High statistics measurement of the K → π0e+ν(Ke3) decay formfactors

The decay K → πeν(Ke3) is studied using in-flight decays detected with ′OKA′ spectrometer. About 6M events are collected for the analysis.The λ+ slope parameter of the decay formfactor f+(t) in the linear approximation (average slope) is measured: λ+ = (29.86 ± 0.2) × 10−3. If the quadratic term is added to the parametrization, the result for the linear slope becomes: λ′+ = (24.6± 0.7)× 10−3, the quadratic coefficient in this fit is λ′′ + = (2.05± 0.3)× 10−3. Several alternative parametrizations are tried: the Pole fit parameter is found to be: MV = 891 ± 3 MeV ; the parameter of the Dispersive parametrization is measured to be Λ+ = (24.6 ± 0.17) × 10−3. All the results are very preliminary, the systematics effects are under study.


Introduction
The decay K → eνπ (K e3 ) provides unique information about the dynamics of the strong interactions. It has been a testing ground for such theories as current algebra, PCAC, Chiral Perturbation Theory (ChPT) etc. Another possible direction is a search for new physics, namely Tensor and Scalar interactions. In this talk we present a high-statistics study( ∼ 6M events) of the Dalitz plot density for this decay.

OKA beam and detector
OKA is the abbreviation for Experiments with Kaons . OKA beam is a RF-separated secondary beam of U-70 Proton Synchrotron of IHEP, Protvino. The beam is described elsewhere [1]. RF-separation with Panofsky scheme is realised. It uses two superconductive Karsruhe-CERN SC RF deflectors [2], donated by CERN. Sophisticated cryogenic system, built at IHEP [3]

Trigger and statistics
Very simple trigger,which is almost minimum bias one, has been used during data-taking: It is a combination of beam Sc counters,Č 1,2 threshold Cerenkov counters(Č 1 sees pions,Č 2 -pions and kaons), S bk -a beam-killer counter located in the beam-hole of the GAMS gamma-detector. Σ GAMS > MIP is a requirement for the analog sum of amplitudes in the GAMS-2000 to be higher than a MIP signal. The OKA is taking data since 2010, the available statistics is shown in Table 3. In the present study we present use part of the statistics taken in 2011 and 2012.

K e3 decay selection.
The data processing starts with the beam particle reconstruction in BPC 1 ÷ BPC 4 , then the secondary tracks are looked for in events with one good positive track are selected. The decay vertex is searched for, and a cut is introduced on the matching of incoming and decay track. The next step is to look for showers in GAMS-2000 and EGS calorimeters. The matching of the charged track and a shower in GAMS is done on the basis of the distance r between the track extrapolation to the ECAL and the shower coordinates (r ≤ 3cm). The electron identification is done using the ratio of the energy of the shower to the momentum of the associated track. The E/p distribution is shown in Fig 2. The particles with 0.8 < E/p < 1.2 are accepted as electrons. The events with one charged track identified as electron and two additional showers in ECAL are selected for further processing. The mass spectrum of γγ is shown in Fig 3. The π 0 peak is situated at M π0 = 134.9MeV with a resolution of ∼ 8.5 MeV. To fight the main background from K π2 decay, the angle between the momentum of the beam kaon p K and that of the eπ-system i.e. p e + p π is considered, see Fig 4. The background is clearly seen as a peak at zero angle.  The cut is α > 2mrad. Further selection is done by the requirement that the event passes 2C K → eνπ 0 fit. The surviving background is estimated from MC to be less than 1%.

Analysis
The event selection described in the previous section results in ∼6M events. The distribution of the events over the Dalitz plot is shown in Fig 5. The variables y = 2E * e /M K and z = 2E * π /M K , where E * e , E * π are the energies of the electron and π 0 in the kaon c.m.s are used. The background events, as MC shows, occupy the peripheral part of the plot. The most general Lorentz invariant form of the matrix element for the decay K + → l + νπ 0 is [4] ]v(p l ) It consists of scalar, vector and tensor terms. f ± are the functions of t = (P K − P π ) 2 . In the Standard Model (SM) the W-boson exchange leads to the pure vector term. The term in the vector part, proportional to f − is reduced(using the Dirac equation) to a scalar form-factor, proportional to (m l /2m K ) f − and is negligible in the case of K e3 . Different parametrizations have been used for f + (t). First is just a Taylor series: . It is usually used to compare with ChPT predictions. Alternative parametrization is the pole one: f The last is a relatively new Dispersive parametrization [5]: H(t))). Here H(t) is a known function. The procedure for the experimental extraction of the parameters λ + , f S , f T , which was developed in [6], starts from the Dalitz plot region y = 0.12 ÷ 0.92; z = 0.55 ÷ 1.075 subdivision into 100 × 100 cells. One can observe that the Dalitz-plot density function ρ(y, z) obeys a property of quasi-factorization, i.e ρ(y, z) = α F α (λ + , λ + , f s , f T ) · K α (y, z). K α are kinematic functions,which does not depend on λ + , λ + , f s , f T . The signal MC is generated with constant matrix element and K α (y, z) used as weights(y,z are MC-truth values). For each α the sums of K α (y, z) over events are accumulated in the Dalitz plot bins (i,j), to which the MC events fall after the reconstruction. As a result, every bin in the Dalitz plot gets weights W α (i, j) and the density function r(i,j) which enters into the fitting procedure is constructed: r(i, j) = α F α (λ + , λ + , f s , f T ) · W α (i, j). This procedure allows to avoid systematic errors due to the "migration" of the events over the Dalitz plot because of the finite experimental resolution. To take into account the finite number of MC events, we minimize a −L function defined as [7]:  ) where the sum runs over all populated cells of the Dalitz plot, and n j , r j , m j are the number of data events, the fitting function, and MC events in j-th cell. The radiative corrections were taken into account by reweighting every MC event, according to [8].

Results and comparison with theory
The results of the fit are summarized in the Table 3.The first line is just a linear fit, it gives average slope of the f + (t) formfactor. The result could be compared to quite old ChPT O(p 4 ) result [9]: λ ChPT + = (31.0 ± 0.6) × 10 −3 . The second line is the "standard fit with two parameters-linear and quadratic slopes. The quadratic term is quite significant, there is a correlation between parameters as it is seen on Fig 6. The ChPT O(p 6 ) prediction for the quadratic slope is [10]: λ + (ChPT ) = (1.1 ± 0.1) × 10 −3 .
The quality of the fit is illustrated by the z and y projections of the Dalitz plot, shown on Fig 7. The third and fourth lines of the Table 3 correspond to the fits, when on top of the standard V-A term the scalar or tensor terms are allowed. It is seen, that f S and f T are not significant, i.e can provide upper limits.  An interpretation of limits on F S and F T is possible in the framework of the scalar LeptoQuark(LQ) model. Then a diagram with LQ exchange should be added to the SM diagram with W (Fig 8). Applying Fiertz transformation to the LQ matrix element we get: (sμ)(νu) = − 1 2 (su)(νμ) − 1 8 (sσ αβ u)(νσ αβ μ). The first term is the scalar, the second one -tensor. The relation between f S , f T and the Leptoquark scale Λ LQ can be set out ( [12]), as a result, a 95% lower limit for the LeptoQuark scale is Λ LQ > 3.5 TeV.