Fourier Studies:Looking at Data

A. Cerri

2

Outline

• Introduction• Data Sample• Toy Montecarlo

– Expected Sensitivity– Expected Resolution

• Frequency Scans:– Fourier– Amplitude Significance– Amplitude Scan– Likelihood Profile

• Conclusions

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Introduction• Principles of Fourier based method presented on

12/6/2005, 12/16/2005, 1/31/2006, 3/21/2006• Methods documented in CDF7962 & CDF8054• Full implementation described on 7/18/2006 at BLM• Aims:

– settle on a completely fourier-transform based procedure– Provide a tool for possible analyses, e.g.:

• J/ direct CP terms• DsK direct CP terms

– Perform the complete exercise on the main mode ()– All you will see is restricted to . Focusing on this mode alone

for the time being• Not our Aim: bless a mixing result on the full sample

4Data Sample• Full 1fb-1

• Ds, main Bs peak only

• ~1400 events in [5.33,5.41] consistent with baseline analysis

• S/B ~ 8:1• Background modeled

from [5.7,6.4]• Efficiency curve

measured on MC• Taggers modeled after

winter ’05 (cut based) + OSKT

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Toy Montecarlo

• Exercise the whole procedure on a realistic case (see BML 7/18)

• Toy simulation configured to emulate sample from previous page

• Access to MC truth:– Study of pulls (see BML 7/18)– Projected sensitivity– Construction of confidence bands to measure

false alarm/detection probability– Projected m resolution

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Toy Montecarlo: sensitivity

• Rem: Golden sample only

• Reduced sensitivity, but in line with what expected for the statistics

• All this obtained without t-dependend fit

• Iterating we can build confidence bands

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Distribution of Maxima• Run toy montecarlo several times

– “Signal”default toy– “Background”toy with scrambled taggers

• Apply peak-fitting machinery• Derive distribution of maxima (position,height)

Max A/: limited separation and uniform peak distribution for background, but not model (&tagger parameter.) dependent

Min log Lratio: improved separation and localized peak distribution for background

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Toy Montecarlo: confidence bandsSignal or background depth of deepest minimum in toys

•Tail integral of distribution gives detection & false alarm probabilities

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Toy Montecarlo: m resolutionTwo approaches:

•Fit pulls distributions and measure width

•Fit two parabolic branches to L minimum in a toy by toy basis

Negative Error

Positive Error

RMS~0.5

DataAll the plots you are going to see are

based on Fourier transform & toy montecarlo distributions, unless

explicitely mentioned

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Data: Fourier and Amplitude

53.2~A

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Compare with standard A-scan

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Data: Where we look for a Peak•Automated code looks for –log(Lratio) minimum

•Depth of minimum compared to toy MC distributions gives signal/background probabilities

Background

Signal

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Data Results

• Peak in L ratio is: -2.84 (A/=2.53)– Detection (signal) probability: 53%– False Alarm (background fake) probability: 25%

• Likelihood profile:

141.055.023.17

psms

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Conclusions

• Worked the exercise all the way through• Method:

– Assessed– Viable– Power equivalent to standard technique

• Completely independent set of tools/code from standard analysis, consistent with it!

• Tool is ready and mature for full blown study• Next: document and bless result as proof-of-

principle

Backup

17Tool Structure

BootstrapToy MC

Ct Histograms

Configuration Parameters

Signal

(ms,,ct,Dtag,tag,Kfactor),

Background

(S/B,A,Dtag,tag, fprompt, ct, prompt, longliv,),

curves (4x[fi(t-b)(t-b)2e-t/]),

Functions:

(Re,Im) (+,-,0, tags)(S,B)

Ascii Flat File

(ct, ct, Dexp, tag dec., Kfactor)

Data

Fourier Transform Amplitude Scan

Re(~[ms=])()Same ingredients as standard

L-based A-scan Consistent framework for:

•Data analysis

•Toy MC generation/Analysis

•Bootstrap Studies

•Construction of CL bands

Validation:•Toy MC Models

•“Fitter” Response

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Ingredients in Fourier space

3/)1(

3

1

i

e bi

/2)()( tebxbx

2

22

1 x

e

Resolution Curve (e.g. single gaussian)

Ct efficiency curve, random example

Ct (ps)

Ct (ps)

m (ps-1)

m (ps-1) m (ps-1)

22

1 e 222 1

1

im

iD

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Toy

Data

Toy Montecarlo

• As realistic as it can get:– Use histogrammed ct,

Dtag, Kfactor

– Fully parameterized curves

– Signal:m, ,

– Background:• Prompt+long-lived• Separate resolutions• Independent curves

Toy

Data

Data+Toy

Realistic MC+Toy

Ct (ps)

Ct (ps)

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Flavor-neutral checks

Re(+)+Re(-)+Re(0) Analogous to a lifetime fit:

•Unbiased WRT mixing

•Sensitive to:

•Eff. Curve

•Resolution

Ct efficiency

Resolution

…when things go wrong

Realistic MC+Model Realistic MC+Toy

m (ps-1)

m (ps-1) Realistic MC+Wrong Model Ct (ps)

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“Lifetime Fit” on Data

Ct (ps) m (ps-1)

Data vs Toy Data vs Prediction

Comparison in ct and m spaces of data and toy MC distributions

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“Fitter” Validation“pulls”

Re(x) or =Re(+)-Re(-) predicted (value,) vs simulated.

Analogous to Likelihood based fit pulls

•Checks:

•Fitter response

•Toy MC

•Pull width/RMS vs ms shows perfect agreement

•Toy MC and Analytical models perfectly consistent

•Same reliability and consistency you get for L-based fits

Mea

nR

MS

m (ps-1)

m (ps-1)

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Unblinded Data• Cross-check against

available blessed results• No bias since it’s all

unblinded already• Using OSTags only• Red: our sample,

blessed selection• Black: blessed event list• This serves mostly as a

proof of principle to show the status of this tool!

Next plots are based on data skimmed, using the OST only in the winter blessing style. No box has been open.

M (GeV)

25From Fourier to Amplitude

•Recipe is straightforward:

1)Compute (freq)

2)Compute expected N(freq)=(freq | m=freq)

3)Obtain A= (freq)/N(freq)•No more data driven [N(freq)]•Uses all ingredients of A-scan•Still no minimization involved though!

•Here looking at Ds() only (350 pb-1, ~500 evts)

•Compatible with blessed results

m (ps-1)

m (ps-1)

Fourier Transform+Error+Normalization

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Toy MC

• Same configuration as Ds() but ~1000 events• Realistic toy of sensitivity at higher effective

statistics (more modes/taggers)

Able to run on data (ascii file) and even generate toy MC off of it

m (ps-1) m (ps-1)

Fourier Transform+Error+Normalization

Confidence Bands

28Peak Search

Two approaches:• Mostly Data driven:

use A/– Less systematic prone– Less sensitive

• Use the full information (L ratio):– More information

needed– Better sensitivity(REM here sensitivity is defined as

‘discovery potential’ rather than the formal sensitivity defined in the mixing context)

• We will follow both approaches in parallel

Minuit-based search of maxima/minima in the chosen parameter vs m

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“Toy” Study

• Based on full-fledged toy montecarlo– Same efficiency and ct as in the first toy– Higher statistics (~1500 events)– Full tagger set used to derive D distribution

• Take with a grain of salt: optimistic assumptions in the toy parameters

• The idea behind this: going all the way through with our studies before playing with data

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Distribution of Maxima• Run toy montecarlo several times

– “Signal”default toy– “Background”toy with scrambled taggers

• Apply peak-fitting machinery• Derive distribution of maxima (position,height)

Max A/: limited separation and uniform peak distribution for background, but not model (&tagger parameter.) dependent

Min log Lratio: improved separation and localized peak distribution for background

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Maxima Heights

•Separation gets better when more information is added to the “fit”

•Both methods viable “with a grain of salt”. Not advocating one over the other at this point: comparison of them in a real case will be an additional cross check

•‘False Alarm’ and ‘Discovery’ probabilities can be derived, by integration

32Integral Distributions of Maxima heights

Linear scale

Logarith. scale

Determining the Peak Position

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Measuring the Peak Position

• Two ways of evaluating the stat. uncertainty on the peak position:– Bootstrap off data sample– Generate toy MC with the

same statistics

• At some point will have to decide which one to pick as ‘baseline’ but a cross check is a good thing!

• Example: ms=17 ps-1

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Error on Peak Position• “Peak width” is our goal (ms)• Several definitions: histogram RMS, core

gaussian, positive+negative fits

• Fit strongly favors two gaussian components• No evidence for different +/- widths• The rest, is a matter of taste…

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Next Steps

• Measure accurately for the whole fb-1 the ‘fitter ingredients’:– Efficiency curves– Background shape– D and ct distributions

• Re-generate toy montecarlos and repeat above study all the way through

• Apply same study with blinded data sample• Be ready to provide result for comparison to main

analysis• Freeze and document the tool, bless as procedure

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Conclusions

• Full-fledged implementation of the Fourier “fitter”

• Accurate toy simulation• Code scrutinized and mature• The exercise has been carried all the way

through– Extensively validated– All ingredients are settled– Ready for more realistic parameters– After that look at data (blinded first)