As can be seen in Figure5.22, theEECLdistribution in the signal decay vanishes for higher val-ues. We define the sideband byEECL>1.35 GeV, to make sure basically no signal contribution is left here.
To show that the simulation describes the data well, both distributions are compared in the sideband region, shown in Figure5.23. For completeness, the MC distributions are not blinded.
No second sideband region is available for this selection as the signal does not show a peak-ing structure in any other kinematic variable or other distribution. As a second check, MC vs data comparison plots are produced for a lot of other variables applying the EECL sideband cut. After measuring, MC vs data plots without the sideband cut are produced, too. Disagree-ments between data and simulation in the lower EECL region should translate differently into different variables and should be visible in these plots, which is not the case. The additional
sideband comparison plots are shown in AppendixA.2.1and the unblinded comparison plots in AppendixA.2.2.
[GeV]
EECL
0 2 4
Events / 0.15 GeV
0
Events / 0.15 GeV
0
Events / 0.15 GeV
0
Events / 0.15 GeV
0
Figure 5.23.: Comparison of data and Monte Carlo simulation in the sideband EECL >
1.35 GeV. The Monte Carlo expectations are drawn in the complete region as-suming aB B0 →πτ ν= 1.0×10−4.
The measurement and the final result of the analysis are presented in this chapter. In order to reduce the systematic uncertainty on theB0 →Xc MC prediction, a fit is first performed. The fit also produces the number of signal events in the Belle data sample, as well as the branching fraction, presented in Section6.1. As no evidence is obtained, the significance level and upper limit are computed in Sections6.4and6.5, respectively, using Monte Carlo techniques explained in Section 4.4. The construction of the likelihood is described in Section 6.2, followed by the description of the systematic uncertainties included in the final result, in Section 6.3.
6.1. Fit
A binned maximum likelihood fit is performed inEECL. The number of signal events is extracted from the fit. Additionally, the fit is used to determine a scaling factor for theb→ccontribution in order to reduce the systematic uncertainty of this contribution due to finite MC sample size.
Please note, that in this chapter,b→c labels only the dominantB0 →Xc decays, not the less relevant B+ → Xc decays. In order to improve readability, the b → c label has been chosen.
Due to low statistics and similar shapes, all background contributions are fixed except for the dominantb→c transitions. The pdfs are built from the Monte Carlo predictions inEECLthat are shown above in Figure5.22. The fit is performed using the RooFit framework.
The likelihood function of a single reconstruction channelcused for the fit can then be described as
L=Ntot·PDFc= (µ·Nc,sigMC)·PDFc,sig + (fc,b→c·Nc,b→cMC )·PDFc,b→c + (fc,const·Nc,constMC )·PDFc,const
(6.1) with the coefficients µ, fb→c and fconst and the number of events per contribution determined from MC simulation NMC. As there are only rough calculation available for the expected branching fraction B B0 →πτ ν in the SM, we define the B B0 →πτ νMC = 1.0×10−4 in this analysis. The coefficients µ, fb→c and fconst are the fitting parameters with fconst = 1.0 fixed. The single PDFs are normalized to 1.
The coefficientµwill be called the signal strength and is defined as µ= B B0 →πτ ν
B(B0 →πτ ν)SM. (6.2)
Same as above, we define
BB0 →πτ νSM= 1.0×10−4, (6.3)
which also allows easy conversion the measured signal strength to the measured branching fraction.
Before the fit is performed on data, the stability of the fit is tested using pull distributions and a linearity test, which are described in Sections 6.1.1and 6.1.2, respectively. The fit result on data is then shown in Section6.1.3.
The final measurement is performed as a simultaneous fit in the threeτ reconstruction channels τ →e,τ →πandτ →ρ, as theτ →µreconstruction does not improve the expected significance.
This will be shown below in Section 6.4. It is already noted here because only stability tests and fit results for the combination of these three modes are shown in this section for better readability. The results of the test as well as the results of the fit are shown in Appendix A.1.
6.1.1. Pull Distribution
Pull distributions allow to investigate the fit results on whether the fit produces a bias and if the errors are estimated correctly. The pull p of a fit is defined as the difference between input valuefin and fit valueffit, normalized to the fit error σfit,
p= fin−ffit σfit
. (6.4)
The pull distribution is obtained by repeating the fit multiple times on pseudo data which is generated based on the fit pdf with the parameter values set tofin. A robust fit usually results in a pull distribution in form of a normal Gaussian distribution with meanµ= 0 and standard deviationσ= 1. A different mean value indicates a bias in the fit results while smaller or higher standard deviations indicates that the errors are over- or underestimated by the fit, respectively.
To avoid confusion due to the multiple meanings of the letterµ, the signal strength parameter will be labeled with an additional superscriptµsig for this section only. The difference between the muon and the mean of the Gaussian distribution should be clear from context.
The pull distributions for µsig and fb→c for all reconstruction modes and the combination is shown in Figure 6.1. The pull distribution for this fit is obtained for µ = fb→c = 1 for all background parameters. Pull distribution parameters have been determined for different input values of the signal strength parameter and background b → c contributions. The results are summarized in Table6.1. The table contains the parameters without a quoted error to improve readability. The size of the error is the same for all values of input parameters because the same number of pulls has been computed. As quoted in Figure 6.1, we obtain σµ = 0.01 and σσ = 0.007. Pull distributions fits in the single modes and in a simultaneous fit in all four reconstruction modes have been computed, too. The results are summarized in AppendixA.1.1.
As can be seen, no substantial bias is observed and all pull distributions fit well to a unit Gaussian.
6.1.2. Linearity
In order to investigate the stability of the fit over a broad range of possible parameter values, the linearity of the fit is tested. For a range of input parameterfinput, the fitted value ffitis plotted against the input value. For a stable fit without bias, this should produce a straight line through the origin with a slope equal to one. The distribution offfit versusfinput is shown in Figure6.2 for all four parameters in the combined fit of τ →e, π, ρ. The parameters of a straight line fit
Pull
Figure 6.1.: Pull distributions in the combined fit of τ → e, π, ρ for the signal (lower right) and the three b→ccontributions.
are shown in the plots and all agree well with ffit(finput) = 0 + 1·finput. The linearity tests for the fits in the single reconstruction modes and in all four reconstruction channels combined are shown in AppendicesA.1.2and A.1.3.
6.1.3. Fit on Data
The fit on data in the combined modes τ → e, π, ρ results in a signal strength parameter of µ= 1.52±0.72, which is equal to 51.9±24.3 signal events. The results are shown graphically in Figure 6.3 and listed in Table 6.2. From the fit, the significance σ of the signal in terms of standard deviations can be computed ignoring systematic uncertainties by
σ = s
−2 ln L0
Lfit = 2.70, (6.5)
whereLfit and L0 are the likelihood values evaluated after the fit and at µ= 0, respectively.
The fit has been performed in the single decay channels, too, the results are listed in Ap-pendix A.1.4.
µsigin µsig fb→c(τ →e) fb→c(τ →π) fb→c(τ →ρ)
[×10−4] µ σ µ σ µ σ µ σ
0.5 0.05 0.983 −0.01 0.991 0.01 0.985 0.02 0.981 1.0 0.02 0.985 0.00 0.991 0.03 0.990 0.00 1.000 1.5 0.04 0.985 0.00 0.997 0.02 1.000 0.02 0.989 2.0 0.05 0.994 0.02 1.010 −0.01 0.988 0.01 0.991 fb→c,in
0.8 0.04 0.969 0.01 0.996 0.01 0.995 0.03 0.974 0.9 0.02 0.996 0.02 0.995 0.03 0.994 0.00 0.994 1.0 0.02 0.994 0.00 1.010 0.02 0.987 0.03 0.990 1.1 0.04 0.993 0.01 0.989 0.01 0.996 0.01 0.989 1.2 0.04 1.010 0.01 0.990 0.02 0.983 0.01 0.987
Table 6.1.: Pull distributions for all fitted parameters in the combined fit of τ → e, π, ρ for different input values of signal strength paramer µsigin in the upper half and the b→ c contributionfb→c,in in the second half. No substantial bias is observed and the errors are estimated correctly in the fit.
, input
Figure 6.2.: Linearity test for signal strength µand fb→c in the combined fitτ →e, π, ρ.
[GeV]
(a)Simultaneous fit to all three modes, results for τ →e.
(b) Simultaneous fit to all three modes, results for τ→π.
(c) Simultaneous fit to all three modes, results for τ→ρ.
Figure 6.3.: EECLdistribution in the combined fitsτ →e, π, ρ.
Combined Fit
As in the fit, a binned likelihood approach is used in this analysis. The likelihood is then used to compute the signal significance and the upper limit of the signal branching fraction, by methods explained in detail in Section4.4. Systematic uncertaintiespare included in the likelihood in the