson rest frame is well-defined (β * ∼ 0.22), the requirement 0.18 < β * < 0.27 is applied. The K L direction obtained from the K L interaction coordinates in the calorimeter allows to determine the K L momentum p K L with good precision, and hence the K S momentum:

Momenta smearing
In order to improve the MC simulation description of the experimental momentum resolution effects, the reconstructed MC track momentum components p i have been smeared using three Gaussian functions: where G(0, σ j ) is the Gaussian distribution with zero mean and standard deviation σ j , f j is its amplitude, while ∆ is the fractional uncertainty on the track curvature.

Event preselection
The selection of K S → πeν decays starts with the reconstruction of a vertex formed by two opposite curvature tracks close to the e + e − interaction point (IP) with ρ vtx < 15 cm and |z vtx | < 10 cm, being ρ vtx and z vtx the transverse distance and the longitudinal coordinate of the vertex, respectively. In the majority of the three-body decays of K S the angle between charged secondaries (α) is contained in the (70 • , 175 • ) range in the K S rest frame, as shown in the left panel of Figure 1. Since the main source of background originates from the K S → π + π − decay, a cut on the invariant mass under the assumption of both particles being charged pions is also applied (300 MeV < M inv (π, π) < 490 MeV), as indicated in the right panel of Figure 1. Both tracks reconstructed in the drift chamber must be associated with clusters in the calorimeter by the Track to Cluster Association (TCA) procedure. This procedure extrapolates each track from the last hit in the DC towards the calorimeter surface and determines the impact point.

Time of flight selection cuts
Further background reduction and final charged state (π ± e ∓ ) identification is based on the difference δ t (X) between the particle time of flight (TOF) from the K S decay vertex to the calorimeter (t cl − T 0 ), and the time calculated from the DC measurement of track length L and particle momentum p under the m X mass hypothesis 2 : Since at this stage the φ decay time (T 0 ) is not known with sufficient precision, the following difference is introduced: where the mass hypothesis m X(Y ) is used for track 1 (2). Since for the correct mass assignments the value of δ t (X, Y ) is close to zero, the condition |δ t (π, π)| > 1.5 ns is applied for further K S → π + π − rejection. The remaining pairs of tracks are tested under pion-electron δ t (π, e) and electron-pion δ t (e, π) hypothesis (see Figure 2). Once particle identification has been performed, the T 0 and the time differences δ t (e) and δ t (π) are reevaluated accordingly. Events are then selected within the circle in the δ t (e) − δ t (π) plane as shown in Figure 3. The best separation between the signal and background components is obtained with the variable: where E K S is computed from the kinematics of the two body decay φ → K S K L , knowing the φ-meson momentum (from Bhabha events) and the reconstructed K L direction, E(π) is evaluated from the measured track momentum in the pion hypothesis, and E ν = | p K S − p(e)− p(π)|. M 2 (e) is calculated according to the TOF particle identification. For the signal events M 2 (e) peaks close to zero (see Figure 4).

Signal extraction
The signal yield is obtained by fitting the M 2 (e) distribution with a superposition of the corresponding simulated distributions for signal and residual background components, with free normalizations, separately for each final charge state, and taking into account the statistical uncertainty of the Monte Carlo sample [22,23]. The remaining residual background components are: • the K S → π + π − decays with one of the pion tracks not correctly reconstructed and classified as an electron by the TOF algorithm (1.6% of the sample after the fit, summing on the two final charge states); • the K S → π + π − decays where one of the pions decays into a muon before entering the drift chamber (18.7%); • radiative K S → π + π − γ decays (2.5%); • other decays mainly originating from φ → K + K − (6.7%) .
The result of the fit for the signal events is 34579±251 for K S → π − e + ν and 36874±255 for K S → π + e −ν , with total χ 2 /ndof = 118/109, summing on the two final charge states (see Figure 4).

K L → πeν control sample selection
A data sample of K L → πeν decay, which is a dominant decay mode of K L meson, is selected and used as a control sample. These events are tagged by the K S → π 0 π 0 decay 3 , identified by a total energy deposition in the calorimeter greater than 300 MeV, single photon deposit in the range from 20 to 300 MeV, and the π 0 π 0 invariant mass in the range from 390 to 600 MeV. The estimated tag efficiency is (60.0 ± 0.3)%. No appreciable contamination is found from other φ meson decays and the beam-induced background is kept at the level of 1%.
Due to the different lifetimes of K S and K L , the vertex distribution of K L decays is weighted to reproduce the K S distribution. The weighting is performed bin-by-bin in the same ρ vtxz vtx acceptance region of the signal K S → πeν. In this way the K L → πeν selected sample accurately mimicks the signal.
The events of the control sample are used to estimate directly from data the efficiencies for positive and negative pions. To this aim a single track selection scheme is developed and applied, after vertex reconstruction and cuts on opening angle in K L rest frame and M inv (π, π), as described in Section 3.3.
At this stage we require that at least one track reaches the calorimeter with TCA. For this track the δ t (e) and δ t (π) variables are constructed (see Equation 3.3). A pure sample of electrons (positrons) is then selected by requiring [(δ t (e) − 0.07 ns)/1.2 ns] 2 + [(δ t (π) + 4 ns)/3.2 ns] 2 < 1. Assuming the other track is a π + (or a π − ), we can test if it is associated to a calorimeter cluster to obtain the TCA efficiency KL   (π ± e ∓ ), respectively.

Efficiency determination
The total K S → πeν selection efficiency is estimated as follows: where T EC stands for trigger and event classification efficiency, while T AG and AN A denote tagging and analysis efficiencies, respectively. The analysis efficiency AN A can be expressed in turn as a product of four contributions: • kinematical cuts ( KC ): cuts on reconstructed vertex fiducial volume, opening angle α, and M inv (π, π) (see Section 3.3); • Track to Cluster Association algorithm ( T CA ); • Time of Flight cuts ( T OF ); • fit range ( F R ) of the M 2 (e) variable.
The efficiency T EC is evaluated using downscaled minimum-bias data samples without event classification and background rejection filters. The estimation of T AG , KC and F R are based on MC simulation; T CA and T OF are determined using the K L → πeν control sample with the method described in Section 4; T CA consists of the product of T CA (π ± ) and T CA (e ∓ ), the first evaluated from the control sample and the second from MC: the MC simulation; a data-MC disagreement could be due both to an imperfect detector simulation and/or to a bias in the estimate of the background induced by the machine or from other physical processes. The contributions from the stability of M 2 (e) distribution fit, momenta smearing, trigger and event classification procedures are also estimated. Unless differently specified in the following, each contribution is calculated as the absolute deviation from the nominal result (5.4) averaged on the two +/− variations. The stability of the A S result is also checked along the running period and against larger variations of the cut values. The resulting values for A S do not exhibit any anomaly; their behaviour is monotone or smooth. The systematic uncertainties are classified into the following groups (see Table 2): • Trigger and event classification: -Systematic effects originating from the trigger and the event classification procedure are estimated in prescaled data samples. The analysis of the prescaled samples follows the standard analysis chain. The systematic contribution (σ T EC ) is estimated to be 0.28 × 10 −3 .
-The z vtx and ρ vtx cuts for the reconstructed K S → πeν decay vertex position are each independently varied by ±0.2 cm (±1σ). The stability of the result is checked against a variation of ±5σ.
-The range of the opening angle α of the charged secondaries in the K S rest frame is enlarged or shrunk by 2 • (1σ) on each side (70 ∓ 2 • < α < 175 ± 2 • ). The stability of the result is checked up to a variation of ±5σ with the constraint of the upper bound not exceeding 180 • .
• Time of flight selection: -The |δ t (π, π)| cut is varied by ±0.1 ns. The stability of the result is checked up to a variation of ±0.4 ns.
-The circular region for selection of the signal in the {δ t (e), δ t (π)} plane is enlarged or shrunk by varying its radius of ±0.1 ns. The stability of the result is checked for variations ranging from −0.3 ns to +0.4 ns.
• Momenta smearing: -The K L → πeν control sample is divided into ten, equal in luminosity subsamples. The momenta smearing parameters are tuned separately for each subsample. From the standard deviation of the results the systematic contribution (σ M S ) is estimated to be 0.58 × 10 −3 .
• Fit procedure: -The systematic uncertainty from the histogram bin width σ HBW is determined by varying the bin width from 0.8 to 1.6 MeV 2 /1000 (this variation corresponds to the M 2 (e) resolution evaluated from MC). σ HBW is estimated to be 0.61 × 10 −3 . The stability of the result is checked for variations of the bin width from 2σ to 5σ. The total systematic uncertainty is estimated as the sum in quadrature of the contributions listed above and reported in Table 2.
As a cross-check, the A L value for the K L → πeν control sample is determined following the same analysis steps as for A S . The result A L = (1.7 ± 2.7 stat ) × 10 −3 is consistent with the KTeV measurement [14].