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August 6, 2007 Charm 2007, Cornell University 1/15
A. Bondar Model-independent φ3 measurement
γ/φ3 model-independentDalitz analysis
(Dalitz+CP tagged Dalitz plots and γ/φ3 extraction)
Alex Bondar, Anton Poluektov
Budker Institute of Nuclear PhysicsNovosibirsk, Russia
August 6, 2007 Charm 2007, Cornell University 2/15
A. Bondar Model-independent φ3 measurement
The relative phase γ (φ3)
Interference between tree-level decays; theoretically clean
Parameters: , (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-
Vcs* Vub: suppressedFavored: Vcb Vus
*
b
u
s
u u
b
u
cD(*)0
K(*)-
B- B-
u
s
u
c
D(*)0f
Common
final state
K(*)-
30
0 iiBKDBA
KDBA eer
August 6, 2007 Charm 2007, Cornell University 3/15
A. Bondar Model-independent φ3 measurement
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
2222 ,, 3
mmfeermmfA ii
B KKB Ds
Sensitivity to in interference term
2-fold ambiguity on 3 : (3, δ)→ (3+π,
δ+π) Model uncertainty ~ 100
decay amplitude
Determine f in flavor-tagged D*+D0π+ decays
sKmm 22 Mirror symmetry between D0 and D0 Dalitz plots
August 6, 2007 Charm 2007, Cornell University 4/15
A. Bondar Model-independent φ3 measurement
ρ-ω interference
Doubly Cabibbosuppressed K*
)(GeV22sK
M
)(GeV22sK
M
)(GeV22sK
M )(GeV22sK
M
)(GeV 22
M )(GeV 22
M
M
(
GeV
2 )
Ksπ
–
2
D0 Ksπ+π– decay model
August 6, 2007 Charm 2007, Cornell University 5/15
A. Bondar Model-independent φ3 measurement
Belle/Babar model-dependent results • HFAG averages for x± = rB cos( δ ± ) , y± = rB sin( δ ± )
• Belle/Babar measurements in good agreement
• Note: σ(3) depends significantly on the value of rB
Contours do not include Dalitz model errors
Contours do not include Dalitz model errors
rB
23
August 6, 2007 Charm 2007, Cornell University 6/15
A. Bondar Model-independent φ3 measurement
Model-independent analysisModel-independent way: obtain D0 decay strong phase from ψ(3770)DD data
22222 |),(|),( mmfmmP DD
CPPPPffmmP DDDDDDCP 2||),( 222
][2|)(|),( 2222 SyCxPPPrPfiyxfmmP DDDBDDDB
)),(),(cos(),( 222222 mmmmmmC DD
)),(),(sin(),( 222222 mmmmmmS DD
22222 |),(|),( mmfmmP DD
)(2
||),,,( 22222
SSCCPPPPPPPP
ffffmmmmP
DDDDDDDD
DDDDCorr
DCPKSπ+π–:
ψ(3770)(KSπ+π–)D (KSπ+π–)D :
where )cos( 3 Brx
)sin( 3 BryFree parameters
Unknown, can be obtained
from charm data:
(1)
(2)
(3)
August 6, 2007 Charm 2007, Cornell University 7/15
A. Bondar Model-independent φ3 measurement
Model-independent analysis
CLEO-c with 750 pb-1 (estimated numbers) ~ 1000 DCPKSπ+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
August 6, 2007 Charm 2007, Cornell University 8/15
A. Bondar Model-independent φ3 measurement
Binned analysis: DCP
[A. Giri, Yu. Grossman, A. Soffer, J. Zupan, PRD 68, 054018 (2003)]
)](2[ 2iiiiiBiBi ysxcKKKrKhN
]2[ iiiiiCPi cKKKKhN
Number of events in B-plot
Number of events in DCP-plot
122 ii sc ( =1 if bin size is small enough)
i
-i
• ci , si can be obtained from B data only
• ci from DCP, si from B data
• si constrained as
21 ii cs
Very poor sensitivity
Poor sensitivity for y
Bias if bin size is large
Number of events in D0-plot: iK
iic DD )(cos iis DD )(sin
(4)
(5)
August 6, 2007 Charm 2007, Cornell University 9/15
A. Bondar Model-independent φ3 measurement
Binned analysis: DCP
[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.
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
rB=0.2
21 ii cs
August 6, 2007 Charm 2007, Cornell University 10/15
A. Bondar Model-independent φ3 measurement
Binned analysis: DCP
[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 21 ii cs Satisfy simultaneously for
binning with
Good approximation:
uniform binning in ΔδD
But the optimal binning depends on
D0 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).
122 ii sc
21 ii cs
August 6, 2007 Charm 2007, Cornell University 11/15
A. Bondar Model-independent φ3 measurement
Binned analysis: (KSπ+π–)2
)](2[ jijijjiiijjiij ssccKKKKKKKKhN
2 correlated Dalitz plots, 4 dimensions:
Can use maximum likelihood technique:
min),(log2log2 ijijPoisson NNpLwith 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.
(6)
(7)
August 6, 2007 Charm 2007, Cornell University 12/15
A. Bondar Model-independent φ3 measurement
Binned analysis: (KSπ+π–)2
ci ,si measured in toy MC (points) and
calculated (crosses) for ΔδD-binning
• 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
21 ii cs
[A. Bondar, A.Poluektov, hep-ph/0703267]
August 6, 2007 Charm 2007, Cornell University 13/15
A. Bondar Model-independent φ3 measurement
MC simulation of the bias in 3 measurement
Model dependence for 2x8 bins in Dcp case:
Model dependence for 2x8 bins in (Ks+-)2 case:
August 6, 2007 Charm 2007, Cornell University 14/15
A. Bondar Model-independent φ3 measurement
Summary of stat. sensitivity
BinningB-stat. error DCP-stat. error (Kππ)2-stat. error
σx σy σx σy 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 0.0114
Uniform, N=20 0.027 0.055 0.0042 0.0112 - -
ΔδD, N=20 0.027 0.035 0.0048 0.0074 - -
Optimal, N=20 0.022 0.029 0.0078 0.0110 - -
Unbinned 0.021 0.028 - - - -
Errors corresponding to 1000 events in B, DCP and (Kππ)2 samples
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)~3° with rB=0.1
August 6, 2007 Charm 2007, Cornell University 15/15
A. Bondar Model-independent φ3 measurement
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
Belle
CLEO
HFAG
BaBar
Super B factory
Super Babar/Belle
C/ factory
Super HFAG
August 6, 2007 Charm 2007, Cornell University 16/15
A. Bondar Model-independent φ3 measurement
Backup slides
August 6, 2007 Charm 2007, Cornell University 17/15
A. Bondar Model-independent φ3 measurement
If is known, parameters are obtained from
the fit to Dalitz distributions of D Ksπ+π– from B±DK± decays
Dalitz analysis method
Using 3-body final state, identical for D0 and D0: Ksπ+π-.
Dalitz distribution density:22222 ||),(
ssss KKKK
dmdmmmd A
222 |),(| ss KKmmA
(assuming СР-conservation in D0 decays)
000~DerDD i
B
A. Giri, Yu. Grossman, A. Soffer, J. Zupan, PRD 68, 054018 (2003)A. Bondar, Proc. of Belle Dalitz analysis meeting, 24-26 Sep 2002.
),( 22 ss KK
mmf ),,( Br
August 6, 2007 Charm 2007, Cornell University 18/15
A. Bondar Model-independent φ3 measurement
Statistical sensitivity of the method depends on the propertiesof 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).
D0 Ksπ+π– decay
August 6, 2007 Charm 2007, Cornell University 19/15
A. Bondar Model-independent φ3 measurement
Binning optimization
Maximizing “binning quality” function allows to
find a binning with the B-stat. sensitivity close to unbinned approach.
i i
iii
N
scNQ
)( 222