Model independent determination of γ from B ± →D(K 0 S π + π − )K ±

30th October 200730th October 2007 LHCb Tuesday MeetingLHCb Tuesday Meeting 11

Model independent Model independent determination of determination of γγ from from BB±±→D(K→D(K00

SSππ++ππ−−)K)K±±

Jim Libby (University of Jim Libby (University of Oxford)Oxford)


OutlineOutline

Current Current ee++ee−− B-factory B-factory and LHCb statusand LHCb status– Why use a model independent method?Why use a model independent method?

Explanation of the model independent Explanation of the model independent methodmethod– Importance of CLEOcImportance of CLEOc

Implementation to LHCb environment Implementation to LHCb environment and resultsand results– BackgroundBackground– AcceptanceAcceptance– Systematic uncertaintiesSystematic uncertainties

OutlookOutlook


cbV

*usV

ubV

*csV

Decays of Decays of DD00 or or DD00 to common final state gives sensitivity to to common final state gives sensitivity to γγ For For BB++→D(K→D(K00ππ++ππ−−)K)K++

Assume isobar model (sum of Breit-Wigners)Assume isobar model (sum of Breit-Wigners)

Fit Fit DD-Dalitz plots from -Dalitz plots from BB-decay to extract-decay to extract γγ, r, rB B and and δδBB

0D)/GeV(

2

2

c

m

BB±±→D(K→D(K00SSππ++ππ−−)K)K±±

BAKDBA )( 0 )(0)( i

BB erAKDBA

),(),(

),(),(22)(22

22)(22

mmfermmfA

mmfermmfAi

B

iB

amplitudes Dalitz ),( and massinvariant 220mmfKm S

iN

jj

ij bemmAeammf j

1

2222 ,,

)(GeV/ 22 cm

Vub e -iγ

K*(892)

(770)


Current resultsCurrent results Current best direct constraints on Current best direct constraints on γγ::

These measurements are theoretically cleanThese measurements are theoretically clean– No penguin No penguin CKM standard candle CKM standard candle – D-mixing ignored in formalism but < 1D-mixing ignored in formalism but < 1° ° correctioncorrection

However, large error from isobar model However, large error from isobar model assumptionsassumptions

BABAR and Belle use large samples of flavour BABAR and Belle use large samples of flavour tagged tagged D*D*++DD00ππ++ events to find parameters of the events to find parameters of the isobar modelisobar model– Excellent knowledge of Excellent knowledge of ||ff||22 but phases less well known but phases less well known

Model uncertainties from assumptions about the Model uncertainties from assumptions about the resonance structures in the modelresonance structures in the model

[BABAR] model))(12)syst(11)stat(4192(

[Belle] model))(9)syst(3)stat(53( 15183


K*0(1430)

Isobar model Isobar model uncertaintyuncertainty

Most challenging Most challenging aspects of the aspects of the model model uncertainty come uncertainty come from Kfrom Kππ and and ππππ S-S-wave wave

=rB sin(δ+γ)

BABARCKM06

BABAR (PRL 95 121802,2005)

Fit to flavour tag sample


Aside: K-matrix Aside: K-matrix Breit Wigner description of broad Breit Wigner description of broad

overlapping resonances overlapping resonances violates violates unitarity and requires non-unitarity and requires non-physical physical σσ

K-matrix description preserves K-matrix description preserves unitarityunitarity

First studies (Lauren Martin/JL) of First studies (Lauren Martin/JL) of LHCb LHCb γγ fit with one K-matrix fit with one K-matrix parameterisation of the parameterisation of the ππππ S-wave S-wave – Difference between assuming K-Difference between assuming K-

matrix and BW model consistent matrix and BW model consistent with B-factory observationswith B-factory observations

– Note with EB – draft available Note with EB – draft available from CPWG webpagefrom CPWG webpage

Explore different physical K-Explore different physical K-matrix parameterisation to matrix parameterisation to evaluate systematic rather than evaluate systematic rather than introduce introduce σσ will reduce model will reduce model uncertaintyuncertainty

5 pole K-matrix


Model uncertainty impact Model uncertainty impact on LHCb on LHCb Recent studies for LHCb (LHCb-048-2007) showed Recent studies for LHCb (LHCb-048-2007) showed

that the model-dependent fit would yield an that the model-dependent fit would yield an uncertainty onuncertainty on γγ between between 7-12° 7-12° for anfor an r rBB=0.1=0.1– Range represents differing assumptions about the backgroundRange represents differing assumptions about the background

However, the best current model uncertainty is 10-However, the best current model uncertainty is 10-15° with an 15° with an rrBB=0.1=0.1– Uncertainties Uncertainties 1/1/rrBB

Without improvements LHCb sensitivity will Without improvements LHCb sensitivity will be dominatedbe dominated by model assumptions within 1 by model assumptions within 1 year of data takingyear of data taking

Motivates a model-independent method that Motivates a model-independent method that relies on a binned analysis of the Dalitz plotrelies on a binned analysis of the Dalitz plot– Disadvantage is that information is lost via binningDisadvantage is that information is lost via binning

Rest of this talk will discuss this method and Rest of this talk will discuss this method and its implementation at LHCbits implementation at LHCb


Average cosine and sine of strong

phase difference between D0 and

D0 decay amplitudes (ΔδD) in this bin

Binned methodBinned method Proposed in the original paper by Giri, Grossman, Soffer and Proposed in the original paper by Giri, Grossman, Soffer and

Zupan and since been extended significantly by Bondar and Zupan and since been extended significantly by Bondar and PoluektovPoluektov– GGSZ, PRD GGSZ, PRD 6868, 054018 (2003), 054018 (2003)– BP, Eur. Phys. J. C47, 347 (2006)+hep-ph/0703267BP, Eur. Phys. J. C47, 347 (2006)+hep-ph/0703267

Bin the Dalitz plot symmetrically Bin the Dalitz plot symmetrically about mabout m−−

22= m= m++2 2 then number of entries in Bthen number of entries in B−−

decay given by: decay given by:

)(),(),(2

),(),(

222222

2222222

iiDD

DBDi

sycxdDmmfdDmmf

dDmmfrdDmmfN

ii

ii

# events in bin of flavour tagged D0 decays

s'coordinateCartesian '

)sin()cos( BBBB ryrx


Binned method Binned method continuedcontinued

Can determine Can determine ssii and and cci i at at the same time as extracting the same time as extracting γγ, , rrB B and and δδB B from from B B datadata– 3 + 3 + NNbins bins free parameters (cfree parameters (ci i =c=c-i -i

and sand si i =−s=−s-i-i) and 2 N) and 2 Nbinsbins

– Huge loss in Huge loss in γγ sensitivity not sensitivity not practical until you have O(10practical until you have O(1066) ) events (2500/fbevents (2500/fb-1-1 @ LHCb) @ LHCb)

However, However, CP-correlated CP-correlated ee++ee−−→ → ψ″→ψ″→DD00DD00 datadata where where oneone decay is todecay is to K KSSππππ and the and the other decays to a CP other decays to a CP eigenstate oreigenstate or K KSSππππ allowsallows ssii and and cci i to be determinedto be determined

0D 0D

0even-CPD 0

odd-CPD


CLEO-c: double tagged CLEO-c: double tagged

ψψ(3770) events(3770) events

0000

21 DDDDee

CLEO-c has collected ~ 800 fb-1 at the ψ (3770)

DDbar produced in quantum entangled state:

Reconstruct one D in decay of interest for γ analysis (eg. Kππ), & other in CP eigenstate (eg. KK, Ksπ0 …) then CP of other is fixed.

Almost background free

Can use KL

From talk by E. Whiteat Charm 07


CLEO-c measurement CLEO-c measurement statusstatus

Studies not complete but projected uncertainties

on c and s will lead to 3-4 degree uncertainty on γ

1/3 of total data


Inkblot testInkblot test Bondar and Poluektov Bondar and Poluektov

show that the rectangular show that the rectangular binning is far from optimal binning is far from optimal for both CLEOc and for both CLEOc and γγ analysesanalyses– 16 uniform bins has 16 uniform bins has

only 60% of the B only 60% of the B statistical sensitivitystatistical sensitivity

– c and s errors would be c and s errors would be 3 times larger from the 3 times larger from the ψ″ψ″

Best B-data sensitivity Best B-data sensitivity when cos(when cos(ΔδΔδDD)) and and sin(sin(ΔδΔδDD)) are as uniform are as uniform as possibleas possible within a bin within a bin

Absolute value of strong phase diff.(BABAR model used in LHCb-48-2007)

Good approximation and the binning that yields smallest s and c errors is equal

ΔδD bins-80% of the unbinned precisionNimmNi D /)(2),(/)(2 2

12221


Implementation at Implementation at LHCbLHCb Generate samples of Generate samples of

BB±±→D(K→D(K00SSππππ)K)K±± with a mean of with a mean of

5000 events split between 5000 events split between the chargesthe charges

Bin according to strong phase Bin according to strong phase difference, difference, ΔδΔδDD

Minimise Minimise χχ22

(γ=60°, rB=0.1 and δB=130°)

data] gflavour ta from [measured ),(

factorion normalizat

2),,(

bin in events )( ofnumber

)),,(()),,((

222

2

th0

8

)0(8

222

dDmmfK

h

ysxcKKKrKhhyxN

iKKDBn

n

hyxNn

n

hyxNn

iDi

iiiiiBii

Si

ii i

ii

i

ii

Ki, ci and si amplitudes calculated from model

In reality from flavour tagged samples and CLEO-c


No background with No background with predicted 2 fbpredicted 2 fb-1 -1 yieldyield

5000 experiments

Input parametersγ=60°, rB=0.1 and δB=130°

The four Cartesian coordinates and normalization are free parameters

All pulls are normal therefore calculate γ, rB and δB with propagated Cartesian uncertainties


No background with No background with predicted 2 fbpredicted 2 fb-1 -1 yieldyield

Model independent average uncertainty 7.7° (c.f. Model dependent 5.9°)


AcceptanceAcceptance Acceptance in each bin calculated as a weighted average of Acceptance in each bin calculated as a weighted average of

the acceptance function used for model dependent studiesthe acceptance function used for model dependent studies– 15% relative difference amongst bins15% relative difference amongst bins

Modifies the fit function:Modifies the fit function:

Average Average γγ uuncertainty increases to 8.1ncertainty increases to 8.1°°

))(08.01(1028.0),( where),(),(

2),,(

22322

22222

2

mmmm

K

dDmmmmfa

ysxcKKKrKhahyxN

i

Di

iiiiiBiii

i

Can be calculatedfrom Dπ


Background Background 3 types of background to consider 3 types of background to consider

– BB→D(K→D(KSSππππ))ππ (DC04 B/S = 0.24)(DC04 B/S = 0.24) rrBB(D(Dππ)) O(10 O(10-3-3) ) so Dalitz plots are likeso Dalitz plots are like DD00 andand DD00 forfor BB−− andand BB++,, respectively respectively

– Combinatoric (DC04 B/S<0.7)Combinatoric (DC04 B/S<0.7)Admixtures of two types consideredAdmixtures of two types considered

1.1.DKDKcombcomb: real : real D→ D→ D(KD(KSSππππ)) combined with a bachelor combined with a bachelor KKDalitz plot a even sum of Dalitz plot a even sum of DD00 andand DD00 decaysdecays

2.2.PSPScombcomb:: combinatoric combinatoric DD with a bachelor with a bachelor KKFollows phase spaceFollows phase space

Integrate background PDFs used in model-dependent Integrate background PDFs used in model-dependent analysis over each bin, then scaled to background level analysis over each bin, then scaled to background level assumed:assumed:

iicomb

iiiiicomb

iii

PPSN

KaKaDKN

KaDN

)(

)()(

)(

21

fractional area of Dalitz space covered by bin


γγ uncertainties with 5000 uncertainties with 5000 toy experimentstoy experiments

ScenarioScenario 2 fb2 fb-1 -1 Mod. Indep.Mod. Indep. 10 fb10 fb-1 -1 Mod. Indep.Mod. Indep. 2 fb2 fb-1 -1 Mod. Dep. Mod. Dep. (LHCb-048-2007)(LHCb-048-2007)

No backgroundNo background 7.97.9°° 3.53.5°° 5.95.9°°

AcceptanceAcceptance 8.18.1°° 3.53.5°° 5.55.5°°

DDππ (B/S = 0.24) (B/S = 0.24) 8.88.8°° 4.04.0°° 7.37.3°°

DKDKcomb comb (B/S=0.7)(B/S=0.7) 12.812.8°° 5.75.7°° 11.711.7°°

PSPScomb comb (B/S=0.7)(B/S=0.7) 12.812.8°° 5.55.5°° 9.19.1°°

DKDKcomb comb (B/S=0.35)(B/S=0.35)

+PS+PScomb comb (B/S=0.35)(B/S=0.35)12.712.7°° 5.45.4°° 9.89.8°°


Systematic related to Systematic related to acceptanceacceptance

The acceptance varies over the Dalitz planeThe acceptance varies over the Dalitz plane The relative acceptance in each bin can be measured using the The relative acceptance in each bin can be measured using the

BB→D→Dππ control sample with DK selection applied without bachelor K control sample with DK selection applied without bachelor K PID PID

With the DC04 selection expect 60k events/2 fbWith the DC04 selection expect 60k events/2 fb -1-1

– Relative relative-efficiency uncertainty 1-4%/Relative relative-efficiency uncertainty 1-4%/ΔδΔδD D bin with 2 fbbin with 2 fb-1-1

– Increased statistics reduces errorIncreased statistics reduces error Toy MC study smearing bin efficiencies in event generation by this Toy MC study smearing bin efficiencies in event generation by this

amount leads to an additional 1amount leads to an additional 1° uncertainty without background ° uncertainty without background and and 3.2° uncertainty3.2° uncertainty with DK with DKcombcomb B/S=0.7 B/S=0.7– Small effect compared to statistical uncertaintySmall effect compared to statistical uncertainty

NB: the efficiency related to the PID of the bachelor NB: the efficiency related to the PID of the bachelor ππ/K can be /K can be factored out and will be determined from the Dfactored out and will be determined from the D**→D(K→D(Kππ))ππ data to data to better than one percent-ignore at presentbetter than one percent-ignore at present

))D(KN(B),(),( 0

S

22222

i

i

i

Di KK

dDmmmmfa i


Asymmetry in efficiency Asymmetry in efficiency in Dalitz spacein Dalitz space

Considered charge asymmetries in the Considered charge asymmetries in the efficiency across the Dalitz plane efficiency across the Dalitz plane – εε(m(m22

++, m, m2 2 −−)≠)≠εε(m(m22

−−, m, m2 2 ++))

Generated with the efficiency biased Generated with the efficiency biased relative to one another depending on relative to one another depending on whether the event hadwhether the event had m m22

++>m>m2 2 − − oror m m22

++<m<m2 2 −−

Maximum bias onMaximum bias on γγ induced wasinduced was <1° <1° forfor 10% 10% relative effect and full relative effect and full backgroundbackground

10% effects would be evident in the D10% effects would be evident in the Dππ samplesample


ResolutionResolution ΔδΔδD D binning has some narrow binning has some narrow

regions in Dalitz spaceregions in Dalitz space Investigation of how resolution on Investigation of how resolution on

the Dalitz variables might affected the Dalitz variables might affected the extraction of the extraction of γγ

10 MeV10 MeV22/c/c4 4 resolution (DC04) on resolution (DC04) on Dalitz variables and generated toy Dalitz variables and generated toy experiments with this smearingexperiments with this smearing

Found that this led to a few bins Found that this led to a few bins with largest (red) and smallest with largest (red) and smallest (dark blue) phase difference (dark blue) phase difference having a 2-3% relative changes in having a 2-3% relative changes in expected yields due to resolution expected yields due to resolution induced migrationinduced migration

Fit results on toy experiments Fit results on toy experiments where resolution included in where resolution included in generation but ignored in fit generation but ignored in fit found no significant bias found no significant bias (<0.5(<0.5°) on °) on γγ


Background fractionsBackground fractions Combinatoric background rate will be Combinatoric background rate will be

determined from determined from B B and and DD mass sidebands which mass sidebands which will cover at least 2-3 times the area of the will cover at least 2-3 times the area of the signal regionsignal region– Use 10Use 10×× in DC04 background studies but this will in DC04 background studies but this will

probably be unrealistic with dataprobably be unrealistic with data If background distributions relatively flat in If background distributions relatively flat in

masses one can estimate that this leads to B/S masses one can estimate that this leads to B/S will be determined absolutely to around 0.01 or will be determined absolutely to around 0.01 or betterbetter

Toy studies suggest that there is no impact Toy studies suggest that there is no impact on on γγ precision with this kind of uncertaintyprecision with this kind of uncertainty

Maybe complications depending on Dalitz space Maybe complications depending on Dalitz space distribution of the PS background but can only distribution of the PS background but can only speculate until we have the data in handspeculate until we have the data in hand


ConclusionConclusion Implemented model independent fit Implemented model independent fit

with binning that yields smallest error with binning that yields smallest error from exploiting CLEO-c data from exploiting CLEO-c data– Binning depends on model - only Binning depends on model - only

consequence of incorrect model is consequence of incorrect model is non-optimal binning non-optimal binning

Such a measurement will be better than model dependent Such a measurement will be better than model dependent method if we cannot improve the model errormethod if we cannot improve the model error– 10 fb10 fb-1 -1 statistical uncertainty 4-6statistical uncertainty 4-6°° depending on backgrounddepending on background

Experimental systematic uncertainties consideredExperimental systematic uncertainties considered– can be controlled from datacan be controlled from data– acceptance determined from acceptance determined from BB→D→Dππ sample contributes ~3° with 2 fb sample contributes ~3° with 2 fb-1-1

Liaison required with CLEO-c to get common binning and to Liaison required with CLEO-c to get common binning and to best understand all uncertainties related to the best understand all uncertainties related to the ψ″ψ″ datadata

Note with EB since beginning of October Note with EB since beginning of October – Available from CPWG webpageAvailable from CPWG webpage

Model independentModel dependent

σ(model)=10°

σ(model)=5°


Additional slidesAdditional slides


Toy experiment results: Toy experiment results: γγ (2 fb (2 fb−1−1))

ScenarioScenario MeanMean RMSRMS Mean Mean σσ Mean pullMean pull Pull RMSPull RMS

No bckNo bck 60.560.5±0.1±0.1 7.97.9 7.87.8 0.0450.045±0.015±0.015 1.051.05

AccAcc 60.760.7±0.1±0.1 8.18.1 7.87.8 0.0750.075±0.015±0.015 1.071.07

DDππ 60.760.7±0.1±0.1 8.88.8 8.88.8 0.0880.088±0.015±0.015 1.041.04

DDππ + + DK DK (B/S=0.7)(B/S=0.7)

60.760.7±0.2±0.2 12.812.8 12.212.2 0.0490.049±0.016±0.016 1.111.11

DDππ + + PS PS (B/S=0.7)(B/S=0.7)

60.860.8±0.2±0.2 12.812.8 12.512.5 0.0640.064±0.015±0.015 1.051.05

DDππ + + DK+ DK+ PS (50:50) PS (50:50) (B/S=0.7)(B/S=0.7)

60.760.7±0.2±0.2 12.712.7 12.612.6 0.0490.049±0.015±0.015 1.041.04


Toy experiment results: Toy experiment results: γγ (10 fb (10 fb−1−1))


No bckNo bck 60.1760.17±0.05±0.05 3.53.5 3.43.4 0.0500.050±0.015±0.015 1.031.03

AccAcc 60.1360.13±0.05±0.05 3.53.5 3.43.4 0.0360.036±0.015±0.015 1.011.01

DDππ 60.2260.22±0.06±0.06 4.04.0 3.93.9 0.0540.054±0.015±0.015 1.031.03

DDππ + + DK DK (B/S=0.7)(B/S=0.7)

60.1860.18±0.08±0.08 5.75.7 5.75.7 0.0300.030±0.015±0.015 1.011.01

DDππ + + PS PS (B/S=0.7)(B/S=0.7)

60.2660.26±0.08±0.08 5.55.5 5.55.5 0.0450.045±0.015±0.015 1.001.00


60.2260.22±0.08±0.08 5.45.4 5.65.6 0.0380.038±0.015±0.015 0.970.97


Toy experiment results: rToy experiment results: rBB (2 fb (2 fb−1−1))


No bckNo bck 0.10170.1017±0.0002±0.0002 0.0130.013 0.0130.013 0.1430.143±0.015±0.015 1.021.02

AccAcc 0.10170.1017±0.0002±0.0002 0.0140.014 0.0130.013 0.1750.175±0.016±0.016 1.131.13

DDππ 0.10150.1015±0.0002±0.0002 0.0140.014 0.0140.014 0.123±0.0150.123±0.015 1.021.02

DDππ + + DK DK (B/S=0.7)(B/S=0.7)

0.10310.1031±0.0003±0.0003 0.0200.020 0.0200.020 0.2150.215±0.016±0.016 1.161.16

DDππ + + PS PS (B/S=0.7)(B/S=0.7)

0.10350.1035±0.0003±0.0003 0.0200.020 0.0190.019 0.1750.175±0.015±0.015 0.990.99


0.10380.1038±0.0003±0.0003 0.0200.020 0.0200.020 0.1860.186±0.015±0.015 0.980.98


Toy experiment results: rToy experiment results: rBB (10 fb (10 fb−1−1))


No bckNo bck 0.10030.1003±0.0001±0.0001 0.0060.006 0.0060.006 0.0560.056±0.015±0.015 1.001.00

AccAcc 0.10030.1003±0.0001±0.0001 0.0060.006 0.0060.006 0.0510.051±0.015±0.015 1.011.01

DDππ 0.10030.1003±0.0001±0.0001 0.0060.006 0.0060.006 0.049±0.0150.049±0.015 0.980.98

DDππ + + DK DK (B/S=0.7)(B/S=0.7)

0.10090.1009±0.0001±0.0001 0.0090.009 0.0090.009 0.1010.101±0.015±0.015 0.970.97

DDππ + + PS PS (B/S=0.7)(B/S=0.7)

0.10080.1008±0.0001±0.0001 0.0090.009 0.0090.009 0.0930.093±0.015±0.015 0.990.99


0.10070.1007±0.0001±0.0001 0.0090.009 0.0090.009 0.0770.077±0.015±0.015 0.980.98

Model independent determination of γ from B ± →D(K 0 S π + π − )K ±

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Transcript of Model independent determination of γ from B ± →D(K 0 S π + π − )K ±