1. Amplitude Analysis of the D 0        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 D 0  K - K + π 0 decay which will be useful in understanding the behavior of Kπ S-wave below K  ’ threshold. The K ± π 0 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 D 0 decay modes in B ±  DK ± decays in order to be able to constrain  (  3 ). At present the only CS mode exploited so far is D 0  π - π + π 0 [ under internal BaBar review ]. 3-body CS decays of D 0 are especially interesting because of their sensitivity to direct CP violation. Such a analysis is already underway.

3. Event Selection Events used to obtain Bkg shape Use events in 1  mass window for DP analysis: ≈ 7000 events with purity ≈ 97 % We use decays D* +  D 0 [  K - K +  0 ]π s + Integrated Lumi 232 fb -1 | m D* - m D 0 - 145.5 | < 0.6 MeV/c 2 P CM > 2.77 GeV/c 2 m 2 (K - π 0 ) m 2 (K + π 0 ) ~ 3 % bkg

4. Isobar Model 2 NR Constant D form factor R form factor spin factor 11 1 2 2 3 3 3 {12} {13} {23} 1 2 3 NR Schematically: Amplitude for the [ij] channel: Each resonance “ R ” (mass M R, width  R ) typically has a form p, q are momenta in ij rest frame. r D, r R 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 : 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 Y l 0 (cos  H ).

6. Dalitz plot and Fit Model o K + π 0 and K - π 0 S-wave: LASS parameters o K + K - S-wave: f 0 (980) : Flatte (with BES parameters) o P- and D- waves: relativistic Breit Wigner PW: K*(892), K*(1410),  (1020) DW: f 2 ’(1525)

7. Kπ s-wave parameterization -Apart from the K* 0 (1430), resonant structure in the S- wave K  system in the mass range 0.6 – 1.4 GeV/c 2 is not well-understood. -A possible  state ~ 800 MeV/c 2 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 ) e i (  B +  B ) Non-resonant Term + R e i  R e 2i (  B +  B ) sin  R e i  R Resonance 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 [ (m 2 R -s)/(m R  R ) ] a = scat. length, r = eff. range, m R = 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 ) e i (  R +  B) We use : B = R = 1;  B = 90,  R = 0 S = sin(  R +  B + π/2 ). e i (  R +  B + π/2 )

9. s – wave from D +  K -  +  + Dalitz Plot Divide m 2 ( 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 ( c k,  k ) points: Model P and D. [ E791 Collaboration, slide from Brian Meadow’s Moriond 2005 talk ] S (“partial wave”)

10. Comparison of Kπ S-wave Models ∆ E791 MIPWA O LASS Original This analysis LASS phase is shifted by -90 0 and phase in our parameterization is shifted by -180 0.

11. S-wave Modeled on D 0  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. D 0  + K - D 0  - K +

12. KK S-wave: f 0 (980) Coupled-channel BW to the K + K - and K S 0 K S 0 states (Flatte) : BW(s) = 1/ [ m r 2 - s - i m r (  π +  K ) ]  π = g π. [ s/4 - m π 2 ] 1/2  K = (g K /2). [ (s/4 - m K 2 ) 1/2 + (s/4 - m K 0 2 ) 1/2 ] BES parameter values for g π and g K : m r = 0.975 ± 0.010 GeV/c 2 g π = 0.165 ± 0.018 g K / g π = 4.21 ± 0.33 BES is the only experiment which has good amount of data on f 0 (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 DataFit (Data-Fit)/Poisson  2 / = 1.03 for = 705 Normalized Residual

14. Nominal Fit Gen. LASS parameterization for Kπ S-wave Fit Components: 1) K* + (892) (fixed amp & phase) 4) K*- (892) 7) K - π 0 S-wave 2) K* + (1410) 5) K* - (1410) 8) f 0 (980) 3)  (1020) 6) K + π 0 S-wave 9) f 2 ’(1525) m 2 (K + π 0 ) m 2 (K - π 0 ) m 2 (K + K - )

15. Fit Results

16.  2 / = 1.05 Fit with Kπ S-wave from E791 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) f 0 (980) : 0.03 4) K*- : 0.17 9) f2’1525 : 0.006 5) K*1410- : 0.05 S-wave Amplitude using S-P interference in D +  K -  +  + m 2 (K + π 0 ) m 2 (K - π 0 ) m 2 (K + K - )

17. Fit with S-wave Modeled on D 0  K decay 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  2 / = 1.35428  + 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 m 2 (K + π 0 ) m 2 (K - π 0 ) m 2 (K + K - )

18. Moments Analysis p q  cos   = p. q K-K- K+K+ 00 Helicity angle  in K -  + system. Similar definitions applies to the two Kπ channels. 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 Y 0 l (cos  H ). These angular moments provide further information on the structure of the decays, nature of the solution and agreement between data and fit.

19. Angular Moments & Partial Waves We notice a strong S-P interference in both Kπ and KK channels, evidenced by the rapid motion of Y 0 1 at the K*(892) and  masses. The Y 0 2 moment is proportional to P 2 which can be seen in the background-free  (1020) signal region. √ 4π = S 2 + P 2 √ 4π = 2 |S| |P| cos  SP √ 4π = 0.894 P 2 Higher moments = 0 In case of S- and P- waves only and in absence of cross-feeds from other channels: 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 Y01Y01 Y00Y00 Y02Y02 Y04Y04 Y06Y06 Y03Y03 Y05Y05 Y07Y07

21. Angular Moments (K - K + ) -wrong Fit with K 2 *(1430) included! Y01Y01 Y00Y00 Y02Y02 Y04Y04 Y06Y06 Y03Y03 Y05Y05 Y07Y07

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

23. Angular moments (K + π 0 ) Nominal Fit : Excellent agreement with data Y01Y01 Y00Y00 Y02Y02 Y04Y04 Y06Y06 Y03Y03 Y05Y05 Y07Y07

24. Angular Moments (K - π 0 ) m 2 (K - π 0 ) [GeV/c 2 ] Nominal Fit : Excellent agreement with data Y00Y00 Y01Y01 Y02Y02 Y04Y04 Y06Y06 Y03Y03 Y05Y05 Y07Y07

25. Strong Phase Difference,  D and r D The strong phase difference  D and relative amplitude r D between the decays D 0  K* - K + and D 0  K* + K - are defined, neglecting direct CP violation in D 0 decays, by the equation : r D e i  D = [a K*-K+ / a K*+K- ] exp[ i(  K*-K+ -  K*-K+ ) ] We find  D = -37.0 o ± 2.2 o (stat) ± 0.7 o (exp syst) ± 4.2 o (model syst) r D = 0.64 ± 0.01 (stat) ± 0.01 (exp syst) ± 0.00 (model syst). These can be compared to CLEO’s recent results:  D = -28 o ± 8 o (stat) ± 2.9 o (exp syst) ± 10.6 o (model syst) r D = 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 +90 0 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 f 2 ’(1525). The D 0  K *+ (892)K - decay dominates over D 0  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 f 0 (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 r D and  D.

28. Backup Slides

29. Resonance Shapes K*(892) + K*(892) -  (1020) 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