Transcript Model independent determination of γ from B+→D(K0Sππ)K+

Determination of γ from
B±→DK±: Input from CLEOc
Jim Libby (University of Oxford)
7th February 2008
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Outline


Measuring γ with B±→DK±
Complementary measurements of D
decay at CLEO-c
– K0ππ
– K±X (X=π,ππ or πππ)

Other modes will be discussed later
today
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Searching for new physics
e
b
B 
d
W
0

e
u 

d
b

B0
d

W
t
t
W

TREE



Non Standard Model particles
contribute within the virtual loops
Differences between tree-level and
loop-level triangles
– Signature of new physics
Complements direct searches
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b
 0
B
d 
LOOP
b

B0
d

~
W
~t
~
W
~t
b
 0
B
d 
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Introduction B±→DK±

Strong phase
difference
B→DK decays involve b→c and b→u transitions
A( B   D0 K  )  AB
Vus*
0
A( B   D K  )  AB rB ei ( g )
Vub
Vcb
Vcs*
Ratio of absolute
amplitudes of
colour/CKM suppressed
to favoured (~0.1)

Access g via interference if D0 and D0 decay to the same final state

These measurements are theoretically clean
– No penguin CKM standard candle
– largest correction is sub-degree from D-mixing

LHCb looking at a number of strategies to study such decays
– B+: Atwood-Dunietz-Soni ('ADS'), 3 and 4 body Dalitz Plot Anal.
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B±→D(K0Sπ+π−)K±

For B+→D(K0π+π−)K+
m2
A  f (m2 , m2 )  rB ei ( g  ) f (m2 , m2 )
D0
(770)
(GeV / c 2 )
A  f (m2 , m2 )  rB ei (g  ) f (m2 , m2 )
m  KS0  invariantmass and f (m2 , m2 ) Dalitzamplitudes

Assume isobar model (sum of Breit-Wigners)
Number of resonances
Rel. BW
N
i
2
2
f m , m   a j e j Aj m2 , m2
 j 1




i

be



Amplitude and phase extracted
from D*+→D0π+ sample at B-factories

K*(892)
m2 (GeV/c 2 )
Non-resonant
Fit D-Dalitz plots from B-decay to extract γ, rB and δB
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±
0
B →D(K
S
+
−
±
π π )K
Absence of CP violation: distributions would be identical
B+
B−
Simulated LHCb data
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Current e+e− results

Current best direct constraints on γ:
PRD 73, 112009 (2006)
hep-ex/0607104
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3  (5318
(stat)  3(syst)  9(model)) [Belle]
g  (92  41(stat)  11(syst)  12(model)) [BABAR]

Based on ~300 events each (~1/3 of final data set)

However, large error from isobar model assumptions

BABAR and Belle use large samples of flavour tagged D*+D0π+
events to find parameters of the isobar model
– Excellent knowledge of |f|2 but phases less well known

Model uncertainties from assumptions about the resonance
structures in the model
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Isobar model uncertainty

BABAR (PRL 95 121802,2005)
Most challenging aspects
of the model uncertainty
come from Kπ and ππ Swave
Fit to flavour tag sample
K*0(1430)
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Model uncertainty impact at LHCb

The model-dependent likelihood fit yields an uncertainty
on γ between 7-12° for an rB=0.1
– One year of data
– Range represents differing assumptions about the background

However, the current model uncertainty is 10-15° with
an rB=0.1
– Uncertainties 1/rB


Without improvements LHCb sensitivity (and
e+e−)will be dominated by model assumptions
within 1 year of data taking
Motivates a model-independent method that relies
on a binned analysis of the Dalitz plot
– Disadvantage is that information is lost via binning
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Binned method

Proposed in the original paper by Giri, Grossman, Soffer and Zupan
and since been extended significantly by Bondar and Poluektov
– GGSZ, PRD 68, 054018 (2003)
– BP, most recently arXiv:0711.1509v1 [hep-ph]

Bin the Dalitz plot symmetrically
about m−2= m+2 then number of entries in B−
decay given by:
 # events in bin of flavour tagged D0 decays
N   f (m , m ) dD  r

i
2

Di
2

2
2
B

Di
2

2
2

f (m , m ) dD
Average cosine and
sine of strong
2
f (m2 , m2 ) dD f (m2 , m2 ) dD( x ci  y si )
Di
Di
phase difference
x  rB cos( B  g ) y  rB sin( B  g )
between D0 and
D0 decay amplitudes
' Cartesiancoordinates'
(ΔδD) in this bin10
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
2

2
CLEO-c measurement status
1/3 of total data
(<1/2 the CP tags)
Studies not complete
but projected uncertainties
on c and s will lead to
3-5 degree uncertainty on γ
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Inkblot test


Absolute value of strong phase diff.
(BABAR model used in LHCb-48-2007)
Bondar and Poluektov show
that the rectangular binning
is far from optimal for both
CLEOc and γ analyses
– 16 uniform bins has only
60% of the B statistical
sensitivity
– c and s errors would be 3
times larger from the ψ″
Best B-data sensitivity when
cos(ΔδD) and sin(ΔδD) are
Good approximation and the binning
as uniform as possible
that yields smallest s and c errors is equal
within a bin
ΔδD bins-80% of the unbinned precision
2 (i  12 ) / N   D (m2 , m2 )  2 (i  12 ) / N
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Implementation at LHCb
(γ=60°, rB=0.1 and δB=130°)
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
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Generate samples of
B±→D(K0Sππ)K± with a mean of
5000 events split between the
charges
Bin according to strong phase
difference, ΔδD
Minimise χ2
 (ni  N i ( x , y , h))2 (ni  N i ( x , y , h))2 
   



n
ni
i  8 ( i  0 ) 
i

ni  number of B   D( K S0   ) K  eventsin i th bin
8
2


N i ( x , y , h)  h K i  rB2 K  i  2 K i K i ci x  si y 
h  normalization factor
K i   f (m2 , m2 ) dD [measuredfromflavour tag data]
2
Ki, ci and si amplitudes
calculated from model

In reality from flavour tagged
samples and CLEO-c

Di
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γ uncertainties with 5000
toy experiments
2 fb-1 Mod. Indep.
10 fb-1 Mod. Indep.
No background
7.9°
3.5°
5.9°
Acceptance
8.1°
3.5°
5.5°
Dπ (B/S = 0.24)
(Best case scenario)
8.8°
4.0°
7.3°
DKcomb (B/S=0.7)
(Worst case scenario)
12.8°
5.7°
11.7°
Scenario
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2 fb-1 Mod. Dep.
(LHCb-048-2007)
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±
0
B →D(K
S

+
−
±
π π )K at
Model independent fit with binning
that yields smallest error
from exploiting CLEO-c data
LHCb
Model independent
Model dependent
σ(model)=10°
σ(model)=5°
– Binning depends on model - only
consequence of incorrect model
is non-optimal binning and a loss
of sensitivity

Measurement has no troublesome and hard-to-quantify systematic
and outperforms model-dependent approach with full LHCb dataset
with currently assigned model error
– 10 fb-1 statistical uncertainty 4-6° depending on background

CLEO-c measurements essential to validation of
assumptions in model dependent measurement

LHCb-2007-141 – Available via CERN document server
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ADS
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ADS method

Look at DCS and CF decays of D to
rates that have enhanced
obtain
interference terms
( B   ( K   ) D K  )  1  (rB rDK ) 2  2rB rDK cos( B   DK  g ),
( B   ( K   ) D K  )  rB2  (rDK ) 2  2rB rDK cos( B   DK  g ),
( B   ( K   ) D K  )  1  (rB rDK ) 2  2rB rDK cos( B   DK  g ),
( B   ( K   ) D K  )  rB2  (rDK ) 2  2rB rDK cos( B   DK  g )
( B   (h  h  ) D K  )  1  rB2  2rB cos( B  g )
( B  (h h ) D K )  1  r  2rB cos( B  g )








2
B
h=π or K
Unknowns : rB~0.1, B, DK, g, NK, Nhh (rD=0.06 well measured)
With knowledge of the relevant efficiencies and BRs, the normalisation
constants (NK, Nhh) can be related to one another
Important constraint from CLEOc σ(cos DK0.10.2
Overconstrained: 6 observables and 5 unknowns
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Four-body ADS

B→D(K πππ)K can also be used for ADS style analysis
– Also Kππ0

However, need to account for the resonant substructure in D→Kπππ
– made up of D→K*ρ, K−a1(1260)+,.,…
– in principle each point in the phase space has a different strong phase
associated with it - 3 and 4 body Dalitz plot analyses exploit this very
fact to extract γ from amplitude fits

Atwood and Soni (hep-ph/0304085) show how to modify the usual
ADS equations for this case
– Introduce coherence parameter RK3π which dilutes interference term
sensitive to γ
(B  (K     ) D K  )  rB2  (rDK 3 )2  2rB rDK 3 RK 3 cos( B   DK 3  g )
 A(s) A(s)e ds
 A(s) ds  A (s) ds
i ( s )
With A( D0  K     ) and A( D0  K     ) RK 3 e

RK3π ranges from


1=coherent (dominated by a single mode) to
0=incoherent (several significant components)
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i DK 3

2
2
Integrating over
phase space
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Determining the
coherence factor

1.
2.
Measurements of the rate of K3π versus different tags at CLEO-c allows
direct access to RK3π and δK3π
Normalisation from CF K−π+π+π− vs. K+π−π−π+ and K−π+π+π− vs.
K+π−
CP eigenstates:
3. K−π+π+π−
4.
vs.



( K 3 : CP )  KCF3 CP 1  rDK 3  2rDK 3 RK 3 cos  DK 3
K−π+π+π−:
K−π+π+π− vs. K−π+:
2

2
(K  3 : K  3 )  KCF3 KDCS
1

R
3
K 3


( K  3 | K  )  KCF3 KDCS
3
 1 

  2
rDK 3
rDK
2
rDK 3
rDK
R K 3 cos( DK   DK 3 )

14.5 
ConstrainδDK fromTQCA.Current value (21.6-16.3
)
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Amplitude models


To fully exploit D→K3π in B-decay an
unbinned fit to the data maybe
optimal
However, need model of DCS decays
– Accessible from CP-tagged data at CLEO-c

Furthermore, model can guide division
of phase space into coherent regions
for binned RK3π analysis
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Conclusion

Focussed on the things that are being done
and how they impact γ
– Apology 1: examples drawn from LHCb because
that is what I know best

Rest of the meeting in three parts:
– status of the UK work on the ADS and four body
fits
– extensions to the current work
– beer

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Apology 2: to those on the phone
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