1. Recent results from NA48/2 on pion scattering lengths using Ke4 decay and cusp in K ± -> π ± π 0 π 0 Stefano Venditti University of Pisa & INFN QCD@WORK 2007 On behalf of the NA48/2 collaboration: Cambridge, CERN, Chicago, Dubna, Edinburgh, Ferrara, Firenze, Mainz, Northwestern, Perugia, Pisa Saclay, Siegen, Torino, Wien

2. QCD@WORK 2007Stefano Venditti17/06/2007 Outline The NA48/2 experiment; Ke4: theory and NA48/2 analysis; Ke4: results for form factors and interpretation in terms of pion scattering lengths (a 0,a 2 ); Cusp: K3pi theory and analysis; Cusp: fit procedure and extraction of (a 0 -a 2 ); Conclusions.

3. NA48/2 beam line K+K+ K−K− P K spectra, 60  3 GeV/c 54 60 66 QCD@WORK 2007Stefano Venditti17/06/2007 Detectors K + /K - flux~1.8 Width ~ 5mm K+/K- ~ 1mm Beams coincide within 1 cm over 114 m of decay volume Incoming SPS 400 GeV protons

4. NA48 DETECTORS Spectrometer: 4 DCHs -> redundancy. σ p /p=1.0%+0.044%× p(p in GeV); Liquid Krypton EM calorimeter: 16000 cells -> high granularity. σ E /E=3.2%/√E + 9%/E + 0.42%; Hodoscopes (charged, neutral): Trigger, time measurement. Muon veto, Hadronic calo, Kabes, photon vetoes. Stefano Venditti17/06/2007QCD@WORK 2007

5. Stefano Venditti17/06/2007 NA48/2 DATA: 2003 run: ~ 50 days 2004 run: ~ 60 days Ke4: 0.68 M events (from 2003 data only) K3pi: ~ 10 8 events (greatest K3pi sample ever collected) EVENTS SELECTED FOR K ± ->π ± π ± e ± ν AND K ± ->π ± π 0 π 0 :

6. QCD@WORK 2007Stefano Venditti17/06/2007 Ke4:Theory 4 body decay->5 independent variables Using Cabibbo-Maksymowicz variables: S π (Mππ 2 ),S e (Meν 2 ),cosθ π,cosθ e,Φ Hadronic ME : =(1/M K )[FP λ +GQ λ +R(K-P) λ +(H/M 2 )ε λμνσ K μ P ν Q σ ] F,G,R: axial form factors H: vector form factor K e+e+ π-π- π+π+ ν θπθπ θeθe Φ Partial wave expansion of amplitude: F = F s e i δ s + F p e i δp cosθ π +d-wave terms G = G p e i δ g + d-wave terms H = H p e i δ h + d-wave terms Fit parameters: F s, F p, G p, H p, δ = δ s -δ p R negligible (relevant in Kμ4) f.f. expansion wrt Se, q 2 : F s =f s +f’ s q 2 +f’’ s q 4 +f e (Se/4m π 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 =(S π /4m π 2 )-1 (P λ =dipion 4V; Q λ =dilepton 4V; K λ =kaon 4V)

7. QCD@WORK 2007Stefano Venditti17/06/2007 Ke4 Analysis: selection and BG rejection Ke4-> 3 charged tracks (2 opposite sign pions),1 ν. BR ~ 4 · 10 -5 K±K± e±e± ν π-π- π+π+ Spectrometer for momenta measurement; LKR info used to tag electron and pions (E/p); Missing energy and Pt (because of neutrino). BG checked with data: wrong sign events have same total charge but wrong electron charge (es:e + π - π - for K- decay):their contribution to total BG is the same or is to be rescaled by a factor 2 wrt “real” BG, depending on the process. Pk Mke4 Background, main sources: π ± π + π -, with π->eν in-flight decay or π misidentified as e; π ± π 0,π ± π 0 π 0,with π 0 ->e + e - γ Dalitz decay and,e misidentified and γ(s) undetected.

8. QCD@WORK 2007Stefano Venditti17/06/2007 Data/MC comparison MππMeν cosθ π cosθ e GeV

9. QCD@WORK 2007Stefano Venditti17/06/2007 Ke4 analysis: fitting procedure Iso-populated 10(Mππ)x5(Meν)x5(cosθe)x5(cosθπ)x12(Φ)=15000 bins in the C-M variables used. Form factor values used to minimize a log-likehood estimator well- suited for small numbers. K + : Data: 435654 events,29 evts/box MC: 10.0 M events,~667 evts/box K - : Data: 241856 events,16 evts/box MC: 5.6 M events,~373 evts/box K + /K - ~1.8 (both data and MC) MC/Data~23 (both K + and K - ) Ten independent fits in Mππ bins, assuming constant f.f. over single bins. no normalization->only relative f.f. and their variation wrt kinematical variables; residual variation (linear slope) observed wrt Mev for Fs: 2-dim fit performed; Fs from bin/bin normalization after fit. Fs 2 ~(1+f’s q 2 + f’’s q 4 +f’e Se/4m π 2 ) 2

10. QCD@WORK 2007Stefano Venditti17/06/2007 f p,h p ≠0, no q 2 dependence g p linear wrt q 2 Correlation: gpgp g’ p -0.914

11. QCD@WORK 2007Stefano Venditti17/06/2007 Ke4: form factors results Relative form factors: f.f./Fs(0); measured separatedly for K +,K - and then combined; Fs obtained from bin/bin normalization,Fp,Gp,Hp deconvoluted from observed Fs(q 2,Se) variation. SYSTEMATIC CHECKS Two independent analyses; Acceptance control; BG level and shape control; Radiative correction included; Possible bin-to-bin correlation considered. 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 value±stat.±syst. All f.f. parameters measured within 5% to 15% relative precision

12. QCD@WORK 2007Stefano Venditti17/06/2007 Ke4: δ phase shift and a 0,a 2 extraction Extraction of pion scattering lengths from δ=(δ 0 0 -δ 1 1 ) phase shift can be done through external experimental and theoretical (e.g. Roy equation) inputs, which relate δ and (a 0,a 2 ); The Universal Band parameterization corresponds to a 1-dimensional fit of δ with a fixed relation between a 0 and a 2.

13. QCD@WORK 2007Stefano Venditti17/06/2007 Ke4: comparison with other experiments Thanks to the independent bin analysis, the scattering length extraction can be performed on old data even if the collaboration doesn’t exist anymore; E865 quotes values ranging from a 0 =0.203 to a 0 =0.237, NA48 seems to obtain slightly higher values; Further checks are ongoing to understand the two different results, expecially in the last M ππ bin. a 0 =0.25 a 0 =0.20

14. QCD@WORK 2007Stefano Venditti17/06/2007 Plane (a0,a2): theory and experiments Several equations relating a 0 and a 2 exist (ACGL,DFGS,…) A 2-dimensional fit (on a 0 and a 2 ) can also be performed (dotted lines in figure,centered on best χ 2 and including 68% of events); E865 and NA48/2 point at slightly different regions of the universal band; removing the last E865 bin brings the two results closer and decreases χ 2. Isospin breaking corrections neglected so far. This contribution was considered negligible until short time ago. Calculations to apply this correction are ongoing.

15. QCD@WORK 2007Stefano Venditti17/06/2007 K3pi:selection D ik Z ij Z kl LKR For each photon pair (e.g. i,j) a decay vertex reconstructed along beam axis assuming π 0 mass: m 0 2 =2E i E k (1-cosβ)~E i E k (D ik ) 2 /(Z ik ) 2 Z ik ; Pair of photons minimizing ΔZ=Z jl -Z ik chosen; Compatibility within ± 6 MeV wrt PDG kaon mass requested. ΔZΔZ γ γ γ γ D jl K m 0 : pion mass Ei,Ej: γ energies Z ik : π 0 vertex distance from LKR ππ invariant mass Excellent at low M ππ values  +  0  0 invariant mass, GeV/c 2 Resolution: 0.9 MeV/c 2 M K PDG ± 6 MeV cut 59.3M K +    contribution 32M K -

16. QCD@WORK 2007Stefano Venditti17/06/2007 Primary goal: asymmetry measurement Dalitz variables: u=(s 3 -s 0 )/π 2 v=(s 2 -s 0 )/π 2 3s 0 =Mk 2 +Mπ 2 +2Mπ 0 2 s i =(Pk-Pi) 2 (i=1,2,3;3=odd pion) Matrix element: |M(u,v)| 2 ~1+gu+hu 2 +kv 2 +... K+K+ π + (even) π - (odd) Ag≠0->direct CP violation v u K ± →  ±  0  0 dalitz plot Comparison between Dalitz plot distribution for K + and K - to look for direct CP violation. SM Ag predictions in range 10 -6 -10 -5 ; Beyond SM models enhance the prediction;

17. QCD@WORK 2007Stefano Venditti17/06/2007 Slope difference: Δg=(2.2±2.1 stat ±0.7 syst )·10 -4 Charge asymmetry: Ag=(1.8±1.7 stat ±0.5 syst )·10 -4 4-ple ratio: R 4 =R US *R UJ *R DS *R DJ ≈ n(1+g/f(u)) 4 Acceptance equalization X Y Jura Saleve Achromats: K + Up B+ BB K+K-K+K-K+K-K+K- In each ratio the charged pions are deflected towards the same side of the detector (left-right asymmetry cancels out) In each ratio the event at the numerator and denominator are collected in subsequent period of data taking (global time variations) The whole data taking is subdivided periods in which all the field configurations are present. 3-fold cancellation: L-R asymmetry; Beam shape asimmetry; Global time variations. N(A+B+K+) N(A+B-K-) R US = N(A+B-K+) N(A+B+K-) RUJ=RUJ= N(A-B+K+) N(A-B-K-) RDS=RDS= N(A-B-K+) N(A-B+K-) R DJ = RESULTS:

18. QCD@WORK 2007Stefano Venditti17/06/2007 “Bonus” goal: cusp in M 2 π 0 π 0 distribution 2003: 16.0 mln events 0.080 0.079 0.078 0.0770.076 0.079 0.078 0.0770.076 80k 60k 40k 20k 0 40k 80k 120k 160k 200k 80k 120k 110k 100k 90k 25k 45k 2004 (80%): 43.6 mln events M 2 (  0  0 ), (GeV 2 ) 70k 30k 35k 40k  +  – threshold Cusp analysis not foreseen at the beginning of the experiment; First cusp observation on 2003 data, 2004 data now included (~80% of the whole statistics); A cusp can be seen in M 2 ( π 0 π 0 ) distribution at 4Mπ + value and is the effect of the interference of (at least) two amplitudes. M 2 (  0  0 ), (GeV 2 )

19. Negative interference under 2m + QCD@WORK 2007Stefano Venditti17/06/2007 Cusp theory: final state rescattering M(K ± π ± π 0 π 0 )=M 1 +M 2 M 0 =direct emission amplitude = A 0 (1+g 0 u/2+h’u 2 /2+k’v 2 /2) Not the same parameterization used for asymmetry! Amplitude parameterized (rather than matrix element) M 1 =rescattering amplitude = -2/3(a 0 -a 2 )m + M + √ 1-(M 00 /2m+) 2 K±K± K±K± π±π± π±π± π0π0 π+π+ π-π- π0π0 π0π0 π0π0 S-wave ππ scattering lengths K->3π ± amplitude at threshold : no rescattering : 1-loop rescattering M 2 π 0 π 0 (GeV 2 ) M 2 =4m 2 π+ N. Cabibbo, PRL 93 (2004) 121801

20. QCD@WORK 2007Stefano Venditti17/06/2007 Cusp theory: higher order diagrams 1 and 2-loop processes included; Five S-vawe scattering lengths (a x, a ++, a +-, a +0, a 00 ), expressed as linear combinations of a 0 and a 2 ; Isospin symmetry breaking (~2%) considered; Radiative corrections missing -> a 0 -a 2 precision >= 5 %. Arbitrary scale 0.074 0.0760.0780.080 Cusp Negative amplitude No rescattering Imaginary amplitude K±K± K±K± K±K± π±π± π±π± π±π± π0π0 π0π0 π0π0 π0π0 π0π0 π0π0 Examples of 2-loop diagrams: N. Cabibbo and G. Isidori, JHEP 503 (2005) 21 π*π* π*π* π*π* π*π* π*π*π*π* π*π* π*π* π*π* π*π* π*π* Two loop effect on cusp No cusp Cusp (2-loop correction) M 2 π 0 π 0 (GeV 2 )

21. QCD@WORK 2007Stefano Venditti17/06/2007 Cusp: fit procedure Detector response computed with a full GEANT MC; resolution matrix (computed on s 3 variable) applied on MC-generated data; Five free parameters (g 0, h’,m + (a 0 -a 2 ),m + a 2,N) in MC data. Use of MINUIT to minimize χ 2 based on difference data-MC. χ 2 (g 0, h’,m + (a 0 -a 2 ),m + a 2,N)=∑ δ F 2 data + δ 2 N 2 F 2 MC bins (F data -NF MC ) 2 Resolution smears MC bins: F i MC =∑ R ij G j G=G(M 00,g 0,h’,m + (a 0 -a 2 ),m + a 2 ) generated MC bin: Generated s 3 =M 2 (  0  0 )(GeV 2 ) Reconstructed s 3

22. QCD@WORK 2007Stefano Venditti17/06/2007 Cusp: dealing with pionium Pionium created when relative velocity between two opposite-sign pions is ~0. It is created by EM interaction, decays (~10 -16 sec) by strong interaction; No Coulomb corrections -> Pionium cannot be accounted for in this model -> 7 bins (±3.5 resolution σ’s around dipion mass) excluded from Data/MC comparison. If excess in data all interpreted as pionium, one gets: R=  (K   + A 2  )/  (K     +  – ) = (1.82  0.21)  10 –5 Theoretical prediction:R=0.8  10 –5 [Z.K. Silagadze, JETP Lett. 60 (1994) 689] 7 bins excluded (0.0775-0.0785 GeV 2 )

23. QCD@WORK 2007Stefano Venditti17/06/2007 Cusp: results (a 0 –a 2 )m + = 0.261  0.006 stat.  0.003 syst.  0.013 ext. a 2 m + = –0.037  0.013 stat.  0.009 syst.  0.018 ext. Using a chiral symmetry constraint [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 (2003+2004 data, 80% of statistics) (a 0 –a 2 )m + = 0.263  0.003 stat.  0.0014 syst.  0.013 ext. Analysis technique; Trigger inefficiency; Resolution; LKR non-linearity; Geometric acceptance; MC sample; LKR showers V-dependence of amplitude. Systematic checks

24. QCD@WORK 2007Stefano Venditti17/06/2007 Conclusions The NA48 results for a 0 and a 2 following two different paths (ke4 and cusp) are consistent; Agreement is also found with DIRAC result, which computes a 0 and a 2 measuring pionium lifetime; Both analyses are still ongoing, room for improvements.