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### Measurement of the CKM angle gwith a D0 Dalitz analysis of the B±→D(*)K± decays at BaBar

Nicola Neri

INFN Pisa

International School of Subnuclear Physics

Erice, 30 Aug - 6 Sept 2006

a

*

*

*

Vtd Vtb

|Vcd Vcb|

Vud Vub

*

|Vcd Vcb|

g

b

(1,0)

(0,0)

CKM matrix and Unitarity Triangle

Unitarity of

quark mixing matrix

- CP violation is proportional to the triangle area

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

CP violation

- Test SM prediction
- with tree-level processes

Nicola Neri - International School of Subnuclear Physics

f

Towards gbc transition

bu transition

If same final state

interference

g measurement

A(B-D0 K-) = AB

A(B-D0 K-) = ABrB e i(dB-g)

f = KSpp (Dalitz Analysis)

f = CP (GLW)

f = DCSD (ADS)

strong phase

in B decay

CKM elements +

color suppression

Critical parameter

Theoretically and experimentally

difficult to determine.

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

Nicola Neri - International School of Subnuclear Physics

(MD0-mp)2

(mKS+mp)2

(mKS+mp)2

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

kinematical Mandelstam variables:

The A(D0 →Kspp) amplitude can be written as AD(s12, s13).

Nicola Neri - International School of Subnuclear Physics

Interference term

Using AD(s12, s13)in B decay amplitude

s12 (GeV2)

s12 (GeV2)

D0 3-body decay Dalitz distribution |AD(s12, s13) |2 (*)

s13 (GeV2)

s13 (GeV2)

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

Assuming CP is conserved in D decays

CP

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

|A(B- )|2 =| AD(s12, s13) |2 +rB2 | AD(s13, s12) |2 +

+2rBRe[AD(s12, s13) AD(s13, s12)* ei(-g+dB)]

The method suffers of a two-fold ambiguity

If rB is large, good precision on g

AD(s12, s13): fitted on

from

with

Nicola Neri - International School of Subnuclear Physics

(*) Def.

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 ADcan 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 pp resonances.

= angular dependence of the amplitude

depends on the spin Jof the resonance r

Relativistic Breit-Wigner with

mass dependent width Gr

where sij=[s12,s13,s23] depending on the resonance Ksp-,Ksp+,p+p-. mr is the mass of the resonance

Nicola Neri - International School of Subnuclear Physics

r (770)

The BaBar Isobar model

BaBar Data with

BaBar isobar model

fit over imposed.

Fit Fraction=1.20

390K sig events

97.7% purity

Good fit in DCS K*(892) region.

K*DCS

Nicola Neri - International School of Subnuclear Physics

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

Mass and widths are fixed to the PDG values.

Except for K*(1430), use E791 values and for s, s`, fit from data.

Nicola Neri - International School of Subnuclear Physics

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

(4S)

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

9.1 GeV

3.0 GeV

50% B0B0

50% B+B-

(4S)

(4S)

=1 for signal events

B-

D*0

K-

B-

D0

K-

D0 p0 ,D0 g

Ksp+ p-

p+

p-

Ksp+ p-

p+

p-

Nicola Neri - International School of Subnuclear Physics

Yields on DATA

347 million of BB pairs at (4S)

D0K

D*0K D*0D0

D*0K D*0D0g

background is >5 times the bkg contribution in each mode. Dp contribution is negligible after all the selection criteria applied in signal region unless for [Dp0]K. The error on the Dp contribution is large and can be explained as a statistical fluctuation (accounted for in systematic error)

Nicola Neri - International School of Subnuclear Physics

Dalitz distributions

D*K (D0p0)K

DK

B-

B+

B-

B+

D*K (D0g)K

Dalitz plot distribution for signal events

after all the selection criteria applied.

B-

B+

Nicola Neri - International School of Subnuclear Physics

CP parameters extraction

Fit for different CP parameters: cartesian coordinates are preferred base. Errors are gaussian and pulls are well behaving.

x= Re[rBexpi(dg))]= rBcos(dg) , y= Im[rBexpi(dg)]= rBsin(dg)

CP parameter

Result

CP parameter

Result

x-=rBcos(d-g)

y-=rBsin(d-g)

x+=rBcos(d+g)

y+=rBsin(d+g)

x*-=rBcos(d*-g)

y*-=rBsin(d*-g)

x*+=rBcos(d*+g)

y*+=rBsin(d*+g)

Main systematics

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

The statistical error dominates the measurement.

Nicola Neri - International School of Subnuclear Physics

2s

Cartesian coordinate resultsD0K

D*0K

B-

B+

B+

d

d

B-

Direct CPV

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

Nicola Neri - International School of Subnuclear Physics

Experimental systematic errors

>>

Experimental systematics

Dalitz model systematics

Statistical error

≳

Nicola Neri - International School of Subnuclear Physics

Frequentist interpretation of the results

rB

r*B

D*0K

D0K

2 s

1 s

1s (2s) excursion

s is to be understood in term of 1D proj of a L in 5D.

Nicola Neri - International School of Subnuclear Physics

Stat Syst Dalitz

rB

Considerations on the resultsy

Dx≈Dy≈rb·Dq

B-

s(g) ≈ Dx/rb

rb

Dq

2g

x

Dq

rb

Experimentally we can improve the measurement of the CP

cartesian coordinates but the improvement on error of g

depends on the true value of the rb parameter.

Similar behavior for statistical and systematic error.

B+

Nicola Neri - International School of Subnuclear Physics

Conclusions and perspectives

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

Toy MC

rb=0.1 assumed

Dalitz model error projection

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

Near Future

Nicola Neri - International School of Subnuclear Physics

Back-up slides

Nicola Neri - International School of Subnuclear Physics

- pp S-wave:
- Use K-matrix pp S-wave model instead of the nominal BW model
- pp P-wave:
- Change r(770) parameters according to PDG
- Replace Gounaris-Sakurai by regular BW
- pp and Kp D-wave
- Zemach Tensor as the Spin Factor for f2(1270) and K*2(1430) BW
- Kp S-wave:
- Allow K*0(1430) mass and width to be determined from the fit
- Use LASS parameterization with LASS parameters
- Kp P-wave:
- Use BJ/psi Ks p+ 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), r(1450)

This is a more realistic and detailed estimate of the model systematics !

Nicola Neri - International School of Subnuclear Physics

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)

Nicola Neri - International School of Subnuclear Physics

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)

Nicola Neri - International School of Subnuclear Physics

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

Nicola Neri - International School of Subnuclear Physics

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 s = 6 MeV/c2 (MC value) and m = 1864.5 MeV/c2 (PDG value).

On Monte Carlo we find the fraction for

true D0 to be:

MC continuum evt

MC BB events

MC BB + qq weighted evt

. DATA (On-Res)

we use this error for conservative systematic error evaluation

Nicola Neri - International School of Subnuclear Physics

Background characterization: true D0 and flavor-charge correlation

D0=KSpp

e-

e+

cc

D0

K- + other particles

estimated on Monte Carlo events

Nicola Neri - International School of Subnuclear Physics

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:

CP parameter

Result

CP parameter

Result

x-=rBcos(d-g)

y-=rBsin(d-g)

x+=rBcos(d+g)

y+=rBsin(d+g)

x*-=rBcos(d*-g)

y*-=rBsin(d*-g)

x*+=rBcos(d*+g)

y*+=rBsin(d*+g)

Statistical error

Experimental systematics

D0 amplitude model uncertainty

- 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:

1s (2s) excursion

Stat Syst Dalitz

s is to be understood in term of 1D proj of a L in 5D.

Nicola Neri - International School of Subnuclear Physics

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

Wolfenstein parameterization: corresponds to a

particular choice of the quark-phase convention

Nicola Neri - International School of Subnuclear Physics

|Vcd Vcb|

Unitarity triangle- CP violation is proportional to the area

CP violation

(r,h)

a

*

*

Vtd Vtb

Vud Vub

*

|Vcd Vcb|

g

b

(1,0)

(0,0)

Nicola Neri - International School of Subnuclear Physics

2

2

B

B

g

g

d

CP violation in decay or direct CP violation CP violation

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

Nicola Neri - International School of Subnuclear Physics

BaBar Detector

1.5T solenoid

DIRC (PID)

144 quartz bars

11000 PMs

EMC

6580 CsI(Tl) crystals

e+ (3.1GeV)

Drift Chamber

40 stereo layers

e- (9.0 GeV)

Silicon Vertex Tracker

5 layers, double-sided sensors

Instrumented Flux Return

iron / RPCs/LSTs (muon / neutral hadrons)

Nicola Neri - International School of Subnuclear Physics

Dalitz: BaBar vs Belle Results HFAG

Experiment Mode γ/φ3 (°) δB (°) rB

Belle‘06 N(BB)=392M

DK–D→KSπ+π–53± +15–18 ± 3 ± 9 146 ± 19 ± 3 ± 23 0.16 ± 0.05 ± 0.01 ± 0.05

D*K–D*→Dπ0D→KSπ+π–53 ± +15–18 ± 3 ± 9 302 ± 35 ± 6 ± 23 0.18 +0.11–0.10 ± 0.01 ± 0.05

BABAR\'06 N(BB)=347M

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 measurement is very important since it stresses one more time

the difficulty to measure g in a regime where the uncertainty on rb is quite large.

Nicola Neri - International School of Subnuclear Physics

BaBar vs Belle experimental results

BABAR\'06 - N(BB)=347M

CP parameter

Result

CP parameter

Result

x-=rBcos(d-g)

y-=rBsin(d-g)

x+=rBcos(d+g)

y+=rBsin(d+g)

x*-=rBcos(d*-g)

y*-=rBsin(d*-g)

x*+=rBcos(d*+g)

y*+=rBsin(d*+g)

Belle’06 - N(BB)=386M

CP parameter

Result

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

given on rb,g,d

given on rb,g,d

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

x-=rBcos(d-g)

y-=rBsin(d-g)

x+=rBcos(d+g

y+=rBsin(d+g)

x*-=rBcos(d*-g)

y*-=rBsin(d*-g)

x*+=rBcos(d*+g)

y*+=rBsin(d*+g)

Experimental measurement of the CP parameters x,y is more precise wrt Belle even

with slightly smaller statistics. Different error on g is due to Belle larger central values.

Nicola Neri - International School of Subnuclear Physics

Statistical error

Experimental systematics

D0 amplitude model uncertainty

Outline

- Theoretical framework
- D0 decay amplitude parameterization
- Selection of the D(*)K events
- CP parameters from the D0 Dalitz distribution
- Systematic errors
- Extraction of g
- Conclusions

Nicola Neri - International School of Subnuclear Physics

Selection criteria for B±D(*)K± decay modes

D0K D*0K (D0p0)D*0K (D0g)

|cos qT| <0.8 <0.8 <0.8

|mass(D0)-PDG| <12MeV <12MeV <12MeV

|mass(Ks)-PDG| <9MeV <9MeV <9MeV

E (g) ----- >30 MeV >100MeV

|mass(p0)-PDG| ----- <15MeV -----

Kaon Tight Selector Yes Yes Yes

|DM-PDG| ----- <2.5MeV <10.0MeV

cos aKs >0.99 >0.99 >0.99

|DE | <30MeV <30MeV <30MeV

---------------------------------------------------------------------------------------- efficiency 15% 7% 9% ----------------------------------------------------------------------------------------

signal events 398±23 97±1393±12

cos aKssuppress fake Ks

|cos qT| suppress jet-like events

Kaon Tight Selector (LH)and |DE|<30 MeV suppress D(*)p events

Nicola Neri - International School of Subnuclear Physics

Likelihood for Dalitz CP fit

A(B-)= |AD(s12,s13) +rBei(-g+dB)AD(s13,s12)|2

|AD(s12,s13)|2

|AD(s13,s12)|2

From MC and D0 sideband data

a=D0,D0

fSig,Dh,Cont,BB from data (extended likelihood yields)

True D0 fraction from MC and data (mES sidebands)

Charge-flavor correlation from MC

Nicola Neri - International School of Subnuclear Physics

(D0g)K – (D0p0)K cross-feed

- From Monte Carlo simulation the cross-feed between the samples is due to events of (D0p0)K where we loose a soft g and we reconstruct it as a (D0g)K.
- Since the cross-feed goes in one direction (D0p0)K (D0g)K, it is correct to assign common events to the (D0p0)K signal sample.
- After all the cuts and after this correction applied we expect <5% of signal (D0g)K from cross-feed.
- A systematic effect to the cross-feed has been assigned adding a signal component according to the (D0p0)K Dalitz PDF and performing the CP fit.
- The systematic bias of the fit with and without (D0p0)K has been quoted as systematic error.
- Negligible wrt the other systematic error sources.

Nicola Neri - International School of Subnuclear Physics

Dalitz model systematic error:K*(1430) parameters

Mass =1.412 +/- 0.006 GeV

Width = 294 +/- 23 MeV

PDG

(from LASS)

Current BaBar model for K*0(1430):

Mass =1.459+/-0.007 GeV

Width = 175 +/- 12 MeV

E791 Isobar model for K*0(1430):

Mass =1.495 +/- 0.01 GeV

Width = 183 +/- 9 MeV

BaBar Isobar model [float] for K*0(1430):

The Isobar model, the fit prefer small value of K*(1430),

both seen in E791 and BaBar, although PDG list the width 294 MeV

Mass and width are not unique parameters, depend on the parameterization and Non-Resonant model

Nicola Neri - International School of Subnuclear Physics

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

Perfect!

Nicola Neri - International School of Subnuclear Physics

Data

Zemach Tensor vs Helicity model

Affects seriously on spin 2

Monte Carlo simulation using f2(1270)pp

MC

MC

In Dsp-p+p+ the non-resonant term is much smaller (5%) in Zemach Tensor while the non-resonant term is (25%) if Helicity model is used

D wave systematics

Nicola Neri - International School of Subnuclear Physics

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) events
- If 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 p 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

Nicola Neri - International School of Subnuclear Physics

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+1s valuethat 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.

Nicola Neri - International School of Subnuclear Physics

s13 (GeV2)

weight =

DCS D0 K0*(1430)+p-

DCS D0 K*(892)+p-

D0 Ksr

s12 (GeV2)

Sensitivity togpoints : weight = 1

Strong phase variation improves the sensitivity to g.

Isobar model formalism reduces discrete ambiguities on the value of g to a two-fold ambiguity.

Nicola Neri - International School of Subnuclear Physics

Reconstruction of exclusive B±D(*)K± decays

Y(4S)

=1 for signal events

B-

K-

D*0

B-

D0

K-

Ksp+ p-

D0 p0 ,D0 g

p+

p-

Ksp+ p-

p+

p-

Nicola Neri - International School of Subnuclear Physics

Background characterization: relative fraction of signal and bkg samples

- Continuum events are the largest bkg in the analysis.

We apply a cut |cos(qT)|<0.8 and we use fisher PDF for the continuum bkg suppression.

Fisher = F [LegendreP0,LegendreP2,|cos qT|,|cos qB*|]

- The Fisher PDF helps to evaluate the relative fraction of BB and continuum events directly from DATA.

D0K

D*0K - D*0D0g

D*0K - D*0D0

Nicola Neri - International School of Subnuclear Physics

We use ~200K D* MC sample D0 Phase Space to compute the efficiency map .

2D 3rd order polynomial function used

for the efficiency map.

Purity 97.7%

Red = perfectly flat efficiency

Blu = 3rd order polynomial fit

Efficiency is almost flat

in the Dalitz plot. The fit

without eff map gives

very similar fit results.

Nicola Neri - International School of Subnuclear Physics

Isobar model formalism

As an example a D0 three-body decay D0ABC decaying through an r=[AB] resonance

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)

D0 three-body amplitude

can be fitted from DATA using a D0 flavor tagged sample from

events selecting with

- We fit for a0 ,ar amplitude values and the relative phase f0 , fr among resonances,
- constant over the Dalitz plot.

Nicola Neri - International School of Subnuclear Physics

- 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 flat

in the Dalitz plot.

CP Fit without eff map to

quote the systematic error

on CP parameters

Nicola Neri - International School of Subnuclear Physics

K-Matrix formalism for pp S-wave

- K-Matrix formalism overcomes the main limitation of the BW model to parameterize large and overlapping S-wave pp resonances. Avoid the introduction of not established s, s´ scalar resonances.
- By construction unitarity is satisfied

SS†=1 S=1+2iT

T=(1-iK·r)-1K

where S is the scattering operator

T is the transition operator

r is the phase space matrix

K-matrix D0 three-body amplitude

F1 = pp S-wave amplitude

Pj(s) = initial production vector

Nicola Neri - International School of Subnuclear Physics

CLEO model 10 resonances + 1 Non Resonant term.

Nicola Neri - International School of Subnuclear Physics

With >10x more data than CLEO, we find that the

model with 10 resonances is insufficient to describe the data.

CLEO model 10 resonances + 1 Non Resonant term.

BaBar Data refitted

using CLEO model.

Nicola Neri - International School of Subnuclear Physics

Belle model 15 resonances + 1 Non Resonant term.

Added DCS K*0,2(1430), K*(1680) and s1 , s2 respect to the

CLEO Model.With more statistics you “see” a more detailed

structure.

Added

Added

Added

Added

Added

Nicola Neri - International School of Subnuclear Physics

BELLE model fit

overimposed.

Not very good fit for DCS K*(892) region.

The BELLE model

DCS region is quite

important for the g

sensitivity. See plot

in the next pages.

Belle model 15 resonances + 1 Non Resonant term.

Added DCS K*0,2(1430), K*(1680) and s1 , s2 respect to the

CLEO Model.

Nicola Neri - International School of Subnuclear Physics

BaBar model does not include the DCS K*(1680) and DCS K*(1410)

because the number of events expected is very small.

K*(1680)

DCS 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!

Nicola Neri - International School of Subnuclear Physics

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.

Nicola Neri - International School of Subnuclear Physics

r (770)

The BaBar K-matrix model

BaBar Data with

BaBar K-matrix model

fit over imposed.

Fit Fraction=1.11

390K sig events

97.7% purity

opens KK channel

K*DCS

Nicola Neri - International School of Subnuclear Physics

K-Matrix parameterization according to Anisovich, Sarantev

V.V. Anisovitch, A.V Sarantev Eur. Phys. Jour. A16, 229 (2003)

Adler zero term to accommodate singularities

where is the coupling constant of the K-matrix pole ma to the ith channel 1=pp 2=KK 3=multi-meson 4= hh 5= hh´.

slow varying parameter of the K-matrix element.

Pj(s) = initial production vector

I.J.R. Aitchison, Nucl. Phys. A189, 417 (1972)

Nicola Neri - International School of Subnuclear Physics

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

pp S-wave term

Nicola Neri - International School of Subnuclear Physics

Overview of Dalitz CP fit strategy

Extract as much as possible information from data

- Step 1: selection PDF shapes and yields

Fit as many as possible component yields and discriminating variables (PDF) shapes from simultaneous fit to DK and Dp 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)

Nicola Neri - International School of Subnuclear Physics

The frequentist method

- Frequentist (classical) method determines a CL regions where the probability that the region will contain the true point is a.
- 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(dg))]= rBcos(dg) , y= Im[rBexpi(dg)]= rBsin(dg)

Nicola Neri - International School of Subnuclear Physics

The Frequentist PDF

Single channel (D0K o D*0K)

Measured parameters (4D): z+=(x+, y+), z-=(x-, y-)

Truth parameters (3D): pt=(rB,g, d)

Easy to include systematic error by replacing sstat→stot

G2(z;x,y,sx,sy,r) is a 2D Gaussian

with mean (x,y) and sigma (sx,sy)

D0K-D*0K combination

Measured parameters (8D): z+, z-, z*+, z*-

Truth parameters (5D): pt=(rB,g, d, r*B, d*)

Nicola Neri - International School of Subnuclear Physics

P(data|pt)

D

.z

Pt

.data

Confidence RegionsExample in 1D

CL=1-a(pt)

Integration domain D

- a(pt): calculated analytically, PDF is product of gaussians.
- We calculate 3D (5D) joint probability corresponding to 1s and 2s CL for a 3D (5D) gaussian distribution.
- Make 1D projections to quote 1s and 2s regions for rb,g,d,rb*,d*

Nicola Neri - International School of Subnuclear Physics

y-

x-

x+

Cartesian coordinates: toy MCLinear correspondence and errors are well behaving.

Fitted parameters vs Generated parameters.

Nicola Neri - International School of Subnuclear Physics

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

CL=1-a(pt)

Integration domain D (CR)

Nicola Neri - International School of Subnuclear Physics

D0K-D*0K confidence intervals

Single channel (D0K o D*0K)

D0K-D*0K combination

Central values are the mean of CL the interval

Nicola Neri - International School of Subnuclear Physics

BD*K decays help in constraining g

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:

Opposite CP

eigenstate

The effective p strong phase shift helps in the determination of g

Nicola Neri - International School of Subnuclear Physics

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