# Presentation: 828893

## Bs Mixing Analysis with the Fourier Method

- Alex, Donatella, Franco, Giuseppe, Marco, Sandro, Stefano

## Contents

- Method
- Quick basics
- Improvements and clean-up
- Connection with the amplitude scan
- (Few) toy examples
- Analysis plans
- Conclusions and outlook

## The Method

- We are looking for a periodic signal: Fourier space is the natural tool
- Even Moser and Roussarie mention this!
- Amplitude approach is seen as an approximation of the Fourier transform
- Amplitude from scan  Re[Fourier]
- Why not go for the real thing?
- Computationally lighter
- As powerful as A-scan
- As is, no need *in principle* for measurements of D,  etc. (however these ingredients add information and tighten the limit)

## From the last episode...

- Re[fourier]
- contains pretty much all the information
- More convenient tool (linearity) than magnitude
- Re[F] and Re[F] can be computed directly from data!
- Sensitivity is qualitatively analogous to the amplitude scan (same dependencies)

## News

- We are improving our toy simulation and “fitter” simultaneously and consistently:
- Ct efficiency uses now same parameterization as A-scan analysis
- Scale factor parameterized at our best (two components ~1 and ~1.7)
- Homogeneously accommodate various samples with different properties (semileptonic vs hadronic, partially reconstructed vs fully reconstructed, etc.)
- Next logical step: compare with the amplitude fit!
- How? Aren’t we looking at (slightly?) different quantities?

## Amplitude from Fourier

- Most A-scan properties derived from this connection (Moser)
- With the proper normalization Fourier gives A-scan plot!
- This additional step requires the same ingredients (D, etc.) of the traditional A-scan approach

## “Fitter” pulls

- Same procedure as for a regular fit:
- Draw N events from toy model with known ,D,S/B,m etc.
- “Measure” Re[F] and error
- Compute (Re[F]-exp.)/

## Example

- “A-scan” a` la fourier

- 1000 toy events
- ms=18
- S/B=2.
- Dsignal2=1.6%
- Dback2=0.4%
- Background and signal parameterized according to standard analyses
- ct=0.0927 (fixed, including 1.44 S.F.)

- Impressively close to our standard toy A-scans!

- Sensistivity:
-   Predicted
-   Measured

- ~1000 pts in A-scan!!!

## Example

- “A-scan” a` la fourier

- 1000 toy events
- ms=18
- S/B=2.
- Dsignal2=1.6%
- Dback2=0.4%
- Background and signal parameterized according to standard analyses
- Histogrammed ct
- Best knowledge on SF parameterization

- This method allows to flexibly study these possibilities (and systematics!) in a matter of minutes!

- Sensistivity:
-   Predicted
-   Measured

## What’s next?

- Obtain from the same toy A-scan histogram and compare point by point
- Further refine our toy model: EbE dilution etc.
- Final proof of principle:
- Process data from last round of analyses and show consistent picture with standard A-scan
- Machinery for systematics (toy + ?)
- Bd Mixing!

## Plans for our method

- Prove viability of our method:
- Full semileptonic and hadronic samples
- Same taggers and datasets as latest blessed A-scans
- Compare results to our method
- Will be ready on time for winter conferences
- Extend:
- 1fb-1
- All possible modes
- State of the art taggers
- We will have a full analysis by Summer conferences

## Backup

## Asymptotic Behavior

- Wider time distributionsnarrower in f-space
- Convolution with narrow distributions in F-space is “small correction” on top of wide ones