15 мая 2009В.Кекелидзе, "Марковские чтения" 1 Фундаментальные параметры рассеяния в распадах заряженных каонов Фундаментальные параметры рассеяния в распадах заряженных каонов эксперимент NA48/2

15 мая 2009В.Кекелидзе, "Марковские чтения" 2 Эксперимент NA48/2 «cusp» эффект и параметры киральной теории сравнение с Ke4 результатами содержание

15 мая 2009В.Кекелидзе, "Марковские чтения" 3 Эксперимент NA48/ run: ~ 50 days 2004 run: ~ 60 days Total statistics in 2 years: K + : ~4·10 9 K 0 0 : ~1·10 8 Rare K ± decays: BRs down to 10 –9 can be measured > 200 TB of data recorded Основная задача - Поиск зарядовой асимметрии в распадах K (3 )

15 мая 2009В.Кекелидзе, "Марковские чтения" 4 Установка NA48 Установка NA48 High resolution magnetic spectrometer ( + ): drift chambers; magnet with momentum kick 265 MeV/c electromagnetic cal. ( 0 0 ): liquid krypton (~10m 3 at 120 K) cells granularity 2x2 cm2 Rate ~10 6 K L decays / spill

15 мая 2009В.Кекелидзе, "Марковские чтения" 5 LKr - э-м жидко-криптоновый калориметр

15 мая 2009В.Кекелидзе, "Марковские чтения" 6 NA48/2 beam line 1cm Front-end achromat Momentum selection Quadrupole quadruplet Focusing sweeping Second achromat Cleaning Beam spectrometer (resolution 0.7%) ~ ppp, 400 GeV K+K+ K Beams coincide within ~1mm all along 114m decay volume focusing beams BM z magnet K+K+ K beam pipe Simultaneous K + and K beams: large charge symmetrization of experimental conditions Be target 2-3M K/spill ( /K~10), decay products stay in pipe. Flux ratio: K + /K – 1.8 P K spectra, 60 3 GeV/c cm 200 vacuum tank not to scale 250 m He tank + spectrometer

15 мая 2009В.Кекелидзе, "Марковские чтения" 7 An anomaly was observed at invariant mass M( 0 0 ) = 2m + i n K 0 0 decays never observed in previous experiments «Сusp» - эффект Cusp - « остриё », особенность в спектре, связанная с изломом; в отличие от пика, достоверное наблюдение такой особенности требует большой статистики и высокого разрешения : 59.6 mln events + – threshold

15 мая 2009В.Кекелидзе, "Марковские чтения" 8 длины пионного рассеяния важнейшим свободным параметром киральной теории (ChPT) является кварковый конденсат, который определяет относительные размеры масс и импульсов в степенном разложении a 0 и a 2 - длины pp рассеяния (S-волны) соответственно с изоспином I=0 и I=2, также входят в амплитуды pp -рассеяния связь с a 0 и a 2 известна из теории с высокой точностью; поэтому измерения a 0 и a 2 позволяют получить важные ограничения для параметров ChPT Pion scattering lengths can be measured in the study of the cusp-effect in K 0 0 decays 2003 result: PLB 633 (2006) 173, Cabibbo-Isidori theoretical framework] «Сusp» - эффект и параметры киральной теории Анализ был проведен независимо 2-мя группами из Пизы и Дубны N. Cabibbo, PRL 93 (2004)

15 мая 2009В.Кекелидзе, "Марковские чтения" 9 m 0 – mass of 0 E i, E k – energy of i, k D ik – distance between i and k on LKr z ik – distance from 0 decay vertex to LKr K 0 0 selection m 0 2 = 2E i E k (1-cos ) E i E k 2 = E i E k (D ik ) 2 (z ik ) 2 M 2 ( 0 0 ), (GeV/c 2 ) 2, MeV/c mass resolution … excellent at low M 00 ! M K -3 mass spectrum (2004 data) 10 3 Background is negligible Events for each photon pair (i,k) a decay vertex is reconstructed along the beam axis under the assumption of 0 decay accidental photon selected photon LKr D ik z z ik z lm K-decay vertex ElEl EiEi EmEm z EkEk

15 мая 2009В.Кекелидзе, "Марковские чтения" 10 No M 1 amplitude M 1 amplitude present: 13% depletion under the threshold Arbitrary scale N. Cabibbo, PRL 93 (2004) M (K 0 0 ) = M 0 + M 1 M 0 = A 0 (1+g 0 u/2+h 0 u 2 /2+k 0 v 2 /2) K+K+ Kaon rest frame: u = 2m K (m K /3 E odd )/m 2 v = 2m K (E 1 E 2 )/m 2 Direct emission: Rescattering amplitude: M 1 = –2/3(a 0 –a 2 )m + M + Combination of S-wave scattering lengths Negative interference under threshold K 3 amplitude at threshold M 2 ( 0 0 ), (GeV/c 2 ) 2 (isospin symmetry assumed here) Объяснение «Cusp» - рассеяние в конечном состоянии 1–( ) 2 M 00 2m +

15 мая 2009В.Кекелидзе, "Марковские чтения" 11 S-wave scattering lengths (a x, a ++, a +–, a +0, a 00 ) expressed as linear combinations of a 0 and a 2 isospin symmetry breaking - following J. Gasser for example, a x = (1+ /3)(a 0 –a 2 )/3, where =(m + 2 –m 0 2 )/m + 2 = isospin breaking parameter all rescattering processes at one- & two-loop level radiative corrections missing: (a 0 – a 2 ) precision ~5% b) irreducible 3 scattering Two-loop diagrams: c) reducible 3 scattering a) 2 scattering N. Cabibbo and G. Isidori (CI), JHEP 503 (2005) 21 Arbitrary scale No rescattering amplitude Subleading effect M 2 ( 0 0 ), (GeV/c 2 ) 2 Cusp point Leading effect Prediction of the two-loop theory One-loop diagrams: Теория: двух-петлевые диаграммы

15 мая 2009В.Кекелидзе, "Марковские чтения" 12 Theory: effective fields Non-relativistic Lagrangian for effective fields; expanding in another small parameters. Valid in the whole decay region. Another (in comparison with CI) part of amplitude is absorbed in the polinomial terms (so another correlations). At two loops, algebraically different formulae for amplitude FORTRAN code written by authors G. Colangelo, J. Gasser, B. Kubis, A. Rusetsky (Bern-Bonn group: BB) Phys.Lett. B638 (2006)

15 мая 2009В.Кекелидзе, "Марковские чтения" 13 процедура фитирования 2 (g 0,h 0,m + (a 0 –a 2 ),m + a 2,N)= (F DATA –NF MC ) 2 F DATA 2 +N 2 F MC 2 s 3 bins Detector response matrix R ij obtained with a GEANT-based MC simulation 5 free parameters 1-dimensional fit of the M 00 projection MINUIT minimization of 2 of data/MC spectra shapes Generated distribution G(M 00 ) = G(g 0,h,a 0,a 2,M 00 ) Generated s 3 =M 2 00, (GeV/c 2 ) 2 Reconstructed s 3 (GeV/c 2 ) 2 420x420 bins Reconstructed distribution: F j MC = R ij G i fit region Log(R ij )

15 мая 2009В.Кекелидзе, "Марковские чтения" 14 Fit quality & pionium signature Combined sample Points excluded from the fit due to absence of EM corrections in the used model 7 data bins skipped around the M( + – ) threshold Excess of events in the excluded interval, if interpreted as due to pionium decaying as A 2 0 0, gives R= (K + A 2 )/ (K + – ) = ( ) 10 –5. Prediction [Z.K. Silagadze, JETP Lett. 60 (1994) 689] : R= –5.

15 мая 2009В.Кекелидзе, "Марковские чтения" 15 Электромагнитные поправки в конечном состоянии (Gevorkian, Tarasov, Voskresenskaya, Phys.Lett. B (2007)) Two contributions from K ± ± decay to the K ± ± cusp region: Pionium formation : atom (negligible width) Additional unbound states with resonance structure resonant structure (no experimental resolution) resonant structure with experimental resolution atoms and resonant structure with experimental resolution In our fit we use a free parameter f atom (relative excess in the bin of width GeV 2 ) to represent Pionium + resonant structure measured value% (6 1) % is in good agreeemen with the prediction 5.8%

15 мая 2009В.Кекелидзе, "Марковские чтения" 16 Effect of radiative corrections Electromagnetic part is included into Lagrangian and all the first-order diagrams are taken into account But the bound states and resonant peak at the very threshold can not be calculated perturbatively (a free parameter f atom is used to represent the peak-like contribution) Could be extracted as a relative effect for amplitude and Implemented also to Cabibbo-Isidori case Bern Bonn radiative correction to pion rescattering amplitude for 3pi decays. (Bissegger M., Fuhrer A., Gasser J., Kubis B., Rusetsky A. Nucl. Phys., B806, 2009, )

15 мая 2009В.Кекелидзе, "Марковские чтения" 17 Uncorrected Corrected 68% - probability ellipses for statistical errors only to illustrate the shift. Effect of radiative corrections

15 мая 2009В.Кекелидзе, "Марковские чтения" 18 Example of correlation coefficients (CI)

15 мая 2009В.Кекелидзе, "Марковские чтения" 19 (a 0 –a 2 )m + = stat syst ext. a 2 m + = – stat syst ext. Результат по данным Additional theoretical uncertainty (higher order terms neglected) is estimated from the difference between results obtained using Bern-Bonn and Cabibbo-Isidori formulae: (a 0 –a 2 )m + = ; a 2 m + = External uncertainty: due to (A ++– /A +00 )| threshold = ( )±0.015; (central value depends on fitted matrix element, error comes from PDG data) ( окончательный ) The relative errors for a 2 are essentially larger than (a 0 -a 2 ) ones, as a 2 alone leads to the second-order contribution to the cusp shape

15 мая 2009В.Кекелидзе, "Марковские чтения" 20 Systematical errors for independent a 0, a 2 in units of 10 -4

15 мая 2009В.Кекелидзе, "Марковские чтения" 21 (a 0 –a 2 )m + = stat syst ext. [Colangelo et al., PRL 86 (2001) 5008]: Theory precision uncertainty for this case is estimated by theorists: (a 0 –a 2 )m + = (2%). Результат по данным учитывающий ограничение из киральной теории ( окончательный ) ChPT prediction : / Experimental precision is compatible with the theoretical one, the only problem is the theoretical uncertainty of measurement !

15 мая 2009В.Кекелидзе, "Марковские чтения" 22 Systematical errors with ChPT constraint (a 2 is a function of a 0 ) In units of 10 -4

23 scattering in K ± e ± decay

K p*(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*( ) a rare decay [ B.R. = (4.09 ± 0.09) x 10 5 ] described by five variables Cabibbo – Maksymowicz variables : s M 2 s e M e 2 e For K K e e K e4 formalism

25 In the partial wave expansion (only S and P waves) the amplitude can be written using form factors: 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 expanded 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 G p =g p +g p q H p =h p +h p q q 2 =(M 2 /4m 2 )-1 K e4 formalism

26 K e4 : selection & background Selection: 3 tracks Missing energy & missing Pt LKr/DCH to electron PID K + ; K - The background is studied using the electron wrong sign events (we assume DQ=DS & total charge ±1) and cross check with MC. The total bkg is at level of 0.5%. Kaon momentum GeV/c Main background sources: ( e e K with misidentified or + (Dalitz) +e misidentified and s outside the LKr

27 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. Evts/bin evts Evts/bin evts MCData 9.8 M M K- K+ 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

28 K e4 : form factors result f s /f s = 0.158±0.007±0.006 f s /f s = ±0.007±0.007 f e /f s = 0.067±0.006±0.009 f p /f s = ±0.003±0.004 g p /f s = 0.869±0.010±0.012 g p /f s = 0.087±0.017±0.015 h p /f s = ±0.014±0.008 f s /f s = 0.158±0.007±0.006 f s /f s = ±0.007±0.007 f e /f s = 0.067±0.006±0.009 f p /f s = ±0.003±0.004 g p /f s = 0.869±0.010±0.012 g p /f s = 0.087±0.017±0.015 h p /f s = ±0.014±0.008 All the Form factors are measured relatively to fs first evidence of fp0 and fe0 The f.f. are measured at level of

а 0 = ± 0.013(stat) ± 0.007(syst) ± 0.002(theor) a 2 = ± (stat) ± (syst) ± (theor) (correlation coefficient 97%) Using ChPT link: a 0 = ± 0.005(stat) ± 0.002(syst) ± 0.006(theor) Ke4 results:

15 мая 2009В.Кекелидзе, "Марковские чтения" 30 Измерение длин пионного рассеяния - фундаментальных параметров киральной теории 68% - probability ellipses, full uncertainties

15 мая 2009В.Кекелидзе, "Марковские чтения" 31 a 0 = % th. uncert.

15 мая 2009В.Кекелидзе, "Марковские чтения" 32 физика каонов, как и 50 лет назад остается актуальной и является потенциальным источником новых открытий заключение

15 мая 2009В.Кекелидзе, "Марковские чтения" 33 SPARE

The Ke4 decay amplitude depends on two complex phases identified with 0 : the scattering phase shift in the I = 0, l = 0 state (s – wave) 1 : the scattering phase shift in the I = 1, l = 1 state (p – wave) The Ke4 decay rate depends on the phase shift difference = 0 – 1 is an increasing function of M ; 0 for M 2m ( from scattering theory : (k) ak at very low centre-of-mass momentum k ; a is the scattering length ; a 0 for s – waves only ) 0 in the Ke4 decay amplitude asymmetric distribution of the charged lepton direction with respect to the plane defined by the two pions ( distribution) (Shabalin 1963) The asymmetry of the distribution increases with M

NA48 / 2 (Preliminary) distributions for four M bins 2m+ < M < GeV < M < GeV < M < GeV GeV < M < m K

versus M (no isospin – breaking corrections) Geneva – Saclay : ~ 30,000 events, p K + = 2.8 GeV/c BNL E865 : 406,103 events (with ~ 4.4% background), p K + = 6 GeV/c NA48/2 : 677,510 events (with ~0.5% background), p K ± = 60 GeV/c NOTE: the isospin – breaking corrections reduce by 0.01 – (J. Gasser) OLD PICTURE !!!

To extract scattering lengths from some external data (I=2 scatteringg at higher energies) and theoretical input are needed, for example: Solution of Roy equations - ACGL, Phys.Rep.353(2001), DFGS, EPJ C24 (2002) that relates and (a 0,a 2 ). The Universal Band centre line parametrisation corresponds to 1-parameter fit with a 2 =f(a 0 ) ChPT constrain gives another relation between (a 0,a 2 ) inside Universal Band. Isospin symmetry breaking is taken into account (new)