New high statistics measurement of Ke4 decay form factors and ππ scattering phase shifts
 
 


We report results from a new measurement of the Ke4 decay K±→π+π-e±ν by the NA48/2 collaboration at the CERN SPS, based on a partial sample of more than 670 000 Ke4 decays in both charged modes collected in 2003. The form factors of the hadronic current (F,G,H) and ππ phase difference (δ=δs-δp) have been measured in ten independent bins of the ππ mass spectrum to investigate their variation. A sizeable acceptance at large ππ mass, a low background and a very good resolution contribute to an improved experimental accuracy, a factor two better than in the previous measurement, when extracting the ππ scattering lengths a00 and a02. Under the assumption of isospin symmetry and using numerical solutions of the Roy equations, the following values are obtained in the plane (a00,a02): a00=0.233±0.016stat±0.007syst,a02=-0.0471±0.011stat±0.004syst. The presence of potentially large isospin effects is also considered and will allow comparison with precise predictions from Chiral Perturbation Theory.


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The NA48/2 Collaboration: K e4 decay form factors and ππ scattering phase shifts Abstract. We report results from a new measurement of the K e4 decay K ± → π + π − e ± ν by the NA48/2 collaboration at the CERN SPS, based on a partial sample of more than 670 000 K e4 decays in both charged modes collected in 2003. The form factors of the hadronic current (F, G, H) and ππ phase difference (δ = δs − δp) have been measured in ten independent bins of the ππ mass spectrum to investigate their variation. A sizeable acceptance at large ππ mass, a low background and a very good resolution contribute to an improved experimental accuracy, a factor two better than in the previous measurement, when extracting the ππ scattering lengths a 0 0 and a 2 0 . Under the assumption of isospin symmetry and using numerical solutions of the Roy equations, the following values are obtained in the plane a 0 0 , a 2 0 : a 0 0 = 0.233 ± 0.016 stat ± 0.007 syst , a 2 0 = −0.0471 ± 0.011 stat ± 0.004 syst . The presence of potentially large isospin effects is also considered and will allow comparison with precise predictions from Chiral Perturbation Theory.

Introduction
The study of K ± → π + π − e ± ν decays (K e4 ) is of particular interest as it gives access to the final state interaction of two pions in absence of any other hadron. The asymmetry of the dilepton system with respect to the dipion system is related to the difference δ between the s-and p-wave ππ scattering phases for isospin states 0 and 1 δ 0 0 − δ 1 1 . Under the assumption of isospin symmetry, the measured variation of the phase shift with the invariant mass M ππ near threshold can be related to a 0 0 and a 2 0 (the ππ s-wave scattering lengths for isospin states 0 and 2) using dispersion relations based on general properties like analyticity, unitarity and crossing symmetry (known as "Roy equations" [1]). Numerical solutions of these relations, including experimental input data at intermediate energies, have been developed in the past years [2,3]. Additional constraints from Chiral Perturbation Theory (ChPT) lead to quite precise predictions for the scattering length values [4] a 0 0 = 0.220 ± 0.005, a 2 0 = −0.0444 ± 0.0010 and for the behavior of the K e4 hadronic form factors near threshold [5].
In the past years, only two experiments collected significant samples of K e4 decays [6][7][8], large enough to study their properties but without reaching yet the same precision level as the theoretical predictions. The NA48/2 experiment exploits the existing detector used for precise measurements of CP violation effects in neutral kaon decays to investigate similar effects in charged kaon decays into 3π [9,10]. The high intensity charged kaon beams allow to record at the same time large samples of rare decay modes like K e4 with branching fraction of order 10 −5 . Data

Beam and detector
Two simultaneous K + and K − beams are produced by 400 GeV protons from the CERN/SPS, impinging on a 40 cm long beryllium target. Opposite charge particles, with a momentum of (60 ± 3) GeV/c, are selected by two systems of dipole magnets (each forming an "achromat"), focusing quadrupoles, muon sweepers and collimators. Kaons then decay in a 114 m long vacuum tank, 192 cm diameter in the first 66 m and 240 cm in the last 48 m. The NA48 detector and its performances are described in detail elsewhere [11]. The important components for the K e4 analysis are recalled here: -Charged particle momenta from K ± decays are measured in a magnetic spectrometer consisting of four drift chambers (DCH) and a large aperture dipole magnet located between the second and third chamber. Each chamber consists of four staggered double planes of sense wires along the horizontal, vertical and ±45 • directions. The momentum resolution of the spectrometer is σ(p)/p = (1.02 ⊕ 0.044p)% (p in GeV/c). -A liquid-krypton calorimeter (LKr) measures the energy of electrons and photons. The transverse segmentation into 13 248 2 cm × 2 cm projective cells and the 27 radiation length thickness result in an energy resolution σ(E)/E = (3.2/ √ E ⊕ 9.0/E ⊕ 0.42)% (E in GeV) and a space resolution for isolated showers σ x = σ y = (0.42/ √ E ⊕ 0.06) cm. This allows to separate electrons (E/p ∼ 1) from pions (E/p < 1).
-A hodoscope consisting of two planes of scintillators segmented into horizontal and vertical strips is used to trigger the detector readout on charged track topologies. Its time resolution is ∼ 150 ps. -A two-level trigger logic selects and flags events with at least 3 tracks forming consistent 2-track vertices with the beam line. -A beam spectrometer (KABES), based on Micromegas amplification in a TPC [12], allows to measure the in-cident kaon momentum with a relative precision better than 1%.

Event selection
The 2003 data were selected for three well reconstructed charged tracks topologies, in time within 6 ns for the chambers signals and within 2 ns for the corresponding hodoscope signals. Two opposite sign pions (E/p < 0.8) and one electron or positron (0.9 < E/p < 1.1) were required. In the first spectrometer chamber, the distance between any two tracks should be at least 2 cm and the minimum distance from the beam line at least 12 cm. The 3-track reconstructed vertex position had to lie within a 5 cm radius transverse to the beam line and within 2 to 95 m downstream of the final collimator. The track impact at the LKr front face was required to fall within the active fiducial region, away from any dead cell. The track-to-track distance had to be larger than 20 cm to prevent shower overlaps. A minimum requirement of 3 GeV/c (resp. 5 GeV/c) for the electron (resp. pion) momenta was applied while the maximum momentum sum had to be less than 70 GeV/c. The reconstructed 3-track invariant mass (assigning a pion mass to each track) and the transverse momentum p t relative to the beam axis had to be outside an ellipse centered on the kaon mass and zero p t , with semi-axes ±20 MeV/c 2 and ±35 MeV/c, allowing missing energy and p t for the undetected neutrino. No more than 3 GeV energy deposits in the calorimeter, not associated to tracks but in-time with the considered track combination, were allowed. A more precise estimate than the 60 GeV/c average beam momentum is obtained by imposing energy-momentum conservation in kaon decay, under the assumption of a missing neutrino (E ν = |p ν |) and fixing the kaon mass and the beam direction to their nominal value. This translates into the solutions of a quadratic equation in p K , the kaon momentum. If a solution exists in the range [50,70] GeV/c, the event is kept and the solution closer to 60 GeV/c is assigned to p K .

Background
The background sources are twofold: K ± → π + π − π ± decays with subsequent π → eν decay or a pion misidentified as an electron; and K ± → π ± π 0 (π 0 ) decays with subsequent Dalitz decay of a π 0 π 0 D → e + e − γ with an electron mis-identified as a pion and photon(s) undetected. The elliptic cut in the plane (M 3π , p t ) rejects K ± → π + π − π ± decays which have very small p t values. Requiring the square missing mass to the (K ± − π ± ) system to be larger than 0.04 (GeV/c 2 ) 2 further rejects K ± → π ± π 0 decays. The invariant mass of the e + e − system (assigning an electron mass to the opposite charge pion) larger than 0.03 GeV/c 2 insures rejection of conversion photons and of some multi π 0 events. Additional rejection against pions mis-identified as electrons is achieved by using a dedicated linear discriminant variable (LDA) based on shower properties (E/p, radial shower width and energy weighted track-cluster distance). The training of this variable has been performed on pion tracks from well reconstructed K 3π events having E/p > 0.9 and electron tracks from K e3 decays. It provides an almost momentum independent high efficiency for electron tracks and additional rejection of pion tracks. The precise rejection level can be adjusted according to the discriminant variable value.
The ratio of background to signal is kept at the ∼ 0.5% relative level. It is estimated from "wrong sign" (WS) events (π ± π ± e ∓ ν), which, assuming the validity of the ∆S = ∆Q rule, can only be background. This has been cross-checked using Monte Carlo simulated events from the various topologies. The contribution of background to signal events has the same magnitude as that measured from WS events if originating from K ± → π ± π 0 (π 0 ) decays but has to be multiplied by a factor 2 if originating from K 3π decays because of the two equal charge pions. A total of 677 510 K e4 candidates (435 654 K + and 241 856 K − ) were selected from ∼ 2 × 10 9 charged triggers recorded in 2003 and kept for analysis. The background was estimated to 3112 (resp. 1994 and 1118) events according to the observed numbers of WS events.

K e4 kinematics
The decay K ± → π + π − e ± ν is conveniently described using 3 different rest frames: the K ± rest frame, the dipion rest frame and the dilepton rest frame. The kinematics is then fully described by the five Cabibbo-Maksymowicz variables [13,14] as shown in the sketch of Fig. 1: -S π = M 2 ππ , the invariant mass squared of the dipion -S e = M 2 eν , the invariant mass squared of the dilepton, -θ π , the angle of the π ± in the dipion rest frame with respect to the flight direction of the dipion in the K ± rest frame, -θ e , the angle of the e ± in the dilepton rest frame with respect to the flight direction of the dilepton in the K ± rest frame, -φ, the angle between the dipion and dilepton rest frames. The transition amplitude for K + can be written as: The NA48/2 Collaboration: K e4 decay form factors and ππ scattering phase shifts In the above expressions, p is the four-momentum of each particle and 0123 = 1. Three axial (F, G, R) and one vector (H) dimensionless complex form factors contribute to the transition amplitude and can be developed in partial wave expansion of s, p, d waves [15].
The third axial form factor R is suppressed by a factor m 2 e /S e . Consequently, there is no way to measure it in K e4 decays.
Neglecting d wave terms and assuming the same phase for F p , G p , H p , only one phase and four real form factors are left (δ = δ s − δ p , and F s , F p , G p , H p ) which are expanded further [5] in powers of dimensionless invariants q 2 = S π /4m 2 π − 1 and S e /4m 2 π : As pointed out in [16], going from K + to K − under CPT conservation, θ e should be replaced by π − θ e , φ should be replaced by π + φ and H p by −H p . Under the assumption of CP conservation, this is equivalent to consider that the φ distribution of K + decays is opposite to the φ distribution of K − decays with the same H p value.

Monte Carlo simulation
Signal and background events were generated in the kaon rest frame according to a realistic decay matrix element and then boosted into the experimental reference frame. The incident kaon trajectory and momentum was generated according to the time variations of the beam geometry for each kaon charge and the decay vertex position according to the exponential decay law. As a precise knowledge of the acceptance and resolution in the five-dimensional space of the kinematic variables is necessary, a detailed GEANT3-based [17] Monte Carlo (MC) simulation was used, including full detector geometry and material description, DCH alignment and local inefficiencies. A large time-weighted MC production was achieved, providing a sample about 25 times larger than the data and reproducing the observed ratio (K + /K − ) = 1.8. The same reconstruction and selection codes as for data were used, except for the timing cuts. The effect of the LDA cut was applied to the simulated electron candidates as a momentumdependent efficiency. This represents the optimal imple- mentation of the cut effect as it avoids to rely on the details of the shower developments, including fluctuations and limited statistics of the simulation. Two independent codes were used for the decay matrix element according to the Pais-Treiman formulation, one with a smooth phase shift variation [18] and constant form factors, the other with a phase shift variation following ChPT prediction [4] and form factors depending on invariant masses. They were used in independent analyses of the data.
The quality of the simulation can be seen from the plots of Fig. 2 where distributions of variables in the laboratory reference frame are shown for data and simulation. The experimental acceptances for the five Cabibbo-Maksymowicz variables are displayed in Fig. 3. The experimental resolutions are respectively: σ(cos θ e ) = 0.064 , σ(S e ) = 0.001408 (GeV/c 2 ) 2 , σ(θ e ) = 80 mrad , σ(φ) = 308 mrad .
Radiative corrections were implemented in the simulation in two successive steps: -First the classical Coulomb attraction between the two opposite charge pions is applied by weighting the ma-trix element, according to the Gamow function: with ω = 2πα/β, where α is the fine structure constant and β the relative velocity expressed as β = The effect is mostly a distortion of the M ππ spectrum near threshold.
-Second, real photons are generated by the program PHOTOS version 2.15 [19][20][21] interfaced to the simulation. Only 10% of the events have photons adding up to more than 1 GeV in the laboratory frame. The event selection furthermore reduces the acceptance for events with energetic photons. For these events, the resulting effect is a bias of the measured M eν and θ e variables.

Analysis method
Following the Pais and Treiman formulation [15], one can write the decay probability as The NA48/2 Collaboration: K e4 decay form factors and ππ scattering phase shifts where ρ(S π , S e ) is the phase space factor: . The function I can be expressed as the sum of 15 terms, each of them being the product of two terms, A i which depends only on the form factor values and B i which is only function of the kinematic variables: In K e4 decays, the electron mass is neglected and the term 1 − m 2 e Se becomes unity. One then defines a grid of equal population boxes in the five-variable space, adapted to the acceptance, resolution and dependence of the decay probability along each dimension. The chosen grid has ten bins in M ππ , five bins in M eν , five bins in cos θ π , five bins in cos θ e and twelve bins in φ, i.e. a total of 15 000 five-dimensional boxes. The number of data events per box is then 29 in the K + sample and 16 in the K − sample which are analyzed separately as the simultaneous beam geometries are not identical. A dedicated estimator [6][7][8] is used in the fitting procedure to take into account Poisson fluctuations of the small number of events per box. This estimator reduces to the "classical" χ 2 definition for the already large number of events per box. In each of the boxes, the simulation allows to compute the average value of B i , and the fitting procedure determines the best set of form factor values which minimizes the difference between the observed and expected number of events computed according to the current decay probability and summed over the 15 000 boxes.
In this analysis, the branching fraction is not measured, so only relative form factors are accessible: F p /F s , G p /F s , H p /F s and the phase shift δ. Neglecting a possible M eν dependence and without prior assumption on the shape of their variation with M ππ , the form factors and phase shift are measured in independent M ππ bins. As pointed out earlier [6], F p /F s and G p /F s are strongly correlated in the fitting procedure. To avoid undesirable large correlation effects, an intermediate quantity, G p = G p + γ/αF p is introduced and used in the fit, where . All correlations are then within ±20% except the correlation between δ and G p /F s which goes up to 40% at large M ππ . Fits are performed in the fourdimensional space, separately for the K + and K − samples but using the same M ππ bins definition. The results are checked for consistency and then combined in each bin according to their statistical weight. The relative normalizations (N Data /N MC ) are proportional to F 2 s and are rescaled to have a mean value equal to unity. Last, values of F p /F s , G p /F s , H p /F s are deconvoluted of the observed F s variation in each bin and plotted against M ππ to investigate a possible further dependence.
In a second stage of the analysis, the observed variations of the form factors and phase shift with M ππ are used to determine other parameter values through specific models.

Systematic uncertainties
Two independent analyses were performed on a large fraction of the 2003 data sample. They were based on different event selection and reconstruction, different detector corrections and different binning and fitting procedures. Consistent results were obtained, ensuring the robustness of the analysis.
Many items were investigated to assess possible biases in the results. For each of them, the analysis was repeated varying one condition at a time and a systematic uncertainty was quoted for each fitted parameter in each M ππ bin. A particular attention was given to possible bin-to-bin correlations, observed for some effects. In most cases, the observed variation has a statistical origin and is conservatively quoted as a systematic uncertainty. When dealing with larger data statistics, these components decrease accordingly.
-Fitting procedure: The number of boxes used in the fitting procedure was varied within a factor 2, keeping the same definition for the 10 M ππ bins. This last number was also extended to 12 and 15 bins. No visible bias was observed. -Trigger efficiency: Two independent attempts to measure the high (∼ 99.3%) trigger efficiency were made. The first one considers K e4 selected candidates satisfying the level 1 trigger condition (downscaled by 100 and thus suffering a lack of statistics) and measures the efficiency from events which satisfy the level 2 trigger. The second approach focuses on K ± → π ± π 0 D events satisfying the level 1 trigger condition, kinematic cuts and loose particle identification. Assigning a pion mass to both π ± and opposite charge electron tracks allows to reach the full M ππ range with sufficient statistics. Both methods have been used to apply the trigger efficiency in the five-dimensional space to the simulation. As the efficiency is almost uniform, the overall effect is marginal.
-Acceptance: The simulation is used to compute the acceptance in boxes of the five-dimensional space. It has been checked to be independent of the detailed physics assumptions provided that the whole phase space is covered. In addition, particular care was taken in controlling the geometrical acceptance and in following the time-dependence of the beam geometry. Many checks were performed to test the stability of the analysis, splitting the data in independent sub-samples according to the kaon charge, achromat polarity, dipole magnet polarity, decay vertex longitudinal position, spatial impact of the electron on the calorimeter front face and data taking time. Results were compared in each bin and found to be consistent within the statistical errors. -Background contamination: The analysis was repeated subtracting the "wrong sign" (WS) events according to their five-dimensional distributions, and scaled by a factor one, two or three. The dependence of each fitted parameter with the WS events scale factor was measured in each M ππ bin. The scale factor for the background subtraction was cross-checked using a detailed simulation of contributing processes and found to be 2.0 ± 0.3. The effect of the 0.3 uncertainty is propagated to each point according to the measured slopes and quoted as systematic uncertainty (labeled background level in Table 1). The effect is bin-to-bin correlated, as expected.
The background in the five-dimensional space is not uniform but accumulates at low M ππ values and around M eν = m π + . Varying the semi-axes of the elliptic cut in the plane (M 3π , p t ) accepts different fractions and shapes of the K 3π background. Results were found to be stable with this cut without bin-to-bin correlation. Residual effects were quoted as systematic uncertainty (labeled background shape in Table 1). -Electron identification: The final rejection against pions mis-identified as electrons (E/p > 0.9) is achieved by a cut on an LDA variable. In the simulation, the cut effect is applied as a momentum dependent efficiency. The cut value was varied from 0.85 to 0.90 (nominal cut) and 0.95. The analysis was repeated in the three conditions and the residual variation quoted as a systematic uncertainty. -Radiative corrections: The effect of radiative corrections has been investigated on simulated data where the PHOTOS generation of photons was switched off. These events were subsequently processed in the same analysis chain as real data. The difference observed for each fitted parameter in each M ππ bin between this simulation without PHOTOS and the same one using PHOTOS was thus evaluated and one tenth of the full effect was quoted as the theoretical uncertainty on the radiative corrections. This is based on detailed comparisons between PHOTOS and KLOR codes available in the K L → π ± e ∓ ν mode where a residual relative difference of 0.1% was observed [23], one tenth of the full PHOTOS effect. -Dependence on S e : As the dependence of the form factors with M eν was not considered in the first stage analysis, its potential effect was estimated by analyzing simulated events as real data, including a reweighting for f e /f s = 0.1. The observed deviation, found bin-tobin correlated, was quoted as a systematic uncertainty. The impact of each considered item on the final set of parameters is shown in Table 1. The main sources of systematic uncertainties are the background contamination and electron identification for the form factors and the acceptance control for the phase measurements.
In addition, a different reconstruction of the Cabibbo-Maksymowicz variables, based on the information of the KABES detector to measure the kaon momentum and incident direction, lead to angular resolutions improved by ∼ 50%. However, as the information was only available for half of the event sample and subject to different systematic uncertainties (mis-tagging rate of few percent), this alternative analysis was used as a cross-check of the standard procedure and found in good agreement with statistical errors improved by 5 to 10%.

Results and interpretation
The detailed numerical results obtained in the ten independent bins of M ππ are given in the Appendix (Tables 2-5). As explained in the previous section, the systematic uncertainties do have a bin-to-bin correlated component but much smaller than the uncorrelated one. In the tables, only the diagonal term of the matrix is quoted but the full error matrix for the ten values of δ is given in the Appendix ( Table 6). The agreement between data and simulation after the fit can be seen in Fig. 4 which shows, for each kaon sign, the evolution of the individual φ distributions with the M ππ bins. The overall agreement between data and simulated distributions is excellent for each of the five kinematic variables as can be seen from Fig. 5.

The F p form factor
No evidence for a mass dependence is observed, thus a constant value is used. The χ 2 of the fit is 9.34 for 9 degrees of freedom. The small f p /f s measured value provides an a posteriori justification for neglecting the d wave terms.

The G p form factor
A degree 1 polynomial in q 2 is used, neglecting a possible S e dependence: G p /f s = g p /f s + g p /f s q 2 .
The correlation between g p and g p is −0.91. The χ 2 of the fit is 8.23 for 8 degrees of freedom.

The H p form factor
No evidence for a mass dependence is observed, thus a constant value is used. The χ 2 of the fit is 12.81 for 9 degrees of freedom.

Numerical results for the form factor measurements
The numerical results for all terms are given below: f s /f s = 0.172 ± 0.009 stat ± 0.006 syst f s /f s = −0.090 ± 0.009 stat ± 0.007 syst f e /f s = 0.081 ± 0.008 stat ± 0.009 syst f p /f s = −0.048 ± 0.004 stat ± 0.004 syst g p /f s = 0.873 ± 0.013 stat ± 0.012 syst g p /f s = 0.081 ± 0.022 stat ± 0.015 syst h p /f s = −0.411 ± 0.019 stat ± 0.008 syst . The phase shift measurements are shown in Fig. 7 together with results from previous experiments [6][7][8]. All data are in agreement (except maybe the highest data point from E865). At this stage of the analysis, the phase shift measurements do not rely on the ππ scattering amplitude through a specific model and can be compared to theoretical predictions already available or still under development.
Under isospin symmetry assumption, the phase shift measurements can be related to the ππ scattering lengths using the analytical properties of the ππ scattering amplitudes and crossing symmetry relating amplitudes with different isospin (Roy equations). Several approaches [2,3,22] have been developed to extract numerical values for a 0 0 and a 2 0 from phase measurements using K e4 data alone or combined with other ππ scattering experimental data. In this paper, we use conservatively (but arbitrarily) the Universal Band (UB) approach [2] and NA48/2 K e4 data alone, extracting first the single variable a 0 0 (expressed in units of m π + ). Figure 8 shows the UB boundaries in the plane a 2 0 , a 0 0 , the width of the band reflecting the uncertainty on the external input data at M ππ > 0.8 GeV/c 2 used in the predictions. Scanning through the band, the preferred values a 0 0 , a 2 0 are found and the fit χ 2 favors the region of low scattering lengths values as it decreases from 14.9 to 9.2 for nine degrees of freedom from top to bottom boundary. At the center line of the Universal Band (1-parameter fit), NA48/2 phase measurements translate as: The external uncertainties correspond to the change in a 0 0 and a 2 0 when moving from the center line to the top (resp. bottom) boundary of the band. Figure 8 shows the The quoted errors correspond to the diagonal terms of the 1-σ error matrix which exhibits a large 96.7% correlation in the non-diagonal term. The universal band interpretation of the K e4 phase measurements favors a much higher value than the quite precise ChPT prediction of [4] a 0 0 = 0.220 ± 0.005, a 2 0 = −0.0444 ± 0.0010 . However, recent work ( [24,25] and references therein) suggests that additional isospin symmetry breaking effects, generated by the pion and quark mass differences and neglected so far in the K e4 phase shift evaluation, are of the order of 10 to 15 mrad over the full accessible range and would thus lead, when taken into account, to substantially smaller a 0 0 and a 2 0 extracted values, as shown in Fig. 8. The isospin corrected values would be decreased from 0.256 to 0.238 (resp. −0.0312 to −0.0347) for a 0 0 (resp. a 2 0 ) at the center line of the universal band. Similar shifts would be observed if other constraints were used. The experimental statistical and systematic uncertainties are unaffected while the additional uncertainty from isospin corrections is expected to be smaller. Such corrections were not applied in the data analysis of previous K e4 experiments. However, as both experiments published their results in the independent-bin analysis, these isospin breaking cor- Fig. 8. Results for the ππ scattering lengths a 0 0 and a 2 0 (in m π + unit) from fits to the NA48/2 K e4 data using different theoretical assumptions. The black solid curves are the universal band boundaries (Roy equations), the narrow blue band is the restricted area using the ChPT constraint. The symbols correspond to the results of 1-parameter fits to the data: the star is the value obtained at the center line of the universal band (dotted line), the square using the ChPT constraint. The error bars are 1-σ errors. The contour corresponds to 68% CL in a two-parameter fit. The whole picture is translated towards lower values of the scattering lengths when (preliminary) isospin symmetry breaking corrections [24,25] are considered rections can be applied a posteriori to all experiments as shown in Fig. 9. The NA48/2 K e4 value is then consistent with values of (a 0 0 − a 2 0 ) and a 2 0 obtained from the study of a cusp-like structure in the K ± → π 0 π 0 π ± decays [26,27] and precise predictions from ChPT. Any attempt to combine measurements from the three K e4 experiments will be dominated by the NA48/2 improved accuracy and larger sensitivity due to the higher accessible M ππ range.  9. Results of the a 0 0 scattering length measurement using the constraint from Universal Band center line after isospin corrections [24,25] have been applied. The error bars correspond to the experimental (first tick mark) and total (second tick mark) errors (obtained by adding in quadrature the experimental and theoretical uncertainties). In the combined result, the experimental errors are uncorrelated between experiments while the theoretical errors are fully correlated

Summary
The axial and vector form factors of the K e4 decays have been measured with an unprecedented relative precision of few percent. The s wave contribution to the F form factor has been measured to be mostly linear with S π with additional slopes with S e and S 2 π . A negative ∼ 5% p wave contribution to F has been established while a linear (resp. constant) behavior with S π for the G (resp. H) form fac-

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The NA48/2 Collaboration: K e4 decay form factors and ππ scattering phase shifts tor has been observed, in good agreement with predictions from ChPT [5]. The phase shift δ has been determined in an independent-bin analysis which allows comparison and combination with earlier measurements. The relation between δ values and the ππ scattering lengths a 0 0 and a 2 0 is subjected to theoretical external inputs whose uncertainties are now larger than the improved NA48/2 experimental accuracy. In particular, isospin symmetry was assumed in the relation to scattering lengths. This opens the way to new interesting developments, competitive with the achieved accuracy of the experimental measurement: a 0 0 = 0.256 ± 0.006 stat ± 0.002 syst +0.018 −0.017 ext , which implies a 2 0 = −0.0312 ± 0.0011 stat ± 0.0004 syst +0.013 −0.012 ext if the constraint between a 0 0 and a 2 0 at the center line of the Universal Band is used; and a 0 0 = 0.233 ± 0.016 stat ± 0.007 syst , a 2 0 = −0.0471 ± 0.011 stat ± 0.004 syst and a 96.7% correlation if both scattering lengths are taken as independent parameters. Promising theoretical work to include isospin symmetry breaking effects suggests that those values could then be decreased by ∼ 0.02 for a 0 0 and by ∼ 0.004 for a 2 0 , bringing the measurement in agreement with other measurements from NA48/2 and with predictions from ChPT.

Appendix: Fit results
for independent M ππ bins The following tables give the definition of the M ππ bins ( Table 2) and the fit results for the four form factors (Tables 3 and 4) and δ phase shift (Table 5).  To take into account correctly the bin-to-bin correlations of some of the systematic uncertainties, a complete treatment using a covariance matrix has been implemented. The statistical errors and the bin-to-bin uncorrelated systematic errors are added in quadrature in the diagonal terms σ 2 ii of a (10 × 10) error matrix. The non-diagonal terms σ 2 ij are filled with ρ σ i σ j , ρ being the correlation coefficient, here taken equal to unity. The error matrix (Table 6) is to be inverted to get the covariance matrix (V ) and used in the χ 2 calculation when extracting values of a set of parameters (a) using a dedicated function y(a): where x is the vector of measured values, y(a) is the vector of fitted values for parameter(s) a and V the covariance matrix of the measurements.