1. A. BondarModel-independent φ 3 measurement August 6, 2007Charm 2007, Cornell University1/15 γ/φ 3 model-independent Dalitz analysis (Dalitz+CP tagged Dalitz plots and γ/φ 3 extraction) Alex Bondar, Anton Poluektov Budker Institute of Nuclear Physics Novosibirsk, Russia

2. A. BondarModel-independent φ 3 measurement August 6, 2007Charm 2007, Cornell University2/15 The relative phase γ (φ 3 ) Interference between tree-level decays; theoretically clean Parameters:  , (r B, δ) per mode Three methods for exploiting interference (choice of D 0 decay modes): Gronau, London, Wyler (GLW): Use CP eigenstates of D (*)0 decay, e.g. D 0  K s π 0, D 0  π + π - Atwood, Dunietz, Soni (ADS): Use doubly Cabibbo-suppressed decays, e.g. D 0  K + π - Giri, Grossman, Soffer, Zupan (GGSZ) / Belle: Use Dalitz plot analysis of 3-body D 0 decays, e.g. K s π + π - V cs * V ub : suppressed Favored: V cb V us * b u s uu b u c D (*)0 K (*)- B-B- B-B- u s u c D (*)0 f Common final state K (*)-

3. A. BondarModel-independent φ 3 measurement August 6, 2007Charm 2007, Cornell University3/15 Dalitz analysis method Measure B + /B - asymmetry across Dalitz plot Includes GLW (D 0  K s ρ 0, CP eigenstate) and ADS (D 0  K * + π -, DCS 2-body decay) regions Sensitivity to   in interference term 2-fold ambiguity on  3 : (  3, δ )  → (  3 +π, δ +π ) Model uncertainty ~ 10 0 decay amplitude Determine f in flavor-tagged D *+  D 0 π + decays Mirror symmetry between D 0 and D 0 Dalitz plots

4. A. BondarModel-independent φ 3 measurement August 6, 2007Charm 2007, Cornell University4/15 ρ-ω interference Doubly Cabibbo suppressed K * M (GeV 2 ) K s π – 2 D 0  K s π + π – decay model

5. A. BondarModel-independent φ 3 measurement August 6, 2007Charm 2007, Cornell University5/15 Belle/Babar model-dependent results HFAG averages for x ± = r B cos( δ ±   ), y ± = r B sin( δ ±   ) Belle/Babar measurements in good agreement Note: σ (  3 ) depends significantly on the value of r B Contours do not include Dalitz model errors rBrB 2323

6. A. BondarModel-independent φ 3 measurement August 6, 2007Charm 2007, Cornell University6/15 Model-independent analysis Model-independent way: obtain D 0 decay strong phase from ψ(3770)  DD data D CP  K S π + π – : ψ(3770)  (K S π + π – ) D (K S π + π – ) D : where Free parameters Unknown, can be obtained from charm data: (1) (2) (3)

7. A. BondarModel-independent φ 3 measurement August 6, 2007Charm 2007, Cornell University7/15 Model-independent analysis CLEO-c with 750 pb -1 (estimated numbers) ~ 1000 D CP  K S π + π – decays with CP-eigenstate tag. ~ 2000 correlated D 0 decays: ψ(3770)  (K S π + π – ) D (K S π + π – ) D ψ(3770)  (K L π + π – ) D (K S π + π – ) D We need to find a way: how to use this data most efficiently how to combine the results of two experiments Binned approach: Divide Dalitz plots into bins Solve the system of equations with constraints

8. A. BondarModel-independent φ 3 measurement August 6, 2007Charm 2007, Cornell University8/15 Binned analysis: D CP [A. Giri, Yu. Grossman, A. Soffer, J. Zupan, PRD 68, 054018 (2003)] Number of events in B -plot Number of events in D CP -plot ( =1 if bin size is small enough ) i -i c i, s i can be obtained from B data only c i from D CP, s i from B data s i constrained as  Very poor sensitivity  Poor sensitivity for y  Bias if bin size is large Number of events in D 0 -plot: (4) (5)

9. A. BondarModel-independent φ 3 measurement August 6, 2007Charm 2007, Cornell University9/15 Binned analysis: D CP [A. Bondar, A.Poluektov, Eur. Phys. J. C47 347-353 (2006), hep-ph/0510246] 50 ab -1 (50000 ev.) at SuperB should be enough for model-indep. φ 3 measurement with accuracy 2-3° ~10 fb -1 (10000 ev.) at ψ (3770) needed to accompany this measurement. Nearest future: ~1000 D CP events at CLEO. Bin size should be large  bias due to r B =0.2

10. A. BondarModel-independent φ 3 measurement August 6, 2007Charm 2007, Cornell University10/15 Binned analysis: D CP [A. Bondar, A.Poluektov, hep-ph/0703267] To use the limited CLEO-c data: Find binning with optimal sensitivity Get rid of bias due to Satisfy simultaneously for binning with Good approximation: uniform binning in Δδ D But the optimal binning depends on D 0 model.  bias if it differs from the actual one (~10° by toy MC). causes unavoidable model sensitivity. Reduces as data increases (by applying finer binning).

11. A. BondarModel-independent φ 3 measurement August 6, 2007Charm 2007, Cornell University11/15 Binned analysis: (K S π + π – ) 2 2 correlated Dalitz plots, 4 dimensions: Can use maximum likelihood technique: with c i and s i as free parameters. For the same binning as in D CP, number of bins is N 2 (instead of N ), but the number of unknowns is the same. With Poisson PDF, it’s OK to have N ij <1. Can obtain both c i and s i. (6) (7)

12. A. BondarModel-independent φ 3 measurement August 6, 2007Charm 2007, Cornell University12/15 Binned analysis: (K S π + π – ) 2 c i,s i measured in toy MC (points) and calculated (crosses) for Δδ D -binning Can use the same binning as in D CP c i,s i measured independently  no model uncertainty due to constraint Only 4-fold ambiguity: c i  -c -i or s i  -s -i. reduces to 2-fold if D CP data are used with the same binning. Stat. sensitivity comparable to D CP [A. Bondar, A.Poluektov, hep-ph/0703267]

13. A. BondarModel-independent φ 3 measurement August 6, 2007Charm 2007, Cornell University13/15 MC simulation of the bias in  3 measurement Model dependence for 2x8 bins in D cp case: Model dependence for 2x8 bins in (K s  +  - ) 2 case:

14. A. BondarModel-independent φ 3 measurement August 6, 2007Charm 2007, Cornell University14/15 Summary of stat. sensitivity Binning B-stat. errorD CP -stat. error ( Kππ ) 2 -stat. error σxσx σyσy σxσx σyσy σxσx σyσy Uniform, N =80.0330.0600.00530.00970.01450.0322 Δδ D, N =80.0270.0370.00420.00720.00500.0095 Optimal, N =80.0230.0320.00580.01140.00820.0114 Uniform, N =200.0270.0550.00420.0112-- Δδ D, N =200.0270.0350.00480.0074-- Optimal, N =200.0220.0290.00780.0110-- Unbinned0.0210.028---- Errors corresponding to 1000 events in B, D CP and ( Kππ ) 2 samples Expected charm data contribution for 750 fb -1 at CLEO-c: (1000 D CP and 2000 ( Kππ ) 2 ) σ x =0.003, σ y =0.007  σ(φ 3 )~3° with r B =0.1

15. A. BondarModel-independent φ 3 measurement August 6, 2007Charm 2007, Cornell University15/15 Conclusion Several approaches are proposed for model-independent analysis: Binned with D CP (original GGSZ): implemented, studied. Bias with limited statistics. Binned with (K S π + π – ) 2 : Twice as much data, similar to D CP sensitivity. No bias, less ambiguity. Getting ready for model-independent measurement together with CLEO. Analysis strategy needs to be discussed Belle CLEO HFAG BaBar Super B factory Super Babar/Belle C/  factory Super HFAG

16. A. BondarModel-independent φ 3 measurement August 6, 2007Charm 2007, Cornell University16/15 Backup slides

17. A. BondarModel-independent φ 3 measurement August 6, 2007Charm 2007, Cornell University17/15 If is known, parameters are obtained from the fit to Dalitz distributions of D  K s π + π – from B ±  DK ± decays Dalitz analysis method Using 3-body final state, identical for D 0 and D 0 : K s π + π -. Dalitz distribution density: (assuming СР-conservation in D 0 decays) A. Giri, Yu. Grossman, A. Soffer, J. Zupan, PRD 68, 054018 (2003) A. Bondar, Proc. of Belle Dalitz analysis meeting, 24-26 Sep 2002.

18. A. BondarModel-independent φ 3 measurement August 6, 2007Charm 2007, Cornell University18/15 Statistical sensitivity of the method depends on the properties of the 3-body decay involved. (For |M| 2 =Const there is no sensitivity to the phase θ ) Large variations of D 0 decay strong phase are essential Use the model-dependent fit to experimental data from flavor-tagged D *  D 0 π sample. Model is described by the set of two-body amplitudes + flat non-resonant term. As a result, model uncertainty in the γ/φ 3 measurement (~10° currently). D 0  K s π + π – decay

19. A. BondarModel-independent φ 3 measurement August 6, 2007Charm 2007, Cornell University19/15 Binning optimization  Maximizing “binning quality” function allows to find a binning with the B-stat. sensitivity close to unbinned approach.