Dalitz Plot Analysis of D0K–K+0 Kalanand Mishra University of Cincinnati BaBar Collaboration

3-Particle Phase Space + 2 Observables Usual choice  Dalitz plot provides information about two-body amplitude structure. 2 Observables From four vectors 12 Conservation laws -4 Final state particle masses -3 Free rotation in decay plane -3 Σ 2 Usual choice Invariant mass squared m212 Invariant mass squared m213 m2+0 + m2-0 + m2+- = m2+ + m2- + m20 + m2D0 D0-+0 Dalitz plot m2(+–) + =0 = π1  π3 D0 m (+0)  cos π2 { π1 , π2, π3 }  { +, 0, - } Hadron07, Frascati, October 8-13 Kalanand Mishra

Isobar Model Formalism three-body decay DABC decaying through an [ rAB ] resonance 1 2 3 {12} {13} {23} NR 2 D decay three-body amplitude NR term(direct 3 body decay) a0, δ0, ar, δr : Free parameters of fit Relativistic Breit-Wigner f0(980) a0(980) Angular distribution D and r form factors Hadron07, Frascati, October 8-13 Kalanand Mishra

Introducing Angular Distributions Schrödinger‘s Equation Angular Amplitude Dynamic Amplitude (BW, Flatte, S-wave) In case only l = 0 (S-wave) and 1 (P-wave) amplitudes are present : For S- and P- waves only, in the absence of cross-feeds from other channels, the amplitudes and the relative phase are given by these relations. Hadron07, Frascati, October 8-13 Kalanand Mishra

D0K–K+0 Dalitz Plot Analysis Interference among three types of singly Cabibbo-suppressed amplitudes Motivation Nature of Kπ S-wave below 1.4 GeV. Is there a charged  state? - Nature of f0/a0 (980): KK S-wave. (I) (II) — D0 decay reconstruction K-  (III)  K+ 0 Color-suppressed D0 +soft Hadron07, Frascati, October 8-13 Kalanand Mishra

Dalitz Plot for D0K-K+0  (1020) Approx. isolated ±1 region: ≈ 11 k events, purity ≈ 98 %  K-K+ channel; can attmpt model ind. analysis. m2(K+0) (GeV2/c4) Events used for bg shape K*+ 385 fb-1 K*- m2(K-0) (GeV2/c4) Define amplitude for the D0K-K+0 decay as: Event Selection PCM ( D0) > 2.77 GeV/c |mD* - m D0 - 145.5| < 0.6 MeV/c2 Dalitz plot intensity  | f |2 Phys. Rev. D74, 091102 (2006) Hadron07, Frascati, October 8-13 Kalanand Mishra

K and K+K- S-wave Amplitudes LASS Nucl. Phys. B296, 493 (1988); For K S-wave E791 Phys. Rev. D73, 032004 (2006);  - The LASS amplitude gives the best fit. E-791 fit worse at low mass.  model yields mass 870 ± 30 MeV/c2 width 150 ± 20 MeV/c2 significantly different from the values reported for 0. -  with E-791 parameters does not give a satisfactory fit. K invariant mass range available in D0K–K+0 decay. Use LASS amplitude for nominal fit and E-791 amplitude for syst. uncertainty. For K-K+ S-wave - f0(980) and a0(980) virtually indistinguishable from each other. Both f0(980) and a0(980) give satisfactory fits. Since they are so similar, we try each as a description of the KK S-wave amplitude. — Hadron07, Frascati, October 8-13 Kalanand Mishra

Fit Results for D0K-K+0 Dalitz Plot : 19 % f0/a0: 7-10% K*(892)- φ(1020) K*(892)+    Ambiguity between a large K+0 S-wave & K*(1410),f2’(1525). 2 Prob = 62 % 2 Prob = 48 % Hadron07, Frascati, October 8-13 Kalanand Mishra

 Partial Wave Analysis in K-K+ channel    Impose   Look into the distributions of the spherical harmonic functions (l=0,1,2,…..). [ in the region where the K cross-channels have little effect ] _ S-wave as in D0K-K+K0. f0(980) a0(980)   (1020)  Solve these equations to extract |S|, |P|, and cos SP. Data Data  Because of the interference from the crossing K channels, the model independent partial-wave analysis performed here seems valid only up to about 1.02 -1.03 GeV/c2. Impose  Breit-Wigner phase for P-wave, (1020)   Data  Model-I  Model-II Choose the solution with increasing S-wave phase (Wigner causality) Hadron07, Frascati, October 8-13 Kalanand Mishra

Back up slides

BaBar: B and charm Factory Electromagnetic Calorimeter 6580 CsI crystals e+ ID, π0 and γ reco Instrumented Flux Return 12-18 layers of RPC/LST μ ID e+ [3.1 GeV] Cherenkov Detector 144 quartz bars K, π, p separation Drift Chamber 40 layers tracking + dE/dx e- [9 GeV] Silicon Vertex Tracker 5 layers (double-sided Si strips) vertexing + tracking (+ dE/dx) 1.5T Magnet Hadron07, Frascati, October 8-13 Kalanand Mishra

Kaon/Pion Discrimination: DIRC LAYOUT Cherenkov angle vs. momentum for pions and kaons >4 s separation at 3 GeV/c Hadron07, Frascati, October 8-13 Kalanand Mishra

D0-+0, K-K+0 Background Sources D* Reconstruction D0h-h+0 Reconstruction Background Sources Charged track combinatoric Mis-reconstructed 0 Real D0, fake s K0 reflection in 0 and KK0 modes h- and h+ tracks are fit to a vertex Mass of 0 candidate is constrained to m0 at h-h+ vertex PCM ( D0 ) > 2.77 GeV/c D* Reconstruction D*+ candidate is made by fitting the D0 and the s+ to a vertex constrained in x and y to the measured beam-spot for the run. |mD* - m D0 - 145.5| < 0.6 MeV/c2 Vertex 2 probability > 0.01 Choose a single best candidate with smallest 2 for the whole decay chain ( multiplicity = 1.03 ). Hadron07, Frascati, October 8-13 Kalanand Mishra

_ BaBar CLEO KK0: Strong-phase Diff. & Amp. Ratio The strong phase difference D and relative amplitude rD between the decays of D0 and D0 to K*(892)+ K- state are defined, neglecting direct CP violation in D decays, by the equation: We find                   BaBar CLEO rD = 0.599  0.013 (stat)  0.011 (syst) δD = -35.5° (stat) ± 1.9° ± 2.2° (syst) rD = 0.52  0.05 (stat)  0.04 (syst) δD = -28° ± 8° (stat) ± 11°(syst) Phys. Rev. D76, 011102 (2007) Phys. Rev. D74, 031108 (2006) These measurements are consistent with each other. Hadron07, Frascati, October 8-13 Kalanand Mishra

KK0: K S-wave Parametrization KpS-wave in mass range 0.6–1.4 GeV/c2 is not well-understood. A possible  state ~ 800 MeV/c2 has been conjectured, but has only been reported in the neutral state. For the K+0 and K-0 S-wave amplitudes, we try three models: Amplitude from LASS K-+ K-+ scattering. K-+ amplitude from a model-independent analysis of D+K-++ data by the E791 collaboration. [ coherent sum of (800) + uniform NR + K*0(1430) ]. Nucl. Phys. B296, 493 (1988); Phys. Rev. D73, 032004 (2006); {No evidence in K elastic scattering.}   E-791 LASS - 800 Normalized to arbitrary scale for m > 1.15 Gev/c2 for easy comparison. Hadron07, Frascati, October 8-13 Kalanand Mishra

LASS K S-wave Parameterization Kπ S-wave amplitude is described by the coherent sum of an effective range term and the K*0(1430) resonance: S(s) = (√s/ p). sin . ei  = cot-1 [ 1/ap + rp/2 ] + cot -1 [ (m2R-s)/(mR R ) ] Effective Range (NR) term K*0(1430) resonance term a = scat. length, r = eff. range, mR = mass of K*0(1430), R= width p = momentum of either daughter in the Kπ rest frame. Phase space factor For Kπ scattering, S-wave is elastic up to K' threshold (1.45 GeV). Hadron07, Frascati, October 8-13 Kalanand Mishra

[ E791 Collaboration, slide from Brian Meadow’s Moriond 2005 talk ] K S-wave from D0K-++ DP [ E791 Collaboration, slide from Brian Meadow’s Moriond 2005 talk ] Divide m2(K-+) into slices Find s–wave amplitude in each slice (two parameters) Use remainder of Dalitz plot as an interferometer For s-wave: - Interpolate between (ck,k). Model P and D waves. S (“partial wave”) Hadron07, Frascati, October 8-13 Kalanand Mishra

Excellent agreement between data & models. Analysis of Angular Moments Excellent agreement between data & models. Each event is weighted by the spherical harmonic functions (l=0,1,2,…..). Large interference between S and P waves. 0.6 0.8 1 1.2 1.4 1 1.5  0.6 0.8 1 1.2 1.4 1 1.5 m2(K-K+) (GeV2/c4) m2(K-K+) (GeV2/c4) m2(K+0) (GeV2/c4) m2(K+0) (GeV2/c4) Higher moments above 1.1 GeV are coming from cross channels. For S- and P- waves only, in the absence of cross-feeds from other channels, the amplitudes and the relative phase are given by: We solve these equations for the K-K+ system in a limited mass range (where the above conditions are satisfied) to extract |S|, |P|, and cos SP. Hadron07, Frascati, October 8-13 Kalanand Mishra

Partial Wave Analysis in K-K+ channel f0(980) Solve the equations on the previous slide to extract |S|, |P|, and cos SP. a0(980)   (1020)  Data Data KK S-wave amplitude, extracted in a model independent analysis of the decay D0K-K+K0. – Phys. Rev. D 72, 052008 (2005) Two solutions for SP  upper one is the physical   Because of the interference from the crossing K channels, the model independent partial-wave analysis performed here is valid only up to about 1.02 -1.03 GeV/c2 Breit-Wigner phase for P-wave, (1020)  Data  Model-I  Model-II  Hadron07, Frascati, October 8-13 Kalanand Mishra