1. ВД в эксперименте по измерению масс
25 мая 2002
γ/φ3 model-independent Dalitz analysis (Dalitz+CP tagged Dalitz plots and γ/φ3 extraction)
Alex Bondar, Anton Poluektov
Budker Institute of Nuclear Physics
Novosibirsk, Russia
Nov 26-27, 2007
BES-Belle-Cleo-BaBar workshop, Beijing
А. Полуэктов

2. The relative phase γ (φ3)
Interference between tree-level decays; theoretically clean
Favored: Vcb Vus*
Vcs* Vub: suppressed s u
Common final state
K(*)-
K(*)- u s u b B- B- c c b u f
D(*)0
D(*)0 u u
Parameters: f3, (rB, δ) per mode
Three methods for exploiting interference (choice of D0 decay modes):
Gronau, London, Wyler (GLW): Use CP eigenstates of D(*)0 decay, e.g. D0  Ksπ0, D0  π+ π -
Atwood, Dunietz, Soni (ADS): Use doubly Cabibbo-suppressed decays, e.g. D0  K+π -
Giri, Grossman, Soffer, Zupan (GGSZ) / Belle: Use Dalitz plot analysis of 3-body D0 decays, e.g. Ks π+ π-
Nov 26-27, 2007
BES-Belle-Cleo-BaBar workshop, Beijing

3. Dalitz analysis method
Measure B+/B- asymmetry across Dalitz plot
Includes GLW (D0  Ks ρ0, CP eigenstate) and ADS (D0  K*+π-, DCS 2-body decay)
regions
decay
amplitude
Mirror symmetry between D0 and D0 Dalitz plots
Sensitivity to φ3 in interference term
Determine f in flavor-tagged D*+D0π+ decays
2-fold ambiguity on f3 : (f3, δ) → (f3+π, δ+π) Model uncertainty ~ 100
Nov 26-27, 2007
BES-Belle-Cleo-BaBar workshop, Beijing

4. BES-Belle-Cleo-BaBar workshop, Beijing
D0  Ksπ+π– decay model
ρ-ω interference
Doubly Cabibbo
suppressed K*
M (GeV 2 )
Ksπ – 2
Nov 26-27, 2007
BES-Belle-Cleo-BaBar workshop, Beijing

5. Belle/Babar model-dependent results
HFAG averages for x± = rB cos( δ ± f3 ) , y± = rB sin( δ ± f3 )
Belle/Babar measurements in good agreement
Note: σ(f3) depends significantly on the value of rB
Contours do not include Dalitz model errors
Contours do not include Dalitz model errors rB
2f3
Nov 26-27, 2007
BES-Belle-Cleo-BaBar workshop, Beijing

6. Model-independent analysis
Model-independent way: obtain D0 decay strong phase from ψ(3770)DD data
(1)
where
Free parameters
Unknown, can be obtained from charm data:
DCPKSπ+π–:
(2)
ψ(3770)(KSπ+π–)D (KSπ+π–)D :
(3)
Nov 26-27, 2007
BES-Belle-Cleo-BaBar workshop, Beijing

7. Model-independent analysis
CLEO-c with 750 pb-1 (estimated numbers) ~ 1000 DCPKSπ+π– decays with CP-eigenstate tag.
~ 2000 correlated D0 decays: ψ(3770)(KSπ+π–)D (KSπ+π–)D
ψ(3770)(KLπ+π–)D (KSπ+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
Nov 26-27, 2007
BES-Belle-Cleo-BaBar workshop, Beijing

8. BES-Belle-Cleo-BaBar workshop, Beijing
Binned analysis: DCP
[A. Giri, Yu. Grossman, A. Soffer, J. Zupan, PRD 68, (2003)]
Number of events in D0-plot: i -i
Number of events in B-plot
(4)
Number of events in DCP-plot
(5)
( =1 if bin size is small enough)
Nov 26-27, 2007
BES-Belle-Cleo-BaBar workshop, Beijing

9. BES-Belle-Cleo-BaBar workshop, Beijing
Binned analysis: DCP
[A. Bondar, A.Poluektov, Eur. Phys. J. C (2006), hep-ph/ ]
ci , si can be obtained from B data only
ci from DCP, si from B data
si constrained as
 Very poor sensitivity
 Poor sensitivity for y
 Bias if bin size is large
rB=0.2
50 ab-1 (50000 ev.) at SuperB
should be enough for model-indep.
φ3 measurement with accuracy 2-3°
~10 fb-1 (10000 ev.) at ψ(3770) needed
to accompany this measurement.
Nearest future: ~1000 DCP events at
CLEO.
Bin size should be large 
bias due to
Nov 26-27, 2007
BES-Belle-Cleo-BaBar workshop, Beijing

10. BES-Belle-Cleo-BaBar workshop, Beijing
Binned analysis: DCP
[A. Bondar, A.Poluektov, hep-ph/ ]
To use the limited CLEO-c data:
Find binning with optimal sensitivity
Get rid of bias due to
Good approximation: uniform binning in ΔδD:
Satisfy simultaneously for binning with
CLEO-c expected sensitivity with 750 pb-1
Nov 26-27, 2007
BES-Belle-Cleo-BaBar workshop, Beijing

11. BES-Belle-Cleo-BaBar workshop, Beijing
Binned analysis: DCP
[A. Bondar, A.Poluektov, hep-ph/ ]
To use the limited CLEO-c data:
Find binning with optimal sensitivity
Get rid of bias due to
Good approximation: uniform binning in ΔδD:
Satisfy simultaneously for binning with
But the optimal binning depends on D0 model.  bias if it differs from actual one (same ~10°).
causes unavoidable model sensitivity. Reduces as data increases (by applying finer binning).
Nov 26-27, 2007
BES-Belle-Cleo-BaBar workshop, Beijing

12. Binned analysis: (KSπ+π–)2
2 correlated Dalitz plots, 4 dimensions:
(6)
Can use maximum likelihood technique:
(7)
with ci and si as free parameters.
For the same binning as in DCP, number of bins is N2 (instead of N),
but the number of unknowns is the same.
With Poisson PDF, it’s OK to have Nij<1.
Can obtain both ci and si.
It’s convenient to use combined likelihood of (Ksp+p-)2 and B data to account for errors in ci and si.
Nov 26-27, 2007
BES-Belle-Cleo-BaBar workshop, Beijing

13. Binned analysis: (KSπ+π–)2
[A. Bondar, A.Poluektov, hep-ph/ , arXiv: ]
Can use the same binning as in DCP
ci ,si measured independently  no model uncertainty due to constraint
Only 4-fold ambiguity: ci -c-i or si -s-i reduces to 2-fold if DCP data are used with the same binning.
Stat. sensitivity comparable to DCP
ci ,si measured in toy MC (points) and
calculated (crosses) for ΔδD-binning
Nov 26-27, 2007
BES-Belle-Cleo-BaBar workshop, Beijing

14. MC simulation of the bias in f3 measurement
Model dependence for 2x8 bins in (Ksp+p-)2 case:
No bias when changing the model, but stat. error degrades.
Model dependence for 2x8 bins in Dcp case:
Nov 26-27, 2007
BES-Belle-Cleo-BaBar workshop, Beijing

15. Summary of stat. sensitivity
Errors corresponding to 1000 events in B, DCP and (Kππ)2 samples
Binning
B-stat. error
DCP-stat. error
(Kππ)2-stat. error σx σy
Uniform, N=8
0.033
0.060
0.0053
0.0097
0.0145
0.0322
ΔδD, N=8
0.027
0.037
0.0042
0.0072
0.0050
0.0095
Optimal, N=8
0.023
0.032
0.0058
0.0114
0.0082
Uniform, N=20
0.055
0.0112 -
ΔδD, N=20
0.035
0.0048
0.0074
Optimal, N=20
0.022
0.029
0.0078
0.0110
Unbinned
0.021
0.028
Expected charm data contribution for 750 fb-1 at CLEO-c:
(1000 DCP and 2000 (Kππ)2)
σx=0.003, σy=0.007  σ(φ3)~5° with rB=0.1
Nov 26-27, 2007
BES-Belle-Cleo-BaBar workshop, Beijing

16. BES-Belle-Cleo-BaBar workshop, Beijing
Conclusion
Several approaches are proposed for model-independent analysis:
Binned with DCP (original GGSZ): implemented, studied. Bias with limited statistics.
Binned with (KSπ+π–)2: Twice as much data, similar to DCP sensitivity. No bias, less ambiguity.
Getting ready for model-independent measurement together with CLEO. Analysis strategy needs to be discussed
BESIII
BaBar
LHCb
Super HFAG
Belle
HFAG
CLEO
Super Babar/Belle
Nov 26-27, 2007
BES-Belle-Cleo-BaBar workshop, Beijing

17. BES-Belle-Cleo-BaBar workshop, Beijing
Backup slides
Nov 26-27, 2007
BES-Belle-Cleo-BaBar workshop, Beijing

18. Dalitz analysis method
A. Giri, Yu. Grossman, A. Soffer, J. Zupan, PRD 68, (2003)
A. Bondar, Proc. of Belle Dalitz analysis meeting, Sep 2002.
Using 3-body final state, identical for D0 and D0: Ksπ+π-.
Dalitz distribution density:
(assuming СР-conservation in D0 decays)
If is known, parameters are obtained from the fit to Dalitz distributions of D Ksπ+π– from B±DK± decays
Nov 26-27, 2007
BES-Belle-Cleo-BaBar workshop, Beijing

19. BES-Belle-Cleo-BaBar workshop, Beijing
D0  Ksπ+π– decay
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 D0 decay strong phase are essential
Use the model-dependent fit to
experimental data from flavor-tagged
D* D0π sample.
Model is described by the set of
two-body amplitudes
+ flat non-resonant term.
As a result, model uncertainty in the
γ/φ3 measurement (~10° currently).
Nov 26-27, 2007
BES-Belle-Cleo-BaBar workshop, Beijing

20. BES-Belle-Cleo-BaBar workshop, Beijing
Binning optimization 
Maximizing “binning quality” function allows to find a binning with the B-stat. sensitivity close to unbinned approach.
Nov 26-27, 2007
BES-Belle-Cleo-BaBar workshop, Beijing