As mentioned before, the variables used to describe an event that passed the selection criteria listed in Section 4.1 are mES, ∆E, MLP, ∆t and the position
Table 4.7: Differences between MC and the data mES and ∆E fit parameters
for B0 →D−π+ control sample.
mES ∆E
µ1data−µ1MC (0.0±0.1)MeV/c2 (−3±2)MeV µ2data−µ2MC (1.2±0.2)MeV/c2 (3.6±0.6)MeV
σdata/σMC 1.03±0.07 0.98±0.01
σdata/σMC 0.98±0.05 1.04±0.03
of the event in the Dalitz plot. In the case of uncorrelated variables the total likelihood function for an event is given as the product of the individual PDFs (see Section 3.6). Previous studies have shown that themES and ∆E variables
are mostly uncorrelated [56], while the results presented in Section 4.5 show that in the case of BB events (both signal and background) no correlation between Dalitz plot position andmES, ∆E and MLP can be seen. Therefore,
the likelihood for aBB event can be written in the following form:
PBB(x, y,∆t, mES,∆E,MLP) = P(x, y,∆t)P(mES)P(∆E)P(MLP).
(4.19) Here,P(x, y,∆t) is the joint PDF for the Dalitz plot coordinates and the time difference ∆t. On the other hand, in the case of continuum background events, a correlations between the MLP distribution with the Dalitz plot coordinates was observed. Because of that, the likelihood for a continuum event has the following form:
Pqq(x, y,∆t, mES,∆E,MLP) = P(x, y,MLP)P(∆t)P(mES)P(∆E)
(4.20) Using the previous expressions for the likelihoods of the signal and background events, the total likelihood for an eventα in tagging categoryc is given by:
L(~n,~a) = e −(nsig+nqq+nB+B−+nBflav+nBCP) N! N Y e=1 Leα, (4.21)
) 2 (GeV/c ES m 5.272 5.274 5.276 5.278 5.28 5.282 5.284 5.2860 1000 2000 3000 4000 5000 6000 7000 8000 χ2/ndf = 0.54 2 GeV/c ES m 5.245.2455.255.2555.265.2655.27 5.2755.285.285 50 100 150 200 250 300 350 /ndf = 0.56 2 χ E (GeV) ∆ -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0 2000 4000 6000 8000 10000 /ndf = 0.62 2 χ E (GeV) ∆ -0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 0 50 100 150 200 250 300 350 /ndf = 0.54 2 χ
Figure 4.27: Comparison between MC (left) and data (right) fit of mES (top)
and ∆E (bottom) for the B0 → D−π+, D− → K0
Sπ− control sample. The
results show good agreement between MC and data fit in both cases, which can be seen from Table 4.7, making unnecessary any correction of the PDF parameters extracted from MC.
where Lc
α is:
Lc
α = nsigfsigc Psig,α (4.22)
+nqqfqqc Pqq,α+ nB+B− classX j=1 njfjcPB+B−,j,α + nBflavclass X k=1 nkfkcPBf lav,k,α+ nBCPclass X l=1 nlflcPBCP,l,α.
Here, theni represent the numbers of events of each species and the sums for
each of the three types of BB background run over the different classes of background channels within that type.
Chapter 5
Analysis Results and
Conclusions
In this chapter results of the fit to the data are presented. The chapter starts with a discussion of results of various toy MC tests and fully simulated MC tests.
5.1
MC tests
5.1.1
Toy MC tests
In order to test the stability of the fit and check for potential errors in the models used to describe different event species a number of toy MC tests are performed. The toy MC events are generated according to the PDFs described in the previous chapter. These generated events are then fitted using the same PDFs. The fit results are compared with the generated values by calculating biases, the differences between the means of the distributions of fitted values and the true values, for the all fitted parameters. Three different sets of toy MC tests are performed: toy MC tests with signal only events, toy MC tests with signal and continuum background events and finally, toy MC tests with signal, continuum background and BB¯ background events. For each toy
MC test 500 samples of the analysed set of events are generated (each one containing the amounts of different event species expected in the on-peak data sample). Because of the large number of the fitted parameters it is expected that the fit will not always converge to the global maximum. Therefore, each of the generated samples is fitted 100 times with randomized initial values of the fitted parameters. Results of the toy tests showed that almost 100% of these fits converge, despite their often highly incorrect starting points, and a majority (often>80%) will converge to the solution with the best likelihood. Examining all the different possible solutions in toy experiments it was found that this most favoured and best-likelihood solution is always the one closest to the generated parameters. The adopted practice for dealing with the multiple solutions behaviour is therefore to perform multiple randomised fits and to extract the solution with the best likelihood value.
In signal only tests it was found that biases among the fitted parameters are either non existent or very small, not larger than 15% of the expected statistical error. In the case of signal, continuum background and BB¯ background MC toy tests, slightly larger biases (20%) are noticed in only 2 out of 32 fitted parameters.
In Figure B.1-B.6, given in Appendix B, pull plots (see Section 3.4) for the signal only and signal, continuum background and BB¯ background toy MC tests are given.
5.1.2
Fully simulated MC tests
Fully simulated MC events are used to check whether any neglected effects, such as self cross feed or correlations between variables, are more important than initially estimated. For this purpose, the existing true MCB0 →K0
Sπ +π− model is used. In this model neutral B mesons decay to the K0
Sπ
+π− fi- nal state via 5 resonances (f0(980)KS0, ρ
0(770)K0
S, K∗(892)π, K0∗(1430)π and f0(1300)KS0) and two different non resonant terms. 250 data samples in which
events are made and each of them is fitted 100 times (with randomized ini- tial values of the fitted parameters) using signal, continuum and BB PDFs described in the previous chapter. As before, the best fit is chosen according to its likelihood function value. Figures C.1 to C.3 in Appendix B show the distributions of the fitted variables and the values used for the generation. It can be seen that the agreement between fitted and generated values is very good. The same conclusion can be made from the isobar coefficients distri- butions shown in Figure 5.1. In this figure the generated values of the isobar coefficients (which are complex numbers) and the values returned by the fits are shown in the complex plane. For each of the resonances, distributions of the best fits for 250 samples are shown (see the colour code) together with the mean fit value (black dot marker) and the generated value (black star marker).