Etude de lintraction trs basse nergie auprs de

Etude de l’intéraction à très basse énergie auprès de l’expérience NA 48/2 au CERN: longueurs de diffusion et formation d’atomes de pionium Luigi Di. Lella Scuola Normale Superiore, Pisa § L’expérience NA 48 / 2 § Sélection et reconstruction d’ événements K º º § Motivation initiale: recherche d’atomes + (“pionium”) § Distribution de masse invariante º º § Interprétation: mesure des longueurs de diffusion – § Comparaison avec les résultats d’autres expériences: mesure du temps de vie du pionium (expérience DIRAC au CERN) § Conclusions Séminaire au DAPNIA, Saclay, 12. 10. 2005

The NA 48 / 2 experiment Cambridge – CERN – Chicago – Dubna – Edinburgh – Ferrara – Firenze – Mainz – Northwestern – Perugia – Pisa – Saclay – Siegen – Torino – Vienna Approved in 2001 to search for direct violation of CP symmetry in the decay of charged K-mesons to three pions: K + (Branching fraction 5. 57%) K (Branching fraction 1. 73%) METHOD: Search for K+ / K difference of “odd pion” energy distribution “Odd pion”: p- in K+ p+p+p- ; p+ in K- p-p-p+ ; p in K p p p Kinematic variables: (i = 3 : odd pion) ; Matrix element: Violation of CP symmetry: ;

NA 48/2 main goal: – Measure Ag in both K p p+p- and K p p p decay modes with accuracies δAg< 2. 2 x 10 -4 and δAg< 3. 5 x 10 -4 , respectively – Required statistics: > 2 x 109 events in “charged” mode; >108 events in “neutral” mode NA 48/2 method: maximal cancellations (robustness) – Two simultaneous K+ and K− beams, superimposed in space – Detect asymmetry only from slopes of ratios of normalized u distributions – Equalize averaged K+ and K– acceptances by frequently changing polarities of relevant magnets

NA 48/2 beam setup PK spectra, 60 Ge. V/c 54 2 ÷ 3 M K / spill (π / K ~ 12) π decay products stay in pipe 60 magnet 66 K+ K+ ~7 1011 ppp BM K K− Front-end achromat • Momentum selection z focusing beams Second achromat Quadrupole quadruplet • Focusing • sweeping • Cleaning • Beam spectrometer (0. 7%) Beams coincide within ~1 mm all along 114 m decay volume, always in vacuum tank 1 cm 50 100 200 He tank + spectrometer 10 cm not to scale 250 m

K decay volume 114 m long vacuum tank Diameter: 1. 92 m (first 66 m) 2. 40 m (last 48 m)

The NA 48 detector (at the end of the decay volume) Main detector components: • Magnetic spectrometer (4 DCHs): 4 views: redundancy efficiency σp/p = 1. 02% + 0. 044% p [Ge. V/c] • Hodoscope fast trigger precise time measurement (150 ps) • Liquid Krypton EM calorimeter (LKr) High granularity, quasi−homogeneous σE/E = 3. 2%/√E + 9%/E + 0. 42% [Ge. V] e/π discrimination • Hadron calorimeter, photon vetos, muon veto counters Beam pipe

Data taking: completed 2003 run: ~ 50 days 2004 run: ~ 60 days Total statistics in 2 years: K + − : ~ 4 x 109 K 0 0 : ~ 1. 5 x 108 ~ 200 TB of data recorded

Liquid Krypton electromagnetic calorimeter ~ homogeneous ionization chamber ~ 10 m 3 liquid Krypton Thickness: 27 radiation lengths 13248 projective cells, 2 x 2 cm 2 No longitudinal segmentation Energy resolution: (E in Ge. V) s(E) ≈ 142 Me. V for E = 10 Ge. V Space resolution: sx = sy ≈ 1. 5 mm for E = 10 Ge. V

Motivation for a measurement of the º º invariant mass (M 00) distribution from K º º decay with optimal M 00 resolution: search for + atoms (pionium) produced in K + decay (I. Mannelli) K p p+p- event topologies with p+p- invariant mass M+- = 2 m+ possibility of pionium formation (Coulomb interaction), followed by pionium decay to pºpº pairs mass First observation of pionium atoms at the 70 Ge. V Serpukhov proton synchrotron L. G. Afanasyev et al. , Phys. Lett. B 308 (1993) 200 Pionium radius in the ground state (n = 1): (R∞ : Bohr radius for Mnucleus = ∞ ) Rpionium >> strong interaction radius ( ~10 -13 cm) rather low decay rate for the strong interaction process p+p- pºpº Pionium mean lifetime: tpionium ≈ 2. 9 x 10 15 s VERY NARROW WIDTH

Example of pionium expectation (from Monte. Carlo simulation) 420 bin M 002 distribution ; 1 bin = 0. 00015 Ge. V 2 Expected spectrum without pionium M 002 (Ge. V 2) Full spectrum with pionium M 002 (Ge. V 2) Details of the pionium region (Pionium mass)2 ≈ 0. 0779 Ge. V 2 Pionium signal covers ~7 bins

Event selection § At least one charged particle with momentum p > 5 Ge. V/c § At least 4 photons with Eg > 3 Ge. V detected in the Liquid Krypton (LKr) calorimeter § Geometrical cuts to eliminate detector edge effects (near beam tube and near outer edges of drift chambers and LKr calorimeter) § Distance between photons at LKr > 10 cm § Distance between photons and charged particle at LKr > 15 cm

Reconstruction of the pair For each photon pair (i, k) reconstruct common vertex along beam axis (zik) under the assumption of gg decay m 0: p mass Ei , Ek : photon energies (measured in LKr) Dik : distance between the two photons on the LKr face zik : distance between LKr and p decay vertex Liquid Krypton electromagnetic calorimeter e. V G 60 am be Among all possible pairs select the pair with minimum difference | Dz | = |zik – zlm | < 500 cm (i , k ≠ l , m)

Dz distribution Dz (cm) Main source of tails in Dz distribution at this stage: wrong photon pairing

Choice of common vertex along beam axis (z coordinate): the middle point between the two vertices 1 3 4 2 z 34 z 12 e. V G 60 am be To first order: Optimal resolution on the invariant mass M 00 (~ perfect resolution for M 00 = 2 m 0)

Distribution of reconstructed vertices along beam axis LKr front face at z = 12109 cm

invariant mass M( ) Origin of the tails in the Dm distribution: p± ± decay in flight Select events with | Dm | = | M( ) m. K(PDG) | < 0. 006 Ge. V Fraction of events with wrong photon pairings ~ 0. 25% (as estimated from Monte. Carlo simulation)

invariant mass resolution and event acceptance (from Monte. Carlo simulation) Expected M 002 distributions for five generated values of Moo and Moo resolution (r. m. s. , Me. V) Moo resolution (r. m. s. ) at pionium mass = 0. 56 Me. V Event acceptance vs Moo Arrow: Moo = 2 m+ m+ : p+ mass

Experimental M 002 distribution for 22. 87 x 106 K± ± decays Sudden change of slope (“cusp”) at Moo = 2 m+

Experimental M 002 distribution “Zoom” on the cusp region M 002 (Ge. V 2) STRUCTURE IS TOO BROAD TO BE CONSISTENT WITH EXPECTED NARROW PEAK FROM PIONIUM

Fits to the experimental Moo 2 distribution METHOD § Generate theoretical Moo 2 distribution Gi (420 bins of 0. 00015 Ge. V 2 ) § From Monte. Carlo simulation derive 420 x 420 matrix Tik = probability that an event generated with Moo in bin i is detected and measured in bin k (Tik includes both acceptance and resolution) § Produce “reconstructed” Moo 2 distribution Rk : § Fit distribution Rk to experimental Moo 2 distribution

Log(Tik) (from Monte. Carlo simulation)

Fit interval: 0. 0741 < Moo 2 < 0. 0967 Ge. V 2 DATA FIT INTERVAL

§ Fit using modified PDG prescription for decay amplitude: where : Very bad fit: c 2 = 9225 / 149 d. o. f. § Move lower limit of fit interval 13 bins above cusp point Reasonable fit: c 2 = 133. 6 / 110 d. o. f.

Data – fit comparison shows important “deficit” of events below cusp point Data: 7. 261 x 105 events; extrapolated fit: 8. 359 x 105 events

D ≡ (data – fit) /data versus Moo 2

Is the observed “deficit” due to detector effects? Study event shape distributions in two equal M 00 intervals below (I-) and above (I+) cusp; Normalize I+ and I- to the same area and compare I+ / I- ratio to Monte. Carlo prediction Variation of shape of photon energy distribution across cusp point Points: data Histogram: MC agrees with Monte. Carlo prediction

Variation of shapes of photon distance distributions across cusp point a) distance between LKr centre and closest photon Points: data Histograms: MC b) distance between LKr centre and farthest photon c) minimum distance between photons at LKr d) minimum distance between photons and tracks at LKr Very good agreement with MC predictions for all distributions

N. Cabibbo Determination of the a 0–a 2 Pion Scattering Length from K+ + decay Phys. Rev. Letters 93 (2004) 121801 Matrix element for K+ p+pºpº: unperturbed amplitude; Real, > 0 Contribution from charge exchange diagram Normalization: M 1 = 0 at M 00 = 2 m+ M 1 : real, < 0 for M 00 < 2 m+ known matrix element for K+ p+p+p + º º scattering length destructive interference imaginary for M 00 > 2 m+ no interference

Assumption: EXACT isospin symmetry a 0 (a 2) : p – p scattering length in isospin I = 0 (I = 2) state (scattering length = scattering amplitude at zero energy) § Relative p momentum at threshold = 0 only S – waves are allowed § Pions are BOSONS Y(p 1, p 2) = Y(p 2, p 1) § The isospin wave function of a pp pair with I = 1 is antisymmetric only I = 0 and I = 2 are allowed Predictions from current algebra and partially conserved axial current (Weinberg 1966) a 0 m+ = 0. 159 ; a 2 m+ = -0. 045 Recent predictions in the framework of Chiral Perturbation Theory (Ch. PT) (Weinberg 1967; Gasser & Leutwyler 1984; Colangelo, Gasser & Leutwyler 1984) a 0 m+ = 0. 220 0. 005 ; a 2 m+ = -0. 0444 0. 0010 ; (a 0 - a 2)m+ Ch. PT : PRECISION STRONG INTERACTION THEORY AT ENERGIES NEAR THRESHOLD = 0. 265 0. 004

Cabibbo’s rescattering model for K+ + º º: only one additional free parameter: (a 0 – a 2)m+ D (data – best fit) / data D M 002 (Ge. V 2) Great c 2 improvement (from 9225 / 149 to 420. 1 / 148 d. o. f. ) but still an unsatisfactory fit (especially in the cusp region)

N. Cabibbo and G. Isidori: Pion – pion scattering and the K 3 p decay amplitudes JHEP 03 (2005) 021 More one-loop diagrams :

. . . and also two-loop and three-pion diagrams

Five scattering lengths in the Cabibbo – Isidori model: Exact I-spin symmetry Scattering length Subprocess Isospin symmetry breaking corrections at tree level: (van Kolck 1993; Maltman and Wolfe 1997; Knecht and Urech 1998) ; ;

Fit to the Cabibbo – Isidori rescattering model Add quadratic term to the unperturbed K+ p+pºpº scattering amplitude: Two free parameters: g 0, h’ + a 0 + a 2 + an overall normalization constant five free parameters D (data – best fit) / data D M 002 (Ge. V 2) (a 0 – a 2)m+ = 0. 284 0. 007 a 2 m+ = -0. 077 0. 015 (statistical errors only)

Add pionium contribution: D M 002 (Ge. V 2) (a 0 – a 2)m+ = 0. 269 0. 009 a 2 m+ = -0. 054 0. 019 (K+ p+ + pionium) / (K+ p+pºpº) = (1. 61 0. 66) x 10 -5 2. 4 s evidence for pionium Compare with theoretical prediction (Pilkuhn and Wycech 1978; Silagadze 1994) (K+ p+ + pionium) / (K+ p+pºpº) = 0. 8 x 10 -5 Fix pionium contribution at theoretical prediction: c 2 = 149. 9 / 146 d. o. f. (a 0 – a 2)m+ = 0. 274 0. 007 a 2 m+ = -0. 063 0. 015

Cabibbo – Isidori’s rescattering model does NOT include radiative corrections, very important near M 00 = 2 m+ and contributing to pionium formation Final fit: exclude 7 bins centred at Moo = 2 m+ D M 002 (Ge. V 2) Two independent analyses with two independent acceptance calculations : Parameter Analysis A Analysis B Arithmetic average (a 0 – a 2)m+ 0. 269 ± 0. 010 0. 268 ± 0. 010 a 2 m+ -0. 053 ± 0. 020 -0. 030 ± 0. 022 -0. 041 ± 0. 022 g 0 0. 643 ± 0. 004 0. 647 ± 0. 004 0. 645 ± 0. 004 h’ -0. 055 ± 0. 010 -0. 039 ± 0. 012 -0. 047 ± 0. 012 Arithmetic average of best fit parameter values parameter measurement ; one half of their difference systematic uncertainty on the acceptance calculation

Systematic uncertainties Parameter Acceptance calculation Trigger efficiency Fit interval upper edge K + / Kdifference p± – g min. distance LKr resolution, nonlinearity Total syst. error (a 0 - a 2)m+ ± 0. 001 ± 0. 0025 - ± 0. 002 ± 0. 001 ± 0. 004 a 2 m+ ± 0. 012 ± 0. 005 ± 0. 006 - - - ± 0. 014 g 0 ± 0. 002 ± 0. 008 - - ± 0. 009 h’ ± 0. 009 ± 0. 003 ± 0. 006 - - - ± 0. 011 Theoretical uncertainty on (a 0 – a 2)m+ = 5% (from neglecting higher – order rescattering digrams and radiative corrections) Final NA 48/2 result: (a 0 – a 2)m+ = 0. 268 0. 010(stat) 0. 004(syst) 0. 013(theor) a 2 m+ = -0. 041 0. 022(stat) 0. 014(syst) Reminder of theoretical predictions: (a 0 – a 2)m+ = 0. 265 0. 004 ; a 2 m+ = -0. 0444 0. 0010

Constraint between a 0 and a 2 from chiral symmetry and analyticity (Colangelo, Gasser, Leutwyler 2001) Use this constraint in the fit: a 0 m+ = 0. 220 0. 006(stat) 0. 004(syst) 0. 011(theor) equivalent to (a 0 – a 2)m+ = 0. 264 0. 006(stat) 0. 004(syst) 0. 013(theor) Compare with measurement of K+ p+p-e+ne (BNL experiment 865): a 0 m+ = 0. 216 0. 013(stat) 0. 002(syst) 0. 002(theor) (also obtained using theoretical constraints)

Measurement of the pionium lifetime in the DIRAC experiment at the CERN PS An independent method to measure |a 0 – a 2| m+ A pionium atom; pionium decay A º º Decay rate in the n = 1, l = 0 state: pº momentum in A rest frame QED and QCD corrections d = 0. 058 0. 012 Cross – section for pionium production in an l = 0 state: Pionium wave function at the origin n: principal quantum number Double inclusive production cross – section for p+p- pairs from short – lived sources without Coulomb interaction

Pionium production in thin targets Two competing processes § pionium decay: A pºpº § pionium break – up (ionization): A p+p- (calculable!) DIRAC (DImeson Relativistic Atom Complex) experiment at the CERN PS § 24 Ge. V protons on thin (94 mm, 98 mm) Ni foils § Pionium Lorentz factor g ≈ 17 on average § Detect + pairs in coincidence § Measure precisely + and momentum Expectations from pionium break – up: within measurement errors

Evidence for pionium production and break – up in the DIRAC experiment: relative momentum (Q) distribution for + pairs with QT < 4 Me. V/c Peak at small Q and QL values is due to pionium formation and break-up

§ Calculate number of produced pionium atoms (NA) § Measure number of observed pionium atoms (n. A) § Break – up fraction Pbr = n. A / NA

CONCLUSIONS § A clear cusp has been observed by NA 48 / 2 in the invariant mass distribution from K± ± decay at Moo = 2 m+ § The new level of precision of the NA 48 / 2 data requires a redefinition of the parameters generally used to describe K± ± decay (e. g. , PDG 2004) § This cusp is the effect of scattering in the final state, dominated by the charge exchange process + § The study of the invariant mass distribution from K± ± decay offers a new, precise method to measure (a 0 – a 2)m+ independently of other methods (e. g. , measurement of pionium lifetime) § Result in excellent agreement with theoretical predictions, precision comparable to (or better than) other experiments (K+ + e+ne , pionium lifetime) § The final K± ± decay sample collected in 2003 04 will contain ~108 events § Need improvements of the rescattering model (higher – order diagrams, radiative corrections) in order to extract values of the scattering parameters from these data with the best possible precision