Measurement of the CKM angle g with a D 0 Dalitz analysis of the B ± →D (*) K ± decays
description
Transcript of Measurement of the CKM angle g with a D 0 Dalitz analysis of the B ± →D (*) K ± decays
Measurement of the CKM angle with a D0 Dalitz analysis of the
B±→D(*)K± decays
July 13th 2006
For the d0k-dalitz team:V.Azzolini, G.Cavoto, J.Garra, M.Giorgi, E. Grauges, Y.P.Lau, L.Li Giogi, N. Lopez-March, F.M-V, D.Milanes, N.N., J.Olsen, S.Pruvot, M.Rama, M.H. Schune, A. Stocchi
Nicola NeriINFN Pisa
July 13th 2006 SLAC Nicola Neri - CW Talk 2
Outline
Theoretical framework D0 decay amplitude parameterization Selection of the D(*)K events CP parameters from the D0 Dalitz distribution Systematic errors Extraction of Conclusions
July 13th 2006 SLAC Nicola Neri - CW Talk 3
ubV
*csV
cbV
*usV
bc transition bu transition
A(B-D0 K-) = AB A(B-D0 K-) = ABrB e i(B-)
D. Atwood, I. Dunietz, A. SoniPhys.Rev. D63 (2001) 036005
A. Giri, Y. Grossman, A. Soffer, J. ZupanPhys.Rev. D68 (2003) 054018
If same final state interference measurement
CKM elements + color suppression
strong phasein B decay
Towards
Critical parameter)(
)(
cbA
ubArB
f
f
f = KS final state
Theoretically and experimentally difficult to determine.
July 13th 2006 SLAC Nicola Neri - CW Talk 4
0D0D
A(B- ) = AD(s12, s13) +rB ei(-+B
) AD(s13, s12)
A(B+ ) = AD (s12, s12) +rB ei(+B AD((s12, s13)
CP
|A(B- )|2 =| AD(s12, s13) |2 + rB2 | AD(s13, s12) |2 +
+2rBRe[AD(s12, s13) AD(s13, s12)* ei(-+B)]
AD(s12, s13): fitted on
If rB is large, good precision on
D0 3-body decay Dalitz distribution |
AD(s12, s13) |2
from the Interference term
The method suffers of a two-fold ambiguity BB ,,
Using AD(s12, s13) in B decay amplitude
Assuming CP is conserved in D decays
ccee 00SKD 0* DD with from
July 13th 2006 SLAC Nicola Neri - CW Talk 5
Model Dependent Breit-Wigner description of 2-body amplitudes Three-body D0 decays proceed mostly via 2-body decays (1 resonance + 1
particle) The D0 amplitude AD can be fit to a sum of Breit-Wigner functions plus a
constant term, see E.M. Aitala et. al. Phys. Rev. Lett. 86, 770 (2001) For systematic error evaluation, use K-Matrix formalism to overcome the
main limitation of the BW model to parameterize large and overlapping S-wave resonances.
),(),( 131201312
0
ssAeaeassA ri
rr
iD
r
rJrr BWMssA ),( 1312
)(
1)(
2ijrrrij
ijr
siMMssBW
Angular dependence of the amplitude
depends on the spin J of the resonance r
Relativistic Breit-Wigner with mass dependent width r
where sij=[s12,s13,s23] depending on the resonance Ks-,Ks+,+-. Mr is the mass of the resonance
July 13th 2006 SLAC Nicola Neri - CW Talk 6
The BaBar Isobar Model
Mass and widths are fixed to the PDG 2004 values. Except for K*(1430), use E791 values and for , `, fit from data.
BaBar model 16 resonances + 1 Non Resonant term. Use same Isobar Model used in previous analysis presented at EPS’05.
July 13th 2006 SLAC Nicola Neri - CW Talk 7
The BaBar Isobar model
Good fit in DCS K*(892) region.
BaBar Data with BaBar isobar model fit over imposed.Fit Fraction=1.20
K
KDCS
390K sig events
97.7% purity
00SKD
0* DD
July 13th 2006 SLAC Nicola Neri - CW Talk 8
Signal events and DATA sample Rel18c BchToD0KstarAll skim cycle not in time for generic Monte Carlo but fine for
data. Run over AllEventsSkim on generic MC with Rel18b.
– DATA ON Peak 316.3 fb-1
– DATA OFF Peak 23.3 fb-1
B+e-e
Y(4S)
B
X
-K
0SK
0D
Ks
0
0 )(cos
DKsKs
DKsKsKs
xxp
xxp
=1 for signal events
D0 0 ,D0
K-B-
+ -
Ks + -
D*0D0 B- K-
+ -
Ks + -
July 13th 2006 SLAC Nicola Neri - CW Talk 9
Selection criteria for B±D(*)K± decay modesD0K D*0K (D00) D*0K (D0)
|cos T| <0.8 <0.8 <0.8 |mass(D0)-PDG| <12MeV <12MeV <12MeV |mass(Ks)-PDG| <9MeV <9MeV <9MeV E ----- >30 MeV >100MeV |mass(0)-PDG| ----- <15MeV ----- Kaon Tight Selector Yes Yes Yes |M-PDG| ----- <2.5MeV <10.0MeV cos
Ks>0.99 >0.99 >0.99
|E | <30MeV <30MeV <30MeV---------------------------------------------------------------------------------------- efficiency 15% 7% 9% ---------------------------------------------------------------------------------------- signal events 398±23 97±13 93±12
cos Ks
suppress fake Ks |cos
T| suppress jet-like events
Kaon Tight Selector (LH) and |E|<30 MeV suppress D(*) events
July 13th 2006 SLAC Nicola Neri - CW Talk 10
Yields on DATA
D0K D*0K D*0 D0 D*0K D*0 D0
BD0K 39823 signal events ~60% purity mES>5.272 GeV/c2
BD*0K D*0 D0 9713 signal events ~80% purity
BD*0K D*0 D0 9312 signal events ~50% purity
347 million of BB pairs at (4S)
Signal D BB qq
background is >5 times the bkg contribution in each mode. D contribution is negligible after all the selection criteria applied in mES>5.272 GeV/c2 and |E|<30 MeV unless for [D0]K. The error on the Dcontribution is large and can be explained as a statistical fluctuation (accounted for in systematic error)
qq BB
July 13th 2006 SLAC Nicola Neri - CW Talk 11
Background parameterization: Dalitz shape for background events
● BB and continuum events are divided in real D0 and fake D0. The real D0 fraction is evaluated on qq and BB Monte Carlo counting:
● cross-check on DATA using the mES<5.272 GeV sidebands and fitting the D0 mass.
● For the bkg real D0 : D0 Dalitz signal shape For the bkg fake D0 (combinatorial) 2D symmetric 3rd order polynomial asymmetric
function :• cross-check on DATA using the mES<5.272 GeV and D0 mass sidebands
.BB combinatorics - MC .qqbar combinatorics – MC
fit function fit function
Asymmetric Asymmetric
July 13th 2006 SLAC Nicola Neri - CW Talk 12
Background parameterization: fraction of true D0
● The real D0 fraction is evaluated directly on DATA using the mES<5.272 GeV sidebands and fitting the D0 mass distribution. The signal is a Gaussian with fixed MeV/c2 (MC value) and MeV/c2 (PDG value).
. DATA (On-Res)
On Monte Carlo we find the fraction for true D0 to be:
MC continuum evt
MC BB events
we use this error for conservative systematic error evaluation
0.2218 0.0097
0.0255 0.0076
0.1588 0.0062
Cont
BB
Bkg
R
R
R
MC BB + qq weighted evt
0.254 0.035dataR
July 13th 2006 SLAC Nicola Neri - CW Talk 13
Background characterization: true D0 and flavor-charge correlation
cc
D0=KS
D0
K- + other particles
e+ e-
estimated on Monte Carlo events
15.064.0 , 018.0164.0
##
#00
0
RSBB
RSCont
RS
RR
KDBKDB
KDBR
July 13th 2006 SLAC Nicola Neri - CW Talk 14
Likelihood for Dalitz CP fit
BBCont,R
fSig,Dh,Cont,BB from data (extended likelihood yields)
A(B-) = |AD(s12,s13) +rBei(-BAD(s13,s12) |2
0
0 0Cont,BB,WrongSig
#
# #RS B D K
RB D K B D K
True D0 fraction from MC and data (mES sidebands)
Charge-flavor correlation from MC
D0,D0
|AD(s12,s13)|2
|AD(s13,s12)|2
From MC and D0 sideband data
July 13th 2006 SLAC Nicola Neri - CW Talk 15
Dalitz distributions
B- B+
DK D*K (D00)K
D*K (D0)K
B- B+
B- B+Dalitz plot distribution for signal events after all the selection criteria applied in mES>5.272 GeV/c2 and |E|<30 MeV.
July 13th 2006 SLAC Nicola Neri - CW Talk 16
CP parameters extractionFit for different CP parameters: cartesian coordinates are preferred base.
Errors are gaussian and pulls are well behaving. x= Re[r
Bexpi(]= r
Bcos() , y= Im[r
Bexpi(= r
Bsin()
The statistical error dominates the measurement.
x*-=rBcos()y*-=rBsin() x*+=rBcos()y*+=rBsin()
CP parameter Result
x-=rBcos()y-=rBsin() x+=rBcos()
y+=rBsin()
CP parameter Result
Dalitz model uncertainty. Estimated using alternative models as explained later.
PDF shapes , Dalitz plot efficiency qq Dalitz shape Charge correlation of (D0,K) in qq
Main systematics
July 13th 2006 SLAC Nicola Neri - CW Talk 17
Cartesian coordinate results
B+
B-
d
D0K
Direct CPV
B+
B-
D*0K
ln(L)=0.5
ln(L)=1.92 Direct CP violation d=2 rb(*)|sin|
d
Likelihood contours from ln(L)=0.5, 1.921
July 13th 2006 SLAC Nicola Neri - CW Talk 18
Experimental systematic errors
Experimental systematics Dalitz model systematics≳Statistical error >>
July 13th 2006 SLAC Nicola Neri - CW Talk 19
(D0)K – (D00)K cross-feed From Monte Carlo simulation the cross-feed between the samples is
due to events of (D00)K where we loose a soft and we reconstruct it as a (D0)K.
Since the cross-feed goes in one direction (D00)K (D0)K, it is correct to assign common events to the (D00)K signal sample.
After all the cuts and after this correction applied we expect <5% of signal (D0)K from cross-feed.
A systematic effect to the cross-feed has been assigned adding a signal component according to the (D00)K Dalitz PDF and performing the CP fit.
The systematic bias of the fit with and without (D00)K has been quoted as systematic error.
Negligible wrt the other systematic error sources.
July 13th 2006 SLAC Nicola Neri - CW Talk 20
Dalitz model systematic error: K*(892) parameters In PDG those measurements are from 1970’s. Very low statistics ~5000 events we have ~200000 K*(892) eventsIf we allow mass and width to float, we get the width 46 +/- 0.5 MeV, mass 893 +/- 0.2 MeV
Partial wave analysis of BJ/psi K decay (BaBar) can use as control sample
Mass=892.9+/-2.5 MeV Width=46.6 +/-4.7 MeV
Their values are consistent with our floatedvalues
No S-wave!Very clean measurement
Consider as systematics compared with PDG
July 13th 2006 SLAC Nicola Neri - CW Talk 21
Dalitz model systematic error: K*(1430) parameters
BaBar Isobar model [float] for K*0(1430): Mass =1.495 +/- 0.01 GeVWidth = 183 +/- 9 MeV
E791 Isobar model for K*0(1430): Mass =1.459+/-0.007 GeVWidth = 175 +/- 12 MeV
The Isobar model, the fit prefer small value of K*(1430), both seen in E791 and BaBar, although PDG list the width 294 MeV
Current BaBar model for K*0(1430): Mass =1.412 +/- 0.006 GeVWidth = 294 +/- 23 MeV
PDG(from LASS)
Mass and width are not unique parameters, depend on the parameterization and Non-Resonant model
July 13th 2006 SLAC Nicola Neri - CW Talk 22
K*(892) and K*(1430) with new parameters
Perfect!
July 13th 2006 SLAC Nicola Neri - CW Talk 23
Zemach Tensor vs Helicity model
Monte Carlo simulation using f2(1270)
In Dsthe non-resonant term is much smaller (5%) in Zemach Tensor while the non-resonant term is (25%) if Helicity model is used
MC MCData
Data Ds
D wave systematics
Affects seriously on spin 2
July 13th 2006 SLAC Nicola Neri - CW Talk 24
Dalitz model systematics S-wave:
Use K-matrix S-wave model instead of the nominal BW model P-wave:
Change (770) parameters according to PDG Replace Gounaris-Sakurai by regular BW
and K D-wave Zemach Tensor as the Spin Factor for f2(1270) and K*2(1430) BW
K S-wave: Allow K*0(1430) mass and width to be determined from the fit Use LASS parameterization with LASS parameters
K P-wave: Use BJ/psi Ks + as control sample for K*(892) parameters Allow K*(892) mass and width to be determined from the fit
Blatt-Weiskopf penetration factors Running width: consider a fixed value Remove K2*(1430), K*(1680), K*(1410), (1450)
This is a more realistic and detailed estimate of the model systematics !
July 13th 2006 SLAC Nicola Neri - CW Talk 25
Procedure for Dalitz model systematics
Generate a high statistics toy MC (x100 data statistics) experiment using nominal (BW) model.
The experiment is fitted using the nominal and each alternative model.
Produce experiment-by-experiment differences for all (x,y) parameters for each alternative Dalitz model.
For each CP coordinates, consider the squared sum of the residuals over all the alternative models as the systematic error.
Since the model error increases with the value of rb we conservatively quote a model error corresponding to ~rb+1valuethat we fit on data.
This is a quite conservative estimate of the systematics and we believe it is fair to consider the covariance matrix among the CP parameters to be diagonal.
July 13th 2006 SLAC Nicola Neri - CW Talk 26
Bias on x-, x+ for alternative Dalitz models
Residual for the x-, x+ coordinates wrt the nominal CP fit.
Yellow band is the nominal fit statistical error (x100 Run1-5 statistics)
July 13th 2006 SLAC Nicola Neri - CW Talk 27
Bias on y-, y+ for alternative Dalitz models
Residual for the y-, y+ coordinates wrt the nominal CP fit.
Yellow band is the nominal fit statistical error (x100 Run1-5 statistics)
July 13th 2006 SLAC Nicola Neri - CW Talk 28
Frequentist interpretation of the results
Stat Syst Dalitz
2
1
D0K D*0K
is to be understood in term of 1D proj of a L in 5D.
1 (2) excursion
July 13th 2006 SLAC Nicola Neri - CW Talk 29
Final results •We have measured the cartesian CP fit parameters for DK (x±,y±) and D*K (x*± ,y*±)using 316 fb-1 BaBar data:
D0 amplitude model uncertainty Experimental systematicsStatistical error
•This measurement supersedes the previous one on 208 fb-1 with significant improvements in the method and smaller errors on the cartesian CP parameters.•Using a Frequentist approach we have extracted the values of the CP parameters:
is to be understood in term of 1D proj of a L in 5D.
x*-=rBcos()y*-=rBsin() x*+=rBcos()y*+=rBsin()
CP parameter Result
x-=rBcos()y-=rBsin() x+=rBcos()
y+=rBsin()
CP parameter Result
Stat Syst Dalitz
1 (2) excursion
July 13th 2006 SLAC Nicola Neri - CW Talk 30
BaBar vs Belle experimental results
Belle’06 - N(BB)=386M
BABAR'06 - N(BB)=347M
Experimental measurement of the CP parameters x,y is more precise wrt Belle evenwith slightly smaller statistics. Different error on is due to Belle larger central values.
x*-=rBcos()y*-=rBsin() x*+=rBcos()y*+=rBsin()
CP parameter Result
x-=rBcos()y-=rBsin() x+=rBcos()
y+=rBsin()
CP parameter Result
x*-=rBcos()y*-=rBsin() x*+=rBcos()y*+=rBsin()
CP parameter Result
x-=rBcos()y-=rBsin() x+=rBcos(y+=rBsin()
CP parameter Result
−0.13 +0.17−0.15 ± 0.02
−0.34 +0.17−0.16 ± 0.03
0.03 ± 0.12 ± 0.01
0.01 ± 0.14 ± 0.01
0.03 +0.07−0.08 ± 0.01
0.17 +0.09−0.12 ± 0.02
−0.14 ± 0.07 ± 0.02
−0.09 ± 0.09 ± 0.01
give
n on
rb
,
give
n on
rb
,
D0 amplitude model uncertainty Experimental systematicsStatistical error
July 13th 2006 SLAC Nicola Neri - CW Talk 31
Considerations on the results
2
B+
B-
x
y
rB
x≈y≈rb·
rb
rb
() ≈ x/rb
Experimentally we can improve the measurement of the CPcartesian coordinates but the improvement on error of depends on the true value of the rb parameter. Similar behavior for statistical and systematic error.
July 13th 2006 SLAC Nicola Neri - CW Talk 32
Dalitz: BaBar vs Belle Results HFAGExperiment Mode γ/φ3 (°) δB (°) rB
DK– D→KSπ+π– 53± +15
–18 ± 3 ± 9 146 ± 19 ± 3 ± 23 0.16 ± 0.05 ± 0.01 ± 0.05
Belle‘06 N(BB)=392M
D*K– D*→Dπ0 D→KSπ+π– 53 ± +15
–18 ± 3 ± 9 302 ± 35 ± 6 ± 23 0.18 +0.11–0.10 ± 0.01 ± 0.05
DK– D→KSπ+π– 92 ± 41 ± 10 ± 13 118 ±64 ±21 ±28 <0.142
D*K– D*→D(π0,γ) D→KSπ+π– 92 ± 41 ± 10 ± 13 298 ±59 ±16 ±13 <0.206
BABAR'06 N(BB)=347M
BaBar measurement is very important since it stresses one more time the difficulty to measure in a regime where the uncertainty on rb is quite large.
July 13th 2006 SLAC Nicola Neri - CW Talk 33
Back-up slides
July 13th 2006 SLAC Nicola Neri - CW Talk 34
Three-body D decays: Dalitz-Fabri plot A point of in a three-body decay phase-space can be determined
with two independent kinematical variables. A possible choice is to represent the state in the Dalitz-Fabri plot
230
1132
20
112 , pKpspKps ss
),( 1312 ss
kinematical Mandelstam variables:
The A(D0 →Ks) amplitude can be written as AD(s12, s13).
(mKS+m2
(MD0-m)2
(mKS+m2
(MD0-m)2
July 13th 2006 SLAC Nicola Neri - CW Talk 35
Sensitivity to points : weight = 1
weight =
2
2
ln( )d L
d
22
2
1( ) ~
ln( )d Ld
D0 Ks
DCS D0 K0*(1430)+-
CA D0 K*(892)- +
DCS D0 K*(892)+-
s12 (GeV2)
s1
3
(G
eV
2)
Strong phase variation improves the sensitivity to . Isobar model formalism reduces discrete ambiguities on the value of to a two-fold ambiguity.
July 13th 2006 SLAC Nicola Neri - CW Talk 36
Reconstruction of exclusive B±D(*)K± decays
D0 0 ,D0 K-B-
+-
Ks + -
B
+e-e
Y(4S)
B
X
-K
0SK
0D
Ks
0
0 )(cos
DKsKs
DKsKsKs
xxp
xxp
D*0D0
=1 for signal events
B- K-
+-
Ks + -
July 13th 2006 SLAC Nicola Neri - CW Talk 37
Background characterization: relative fraction of signal and bkg samples
D0K D*0K - D*0 D0 D*0K - D*0 D0
Signal D BB qq
● Continuum events are the largest bkg in the analysis.We apply a cut |cos(
T)|<0.8 and we use fisher PDF for the continuum bkg
suppression.
Fisher = F [LegendreP0,LegendreP2,|cos T|,|cos *|]
● The Fisher PDF helps to evaluate the relative fraction of BB and continuum events directly from DATA.
July 13th 2006 SLAC Nicola Neri - CW Talk 38
Efficiency Map for D*→D0
SvOutPlaceObjectRed = perfectly flat efficiency Blu = 3rd order polynomial fit
Efficiency is almost flatin the Dalitz plot. The fitwithout eff map givesvery similar fit results.
Purity 97.7%
We use ~200K D* MC sample D0 Phase Space to compute the efficiency map .
2D 3rd order polynomial function usedfor the efficiency map.
July 13th 2006 SLAC Nicola Neri - CW Talk 39
Isobar model formalismAs an example a D0 three-body decay D0 ABC decaying through an r=[AB] resonance
D0 three-body amplitude
•We fit for a0 ,ar amplitude values and the relative phase 0 , r among resonances, constant over the Dalitz plot.
can be fitted from DATA using a D0 flavor tagged sample from
events selecting with ccee 00SKD 0* DD
In the amplitude we include FD, Fr the vertex factors of the D and the resonance r respectively.H.Pilkuhn, The interactions of hadrons, Amsterdam: North-Holland (1967)
),(),( 131201312
0
ssAeaeassA rri
rr
iD
July 13th 2006 SLAC Nicola Neri - CW Talk 40
Efficiency Map for B->DK (*)
● Because of the different momentum range we use a different parametrization respect to the D* sample.We use ~1M B->D(*)K signal MC sample with Phase Space D0 decay to compute the efficiency mapping.
● Fit for 2D 3rd order polynomial function to parametrize the efficiency mapping:
Efficiency is rather flatin the Dalitz plot. CP Fit without eff map to quote the systematic erroron CP parameters
)()()(),( 2131213
212
313
31231312
213
21221212101312 ssssssessssesseess
July 13th 2006 SLAC Nicola Neri - CW Talk 41
K-Matrix formalism for S-wave1. K-Matrix formalism overcomes the main limitation of the BW model to
parameterize large and overlapping S-wave resonances. Avoid the introduction of not established ´ scalar resonances.
2. By construction unitarity is satisfied
K-matrix D0 three-body amplitude
),( ),( 13120 ,0
2311312 ssAeasFssA rri
spinkspinrrD
SS†=1 S=1+2iT
T=(1-iK·)-1K where S is the scattering operatorT is the transition operatoris the phase space matrix
j
23-11j2323231 s si-I jssF PρK F1 = S-wave amplitude
Pj(s) = initial production vector
July 13th 2006 SLAC Nicola Neri - CW Talk 42
The CLEO model
CLEO model 10 resonances + 1 Non Resonant term.
July 13th 2006 SLAC Nicola Neri - CW Talk 43
The CLEO modelWith >10x more data than CLEO, we find that the model with 10 resonances is insufficient to describe the data.
BaBar Data refitted using CLEO model.
CLEO model 10 resonances + 1 Non Resonant term.
July 13th 2006 SLAC Nicola Neri - CW Talk 44
The BELLE modelBelle model 15 resonances + 1 Non Resonant term. Added DCS K*
0,2(1430), K*(1680) and
1 ,
2 respect to the
CLEO Model. With more statistics you “see” a more detailed
structure.
Added
Added
Added
AddedAdded
July 13th 2006 SLAC Nicola Neri - CW Talk 45
The BELLE model
Not very good fit for DCS K*(892) region.
BELLE Data with BELLE model fitoverimposed.DCS region is quite
important for the sensitivity. See plotin the next pages.
Belle model 15 resonances + 1 Non Resonant term. Added DCS K*
0,2(1430), K*(1680) and
1 ,
2 respect to the
CLEO Model.
July 13th 2006 SLAC Nicola Neri - CW Talk 46
The BaBar Model
BaBar model does not include the DCS K*(1680) and DCS K*(1410) because the number of events expected is very small.
DCS K*(1680)K*(1680)
Moreover the K*(1680) and the DCS K*(1680) overlap in the same Dalitz region: the fit was returning same amplitude for CA and DCS!
July 13th 2006 SLAC Nicola Neri - CW Talk 47
The BaBar model
2 fit evaluation of goodness of fit: 1.27/dof(3054).CLEO model is 2.2/dof(3054)Belle model is 1.88/dof(1130)
Total fit fraction: 125.0% CLEO model is 120%Belle model is 137%
The 2 is still not optimal but much better respect to all Dalitz fit published so far.
July 13th 2006 SLAC Nicola Neri - CW Talk 48
The BaBar K-matrix modelBaBar Data with BaBar K-matrix model fit over imposed.Fit Fraction=1.11
K
KDCS
opens KK channel
390K sig events
97.7% purity
00SKD
0* DD
July 13th 2006 SLAC Nicola Neri - CW Talk 49
K-Matrix parameterization according to Anisovich, Sarantev
2/
0.10.1 223
023
0
023
0
23223
mssss
s
ss
sf
sm
ggs A
A
Ascatt
scattscatt
ijr
jiij
K
scatt
scattprodj
jj
ss
sf
sm
g
023
01
232
0.1s
P
igwhere is the coupling constant of the K-matrix pole mto the ith channel 1= 2=KK
3=multi-meson 4= 5= ´.
Adler zero term to accommodate singularities
scattscattij sf 0 , slow varying parameter of the K-matrix element. 1 if 0 if scatt
ij
Pj(s) = initial production vector
I.J.R. Aitchison, Nucl. Phys. A189, 417 (1972)
V.V. Anisovitch, A.V Sarantev Eur. Phys. Jour. A16, 229 (2003)
July 13th 2006 SLAC Nicola Neri - CW Talk 50
The BaBar K-Matrix Model
BaBar K-Matrix model 9 resonances + S-wave term. Total fit fraction is 1.11.
S-wave term
July 13th 2006 SLAC Nicola Neri - CW Talk 51
Extract as much as possible information from data
●Step 1: selection PDF shapes and yieldsFit as many as possible component yields and discriminating variables (PDF) shapes from simultaneous fit to DK and D data
Fix the remaining to MC estimates Selection PDFs: mES, Fisher
●Step 2: Dalitz CP fit to extract CP parameters from the D0 Dalitz
distribution Fix shape PDF parameters obtained in step 1 and perform
Dalitz CP fit alone with yields re-floated Impact of fixing shape PDFs on CP violation parameters is small (systematic error taken into account)
Overview of Dalitz CP fit strategy
July 13th 2006 SLAC Nicola Neri - CW Talk 52
The frequentist method Frequentist (classical) method determines CL regions where the probability
that the region will contain the true point is Determine PDF of fitted parameters as a function of the true parameters:
In principle, fitted-true parameter mapping requires multi-dimensional scan of the experimental (full) likelihood:
Prohibitive amount of CPU, limited precision (granularity of the scan).
Make optimal choice of fitted parameters and try analytical construction for the PDF. Gaussian PDF are easy to integrate!
Cartesian coordinates (x,y)
x= Re[r
Bexpi(]= r
Bcos() , y= Im[r
Bexpi(=
rBsin()
July 13th 2006 SLAC Nicola Neri - CW Talk 53
The Frequentist PDFSingle channel (D0K o D*0K) Measured parameters (4D): z+=(x+, y+), z-=(x-, y-)Truth parameters (3D): pt=(r
B,
D0K-D*0K combination Measured parameters (8D): z+, z-, z*+, z*-
Truth parameters (5D): pt=(rB, r*
B, *)
Easy to include systematic error by replacing
stat→
tot
G2(z;x,y,
x,
y,r) is a 2D Gaussian
with mean (x,y) and sigma (x,
y)
July 13th 2006 SLAC Nicola Neri - CW Talk 54
Confidence Regions
Integration domain D
●(pt): calculated analytically, PDF is product of gaussians.●We calculate 3D (5D) joint probability corresponding to 1 and 2CL for a 3D (5D) gaussian distribution.●Make 1D projections to quote 1 and 2 regions for r
b, r
b*,*
CL=1-(pt)
P(data|pt)
Integral
.data
D
.zPt
Example in 1D
July 13th 2006 SLAC Nicola Neri - CW Talk 55
Cartesian coordinates: toy MC
x-
y-
x+
y+
Linear correspondence and errors are well behaving.
Fitted parameters vs Generated parameters.
July 13th 2006 SLAC Nicola Neri - CW Talk 56
Confidence Regions (CR):D0K-D*0K combination
Integration domain D (CR)
CL=1-(pt)
July 13th 2006 SLAC Nicola Neri - CW Talk 57
D0K-D*0K confidence intervalsSingle channel (D0K o D*0K)
D0K-D*0K combination
Central values are the mean of CL the interval
July 13th 2006 SLAC Nicola Neri - CW Talk 58
BD*K decays help in constraining
0*)*(*0**~DerDD i
B
)(
)(0*
0**
KDBA
KDBArB
2
0(*)0(*)0(*) DD
DCP
As pointed out in Phys.Rev.D70 091503 (2004) for the BD*K decay we have:
From the momentum parity conservation in the D* decay:
1)γCP(
1)πCP( 0
DD
DD
*
0*
The effective strong phase shift helps in the determination of
cos21 220*BB rraKDDBBR
cos21 22*BB rraKDDBBR
Opposite CP
eigenstate