# Presentation: 415385

## Fourier Studies:Looking at Data

- Fourier Studies:Looking at Data

- A. Cerri

## Outline

- Introduction
- Data Sample
- Toy Montecarlo
- Expected Sensitivity
- Expected Resolution
- Frequency Scans:
- Fourier
- Amplitude Significance
- Amplitude Scan
- Likelihood Profile
- Conclusions

## 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

## Data 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

## 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

## 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

## 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**

## Toy Montecarlo: confidence bands

- Signal or background depth of deepest minimum in toys
- Tail integral of distribution gives detection & false alarm probabilities

## Toy Montecarlo: m resolution

- Two 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

## Data

- All the plots you are going to see are based on Fourier transform & toy montecarlo distributions, unless explicitely mentioned

## Data: Fourier and Amplitude

## Compare with standard A-scan

## 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

## Data Results

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

## 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

## Tool Structure

- Ct Histograms

- 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

## Ingredients in Fourier space

- Resolution Curve (e.g. single gaussian)

- Ct efficiency curve, random example

- Ct (ps)

- Ct (ps)

- m (ps-1)

- m (ps-1)

- m (ps-1)

## Toy Montecarlo

- Toy

- Data

- 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)

## 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)

## “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

## “Fitter” Validation“pulls”

- “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

- Mean

- RMS

- m (ps-1)

- m (ps-1)

## 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)

## From Fourier to Amplitude

- Recipe is straightforward:
- Compute (freq)
- Compute expected N(freq)=(freq | m=freq)
- 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

## 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

## Peak 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

## “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

## 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**

## 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

## Integral Distributions of Maxima heights

- Linear scale

- Logarith. scale

## Determining the Peak Position

## 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

## 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...

## 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

## 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)