1. Fourier Studies: Looking at Data A. Cerri

2. 2 Outline Introduction Data Sample Toy Montecarlo –Expected Sensitivity –Expected Resolution Frequency Scans: –Fourier –Amplitude Significance –Amplitude Scan –Likelihood Profile Conclusions

3. 3 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 D s K 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

4. 4 Data Sample Full 1fb -1 D s , 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

5. 5 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

6. 6 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

7. 7 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 L ratio : improved separation and localized peak distribution for background

8. 8 Toy Montecarlo: confidence bands Signal or background depth of deepest minimum in toys Tail integral of distribution gives detection & false alarm probabilities

9. 9 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

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

11. 11 Data: Fourier and Amplitude

12. 12 Compare with standard A-scan

13. 13 Data: Where we look for a Peak Automated code looks for –log(L ratio ) minimum Depth of minimum compared to toy MC distributions gives signal/background probabilities Background Signal

14. 14 Data Results Peak in L ratio is: -2.84 (A/  =2.53) –Detection (signal) probability: 53% –False Alarm (background fake) probability: 25% Likelihood profile:

15. 15 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

16. Backup

17. 17 Tool Structure Bootstrap Toy MC Ct Histograms Configuration Parameters Signal (  m s, ,  ct,D tag,  tag,K factor ), Background (S/B,A,D tag,  tag, f prompt,  ct,  prompt,  longliv,),  curves (4x[f i  (t-b)  (t-b) 2  e -t/ ]), Functions: (Re,Im)  (+,-,0,  tags)  (S,B) Ascii Flat File (ct,  ct, D exp, tag dec., K factor ) Data Fourier Transform Amplitude Scan Re(~  [  m s = ])( ) Same ingredients as standard L-based A-scan Consistent framework for: Data analysis Toy MC generation/Analysis Bootstrap Studies Construction of CL bands

18. Validation: Toy MC Models “Fitter” Response

19. 19 Ingredients in Fourier space Resolution Curve (e.g. single gaussian) Ct efficiency curve, random example Ct (ps)  m (ps -1 )

20. 20 Toy Data Toy Montecarlo As realistic as it can get: –Use histogrammed  ct, D tag, K factor –Fully parameterized  curves –Signal:  m, ,  –Background: Prompt+long-lived Separate resolutions Independent  curves Toy Data Data+Toy Realistic MC+Toy Ct (ps)

21. 21 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 ) Realistic MC+Wrong Model Ct (ps)

22. 22 “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

23. 23 “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  m s 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 )

24. 24 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)

25. 25 From 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 ) Fourier Transform+Error+Normalization

26. 26 Toy MC Same configuration as D s (  )  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 ) Fourier Transform+Error+Normalization

27. Confidence Bands

28. 28 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

29. 29 “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

30. 30 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 L ratio : improved separation and localized peak distribution for background

31. 31 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

32. 32 Integral Distributions of Maxima heights Linear scale Logarith. scale

33. Determining the Peak Position

34. 34 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:  m s =17 ps -1

35. 35 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…

36. 36 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

37. 37 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)