1. 1 Measurement of  scattering lengths in Kaon decays by NA48/2 EPS-HEP 2007 Manchester 19-25 july 2007 Gianluca Lamanna (Università & INFN di Pisa) on behalf of NA48/2 collaboration

2. 2Gianluca Lamanna – HEP0719.07.2007 Outline Introduction Ke4 (K ± →     e ± ) Form factors and pion scattering lengths Data 2003: Preliminary results Cusp (in K ± →  ±     ) A new method to extract pion scattering lengths through the strong rescattering process     →     Data 2003+2004: Preliminary results Conclusions

3. 3Gianluca Lamanna – HEP0719.07.2007 Experimental setup:The Beams K+K+ K−K− BM P K spectra, 60  3 GeV/c 54 60 66 Width ~ 5mm K+/K- ~ 1mm SPS protons @ 400 GeV Simultaneus, unseparated, focused beams

4. 4Gianluca Lamanna – HEP0719.07.2007 NA48/2 detector Spectrometer: σ p /p = 1.0% + 0.044% p [p in GeV/c] LKR calorimeter: σ E /E = 3.2%/√E + 9%/E + 0.42% [E in GeV] CHOD, HAC,MUV, vetos Kabes Beam Monitor Only the spectrometer and LKr are involved in the analysis. The CHOD is used at the trigger level.

5. 5Gianluca Lamanna – HEP0719.07.2007 NA48/2 data & results 2003 run: ~ 50 days 2004 run: ~ 60 days Total statistics 2 years: K ± →  ±  0  0 : ~1·10 8 K ± →  ±  +  - : ~3·10 9 Greatest amount of K →3  ever collected u v Beam Pipe CP violation The main goal of NA48/2 was to measure the CP violation in charged kaon decays through the study of the asymmetry in three pion decays The goal to reach a precision of 10 -4 in the CP violation parameters Ag has been obtained after 2 years of data taking (2003 and 2004) No signal of CP violation outside the SM at our level of precision A g =(-1.5+1.5 stat +0.9 trig +1.1 syst )·10 -4 A g 0 =(1.8+1.7 stat +0.5 syst )·10 -4 A g =(-1.5+1.5 stat +0.9 trig +1.1 syst )·10 -4 A g 0 =(1.8+1.7 stat +0.5 syst )·10 -4 Phys.Lett.B 634:474-482,2006 Phys.Lett.B 638:22-29,2006 CERN-PH-EP-2007-021

6. 6Gianluca Lamanna – HEP0719.07.2007 K e4 : formalism M  2, M e 2, cos  , cos  e and  The Ke4 dynamics is fully described by 5 (Cabibbo-Maksymovicz) variables: M  2, M e 2, cos  , cos  e and  In the partial wave expansion the amplitude can be written using 2 axial and 1 vector form factors (the axial form factor R is suppressed in K e4 but accessible in K  4 ): F=F s e i  s +F p e i  p cos   G=G p e i  p H=H p e i  p The form factors can be expansed as a function of M  2 and M e 2 : F (F p,F s ), G, H and  =  p -  s will be used as fit parameters F s =f s +f s ’q 2 +f s ’’q 4 +f e ’(M e 2 /4m  2 )+... F p =f p +f p ’q 2 +... G p =g p +g p ’q 2 +... H p =h p +h p ’q 2 +... q 2 =(M  2 /4m  2 )-1

7. 7Gianluca Lamanna – HEP0719.07.2007 K e4 : selection & background Selection: 3 tracks Missing energy and missing Pt LKr/DCH energy to electron PID 677500 decays The background is studied using the electron “wrong” sign events (we assume  Q=  S and total charge ±1) and cross check with MC. The total bkg is at level of 0.5%. Main background sources:  +  →e  with  misidentified     or   +   (Dalitz) +e misidentified and  s outside the LKr    e K Kaon momentum GeV/c

8. 8Gianluca Lamanna – HEP0719.07.2007 K e4 : Fitting procedure The form factors (F,G,H and  ) are extracted minimizing a log-likehood estimator in each of 10(M  )x5(Me )x5(cos  e)x5(cos  )x12(  )=15000 equi-populated bins. In each bin the correlation between the 4+1 parameters is taken into account. DataMC K+ evts 435654 29 10.0 M 667 Evts/bin K- evts 241856 16 5.6 M 373 Evts/bin The form factors structure is studied in 10 bins of M , assuming constant form factors in each bins A 2D fit (M , Me ) is used to study the Fs expansion All the results are given wrt to F s (q=0) constant term, due to the unspecified overall normalization (BR is not measured) M  ● Data ▬ MC

9. 9Gianluca Lamanna – HEP0719.07.2007 K e4 : Fitting procedure and results Fp(q 2 ) Fs(q 2 ) Gp(q 2 ) Hp(q 2 ) Fs is quadratic in q 2 First measurement of Fp≠0 Linear in q 2 No linear term (h p ’)

10. 10Gianluca Lamanna – HEP0719.07.2007 K e4 : form factors result f’ s /f s = 0.165±0.011±0.006 f’’ s /f s = -0.092±0.011±0.007 f’ e /f s = 0.081±0.011±0.008 f p /f s = -0.048±0.004±0.004 g p /f s = 0.873±0.013±0.012 g’ p /f s = 0.081±0.022±0.014 h p /f s = -0.411±0.019±0.007 f’ s /f s = 0.165±0.011±0.006 f’’ s /f s = -0.092±0.011±0.007 f’ e /f s = 0.081±0.011±0.008 f p /f s = -0.048±0.004±0.004 g p /f s = 0.873±0.013±0.012 g’ p /f s = 0.081±0.022±0.014 h p /f s = -0.411±0.019±0.007 All the Form factors are measured relatively to fs first evidence of fp≠0 and fe’≠0 The f.f. are measured at level of <5% of precision while the slopes at ~15% (factor 2 or 3 improvement wrt previous measurements) Separately measured on K+ and K- and then combined (different statistical error) Systematics checks: Acceptance Background PID Radiative corrections Evaluation of the sensitivity of the form factors on the Me dependence of the normalization Preliminary (2003 data)

11. 11Gianluca Lamanna – HEP0719.07.2007 K e4 :  dependence The extraction of the pion scattering lengths from the  =  s-  p phase shift needs external theoretical and experimental data inputs. The Roy equations, for instance, provide this relation between  and a 0,a 2 near threshold, extrapolating from the M  >0.8 GeV region. The precision of these data defines the width of the Universal Band in the (a 0,a 2 ) plane. The fit of the experimental points using the Roy equations in the universal band allows to extract the a 0 and a 2 values

12. 12Gianluca Lamanna – HEP0719.07.2007 K e4 : (a 0,a 2 ) plane: result and comparison Minimizing the  2 in the 2D fit it’s possible to identify the favoured solution (and the corresponding ellipse) The E865 and NA48/2 results agreement is marginal (manly due to the last  point in E865) (work ongoing (see Gasser talk at Kaon07) ) The correlation between a 0 and a 2 is ~96% (similar for both experiment)

13. 13Gianluca Lamanna – HEP0719.07.2007 K e4 : “neutral” Selection: Selection: one electron track in the DCH, 4 photons in the LKr,  0 mass constraints, missing Pt. 9642 events in 2003 (previous exp. 216 events) ~30000 events in 2004 Background: Background:     with a misidentified , ke3  +1 accidental ~ 3% in 2003 (276 events) ~ 2% in 2004 Due to the     symmetry only the s-wave is present (fs’, fs’’) f’e has been measured consistent with 0 within the present statistics BR(Ke4 00 ) prel = (2.587±0.026 stat ±0.019 syst ±0.029 ext )·10 -5 f’ s /f s =0.129±0.036±0.020 f’’ s /f s =-0.040±0.034±0.020 f’ s /f s =0.129±0.036±0.020 f’’ s /f s =-0.040±0.034±0.020 Preliminary

14. 14Gianluca Lamanna – HEP0719.07.2007 Cusp: K ± →  ±     selection Offline selection Offline selection: among all the possible  pairings, the couple for which   is smallest is selected The K-decay vertex is the average between the two decay vertices After associating a charged track to the 2  0 s the compatibility with the PDG kaon mass is requested to be ± 6 MeV.  +  0  0 invariant mass, GeV/c 2 Resolution: 0.9 MeV/c 2 M K PDG ± 6 MeV/c 2 cut    contribution LKr zz d ij i j Z(i,j) Z(k,l) Vertex

15. 15Gianluca Lamanna – HEP0719.07.2007 Cusp: Dalitz plot distribution The high statistics and the good resolution allow to see a “cusp” in the U (or M 2  ) distribution in the position of 2m  16.0 M events in 2003 + 43.6 M events in 2004 data taking ~65% of the whole statistics M 2 

16. 16Gianluca Lamanna – HEP0719.07.2007 Cusp: one loop rescattering M K±K± ±±   = M0M0 K±K± ±±   + M1M1 K±K± ±±   ++ -- The M 1 contribution is real below and immaginary above  threshold M 0 = A 0 (1+g 0 u/2+h’u 2 /2+k’v 2 /2) 1– ( ) 2 M 1 = –2/3(a 0 –a 2 )m + M + M 00 2m + Below the threshold the (negative) interference term gives a “depletion” in the     mass distribution The cusp is proportional to (a0-a2) Cabibbo Phys. Rev. Lett. 93, 121801 (2004)  threshold 13% of depletion

17. 17Gianluca Lamanna – HEP0719.07.2007 Cusp: two loops M K±K± ±±   = M0M0 K±K± ±±   + M1M1 K±K± ±±   ++ -- K±K±        ++... Including 2-loops diagrams other terms appear in the amplitude All the S-wave amplitudes (5 terms) can be expressed as linear combination of a0 and a2 The isosping breaking effect is taking in to account The radiative correction (most relevant near threshold) are still missing A deviation from the no rescattering amplitude behaviour appears also above threshold Cabibbo,Isidori JHEP 0503 (2005) 21 M 2 (  0  0 ), (GeV/c 2 ) 2 0.074 0.0760.0780.080 Leading effect Sub- Leading effect No rescattering Cusp

18. 18Gianluca Lamanna – HEP0719.07.2007 Cusp: fit procedure & result The detector acceptance correction is obtained with a full GEANT simulation 7 bins The 1-D fit is performed excluding 7 bins around the threshold position pionium The excess of events in this region is interpreted as pionium signature Pionium : R=  (K   + A 2  )/  (K     +  – ) = (1.82  0.21)  10 –5. Prediction: R=0.8  10 –5 (Silagadze, 94) (a 0 –a 2 )m + = 0.261  0.006 stat.  0.003 syst.  0.0013 ext. a 2 m + = –0.037  0.013 stat.  0.009 syst.  0.0018 ext. (a 0 –a 2 )m + = 0.261  0.006 stat.  0.003 syst.  0.0013 ext. a 2 m + = –0.037  0.013 stat.  0.009 syst.  0.0018 ext. Using ChPt constraints [ Colangelo et al., PRL 86 (2001) 5008] a 2 = –0.0444 + 0.236(a 0 –0.22) – 0.61(a 0 –0.22) 2 – 9.9(a 0 –0.22) 3 (a 0 –a 2 )m + = 0.263  0.003 stat.  0.0014 syst.  0.0013 ext (Phys.Lett. B633:173- 283,2006) This result is fully compatible with our previous measurement on partial sample (Phys.Lett. B633:173- 283,2006) Preliminary

19. 19Gianluca Lamanna – HEP0719.07.2007 Cusp: systematics & “neutral” slopes Systematic effect (a 0 –a 2 )  10 2 a 2  10 2 (a 0 –a 2 )  10 2 ChPt Analysis technique±0.10±0.20±0.08 Trigger inefficiencynegl.±0.50negl. Description of resolution±0.06±0.11±0.06 LKr non-linearity±0.06±0.26±0.05 Geometric acceptance±0.02±0.01±0.02 MC sample±0.03±0.21±0.06 Simulation of LKr showers±0.17±0.50±0.04 V-dependence of amplitude±0.17±0.38±0.02 Total±0.28±0.90±0.14 The external error comes from A 00 /A +- = 1.975  0.015 A theoretical error of 0.013 (in a 0 -a 2 ) have to applied to take in to account the still missed radiative correction and the high order terms Standard expansion is not enough to describe the K→3  dynamics The slopes has been remeasured as (slightly different definition wrt to the PDG definition): g = (64.9  0.3 stat.  0.4 syst. )% h’ = (–4.7  0.7 stat  0.5 syst. )% k’ = (0.97  0.03 stat.  0.08 syst. )% g = (64.9  0.3 stat.  0.4 syst. )% h’ = (–4.7  0.7 stat  0.5 syst. )% k’ = (0.97  0.03 stat.  0.08 syst. )% First evidence of k≠0 Preliminary

20. 20Gianluca Lamanna – HEP0719.07.2007 Conclusions NA48/2 NA48/2 exploited two different procedure to measure the  scattering lengths. Ke4 Ke4: the  phase shift can be related to the a 0 and a 2 using theoretical input (e.g. Roy equations) K→     K→     : the  scattering lengths are extracted from the study of  rescattering contribution in the m  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 NA48/2 Ke4 NA48/2 Cusp DIRAC band (prel. 2007) Isospin breaking corrections applied both in Cusp and in Ke4 (work ongoing (see Gasser talk at Kaon07) )

21. 21Gianluca Lamanna – HEP0719.07.2007 Spares

22. 22Gianluca Lamanna – HEP0719.07.2007 Me slope: 2D fit F s =f s +f s ’q 2 +f s ’’q 4 +f e ’(M e 2 /4m  2 )+... f’’s f’s f’s -0.96 0.03 f’’s -0.06 In the 1D fit a residual variation is observed with respect to M e 2D in (M  , M e  ) performed Linear depence with M e  f’ e /f s = 0.081±0.011±0.008

23. 23Gianluca Lamanna – HEP0719.07.2007 Ke4: isospin breaking correction Kaon 2007 See Gasser’s talk @ Kaon 2007 Thanks to the indipendent bin analysis the correction can be applied also to old data coming from previous experiment The results become compatible with the cusp’s results

24. 24Gianluca Lamanna – HEP0719.07.2007 Fit results K- K+

25. 25Gianluca Lamanna – HEP0719.07.2007 Cusp fit (in 2003)    420/148    155/146    149/145    145/139 One loop Two loops Pionium Excluding 7 bins Standard Dalitz plot parameterization  =(data-fit)/data |M(u,v)| 2 ~1+gu+hu 2 +kv 2 +...

26. 26Gianluca Lamanna – HEP0719.07.2007 Dirac experiment The |a0-a2| quantity can be extracted from the measurement of the lifetime of pionic atoms in a model independent way The ionized exotic atoms are produced in a fixed target Lifetime in the order of 3 fs ChPt predicts with high accuracy this lifetime p is the  0 momentum and  corrections Physics Letters B 619 (2005) 50 |a0-a2|=0.264 +0.033 -0.020 expected error in 2007: +7.4%, -4.2%

27. 27Gianluca Lamanna – HEP0719.07.2007 Spectrometer alignment P = P 0 ∙(1+β)∙(1+qb  P 0 ) B signKaon sign Raw momentum Eq. Sensitivity (on DCH4): M/x  1.5 keV/m The kaon mass depends from the time variation of the spectrometer alignment The mis-alignment gives a mis- measurement of the charged pion momentum The reconstructed invariant K mass is used to fine tune the spectrometer by imposing (  correction ) : M K+ =M K- The non-perfect field alternation is tuned by imposing (  correction): M K+- =M Kpdg

28. 28Gianluca Lamanna – HEP0719.07.2007 Beam movements DCH1 (upstream magnet) K+ KK X, cm Large time scale movement: the beam positions change every run Acceptance largely defined by central beam hole edge (~10 cm radius) The cut is defined around the actual beam position obtained with the c.o.g. measured run by run, for both charges as a function of the K momentum (“virtual pipe” cut) Short time scale movement: the beam moves during the SPS spill Monitored with an high resolution beam monitor on the beams The 2 beam movement is “coherent” No effect in the 4-uple ratio

29. 29Gianluca Lamanna – HEP0719.07.2007 “blue field”  Δ<10 -5 The Earth field (Blue Field) was directly measured and used at the vertex recostruction level. The residual systematics is  Δ<10 -5 P kick(stray field) P kick(spectrometer)  10 -4