international school of subnuclear physics erice, 30 aug - 6 sept 2006

Measurement of the CKM angle with a D0 Dalitz analysis of the

B±→D(*)K± decays at BaBar

International School of Subnuclear PhysicsErice, 30 Aug - 6 Sept 2006

Nicola NeriINFN Pisa

September 1st 2006 Erice

Nicola Neri - International School of Subnuclear Physics

2

CKM matrix and Unitarity Triangle

0*** tbtdcbcdubud VVVVVV

CP violation is proportional to the triangle area

,0argcbcd

ubud

VVVVCP violation

Standard Model fits predicts •(64±5 )˚ UTFit - Bayesian•(60±5 )˚CKMFit - Frequentist

•Test SM prediction with tree-level processes

(0,0) (1,0)

(,)

Vtd Vtb*

|Vcd Vcb|*|Vcd Vcb|*

Vud Vub*

tbtstd

cbcscd

ubusud

VVVVVVVVV

VUnitarity of

quark mixing matrix



3

ubV

*csVcbV

*usV

bc transition bu transition

A(B-D0 K-) = AB A(B-D0 K-) = ABrB e i(B-)

Gronau, Wyler, Phys. Lett. B265,172 (1991)

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)()(

sucbAscubArB

f

f

f = KS (Dalitz Analysis)f = CP (GLW)f = DCSD (ADS)

Theoretically and experimentally difficult to determine.



4

Three-body D decays: Dalitz 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 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



5

0D0D

A(B- ) = AD(s12, s13) +rB ei(-+B) AD(s13, s12)

A(B+ ) = AD (s13, 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

s13 (GeV2)

230

12

132

20

12

12 ,

pKpmspKpms ss(*) Def.

s13 (GeV2)

s 12 (G

eV2 )

s 12 (G

eV2 )



6

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.

),(),( 1312013120

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

JrM



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



8

The BaBar Isobar Model

Mass and widths are fixed to the PDG values. Except for K*(1430), use E791 values and for , `, fit from data.

BaBar model 16 resonances + 1 constant term (Non-resonant).



9

Signal events and DATA sample– DATA at (4S) peak 10.580 GeV 316.3 fb-1 (347 M BB

events)– DATA below peak 23.3 fb-1

B

+e-e (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 + -

9.1 GeV 3.0 GeVrate = L·(bb) ~ 1.2·1034cm-2s-1 ·1.1 nb ≳ 13 BB evt/sec

(4S)50% B0B0

50% B+B-

(4S)



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 signal region 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



11

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.



12

CP parameters extractionFit for different CP parameters: cartesian coordinates are preferred base. Errors are gaussian and pulls are well behaving. x= Re[rBexpi(]= rBcos() , y= Im[rBexpi(= rBsin()

The statistical error dominates the measurement.

x*-=rBcos()y*-=rBsin() x*+=rBcos()y*+=rBsin()

CP parameter Resultx-=rBcos()y-=rBsin() x+=rBcos()y+=rBsin()

CP parameter Result

Dalitz model error. Account for phenomenological D amplitude parameterization uncertainty

PDF shapes , Dalitz plot efficiency, qq Dalitz shape Charge correlation of (D0,K) in qq

Main systematics



13

Cartesian coordinate results

B+

B-

d

D0K

Direct CPV

B+

B-

D*0K

Direct CP violation d=2 rb(*)|sin|

d



14

Experimental systematic errors

Experimental systematics Dalitz model systematics≳Statistical error >>



15

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

rB r*B



16

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.



17

Conclusions and perspectives• We demonstrated that the measurement of is possible and compatible with SM predictions.•Dalitz method gives the best sensitivity to but…more statisticsis crucial.•If rB≥0.1 we will know the value ≤15% precision with 1 ab-1.

Near Future=73±29 ([15,136]@95%CL) =-107±29 ([-165,-44]@95%CL)

Toy MCrb=0.1 assumed

Dalitz model error projection



18

Back-up slides



19

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 !



20

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)



21

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)



22

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 .qq combinatorics – MC

fit function fit function

Asymmetric Asymmetric



23

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 0.00970.0255 0.00760.1588 0.0062

Cont

BB

Bkg

RRR

MC BB + qq weighted evt

0.254 0.035dataR



24

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

RRKDBKDB

KDBR



25

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.



CP parameter Result

Stat Syst Dalitz

1 (2) excursion



26

CP Violation in the Standard Model CP symmetry can be violated in any field theory with at least one

CP-odd phase in the Lagrangian This condition is satisfied in the Standard Model through the three-

generation Cabibbo-Kobayashi-Maskawa (CKM) quark-mixing matrix

b

u

W

ubV bt

sc

du

CP-violating phase

23.0

Wolfenstein parameterization: corresponds to a

particular choice of the quark-phase convention



27

Unitarity triangle

0*** tbtdcbcdubud VVVVVV(0,0) (1,0)

Vtd Vtb*

|Vcd Vcb|*

CP violation is proportional to the area

,0argcbcd

ubud

VVVV

CP violation

|Vcd Vcb|*

Vud Vub*

()



28

CP violation in decay or direct CP violation

BHfA || BHfA ||

For example A=A1+A2: two amplitudes with a relative CP violating phase (CP-odd) and a CP conserving phase (CP-even)

A 2 2

f fA

B B CP violation



29

BaBar Detector

DIRC PID)144 quartz bars

11000 PMs

1.5T solenoid

EMC6580 CsI(Tl) crystals

Drift Chamber40 stereo layers

Instrumented Flux Returniron / RPCs/LSTs (muon / neutral hadrons)

Silicon Vertex Tracker5 layers, double-sided sensors

e+ (3.1GeV)

e (9.0 GeV)



30

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.



31

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.



CP parameter Result


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



32

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



33

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



34

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 KRB 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



35

(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.



36

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



37

K*(892) and K*(1430) with new parameters

Perfect!



38

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



39

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



40

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.



41

Sensitivity to points : weight = 1

weight = 2

2

ln( )d Ld

22

2

1( ) ~ln( )d L

d

D0 Ks

DCS D0 K0*(1430)+-

CA D0 K*(892)- +

DCS D0 K*(892)+-

s12 (GeV2)

s

13

(GeV

2 )

Strong phase variation improves the sensitivity to . Isobar model formalism reduces discrete ambiguities on the value of to a two-fold ambiguity.



42

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 + -



43

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.



44

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.



45

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 00

SKD 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)

),(),( 1312013120

ssAeaeassA rri

rr

iD



46

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



47

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 operator

T 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



48

The CLEO model

CLEO model 10 resonances + 1 Non Resonant term.



49

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.



50

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 detailedstructure.

Added

Added

Added

AddedAdded



51

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.



52

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!



53

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.



54

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



55

K-Matrix parameterization according to Anisovich, Sarantev

2/

0.10.1 223

023

0

023

0

23223

msssss

sss

fsm

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)



56

The BaBar K-Matrix Model

BaBar K-Matrix model 9 resonances + S-wave term. Total fit fraction is 1.11.

S-wave term



57

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



58

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[rBexpi(]= rBcos() , y= Im[rBexpi(= rBsin()



59

The Frequentist PDFSingle channel (D0K o D*0K) Measured parameters (4D): z+=(x+, y+), z-=(x-, y-)Truth parameters (3D): pt=(rB,

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 Gaussianwith mean (x,y) and sigma (x,y)



60

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 rb,rb*,*

CL=1-(pt)

P(data|pt)

Integral

.data

D

.zPt

Example in 1D



61

Cartesian coordinates: toy MC

x-

y-

x+

y+ Linear correspondence and errors are well behaving.

Fitted parameters vs Generated parameters.



62

Confidence Regions (CR):D0K-D*0K combination

Integration domain D (CR)

CL=1-(pt)



63

D0K-D*0K confidence intervalsSingle channel (D0K o D*0K)

D0K-D*0K combination

Central values are the mean of CL the interval



64

BD*K decays help in constraining

0*)*(*0**~ DerDD iB

)(

)(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

international school of subnuclear physics erice, 30 aug - 6 sept 2006

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s13 ads13

ad s13

babar model

body decays

babar data

s13 rb eig db ads13

eig db ads12

d0 dalitz analysis