Bernard Peyaud CEA Saclay, IRFU -SPP

Measurements of pp scattering lengths from K ± decays Experimental studies of K 3p and K e4 decays with the NA48/2 experiment at CERN Bernard Peyaud CEA Saclay, IRFU -SPP On behalf of the NA48/2 collaboration: Cambridge, CERN, Chicago, Dubna, Edinburgh, Ferrara, Firenze, Mainz, Northwestern, Perugia, Pisa, Saclay, Siegen, Torino, Wien Bernard Peyaud 1

Outline what can we learn from pp scattering lengths? the NA48/2 experimental setup measuring a 0, a 2 with: cusp in K ± p ± p 0 p 0 decays phase difference d pp in K ± p ± p - e ± n decays conclusion Bernard Peyaud 2

R pp scattering lengths: why interesting? p p angular wave number: k 2mE at low energy kr <<1 S-wave dominates the total cross section Bose statistics isospin I = 0, 2 allowed scattering matrix S pp> = exp(2id) pp> i.e. 2 phases: d 0,2 = a 0,2 k related to scattering lengths a 0, a 2 NA48/2 uses charged kaons to measure scattering lengths a 0, a 2 - cusp-effect in K p p 0 p 0 decays - d pp and form factors in K p p - e + n decays study of pp system near threshold S-wave scattering lengths a 0, a 2 are essential parameters of cpt spontaneous symmetry breaking from <q q> condensate? Bernard Peyaud 3

P K spectra, 60 3 GeV/c 54 60 66 K ± NA48/2 beam line 2-3M K ± /spill (p/k~10), p decay products stay in pipe. Flux ratio: K + /K 1.8 Simultaneous K + and K - beams: large charge symmetrization of experimental conditions magnet K + Be target K + K - focusing beams beam pipe BM z ~7 10 11 ppp, 400 GeV K - Second achromat 1cm Front-end achromat Momentum selection Quadrupole quadruplet Focusing sweeping Cleaning Beam spectrometer ( 0.7% (resolution Bernard Peyaud not to scale 4 50 100 Beams coincide within ~1mm all along 114m decay volume vacuum tank 200 He tank + spectrometer 250 m 10 cm

Main detector components: The NA48 detector Magnetic spectrometer (4 DCHs): 4 views/dch: redundancy efficiency; used in trigger logic; Δp/p = 1.0% + 0.044%*p [GeV/c]. Hodoscope fast trigger; precise time measurement (150ps). ( LKr ) Liquid Krypton EM calorimeter High granularity, quasi-homogenious; E /E = 3.2%/E 1/2 + 9%/E + 0.42% [GeV]; x = y =0.42/E 1/2 + 0.6mm (1.5mm@10GeV). Hadron calorimeter, muon veto counters, photon vetoes. Beam pipe Beams Bernard Peyaud 5

NA48 data Prime goal of NA48/2: measurement of CP-violating charge asymmetry in K 3 p decays both modes with large BR s of (2-5) 10-2 View of the NA48/2 beam line 2003 run: ~ 50 days 2004 run: ~ 60 days Total statistics in 2 years: K ± p ± p - p + : ~4 10 9 evts K ± p ± p 0 p 0 : ~1 10 8 evts Rare K ± decays with BR s down to 10 9 also measured and compared to cpt predictions cusp in K ± p ± p 0 p 0 decays d pp in K e4 decays >200 TB of data recorded Bernard Peyaud 6

Evidence of cusp structure in K ± p ± p 0 p 0 decays 2003: ~16x10 6 events 2004: ~44x10 6 events First observation of the cusp made with 2003 data Addition of 2004 data: statistics x 3.7 0 M 2 (p 0 p 0 ), (GeV/c 2 ) 2 M 2 (p 0 p 0 ), (GeV/c 2 ) 2 Homogeneous selection conditions + MC/Data statistics allows global analysis of all data p + p - threshold M 2 (p 0 p 0 ), (GeV/c 2 ) 2 M 2 (p 0 p 0 ), (GeV/c 2 ) 2 Bernard Peyaud 7

Arbitrary scale Theory: final state rescattering. N. Cabibbo, PRL 93 (2004) 121801 Direct emission (k,h << g): M(K ± p ± p 0 p 0 ) = M 0 + M 1 ( 2 / M 0 = A 0 (1+g 0 u/2+h 0 u 2 /2+k 0 v 2 M + ( 2 / = A + (1+g + u/2+h + u 2 /2+k + v 2 Rescattering amplitude: M 1 = 2/3(a 0 a 2 )m + M M 00 + 1 ( ) 2 2m + Kaon rest frame: u = 2m K (m K /3-E odd )/m 2 p v = 2m K (E 1 -E 2 )/m 2 p Negative interference under threshold p + K + p 0 p 0 Combination of S-wave pp scattering lenghts K ± 3p ± amplitude at threshold No M 1 amplitude (isospin symmetry M 1 amplitude present: ( here assumed 13% depletion under the threshold Bernard Peyaud 8 M 2 (p O p 0 ), (GeV/c 2 ) 2

Arbitrary scale Theory: 2 loops diagrams: N. Cabibbo and G. Isidori (CI), JHEP 503 (2005) 21 S-wave scattering lengths (a x, a ++, a +, a +0, a 00 ) are linear One-loop diagrams: combinations of a 0,a 2 Isospin symmetry breaking following J. Gasser. Radiative corrections missing; (a 0 a 2 ) precision ~5% V-dependent terms ~ (k /2)V 2 introduced recently in unperturbed K ± p ± p + p - and K ± p ± p 0 p 0 amplitudes. Two-loop diagrams: a) 2p scattering Prediction of the two-loop theory Cusp point b) irreducible 3p scattering c) reducible 3p scattering No rescattering amplitude Subleading effect Leading effect Bernard Peyaud 9 M 2 (p 0 p 0 ), (GeV/c 2 ) 2

Theory: effective field G. Colangelo, J. Gasser, B. Kubis, A. Rusetsky (CGKR)Phys.Lett. B638 (2006) 187-194 polynomial parts of amplitudes expressed in terms of (U,V)-slopes g,h,k numerically different from CI ones, and faster integration of the amplitude over V. Non-relativistic Lagrangian for effective fields Validity in the whole decay region. different part of amplitude (wrt CI) absorbed in the polynomial terms ( different correlations). At two loop level different formulae for amplitude FORTRAN code written by authors CI fit uses our recently measured parameters of the K ± 3p ± amplitude M + = A + (1+g + u/2+h + u 2 /2+k + v 2 /2): ( 0.00477(8 - = + k g + = - 0.21117(15); h + = 0.00671(26); CGKR has similar parameters but numerically different due to additional rescattering terms in the in M + amplitude. simultaneous fit of the NA48/2 K ± 3p ± and K ± p ± p 0 p 0 Dalitz plots for CGKR fit: g + = - 0.1837; h + = 0.00043; k + = - 0.0059 Bernard Peyaud 10

Pionium signature 7 Points around p + p - threshold are excluded from the fit due to absence of EM corrections in the model Combined 2003+2004 samples Excess of events in the excluded interval (CI fit), if interpreted as due to pionium decaying as A 2p p 0 p 0, gives R G(K ± p ± A 2p )/ R G(K ± p ± p + p - ) = (1.8±0.3)x10 5. Prediction [Z.K. Silagadze, JETP Lett. 60 (1994) 689]: R=0.8x10 5. Bernard Peyaud 11

( 1 ) Uncertainties & results Experimental constants are needed in the amplitude calculation external uncertainty dominated by R = (A ++ /A +00 ) threshold =1.975 0.015 Using a chiral symmetry constraint [Colangelo et al., PRL 86 (2001) 5008]: a 2 = ( 0.0444 +/- 0.0008) + 0.236(a 0 0.22) 0.61(a 0 0.22) 2 9.9(a 0 0.22) 3 CI :(a 0 a 2 )m + = 0.268 0.003 stat. 0.002 syst. 0.001 ext. CGKR: (a 0 a 2 )m + = 0.266 0.003 stat. 0.002 syst. 0.001 ext. ( preliminary ) Theory precision uncertainty for CI case: d(a 0 a 2 )m + = 0.013. Bernard Peyaud 12

( 2 ) Uncertainties and results With both a 0 and a 2 as free parameters in the fit CI case: ( preliminary ) (a 0 a 2 )m + = 0.266 0.005 stat. 0.002 syst. 0.001 ext. a 2 m + = 0.039 0.009 stat. 0.006 syst. 0.002 ext. Theoretical uncertainty: 5%. CGKR case: (a 0 a 2 )m + = 0.273 0.005 stat. 0.002 syst. 0.001 ext. a 2 m + = 0.065 0.015 stat. 0.010 syst. 0.002 ext. Theoretical uncertainty not given. CGKR case has large correlations between a 2 and polynomial terms of amplitude. CIparametersveryclosetoPDG s,a 0,a 2 better disentangled from g and h slopes. Bernard Peyaud 13

NA48/2: cusp fit results ( preliminary ) theoretical limits (Universal Band) Universal Band CI cpt band 68.3% prob. CI fit results: -stat. error only -stat. + syst. error -stat. + syst. + external -all including theoretical error a CI fit with cpt constraint CGKR 68.3% prob. CGKR fit results: -statistical only error; -statistical and systematical; -stat., syst. and external; CGKR fits with cpt constraint and lower limit of Universal Band constraint Bernard Peyaud 14

K e4 (K -> e n p + p - ) Results from 2003 Data analysis published (EPJC 54 3 (2008) 411) Event selection, reconstruction and FF extraction K e4 decays : kinematic variables and Form Factors Understanding d pp in terms of p p scattering lengths the pp scattering process is described theoretically by dispersion relations (Roy equations) that relate amplitudes with different isospin. As a result d pp depends essentially on two parameters, the scattering lengths a 00 and a 02. at low energy cpt constrains a 00 and a 02 through the size of the quark condensate Bernard Peyaud 15

Ke4 decays: event selection and background rejection Topology of p + p - e n signal : 3 charged tracks, 2 opposite sign pions, 1 electron (LKr info E/p ~ 1), some missing energy and p T (neutrino) Background : main sources π π + π - decay followed by π e n (dominant) or π misidentified as e π π 0 (π 0 ) decay + π 0 Dalitz decay (e + e g) + e misidentified as π and g (s) undetected Total background level is kept at ~ 0.5 % relative level Control from data sample : Wrong Sign -total charge (±1) as Right Sign events -electron charge opposite to total charge Log scale P Kaon In RS events: twice the rate if coming from K3p same rate if coming from K2p(p0) GeV/c Bernard Peyaud 16

Ke4 charged decays : 4-body decay formalism Five kinematic variables (Cabibbo-Maksymowicz): 2 invariant masses: S p M 2 pp, S e =M 2 en 3 angles : cosq p, cosq e and f. partial wave expansion of the amplitude: F, G = Axial Form Factors F = F s e ids + F p e idp cosq p + d-wave term G = G p e idg + d-wave term H = Vector Form Factor H = H p e idh + d-wave term expansion in powers of q 2, Se/4mp 2 (q 2 (S p /4m p2-1)) F F G s H p p p f f s g p h + p p + + f ' s f g + h q ' p 2 q ' p ' p q 2 q + 2 2 f '' s +.. +.. +.. q 4 + f e ( 4 ) 2 S / m e p +.. In each m pp bin the fit parameters are F s, F p, G p, H p and d = d s - d p Bernard Peyaud 17

Ke4 charged decays : Total Data sample (2003 + 2004) = 1.15x10 6 events! The 5 dimensional space spanned by C.M. variables, (M pp, M en, cosq p, cosq e and f) is binned with 10x5x5x5x12=15000 boxes with equal population The set of Form Factor values is used to minimize a log-likelihood estimator taking into account the statistics of the data and MC in each box. K + sample (739 500 events) K - sample (411 600 events) MC K+ sample (17.7 Millions events) MC K- sample (9.8 Millions events) 49 events/box 27 events/box ~1180 events/box ~653 events/box Ratio K+/K- ~ 1.8 both in Data and MC (run by run basis) Ratio MC/Data ~ 24. both for K+ and K- (run by run basis) Bernard Peyaud 18

Ke4 charged decays : the mass and cosq distributions K + and K - samples fitted separately, results combined data (symbols), simulation after fit (hist) and background ( x 10 to be visible) Background flat in cosq p, cosq e, f, concentrated at low M pp and M en = m p Mpp (GeV/c2) Men (GeV/c2) cosqp cosqe Bernard Peyaud 19

Ke4 charged decays : the f distributions CP symmetry : [d 5 N(K + )] at q p,q e,f = [d 5 N(K - )] at q p,q e,-f K - K + f (rad) Ratio K+/K- = 1.8 as expected from beam composition Bernard Peyaud 20

Ke4 joint distribution (cosqe,f) à la Pais-Treiman Distribution has 9 intensity functions:i i = I i (m pp,m en ) N( cosq e,f) = I 1 +I 2 cos2q e +I 3 sin 2 q e cos2f +I 4 sin2q e cosf +I 5 sinq e cosf +I 6 cosq e +I 7 sinq e sinf + I 8 sin2q e sinf + I 9 sin 2 q e sin2f With I 1 = f s 2 g 2 + f p2 g 2 /3 + g 2 (b 2 +a 2 /3)+ h 2 b 2 g 2 + 2/3 f p g cose f ag I 2 = -f s2 g 2 + g 2 /3 (b 2 -a 2 ) - f p2 /3 g 2 + h 2 /3 b 2 g 2-2/3 f p g cose f ag I 3 = -2/3 g 2 b 2 + 2/3 h 2 b 2 g 2 I 4 = p/2 f s g cos(d) bg I 7 = p f s g sin(d) bg I 5 -p f s h cos(d-e 2 ) bg 2 I 8 -p/2 f s h sin(d-e 2 ) bg 2 I 6-8/3 g h cos(e 2 ) b 2 g I 9-8/3 g h sin(e 2 ) b 2 g Intensity functions d=atan(i 7 /2I 4 ) d-e 2 =atan(2i 8 /I 5 ) e 2 =atan(2i 9 /2I 6 ) Bernard Peyaud 21

d pp signature in joint distribution (cosq e,f) m pp bin 1 m pp bin 5 m pp bin 10 COSq e > 0 data fit COSq e < 0 data fit COSq e > 0 data fit COSq e < 0 data fit f COSq e < 0 data fit f COSq e < 0 data fit f -p 0 p d pp 0.03 asym(cosq e ) 0 asym(f) 0 -p 0 p d pp 0.15 asym(cosq e ) > 0 asym(f) > 0 -p 0 p d pp 0.30 asym(cosq e ) >> 0 asym(f) >>0 Bernard Peyaud 22

Getting Fp, Gp, Hp fp 0 No q2 dependence Linear with q2 Correlation g p g p (0) -0.914 Bernard Peyaud 23

Ke4 charged decays : d pp and (a 0 0,a 02 ) Extracting a 00 and a 02 from d =d 00 - d 11 variation requires I=2 pp data at m pp > 0.8 + theoretical work numerical solutions of Roy equations (ACGL Phys. Rep.353 (2001), DFGS EPJ C24 (2002) ) to connect d with (a0, a2 ) Universal Band where CL uses a relation a 2 = f(a 0 ) the cpt constrain band (CGL NPB603(2001)): reduced uncertainty(± 0.008) a 2-parameter fit can also be performed with 2 free parameters a 00 and a 0 2 Isospin symetry breaking to be accounted for ( Bern arxiv:hep-ph/0710.3048 by J. Gasser) 2 free parameters isospin corrections ON Da0 = ± 0.013 (stat) Da2 = ± 0.0084 (stat) Bernard Peyaud 24

Ke4 charged decays : isospin corrections to d Using Bern corrections: 11 to 15 mrad over the fitted Mpp range Uncertainty R ± DR = 37 ± 4 ( R=m s -m ud /(m d -m u )) DR translates into Ddcorr = at most ± 0.5 mrad @ 0.390 GeV/c 2 quoted as theoretical precision (though marginal effect) Well matched to the NA48/2 analysis where radiative and isospin symmetry breaking effects factorize: Gamow factor x PHOTOS generator x Isospin corrections Formula coded from arxiv 0710.3048 Fit range [0.285,0.390] Bernard Peyaud 25

Ke4 charged decays : isospin corrections to d All Data K+ and K- combined= = Isospin corr OFF = Isospin corr ON Correction (~10mrad) is larger than the statistical error on each point above 0.3 GeV/c 2 (7-8 mrad) 2p fit a 0 0 a 0 2 Isospin corr OFF 0.244 ± 0.013-0.0385 ± 0.0084 Isospin corr ON 0.218 ± 0.013-0.0457 ± 0.0084 Bernard Peyaud 26

relative Form Factors = FF/Fs(0) - measured separately for K + and K -, - combined according to stat. errors, - Fs obtained from bin to bin norm. - Fp, Gp, Hp de-convoluted from observed Fs(q2,Se) variation. Scattering length a 0 now measured with ~2% relative precision (cpt 1p fit), Improved precision for a 0 (~6%) and a 2 (~18%) in a 2-free parameter fit Systematic uncertainty as for 2003 (conservative, will be revisited) but~0.5 x (stat) for a 0,a 2 Charged Ke4: results from 1 151 100 decays 2003 EPJC 54(2008) 2003 + 2004 preliminary Bernard Peyaud 27 f s /f s f s /f s f e /f s 0.172 ± 0.009-0.090 ± 0.009 0.081 ± 0.008 0.158 ± 0.007-0.078 ± 0.007 0.067 ± 0.006 f p /f s -0.048 ±0.004-0.049 ± 0.003 g p /f s g p /f s 0.873 ± 0.013 0.081 ± 0.022 0.869 ± 0.010 0.087 ± 0.017 h p /f s -0.411± 0.019-0.402 ±0.014 a 0 ChPT 1p fit a 2 =f(a 0 ) a 0 free a 2 free 0.223 ±0.006 (-0.0437 ±0.0015) 0.209 ± 0.016-0.0529±0.0105 value stat. 0.220 ± 0.005 (-0.0444 ±0.0011) 0.218 ± 0.013-0.0457±0.0084

scattering lengths measurements by NA48 Two statistically independent measurements by NA48/2: CI Cusp in K (p ± p 0 p 0 ) : 1p fit within 2 models (CI, CGKR) Ke4 with isospin corrections 2p fit using Roy equations 1p fit with cpt constraint remarkable agreement with cpt CGKR Precise cpt predictions : a 2 = -0.0444 ± 0.0008 and a 0 = 0.220 ± 0.005 Bernard Peyaud 28

Comparing with other measurements Ke4 : isospin corrections to published d pp of all 3 experiments + 1p cpt fit Note : E865 result dominated by highest energy data point, otherwise compatible Cusp : use 1p cpt fit and 2 models DIRAC : a 0 -a 2 non-symmetric errors from PLB619 (2005), use cpt constraint, still being revisited Yellow band is cpt prediction NA48/2 experimental precision now at the same level as theory! Events 1.15 10 6 4 10 5 3 10 4 100 10 6 6.5 10 3 a 0 from 1p ChPT fit, isospin corr ON Bernard Peyaud 29

Epilogue arxiv hep-ph/0212323v1 20 Dec 2002 2003+04 K e4 and cusp data from NA48/2 has allowed to measure (a 0,a 2 ) with precision equal to cpt data and theory agree remarkably well and it demonstrates that the pion mass is dominated (94%) by the quark condensate i.e. spontaneous breakdown of cpt (G. Colangelo, Garda workshop Feb. 2008) M 2 p - qq F 2 p ( m u + m d ) + ( m 2 q ) M. Gell-Mann, R. J. Oakes and B. Renner, Phys. Rev. 175 (1968) 2195 Bernard Peyaud 30

SPARES Bernard Peyaud 31

K e4 amplitudes and phases with T and CP violation f p e f nominal Ke4 parameters <g>=0.92, <g >=0.8, h =-0.37 e 1 =0, e h =p f s e fp h p K + e h g p gf p /a e f d e 2 e 1 g p -e fp CPT /CP /T f s f p -e f e 1 (K - ) = -e 1 (K + ) and e 2 (K - ) = -e 2 (K + ) K - -e 2 h p -e 1 g p d gf p /a Bernard Peyaud 32 g p -e f