Measurement of the ckm angle g with a d 0 dalitz analysis of the b d k decays at babar
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Measurement of the CKM angle g with a D 0 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. ( r , h ). a. *. *. *. V td V tb. |V cd V cb |. V ud V ub. *. |V cd V cb |. g. b.

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Measurement of the ckm angle g with a d 0 dalitz analysis of the b d k decays at babar

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


International school of subnuclear physics erice 30 aug 6 sept 2006

(r,h)

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


Towards g

f

f

Towards g

bc transition

bu 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


Three body d decays dalitz plot

(MD0-mp)2

(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


International school of subnuclear physics erice 30 aug 6 sept 2006

g from the

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

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


International school of subnuclear physics erice 30 aug 6 sept 2006

K*(892)

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


International school of subnuclear physics erice 30 aug 6 sept 2006

The BaBar Isobar Model

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

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

Signal DpBBqq

Yields on DATA

347 million of BB pairs at (4S)

D0K

D*0K D*0D0

D*0K D*0D0g

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

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

CP parameters extraction

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

x= Re[rBexpi(dg))]= rBcos(dg) , y= Im[rBexpi(dg)]= rBsin(dg)

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


Cartesian coordinate results

1s

2s

Cartesian coordinate results

D0K

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 systematic errors

>>

Experimental systematics

Dalitz model systematics

Statistical error

Nicola Neri - International School of Subnuclear Physics


Frequentist interpretation of the results

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


Considerations on the results

()

rB

Considerations on the results

y

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

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

Back-up slides

Nicola Neri - International School of Subnuclear Physics


International school of subnuclear physics erice 30 aug 6 sept 2006

Dalitz model systematics

  • 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 BJ/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

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

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

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 d 0

    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 d 0 and flavor charge correlation

    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

    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


    International school of subnuclear physics erice 30 aug 6 sept 2006

    CP-violating phase

    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


    Unitarity triangle

    *

    |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


    Cp violation in decay or direct cp violation

    A

    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

    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


    International school of subnuclear physics erice 30 aug 6 sept 2006

    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 ± 9146 ± 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


    International school of subnuclear physics erice 30 aug 6 sept 2006

    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

    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

    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 YesYes

    |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

    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


    D 0 g k d 0 p 0 k cross feed

    (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

    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


    International school of subnuclear physics erice 30 aug 6 sept 2006

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

    Perfect!

    Nicola Neri - International School of Subnuclear Physics


    International school of subnuclear physics erice 30 aug 6 sept 2006

    Data Dsppp

    Data

    Zemach Tensor vs Helicity model

    Affects seriously on spin 2

    Monte Carlo simulation using f2(1270)pp

    MC

    MC

    In Dsp-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

    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 BJ/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

    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


    Sensitivity to g

    CA D0 K*(892)- p+

    s13 (GeV2)

    weight =

    DCS D0 K0*(1430)+p-

    DCS D0 K*(892)+p-

    D0 Ksr

    s12 (GeV2)

    Sensitivity tog

    points : 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

    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

    Signal DpBBqq

    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*0D0g

    D*0K - D*0D0

    Nicola Neri - International School of Subnuclear Physics


    International school of subnuclear physics erice 30 aug 6 sept 2006

    Efficiency Map for D*→D0p

    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

    Isobar model formalism

    As an example a D0 three-body decay D0ABC 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


    International school of subnuclear physics erice 30 aug 6 sept 2006

    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 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 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


    International school of subnuclear physics erice 30 aug 6 sept 2006

    The CLEO model

    CLEO model 10 resonances + 1 Non Resonant term.

    Nicola Neri - International School of Subnuclear Physics


    International school of subnuclear physics erice 30 aug 6 sept 2006

    The CLEO model

    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


    International school of subnuclear physics erice 30 aug 6 sept 2006

    The BELLE model

    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


    International school of subnuclear physics erice 30 aug 6 sept 2006

    BELLE Data with

    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


    International school of subnuclear physics erice 30 aug 6 sept 2006

    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.

    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


    International school of subnuclear physics erice 30 aug 6 sept 2006

    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.

    Nicola Neri - International School of Subnuclear Physics


    International school of subnuclear physics erice 30 aug 6 sept 2006

    K*(892)

    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

    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


    International school of subnuclear physics erice 30 aug 6 sept 2006

    The BaBar K-Matrix Model

    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

    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

    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(dg))]= rBcos(dg) , y= Im[rBexpi(dg)]= rBsin(dg)

    Nicola Neri - International School of Subnuclear Physics


    The frequentist pdf

    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


    Confidence regions

    Integral a

    P(data|pt)

    D

    .z

    Pt

    .data

    Confidence Regions

    Example 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


    Cartesian coordinates toy mc

    y+

    y-

    x-

    x+

    Cartesian coordinates: toy MC

    Linear 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

    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

    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


    B d k decays help in constraining g

    BD*K decays help in constraining g

    As pointed out in Phys.Rev.D70 091503 (2004) for the BD*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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