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Amplitude Analysis of the D 0        Dalitz Plot

This study focuses on analyzing the Dalitz plot of the D0 decay to KKπ0 to understand Kπ S-wave behavior and investigate the presence of the κ(800) state. It aims to provide insights useful for CP violation analysis in B±→DK± decays. By applying an isobar model and different amplitude parameterizations, including LASS and E791 models, the study explores the resonant and non-resonant components of the decay process. The analysis involves event selection using D*+ decays and fitting models for S-wave, P-wave, and D-wave components. The investigation includes parameterizing the Kπ S-wave amplitude, utilizing the generalized LASS notation and exploring different models for the Kπ scattering process.

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Amplitude Analysis of the D 0        Dalitz Plot

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  1. Amplitude Analysis of the D0 Dalitz Plot G. Mancinelli, B.T. Meadows, K. Mishra, M.D. Sokoloff University of Cincinnati BaBar Coll. Meeting, 9/12/2006

  2. Motivation • Theorist community has expressed interest [ see J.L. Rosner, hep-ph/0608102 ] in an amplitude analysis of D0K-K+π0 decay which will be useful in understanding the behavior of Kπ S-wave below K’ threshold. • The K±π0 system from this decay can also provide information relevant to the existence of (800). Evidence for such a state has been reported only for the neutral state. If  is an I = 1/2 particle, then it should also be observed in the charged state. • These decays are also interesting because one needs to analyze several D0 decay modes in B±DK± decays in order to be able to constrain  (3). At present the only CS mode exploited so far is D0π-π+π0 [ under internal BaBar review ]. • 3-body CS decays of D0 are especially interesting because of their sensitivity todirect CP violation. Such a analysis is already underway.

  3. Event Selection • We use decays D*+D0 [K-K+0]πs+ • Integrated Lumi 232 fb-1 • | mD* - mD0- 145.5 | < 0.6 MeV/c2 • PCM > 2.77 GeV/c2 ~ 3 % bkg m2(K+π0) m2(K-π0)

  4. {12} {23} {13} 1 1 1 2 2 2 3 3 3 1 3 Isobar Model Schematically: NR 2 Amplitude for the [ij] channel: D form factor R form factor spin factor NR Constant Each resonance “R” (mass MR, width R) typically has a form p, q are momenta in ij rest frame. rD, rR meson radii

  5. S-, P-, D- wave Amplitudes The Decay Processes are of type : Parent [P]  bachelor [b] + Resonant System [R] Write amplitude schematically as : <(R)L | P b > L = angular momentum Introduce a complete set of intermediate states for each L : for L = 0, S-wave for L = 1, P-wave for L = 2, D-wave, ….. The interference between these waves can be viewed as the addition of angular momenta and can be described by spherical harmonics Yl0 (cos H).

  6. Dalitz plot and Fit Model • K+π0 and K-π0 S-wave: LASS parameters • K+K- S-wave: f0(980) : Flatte (with BES parameters) • P- and D- waves: relativistic Breit Wigner • PW: K*(892), K*(1410), (1020) DW: f2’(1525)

  7. Kπ s-wave parameterization • Apart from the K*0(1430), resonant structure in the S-wave Kp system in the mass range 0.6 – 1.4 GeV/c2 is not well-understood. • A possible  state ~ 800 MeV/c2 has been conjectured, but this has only been reported in the neutral state. Its existence is not established and is controversial. • The best results on Kπ S-wave parameters come from the LASS experiment. Recently, the E791 collaboration has come up with a model independent parameterization of Kπ S-wave. • We try three different models: LASS Kπ scattering results, E791 shape and  model.

  8. Generalized LASS Parameterization(W. M. Dunwoodie notation) • Kπ S-wave amplitude is described by: S = B sin(B+ B) ei(B + B)Non-resonant Term +R eiR e2i (B + B) sinR eiRResonance Term B, B, R, R are constants, phases B and R depend on Kπ mass. • B = cot-1 [ 1/aq + rq/2 ], R= cot -1 [ (m2R-s)/(mR R ) ] a = scat. length, r = eff. range, mR = mass of K*0(1430), R= width For Kπ scattering, S-wave is elastic up to K' threshold (1.45 GeV). • Original LASS parameterization: B = R =1; B = R =0 S = sin(R+B) ei (R + B) We use :B = R = 1; B = 90, R = 0 S = sin(R+B+ π/2 ) . ei (R + B + π/2)

  9. s–wave fromD+K-p+p+ Dalitz Plot [ 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) points: • Model P and D. S (“partial wave”)

  10. Comparison of Kπ S-wave Models ∆ E791 MIPWA O LASS Original This analysis

  11. S-wave Modeled onD0K decay • The E791 collaboration needed a broad scalar resonance to get a good fit in their first D+K-π+π+ DP analysis (2002). • We formulate  as a I = 1/2 particle with parameters taken from E791, mass = 797 ± 47 MeV and  = 410 ± 97 MeV. • The parameterization of  as a BW is an ad hoc formulation. D0+K- D0-K+

  12. KK S-wave: f0(980) • Coupled-channel BW to the K+K- and KS0KS0 states (Flatte) : BW(s) = 1/ [ mr2 - s - i mr (π + K) ] π = gπ . [ s/4 - mπ2 ]1/2 K = (gK /2). [ (s/4 - mK2 )1/2 + (s/4 - mK02 )1/2 ] • BES parameter values for gπ and gK: mr = 0.975 ± 0.010 GeV/c2 gπ = 0.165 ± 0.018 gK / gπ = 4.21 ± 0.33 BES is the only experiment which has good amount of data on f0(980) decays to both π+π- (from J/π+π-) and K+K- (from J/K+K-) . The BES measurements of these parameters have made E791 and WA76 measurements obsolete.

  13. Nominal Fit Data Fit Normalized Residual Normalized Residual (Data-Fit)/Poisson (Data-Fit)/Poisson 2/= 1.03 for  = 705

  14. Nominal Fit Gen. LASS parameterization for Kπ S-wave m2(K+K-) m2(K-π0) m2(K+π0) Fit Components: 1) K*+(892) (fixed amp & phase) 4) K*- (892) 7) K-π0 S-wave 2) K*+(1410) 5) K*-(1410) 8) f0(980) 3)  (1020) 6) K+π0 S-wave9) f2’(1525)

  15. Fit Results

  16. Fit with Kπ S-wave from E791 S-wave Amplitude using S-P interference in D+K-p+p+ m2(K+K-) m2(K-π0) m2(K+π0) FIT FRACTIONS: 1) K*+ : 0.41 6) K+pi0 SW : 0.08 2) K*1410+ : 0.006 7) K-pi0 SW : 0.07 3) Phi : 0.19 8) f0(980) : 0.03 4) K*- : 0.17 9) f2’1525 : 0.006 5) K*1410- : 0.05 2 / = 1.05

  17. Fit with S-wave Modeled onD0K decay m2(K-π0) m2(K+π0) m2(K+K-) K*-_amp 0.57 ± 0.02 K*- phase -28.5 ± 3.1 K*1410+ amp 1.41 ± 0.12 K*1410+ phase -136.2 ± 11.0 K*1410- amp 1.80 ± 0.22 K*1410- phase 186.6 ± 7.3 Fit Fractions K*+ : 0.43 + : 0.16 K*(1410)+ : 0.01 Phi : 0.2 K*- : 0.14 - : 0.13 K*(1410)- : 0.02 + amp 1.60 ± 0.08 + phase 104.0 ± 3.2 - amp 1.46 ± 0.08 - phase 174.0 ± 3.4  amp 0.68 ± 0.01  phase -0.4 ± 4.7 2 / = 1.35428

  18. Moments Analysis • Several different fit models provide good description of data in terms of 2/ and NLL values. • We plot the moments of the helicity angles, defined as the invariant mass distributions of events when weighted by spherical harmonic functions Y0l (cosH). • These angular moments provide further information on the structure of the decays, nature of the solution and agreement between data and fit. K- 0 q p q cosq = p.q K+ Helicity angle q in K-+ system. Similar definitions applies to the two Kπ channels.

  19. Angular Moments & Partial Waves In case of S- and P- waves only and in absence of cross-feeds from other channels: • We notice a strong S-P interference in both Kπ and KK channels, evidenced by the rapid motion of Y01 at the K*(892) and  masses. • The Y02 moment is proportional to P2 which can be seen in the background-free (1020) signal region. With cross-feeds or presence of D-waves, higher moments ≠ 0 . Wrong fit models tend to give rise to higher moments, as seen in the moments plots earlier, thus creating disagreement with data.

  20. Angular Moments (K-K+) Nominal Fit : Excellent agreement with data Y01 Y00 Y02 Y03 Y05 Y04 Y06 Y07

  21. Angular Moments (K-K+) -wrong Y01 Y00 Y03 Fit with K2*(1430) included! Y02 Y04 Y05 Y06 Y07

  22. Angular Moments (K-K+) - wrong No KK SW !

  23. Angular moments (K+π0) Nominal Fit : Excellent agreement with data Y01 Y00 Y03 Y02 Y05 Y04 Y06 Y07

  24. Angular Moments (K-π0) Nominal Fit : Excellent agreement with data Y00 Y01 Y02 Y03 Y04 Y05 Y06 Y07 m2(K-π0) [GeV/c2 ] m2(K-π0) [GeV/c2 ]

  25. Strong Phase Difference, D and rD • The strong phase difference D and relative amplitude rD between the decays D0K*-K+ and D0K*+ K- are defined, neglecting direct CP violation in D0 decays, by the equation : rD eiD = [aK*-K+/ aK*+K-] exp[ i(K*-K+ - K*-K+) ] • We find D = -37.0o ± 2.2o (stat) ± 0.7o (exp syst) ± 4.2o (model syst) rD = 0.64 ± 0.01 (stat) ± 0.01 (exp syst) ± 0.00 (model syst). • These can be compared to CLEO’s recent results: D = -28o ± 8o (stat) ± 2.9o (exp syst) ± 10.6o (model syst) rD = 0.52 ± 0.05 (stat) ± 0.02 (exp syst) ± 0.04 (model syst).

  26. Summary • The resonance structure is largely dominated by various P-wave resonances, with small but significant contributions from S-wave components. • The Kπ S-wave modeled by a ±(800) resonance does not fit the data well, 2/ being 1.35 for  = 706. • The E791 model-independent amplitude for a Kπ system describes the data well except near the threshold. • The generalized LASS parameterization shifted by +900 gives the best agreement with data and we use it in our nominal fits. • A small but statistically significant contribution comes from KK D-wave component f2’(1525). • The D0K*+(892)K- decay dominates over D0K*-(892)K+. This may be related to the dominance of the external spectator diagram. • But the order is reversed for the next p-wave state K*(1410).

  27. Summary continued …. • The f0(980) with Flatte shape and the BES parameters is enough to parameterize the KK S-wave. • A good 2 value does not guarantee a robust fit. One needs to also look at angular moments to understand localized effects produced by interference from cross-channels. • We have measured rD and D.

  28. Backup Slides

  29. Resonance Shapes (1020) K*(892)+ K*(892)- NR K*(1410)+ K(1410)*- Kappa+ Kappa- P-wave NR(+) P-wave NR(-) P-wave NR(0) K*0(1430)+ K*0(1430)-

  30. Fit with CLEO PDF 1 Nonres_amp 4.80848e+00 8.76759e-02 (5.6 in CLEO results) 2 Nonres_phase 2.45715e+02 1.41802e+00 (220 in CLEO results) 3 K*- amp 5.21620e-01 1.26111e-02 4 K*-_phase -2.51342e+01 2.09421e+00 5  amp 6.03842e-01 1.11649e-02 6  phase -3.30354e+01 2.89297e+00 2 / = 1.83342

  31. Fit with p-wave NR 1 K*-_amp 6.13060e-01 1.98369e-02 2 K*-_phase -4.28001e+01 3.65266e+00 3 K*1410+_amp 3.46743e+00 4.76307e-01 4 K*1410+_phase 3.99550e+01 8.05654e+00 5 K*1410-_amp 2.67283e+00 4.14485e-01 6 K*1410-_phase 1.65986e+02 1.19152e+01 7 Kappa+_amp 7.30570e-01 2.10914e-01 8 Kappa+_phase 8.81885e+01 1.80236e+01 9 Kappa-_amp 6.05465e-01 1.68914e-01 10 Kappa-_phase 1.08270e+02 2.16174e+01 11 NRPW_P_amp 4.88345e+00 1.64838e+00 12 NRPW_P_phase 8.97154e+01 2.37566e+01 13 NRPW_M_amp -4.66088e+00 1.66335e+00 14 NRPW_M_phase -1.02777e+02 2.27370e+01 15 NRPW_0_amp 1.23893e+01 2.76792e+00 16 NRPW_0_phase 7.53007e+01 1.38116e+01 17 Nonres_amp 2.60086e+00 2.58137e-01 18 Nonres_phase 2.80830e+02 7.04073e+00 19 Phi_amp 6.49647e-01 1.52032e-02 20 Phi_phase 7.74845e+01 7.16402e+00 Fit Fractions K*+ : 0.45507 K*1410+ : 0.090682 Kappa+ : 0.035070 P-wave NR+ : 0.15697 Phi : 0.19792 P-wave NR0 : 0.63210 K*- : 0.17685 K*1410- : 0.053947 Kappa- : 0.023975 P-wave NR- : 0.14484 Nonres : 0.090031 2 /nDOF = 1.00708

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