1. Measurements of K ±  π + π - e ± ν (Ke4) and K ±  π ± π 0 π 0 decays at NA48/2. Silvia Goy López (CERN) For the NA48/2 collaboration HS07, September 3th 2007

2. 9/3/2007Silvia Goy Lopez HS072  Summary and Conclusions Overview  K ±  π + π - e ± ν (Ke4) decay Introduction Selection, reconstruction Form factor and ππ scattering length measurements  K ±  π ± π 0 π 0 decay Selection, reconstruction Cusp effect and interpretation Form factors and ππ scattering length measurements  Introduction to NA48  The NA48/2 experiment: goal, beam line, detector  Introduction to  scattering lengths

3. 9/3/2007Silvia Goy Lopez HS073 Introduction  NA48: 1997-2000. DCPV in neutral K Re(  /  ) = (14.7  2.2) 10 -4  NA48/1: 2002. Rare Ks decays Br(Ks   0 e  e - )=(5.8 +2.8 -2.3 ± 0.8) 10 -9 Br(Ks   0     - )=(2.8 +1.5 -1.2 ± 0.2) 10 -9  NA48/2: 2003-2004. DCPV in K   3  Ag=(-1.5+1.5 stat +0.9 trig +1.1 syst )·10 -4 Ag=(-1.5+1.5 stat +0.9 trig +1.1 syst )·10 -4 Ag 0 =(1.8+1.7 stat +0.5 syst )·10 -4 Ag 0 =(1.8+1.7 stat +0.5 syst )·10 -4  In addition many results on rare kaon decays and on hyperon decays.  NA62 K + -> . Test beam in 2006, 2007. Ke2/K  2 in 2007

4. 9/3/2007Silvia Goy Lopez HS074 The N A48/2  2003 run: ~ 50 days  2004 run: ~ 60 days  Total statistics 2 years: K±→  ±  0  0: ~1·10 8 K±→  ±  0  0: ~1·10 8 K±→  ±  +  -: ~3·10 9 K±→  ±  +  -: ~3·10 9 K±  π+π-e±ν: ~1.10 6  Greatest amount of K→3  ever collected  For these results 2003 for Ke4 and 90% of all data for cusp

5. 9/3/2007Silvia Goy Lopez HS075 The NA48/2 Beam Line  Simultaneous, coaxial and focused K + and K - beams  Central kaon momentum of 60 GeV/c and a spread of ± 3.8% (rms)  Beam line and spectrometer magnet polarities periodically inverted

6. 9/3/2007Silvia Goy Lopez HS076 The NA48 Detector  Magnetic spectrometer 4 DCHs, 4 views, 2 planes/view + dipole magnet  (p)/p=(1.02  0.044p)% P t kick=120 MeV/c  (t) = 1.4 ns  Charged hodoscope  (t) = 200 ps  LKr electromagnetic calorimeter 27 X 0 with 13212 2x2 cm 2 cells  (E)/E=(3.2/  E  9.0/E  0.42)%  (x)=  (y)=0.42 /  E  0.06 cm  (t) = 265 ps  Muon Counters 2+1 planes 25x25 cm 2  (t) = 350 ps Mass resolutions  (      - )= 1.7 MeV/c 2  (      0 )=1.4 MeV/c 2

7. 9/3/2007Silvia Goy Lopez HS077 Introduction to  scattering lengths   scattering amplitude can be expanded in partial waves. Scattering lengths are parameters in the expansion  At low energies S-wave dominates  Due to Bose statistics only I=0,2 allowed  Scattering lengths a0,a2 are the relevant parameters describing  interactions at low energies. a0,a2

8. 9/3/2007Silvia Goy Lopez HS078 Introduction to  scattering lengths  Theory Dispersion relations: based on causality and analiticity of scattering amplitude. Using  scattering data at intermediate energies a relation can be found a2=f(a0) Ej: UB from Roy eqn. ChPT: a2=f ChPT (a0). Predict a0 with <2% error, related to size of condensate. Recently lattice calculations.  Measurements Ke4 form factors (  variation with M  )  Geneva-Saclay (~30 10 3 evts)  BNL E865 (~390 10 3 evts)  NA48/2 (~677 10 3 evts) Cusp effect (NA48/2) Pionium lifetime (DIRAC)  Prediction for a0-a2 in ChPT a0 -a2 = 0.265 ± 0.004, a0=0.220 ± 0.005 (CGL, PLB 488 (2000) 261) ej using disper. rel. = 0.278 ± 0.016, a0=0.230 ± 0.015 (PY, PRD 71 (2005) 074016)  NA48/2 provides measurement from two independent channels

9. 9/3/2007Silvia Goy Lopez HS079 The K ± e4 decay: Introduction KK  p*(e  ) ee e Direction of the total e + e momentum in the K + rest frame Direction of the total     momentum in the K + rest frame p*(   )    Rare decay: Br( K ±  π + π - e ± ν )=(4.08±0.09) 10 -5 (PDG06)  Kinematic variables (Cabibbo – Maksymowicz ) S π (M 2 ππ ) (related to q 2 ) S e (M 2 eν ) cos  π cos  e  related to  scattering phases  related to scattering lengths 

10. 9/3/2007Silvia Goy Lopez HS0710 The K ± e4 decay: Introduction  Form factors F,G,R (axial, R relevant only for K  4 ) and H (vector) determined by fitting the data distributions.  Because of small values of s  in Ke4 expansion restricted to s&p waves. Fit F s, F p, G p, H p and  =  s -  p  Expanding in terms of q 2 (  S  ), Se  s : the  scattering phase shift in the I = 0, l = 0 state (s – wave)  p : the  scattering phase shift in the I = 1, l =1 state (p – wave)

11. 9/3/2007Silvia Goy Lopez HS0711 The K ± e4 decay: Selection  Selection: Three tracks, two opposite charged pions, one electron, missing energy and p t (separation  /e given by LDA: linear discriminator analysis)  Background:         - with  e is dominant  misidentified as electron         0 ) with   Dalitz decay (e  e   ) with e misid. as , and  undetected Estimation using MC and data (‘Wrong sign’ events) in agreement  In data ‘Wrong sign’ events (e different sign than kaon) can be only due to background. Scaling factor depending on process. Total background level  0.5%

12. 9/3/2007Silvia Goy Lopez HS0712 The K ± e4 decay: Reconstruction  Reconstruction. Two approaches: Use constrain given by to solve energy-momentum conservation equations and get kaon momentum. Take solution closest to 60 GeV/c Assume a 60 GeV/c kaon along the z axis, assign the missing p t to the and compute the resulting kaon invariant mass. Take events within ± 20 MeV/c 2  Statistics: 2003 data sample, ~677500 events P K (GeV/c)m K (GeV/c 2 )

13. 9/3/2007Silvia Goy Lopez HS0713 The K ± e4 decay: Fitting of form factors  Define 10x5x5x5x12 iso-populated bins in (M ππ, M eν, cos  π, cos  e,  )  The form factors are extracted from the data using simulated events by minimizing a log-likehood estimator in each of the M  bins: In each M  bin the form factors are assumed to be constant 10 independent fits (one fit per M  bin) of 4 parameters (Fp, Gp, Hp and  ) plus free normalization (related to Fs) in 4D space. The correlation between the 4+1 parameters is taken into account. K+ and K- fitted separately and combined. Statistics DataMC K+ evts 435654 29 10.0 M 667 Evts/bin K- evts 241856 16 5.6 M 373 Evts/bin K+/K-~1.8 MC/Data ~23

14. 9/3/2007Silvia Goy Lopez HS0714 The K ± e4 decay: Kinematic Distributions  Distribution of kinematic variables for data (symbols) and MC after fit (hist.)  Background distribution taken from data (hardly visible) M  (GeV ) Me  (GeV)  rad  cos  cos  e K+ K-  distribution shown separately for K+ and K- (CP symmetry  )

15. 9/3/2007Silvia Goy Lopez HS0715 The K ± e4 decay: Kinematic Distributions:    of the Ke4 decay amplitude is extracted from the measured asymmetry of the  distribution as a function of M   The asymmetry of the  distribution increases with M   increaing sensitivity to      0.309 < M  < 0.318 GeV 0.335 < M  < 0.345 GeV 0.373 GeV < M  < m K 2m+ < M  < 0.291 GeV

16. 9/3/2007Silvia Goy Lopez HS0716  No absolute normalization available for the moment (no BR measurement)  can only get relative form factors and variation with q 2 and with S e (all results are given wrt to Fs(q=0) constant term)  F 2 s obtained from relative bin to bin normalization of Data/MC after fit  Once q 2 dependency is extracted residual Me variation can be observed  A 2D fit (M , Me ) is used to get the slopes of the Fs expansion The K ± e4 decay: Form Factors: F 2 s (normalization) First measurement of f’’e≠0

17. 9/3/2007Silvia Goy Lopez HS0717 The K ± e4 decay: Form Factors: Fp,Gp, Hp  Once the normalization variation has been deconvoluted can extract variation of Fp,Gp and Hp with q 2 First measurement of fp≠0

18. 9/3/2007Silvia Goy Lopez HS0718 The K ± e4 decay: Form Factors  The  s  p  variation with M  was fitted using Universal Band (numerical solution of Roy equations (A, Phys. Rep. 353 (2001) 207)  1 parameter function to extract a 0 with a 2 =f(a 0 ) in center of UB  High sensitivity to scattering length due to good acceptance for high M 

19. 9/3/2007Silvia Goy Lopez HS0719 The K ± e4 decay: Results  Systematic checks Two analysis with different reconstructions, acceptance corrections, fitting methods Beam simulation (acceptance changes) Background level Electron identification Radiative corrections Possible S e dependence Possible bin to bin correlations investigated and taken into account f’ s / f s = 0.165± 0.011 stat ± 0.006 syst f’’ s / f s = -0.092± 0.011 stat ± 0.007 syst f’ e / f s = 0.081± 0.011 stat ± 0.008 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 ± 0.014 syst h p /f s = -0.411± 0.019 stat ± 0.007 syst NA48/2 Preliminary: 2003 data First observation of f’e and fp All form factors measured to 5 to 15% accuracy Error dominated by statistics

20. 9/3/2007Silvia Goy Lopez HS0720 The K ± e4 decay: Results comparison   points from different experiments can be fitted together to the center of UB.  Value for a0 is lower than given by NA48 data alone  Mainly due to last point of BNL (under investigation)  Superposed are the predictions for  vs M  for a0=0.26 and a0=0.22 Fit to all experimental points Predictions for a0=0.26 and a0=0.22

21. 9/3/2007Silvia Goy Lopez HS0721 The K ± e4 decay: Results comparison  Different methods can be used to relate  measurements to a0,a2 Fit 1 parameter a0 with a2=f(a0) by Roy eqn. Fit 1 parameter a0 with a2=f(a0) by ChPT Fit 2 parameters a2,a0  NA48/2 and E865 favor slightly different regions of UB

22. 9/3/2007Silvia Goy Lopez HS0722 The K ± e4 decay: Results comparison  Theory improvements (Gasser): Measured  has to be corrected by isospin symmetry breaking effects before extracting a0  After this correction (preliminary) a0 values are in good consistency with ChPT predictions  Using different methods a0 are in the range [0.209,0.255] ±0.006 stat ±0.005 syst and values of a2 around -0.045

23. 9/3/2007Silvia Goy Lopez HS0723 The K ±  π ± π 0 π 0 decay: Cusp effect  2003: Observation of sudden slope variation in the  invariant mass distribution seen at M 00 2 =(2m  + ) 2 An unexpected discovery from the NA48/2  Initial motivation for this study: use large statistics and good M 00 2 resolution to try to detect  +  - atom formation in K± →  ±  +  - decays, followed by pionium annihilation to   Standard Dalitz plot parameterization shows deficit in data before cusp Standard Dalitz plot parameterization Whole region   2 /ndf=9225/149! Above cusp:  2 /ndf=133/110 Subsample of 2003

24. 9/3/2007Silvia Goy Lopez HS0724 The K ±  π ± π 0 π 0 decay: Cusp effect  Now 2003+80% of 2004 combined: 59,624,170 fully reconstructed K ± →  ±  4m+24m+2 4m+24m+2 Zoom in cusp region

25. 9/3/2007Silvia Goy Lopez HS0725 The K ±  π ± π 0 π 0 decay: Reconstruction  Reconstruction : At least 4 clusters 15 cm away from any track and 10 cm away from other clusters. Among all the possible  pairings, the couple for which  2 is smallest is selected M 00 2 computed using average vertex of two  0 m 0 2 = 2E i E k (1-cos  ) ≈ E i E k  2 = E i E k (D ik ) 2 /(z ik ) 2  No physical background LKr zz d ij i j Z(i,j) Z(k,l) Vertex

26. 9/3/2007Silvia Goy Lopez HS0726 Cusp on K ±  π ± π 0 π 0. Checks for instrumental effects  Good resolution near cusp region  Acceptance linear near the cusp M 00 2 reconstructed distributions for three generated values of M 00 2 Acceptance  = 0.56 MeV at M 00 = 4m+

27. 9/3/2007Silvia Goy Lopez HS0727 Cusp on K ±  π ± π 0 π 0.Checks for instrumental effects  Data-MC comparisons above and below cusp  MC simulation describes well the of detector efficiency around cusp. Event deficit is real effect

28. 9/3/2007Silvia Goy Lopez HS0728 Cusp on K ±  π ± π 0 π 0. Interpretation  Two amplitudes contribute to K ±  π ± π 0 π 0 Direct emission  Given by M 0 Charge exchange (π + π -  π 0 π 0 ) in final state of K ±  π ± π + π  Given by M 1, proportional to M + and to one extra parameter a x =( a 0 - a 2 ) /3 in limit of exact isospin symmetry  These amplitudes interfere destructively below threshold  Rescattering model at one-loop (C, PRL 93 (2004) 121801) (k’ first observed by NA48/2) DE CE

29. 9/3/2007Silvia Goy Lopez HS0729 Cusp on K ±  π ± π 0 π 0. Interpretation  More complete formulation of the model includes all re-scattering processes at one-loop and two-loop level (CI, JHEP 0503 (2005) 21) has been used to extract NA48/2 result. More parameters than just a x Isospin correction applied in order to extract a 0 and a 2 from fits to data. Theoretical uncertainty of ~ 5% but radiative corrections and three-loops can be computed to reach ~1%

30. 9/3/2007Silvia Goy Lopez HS0730 Cusp on K ±  π ± π 0 π 0 : Fitting procedure  One dimensional fit to M 00 2 distribution  MINUIT minimization of  2 of data/MC spectra shapes  Fitting up to 0.097 (GeV/c 2 )  Fits to 5 parameters: norm, g, h’,a 0 - a 2 and a 2 () (k’ fixed)  For final result 7 bins around cusp excluded from the fit in order to reduce sensitivity to Coulomb corrections and pionium. The excess of events in this region is interpreted as pionium signature (E.M. corrections?) R=  (K   + A 2  )/  (K     +  – ) = (1.82  0.21)  10 –5 (th. prediction 0.8 10 -5 )

31. 9/3/2007Silvia Goy Lopez HS0731 The K ±  π ± π 0 π 0 decay. Quadratic term  Matrix element for DE (different from PDG parameterization)  Technique: Consecutive 1D-fits in both data projections (a0,a2,g0,h’) measurement (fixed k’=0) [2003 data]  k’ measurement (fixed a0,a2,g0,h’) [2003 data]  (a0,a2,g0,h’) second iteration (fixed k’) [2004 data]  a0,a2 are not significantly affected by neglecting the k’v 2 term, but g0,h’ are biased by  g0=–1.5%,  h=–1.2% M 0 ~ (1+g 0 u/2+h’u 2 /2+k’v 2 /2)

32. 9/3/2007Silvia Goy Lopez HS0732 The K ±  π ± π 0 π 0 decay. Quadratic term  Change of Dalitz variables, from (s 3, s 2 -s 1  to (s 3, cos  k’ = 0.0097 ± 0.0003 stat ± 0.0008 syst NA48/2 Preliminary: 2003 data  Define  as  angle between π ± and π 0 in π 0 π 0 COM  Fitting new Dalitz plot above cusp  Data-MC comparison for cos  for different k’ values  Change in value and meaning of g and h’ with respect to previous matrix element  No change of a 0 -a 2 and a 2

33. 9/3/2007Silvia Goy Lopez HS0733 Cusp on K ±  π ± π 0 π 0. Results  Systematic effects: analysis technique, acceptance determination, trigger efficiency, resolution, fitting interval, electromagnetic shower simulation, v-dependence of amplitude g = 0.649 ± 0.003 stat ± 0.004 syst h’ = -0.047 ± 0.007 stat ± 0.005 syst k’ = 0.097 ± 0.003 stat ± 0.008 syst First evidence of k≠0

34. 9/3/2007Silvia Goy Lopez HS0734 Cusp on K ±  π ± π 0 π 0. Results  External uncertainty: from the uncertainty on the ratio of K+ → p+p+p- and K+ → p+p  p  decay widths  Theoretical uncertainty on (a0 – a2): ± 5% (estimated effect from neglecting higher order diagrams and radiative corrections)  Fit imposing ChPT constrain between a 0 and a 2 (CGL, PRL 86 (2001) 5008)  From (a0 – a2) and a2 extract a0 (must take into account the statistical error correlation coefficient ≈ -0.92) a 0 - a 2 = 0.261 ± 0.006 stat ± 0.003 syst ± 0.0013 ext ± 0.013 th a 2 = -0.037 ± 0.013 stat ± 0.009 syst ± 0.002 ext  a 0 - a 2 = 0.263 ± 0.003 stat ± 0.014 syst ± 0.0013 ext ± 0.013 th a 0 = 0.224 ± 0.008 stat ± 0.006 syst ± 0.003 ext ± 0.013 th

35. 9/3/2007Silvia Goy Lopez HS0735 Comparison Cusp and DIRAC  DIRAC: Pionium lifetime related to scattering lengths. Proportional to |a0-a2| 2  The yellow area represents theoretical uncertainty (assumed Gaussian) Also dashed bars.

36. 9/3/2007Silvia Goy Lopez HS0736 Comparison Cusp and Ke4 results.

37. 9/3/2007Silvia Goy Lopez HS0737 Comparison Cusp and Ke4 results.

38. 9/3/2007Silvia Goy Lopez HS0738 Summary and Conclusions  NA48/2  NA48/2 exploited two different procedure to measure the  scattering lengths.  Ke4  Ke4: the  phase shift can be related to the a0 and a2 using theoretical input (e.g. Roy equations)  K→  0  0  K→  0  0 : the  scattering lengths are extracted from the study of  rescattering contribution in the M 00 mass distribution (the error is dominated by the theoretical error)  Applying the isosping breaking corrections the two results are fully compatible.  The results are compatible with the DIRAC experiment results and with ChPT predictions.

39. 9/3/2007Silvia Goy Lopez HS0739 SPARES

40. 9/3/2007Silvia Goy Lopez HS0740 Cusp on K ±  π ± π 0 π 0. Results  NA48/2 results on partial sample of 2003 data (PLB 633 (2006))  Systematic effects: Acceptance determination, trigger efficiency, fitting interval  Prediction for a2 in ChPT a2 = -0.0444 ± 0.0010 (CGL, PLB 488 (2000) 261)  Prediction for a0-a2 in ChPT a0 -a2 = 0.265 ± 0.004 (CGL, PLB 488 (2000) 261) using disper. rel. = 0.278 ± 0.016 (PY, PRD 71 (2005) 074016)  Fit imposing ChPT constrain between a 0 and a 2 (CGL, PRL 86 (2001) 5008) g = 0.645 ± 0.004 stat ± 0.009 syst h’ = -0.047 ± 0.012 stat ± 0.011 syst a 0 - a 2 = 0.261 ± 0.006 stat ± 0.003 syst ± 0.013 ext ± 0.013 th a 2 = -0.037 ± 0.013 stat ± 0.009 syst ± 0.002 ext a 0 = 0.220 ± 0.006 stat ± 0.004 syst ± 0.011 ext  a 0 - a 2 = 0.263 ± 0.003 stat ± 0.014 syst ± 0.013 ext ± 0.013 th

41. 9/3/2007Silvia Goy Lopez HS0741 The K ± e4 decay: Results before Isospin symmetry breaking corrections a 0 (UB)= 0.256± 0.008 stat ± 0.007 syst ± 0.018 th  a 2 = -0.031± 0.015 stat ± 0.015 syst ± 0.019 th NA48/2 Preliminary: PARTIAL Sub sample 2003 data  Prediction for a 0 in ChPT a 0 = 0.220 ± 0.005 (CGL, PLB 488 (2000) 261)

42. 9/3/2007Silvia Goy Lopez HS0742 Summary and conclusions  Pion scattering lenghts have been measured in NA48/2 with two independent channels: From K ± e4 From cusp on K ±  π ± π 0 π 0 But with different theoretical frameworks!  Form factors for K ± e4 and K ±  π ± π 0 π 0 have been measured Non zero quadratic term in K ±  π ± π 0 π 0 k’ = 0.0097 ± 0.0003 stat ± 0.0008 syst a 0 (UB)= 0.256± 0.008 stat ± 0.007 syst ± 0.018 th a 0 - a 2 = 0.268 ± 0.010 stat ± 0.004 syst ± 0.013 ext For comparison evaluation of a 0 for previous experiments using center of UB: Geneva Saclay: a 0 (UB) = 0.253 ± 0.037 (stat+syst) ± 0.014 th E865: a 0 (UB) = 0.229 ± 0.012 stat ± 0.004 syst ± 0.014 th

43. 9/3/2007Silvia Goy Lopez HS0743 Correlations  PARAMETER CORRELATION COEFFICIENTS for the normalisation fit fs' 1.000 -0.955 0.085 fs" -0.955 1.000 0.015 fe 0.085 0.015 1.000 which gives,once reduced to 2 significant digits : -0.96 0.08 0.02 Including fe almost do not alter the value of the other parameters, very little correlation.  For gp(0)&g’p = -0.914

44. 9/3/2007Silvia Goy Lopez HS0744  1 entry per event 4m+24m+2 CUSP STRUCTURE MUCH LESS VISIBLE THAN IN K ± →  ±  DECAY 87,950,601 events (3 entries per event) Cusp in KL → 

45. 9/3/2007Silvia Goy Lopez HS0745 two possible     pairs Calculate matrix elements at cusp point (M  = 2m + ) from measured partial width ratios and slope parameters: R (K + ) ≈ 6.1 ; R (K L ) ≈ 0.47 R (K+) R (K L ) ≈ 13 Cusp “visibility” is ~ 13 times higher in K + →  +  decays than in K L →  decays M   :  K   →        matrix element M   :  K   →    matrix element M   :  K L  →      matrix element M   :  K L  →  matrix element K+ →  + , KL→  comparison

46. 9/3/2007Silvia Goy Lopez HS0746 Summary and conclusions  Pion scattering lenghts have been measured in NA48/2 with two independent channels: From K ± e4 From cusp on K ±  π ± π 0 π 0 But with different theoretical frameworks!  Form factors for K ± e4 and K ±  π ± π 0 π 0 have been measured Non zero quadratic term in K ±  π ± π 0 π 0 k’ = 0.0097 ± 0.0003 stat ± 0.0008 syst a 0 (UB)= 0.256± 0.008 stat ± 0.007 syst ± 0.018 th a 0 - a 2 = 0.268 ± 0.010 stat ± 0.004 syst ± 0.013 ext For comparison evaluation of a 0 for previous experiments using center of UB: Geneva Saclay: a 0 (UB) = 0.253 ± 0.037 (stat+syst) ± 0.014 th E865: a 0 (UB) = 0.229 ± 0.012 stat ± 0.004 syst ± 0.014 th