The aims for the asymmetry fitter are the following

- Perform basic checks of current fitting package with toy data for Bd and Bs asymmetries. In process of checking by Alison and will copy below once completed..
- Implement an automatic method such that when the mass region is changed, the normalisation for the pdfs is changed. DONE by Paul.
- Add proper time pdf to the fitting package. Stand alone model of pdfs done - see below.
- Alter the parameters that are being varied by the minimiser to d, θ and γ.
- Think about adding the other background pdfs (3 body, Λ's).
- Move to MC data/ real data (Bd) for first looks.

The basic pdf that was implemented in the asymmetry fitter had the following form

f(t,tmin, signal)=N*exp(-t/τ)[1+q(1-2ω)(Adir.cos(Δmt)-Amix.cos(Δmt))].

This PDF has been plotted with standard values for the Bs→KK decay for the tags of q=+1 (red) and q=-1 (blue) and shown here Old PDF distribution. The integrated area from tmin (nominally set to 1.0) to 10.0 is 0.994402.

I am studiying the pdf obtained from the following distribution

f(t|tmin,signal)=N.1/τ.exp(-t/τ).[cosh(ΔΓ.t/2)-A_ΔΓ.sinh(ΔΓ/2)].[1+q(1-2ω).(Adir.cos(Δmt)-Amix.cos(Δmt))].Θ(t) conv 1/√2πexp-(t'-t)^2/2σ^2.

In the solution for this distribution the resultant pdf needs an error function of a complex number (x+iy), which there is no such routine in root. Hence we wrote a routine to use the infinite series approximation for complex error function from Abramozitz and Stegun (page 299, equation 7.1.29) which yields the folowing results for up to 7 terms of the series summation.

The new PDF has been plotted showing the real and imaginary parts, along with the old PDF in the following new PDF. The imaginary part is, as expected, zero with only the entry at time t= tmin giving a delta function.

A comparision of the normalised old PDF (blue oscillations) and the new UNnormalised PDF (red oscillations) is shown here.

Next the normalisation was implemented and again a comparision of the normalised old PDF (blue oscillations) and the new normalised PDF (red oscillations) is shown here with the real and imaginary PDFs and the old PDF is shown here. The area under the curves (summed over the plus and minus tag states) were calculated numerically using a step size of 0.5fs and in a range of tmin to 50ps and the following was found:

Old PDF area = 0.99983 (0.02% from 1.0)

New PDF area = 0.985813 (1.4% from 1.0)

Now need to understand if this 1.4% is significant (trawled through normalisation calculation and can not find any errors). Next task is to move all this code over to G-Fact so can change to use this new PDF for Bs2KK.

Toy data was generated with a mix of 56.1% Bd → π π, 10.3% Bd → K π and 33.6% combinatoric background with ssb of 0.664 (ssb is S/S+B) generated in a mass region of 5-5.8 GeV. The mistag rates in generation and fitting was ωBs = 0.361and ωBd = 0.355 and ACPBd→Kπ = 0.1. 100 data sets with 100,000 events per data set was used here but Lars ran with 1000 data sets with 100k events per set and found no bias on the pulls and much improved uncertainties. I just show the plots below (100 data sets) for completion.

Adir Bd - input was **0.38** and fitted was ** 0.3814 ± 0.0043 **( A^{dir}_{Bd})

Amix Bd - input was **0.61** and fitted was **0.6089 ± 0.0026** ( A^{mix}_{Bd})

Next I generated a data sample with the following composition (calculating for our mass range...) and simultaneously fitted for the Bd2pipi asymmetries and the Bs2KK asymmetries (for now there is still no proper time description in the fitting pdf but there is Gaussian smear on the data generation). Note the mistag rates will change for simultaneous fitting to ωBs = 0.347and ωBd = 0.370.

Everything below is out of date and will be removed as soon as the above is updated with the newest and unbiased results. Jobs running now.

50 datasets were generated with 6099 events per data set using 0.68 b2pipi (sf1=1 and ssb = 0.68) and 0.32 combinatoric background (MassDistribution) mass distribution. The asymmetry fitter was used to fit for Bd asymmetries only with sf1'=0.68 and obtained the following:

Adir Bd - input was **0.38** and fitted was **0.3953 ± 0.0476** ( A^{dir}_{Bd})

Amix Bd - input was **0.61** and fitted was **0.6503 ± 0.0311 **( A^{mix}_{Bd})

Then re-ran same job with 150 data sets with 2000 events per data set and fitted for the following:

Adir Bd - input was **0.38** and fitted was ** 0.4338 ± 0.0196 **( A^{dir}_{Bd})

Amix Bd - input was **0.61** and fitted was ** 0.6619 ± 0.0197 **( A^{mix}_{Bd})

The asymmetry plot is shown here .

******240 data sets with 6k (running just now will update when gets to 300) events per data set using 0.68 b2pipi (sf1=1 and ssb = 0.68) and 0.32 combinatoric background and mass window of 5229 - 5329 MeV, I fit the following asymmetries:

Adir Bd - input was **0.38** and fitted was ** 0.4175 ± 0.0080 **( A^{dir}_{Bd})

Amix Bd - input was **0.61** and fitted was **0.675 ± 0.006** ( A^{mix}_{Bd})

50 datasets were generated with 6099 events per data set using 0.68 bs2KK (sf3=1 and ssb = 0.68) and 0.32 combinatoric background ( mass distribution ). The asymmetry fitter was used to fit for Bs asymmetries only with sf3'=0.68 and obtained the following Adir for Bd and Amix for Bs which in numbers are:

Adir Bs - input was **0.1** and fitted was ** 0.0811 ± 0.0275 **

Amix Bs - input was **0.25** and fitted was **0.1799 ± 0.0346**

Ran a job with 100 data sets with 6000 events per data set using 0.664 Bs2KK and 0.336 combinatoric background.

Adir Bs - input was **0.1** and fitted was ** 0.06436 ± 0.00630 **( A^{dir}_{Bs})

Amix Bs - input was **0.25** and fitted was ** 0.1721 ± 0.0076 **( A^{mix}_{Bs})

Now ran the same job as above but removed the Gaussian smearing of the proper time in the toy data generation and re-ran the job and got the exact same asymmetries and uncertainties. This should make a difference surely? Still looking in to it.

50 data sets were generated with 6099 events per data set using 0.68 bd2pipi and bd2Kpi (sf1=sf2=0.5 and ssb = 0.68) and 0.32 combinatoric background ( mass distribution ).The asymmetry fitter was used to fit for Bd asymmetries only with sf1'=sf2'=0.34 and obtained the following Adir for Bd and Amix for Bd which in numbers are:

Adir Bd - input was **0.38** and fitted was ** 0.298 ± 0.206 **

Amix Bd - input was **0.61** and fitted was **0.660 ± 0.062**

Now using 150 data sets with 100k events per data set using bd2pipi and bd2Kpi (sf1= 0.845 and sf2=0.155 and ssb = 0.664, see Laurence's thesis, page 183). The asymmetry fitter was used to fit for Bd asymmetries with sf1'=0.561 and sf2'=0.103 and obtained the following:

Adir Bd - input was **0.38** and fitted was **0.4241 ± 0.0017** ( A^{dir}_{Bd}) and Laurence got ** 0.407 ± 0.033 **

Amix Bd - input was **0.61** and fitted was ** 0.6818 ± 0.0018 **( A^{mix}_{Bd}) and Laurence got ** 0.655 ± 0.024 **

The asymmetry plot is shown here . The pull plots show a large bias. The only difference was the large mass window I used (4800 - 5800 MeV) compared with Laurence (5229 - 5329 MeV). Hence I have re-run the same signal fractions in a 5fb-1 job (288 data sets with 100k events per data set) with the much tighter mass window and found the following:

Adir Bd - input was **0.38** and fitted was **0.4245 ± 0.0022** ( A^{dir}_{Bd}) and Laurence got **0.407 ± 0.033**

Amix Bd - input was **0.61** and fitted was ** 0.6811 ± 0.0019 **( A^{mix}_{Bd}) and Laurence got **0.655 ± 0.024**

The asymmetry plot is shown here . However the pull plots are still showing a large bias in the mean of the pull plot (the widths are approximately ok). What's causing the bias in the pull plots? It is not present in the first case when only B2pipi and combinatoric background are used....looking in to this now. Interesteingly, in an earlier version of Laurence's thesis he has numbers like mine then they suddenly got really good. Waiting to hear back from him to see if he can remember what he changed.

50 datasets were generated with 6099 events per data set using all 3 channels with sf1=sf2=sf3=0.333 and ssb = 0.68) and 0.32 combinatoric background ( mass distribution ). The asymmetry fitter was used to fit for Bd asymmetries only with sf1'=sf2'=sf3'=0.226 and obtained the following Adir for Bd and Amix for Bd which in numbers are:

Adir Bd - input was **0.38** and fitted was **0.4917 ± 0.0638 **

Amix Bd - input was **0.61** and fitted was **0.6852 ± 0.0456 **

Then re-ran the asymmetry fitter for the B_{s} asymmetries and found;

Adir Bs - input was **0.1** and fitted was ** 0.0416 ± 0.0606 **( A^{dir}_{Bs})

Amix Bs - input was **0.25** and fitted was ** 0.1505 ± 0.0598**( A^{mix}_{Bs})

-- AlisonBates - 2010-08-06

Topic revision: r15 - 2010-10-12 - AlisonBates

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