In our study, both counting and fitting methods are used to estimate the upper limit of branching fraction. Before box opening, we can estimate the expected limits by assuming that the signal yield is 0.
8.1 Counting Method
Feldman-Cousins’ method is used in counting method to obtain the upper limit under 90%
confidence level, and a limit estimation package named TRolke is used [73]. In addi-tion to the tradiaddi-tional confidence level belt construcaddi-tion in the Feldman-Cousins’ method, Gaussian constraints on signal efficiency and background systematic uncertainties are also considered in this approach.
8.1.1 Calibration on Signal Box Efficiency
In our counting method, we count the number of events in the signal region. However, from MC results we know that the true events do not always lie in the signal region. This causes an efficiency drop and should be calibrated.
We use the control sample B0 → KS0π+π−γ to calibrate the ratio R = Nsig, signal region
Nsig, whole region of true event number between signal region and whole region, by comparing the MC counting result and data fitting result. For MC samples, we obtain the ratio RMC= 0.881± 0.001, where the error is binomial. As for data, we obtain the ratio Rdata = 0.833± 0.009+0.027−0.023,
where the symmetric error is binomial and the asymmetric error comes from the uncer-tainty of PDF mean shift and width expansion. By comparing the two value, we obtain the calibration factorRRdata
MC = 0.945+0.033−0.028, where the errors corresponds to +3.47% and -3.00%.
SinceTRolke only accepts symmetrical error, we choose the larger (3.47%) one.
8.1.2 Expected Background in Signal Region
The expected background number in signal region can be estimated by using the sideband data and the box-sideband ratio in the signal MC, which is:
Expected background = # of events in sideband (data)×# of events in signal region (MC)
# of events in sideband (MC) (8.1)
In the equation above, the first term is obtained from data. Therefore, only statistical error is considered. The second term is obtained from MC, and the systematic error is estimated by comparing the number of events in sideband for data and MC samples, as listed in Table. 8.1.2. In the MC signal region/sideband determination, B ¯B generic MC, q ¯q continuum MC, and the B → J/ψX decay MC is used.
# of MC events # of data events Systematic Channel in sideband in sideband error
Dimuon 272.67 306 10.89%
Dielectron 341.53 413 17.31%
Table 8.1: Systematic error on background level
(a) Dimuon channel (b) Dielectron channel
Figure 8.1: Background MC and sideband data histogram
8.1.3 Expected Counting Results
Here we assume the observed number of event = the nearest integer to the expected back-ground number. We combine the dimuon and dielectron channel by adding up the expected background and observed number of event.
Expected Upper limit B(B0 → X(3872)γ) Channel background (90% c.l.) ×B(X(3872) → J/ψπ+π−)
Dimuon 9.3 5.75 9.21× 10−7
Dielectron 12.1 7.35 1.38× 10−6
Total 21.4 9.41 8.13× 10−7
Table 8.2: Expected Counting Results
8.2 Fitting Method
Fitter we construct for B0 → X(3872)γ can be used to estimate the upper limit of branch-ing fraction in fittbranch-ing method. We can compute the Bayesian upper limits with 90% con-fidence level. The upper limit on signal yield N is obtained by integrating the likelihood function:
Z N
0
L(n)dn = 0.9 Z ∞
0
L(n)dn (8.2)
whereL(n) denotes the likelihood of the fitting result with N fixed to the value n. The systematic uncertainties on signal efficiency and PDF modeling are taken into account by replacingL(n) with a smeared likelihood function:
Lsmear(n) = Z ∞
−∞L(n′)e−(n−n′)2/2σ2syst
√2πσsyst dn′ (8.3)
Before box opening, we can estimate by fitting on MC events with all backgrounds normalized to 1 stream.
8.2.1 Uncertainty on PDF Modeling
In the data fit of our control sample B0 → KS0π+π−γ, mean shift and width expansion parameters are floated to obtain the calibration factors as listed in Table. 6.6.
The shape-fixed signal PDF in the B0 → X(3872)γ nominal data fit is calibrated by these factors. We estimate the uncertainty on PDF modeling by varying those parameters with Gaussian random numbers, where σ of the Gaussians are the error of the calibration factors. Then, we perform 10,000 data fits with 10,000 randomized parameter sets by a Gaussian. The fit is directly on B0 → X(3872)γ data. To avoid looking into the box, the uncertainty can be obtained by the ratio of the Gaussian’s width and mean without knowing the central nominal value of the yield. We get the uncertainty as 89.0% for dimuon channel and 33.2% for dielectron channel.
8.2.2 Fitting bias
The fitting bias can be obtained from the peak value of the Noutsig distribution for Gsim ensemble test when generated Nsig = 0. We use a Gaussian + Crystal Ball function to fit the distribution, and then obtain the peak value with its error.
(a) Dimuon channel (b) Dielectron channel
Figure 8.2: Noutsigdistribution for Gsim ensemble test when generated Nsig = 0.
We get the peak value (fitting bias) as 0.549+0.050−0.051 and 0.253+0.072−0.070 for dimuon and dielectron channel, respectively. Then, we deal with such fitting bias by directly shifting the fitting signal yield back.
8.2.3 Expected Fitting Results
We combine the dimuon and dielectron channel by multiplying both likelihood functions with the x-axis normalized to the branching fraction.
−10 −5 0 5 10 15 20 25 30 (Dimuon channel) Nsig
Likelihood (max = 1)
−10 −5 0 5 10 15 20 25 30 (Dielectron channel) Nsig
Likelihood (max = 1)
−1 0 1 2 3 4
Likelihood (max = 1)
Likelihood fcn
likelihood fcn Smeared
Figure 8.3: Likelihood function, where the maximum value is set as 1.
Upper limit B(B0 → X(3872)γ) Channel (90% c.l.) ×B(X(3872) → J/ψπ+π−)
Dimuon 9.33 1.41× 10−6
Dielectron 13.04 2.31× 10−6
Total - 1.10× 10−6
Table 8.3: Expected Fitting Results
8.3 Dicision to use counting or fitting method
To decide what method to use when deal with real data, we generate toy events by PDF shape, and then compare the upper limits obtained by the two methods. We vary the generated number of signal and background samples with Poisson. The mean of Nsig is set as 0, 10, and 20, and the mean of background numbers of each type are set as the expected value. We can see that counting method is more likely to obtain smaller upper limit when Nsig = 0. and the fitting method is more likely to obtain smaller upper limit when Nsig becomes larger. The results of toy test shows that if the signal yield is small (i.e. Nsig < 10), counting method may be a better choice. Otherwise, we should use the fitting method.
Nsig = 0
Channel U Lfit < U Lcount U Lfit> U Lcount
Dimuon 31.2% 68.8%
Dielectron 14.4% 85.6%
Total 43.2% 56.8%
Nsig = 10
Channel U Lfit < U Lcount U Lfit> U Lcount
Dimuon 43% 57%
Dielectron 38% 62%
Total 46% 54%
Nsig = 20
Channel U Lfit < U Lcount U Lfit> U Lcount
Dimuon 86% 14%
Dielectron 75% 25%
Total 85% 15%
Table 8.4: Compare the upper limits obtained by the two methods
In the next chapter, we will show that our box opening result shows no signal. There-fore, we decide to use counting method to obtain the upper limit of BF, and the fitting method result can be used as comparison.