2.4 (w <2 cm) 2.4 (w <2 cm) 2.4 (w <2 cm)
uγ,vγ (mm) 5 5 5
wγ (mm) 6 6 6
tγ (ps) 96 67 67
Positron Resolutions
Ee (MeV) 0.31 0.32 0.31
φe (mrad) 6.6 7.2 7.5
θe (mrad) 9.4 11.0 10.6
ye (mm) 1.1 (core) 1.1 (core) 1.2 (core)
ze (mm) 1.1 1.7 1.9
te (ps) 107 107 107
Combined Resolutions
φeγ (mrad) 8.9 9.0 8.9
θeγ (mrad) 15.0 16.1 16.2
teγ (ps) 156 123 127
Efficiency
γ (%) 63 63 63
e (%) 28 35 31
trg (%) 91 92 97
A maximum likelihood analysis is performed to estimate the number of signal events (Nsig) since only the small number of events are expected to be observed in the signal region. A maximum likelihood function is defined with probability density functions (PDFs). In order to avoid possible human biases, we use a blind analysis procedure. The events close to the µ+→e+γ signal region were hidden until calibrations and analysis have been fixed for 2011 data. The datasets of 2009 and 2010 are re-analyzed in the same analysis procedure which is used for the 2011 dataset. Although the signal region is closed, all necessary parameters for the analysis can be extracted from events outside of the blind box, the data from the calibration runs, the data taken by the different trigger and Monte Carlo simulation.
The sensitivity is calculated in the same way as our data analysis by using many pseudo experiments assuming null-signal-hypothesis.
In order to translate the number of events to the branching ratio, the normalization factor is calculated by combining results from two different ways, which are called Michel normalization and RMD normalization. The normalization factors for 2009 and 2010 datasets are also calculated again, because of the efficiency improvements due to the offline noise filtering, new Kalman and the pileup unfolding.
In order to improve the sensitivity, the per-errors of observables associated to the positron measurement, which is introduced in Sec. 3.2.6.2, are incorporated into the PDFs. The detailed analysis using the per-errors is described in Sec. 6.3.
6.1 Datasets
We analyzed combined dataset taken in 2009–2011 with the MEG trigger. The datasets taken in 2009 and 2010 are re-analyzed with the new reconstruction methods in order to gain the performance and the experimental sensitivity as already mentioned. The analysis using several sidebands are also performed.
• −6.875< tγ−tTC <4.375 ns,
• |tTrackCandidate−tTC|<50 ns,
wheretTrackCandidateis the best estimated time of the hit calculated by the track candidate.
Since the time difference betweentTrackCandidateis set to wide enough, this cut corresponds to the requirement of which at least one track candidate is reconstructed in an event. The events which are not selected by the pre-selection are used only for the calibrations.
6.1.2 Blind Box
The blind box is defined by using the gamma ray energy and the time difference between a gamma ray and a positron as
• 48≤Eγ ≤58 MeV,
• |teγ| ≤1 ns,
Before opening the blind box, we fix the calibrations and the analysis. The analysis on the datasets in time sidebands, angle sidebands and the reliability check for the newly implemented PDFs by using full Monte Carlo simulation (full-MC) are also done before the un-blinding.
6.1.3 Analysis Region
In order to perform the maximum likelihood fitting, the analysis region, which includes the µ+→e+γ like gamma-positron pair event, is defined as follows;
• 48≤Eγ ≤58 MeV,
• 50≤Ee ≤56 MeV,
• |φeγ| ≤50 mrad and|θeγ| ≤50 mrad,
• |teγ| ≤0.7 ns,
• a pair of a positron and a gamma is selected by using thePositronSelection and the GammaSelection.
0 20 40 60 80
(nsec)
γ
Te
-4 -2 0 2 4
42 44 46 48 50
Figure 6.1: Two dimensional event distribution in teγ vs Eγ taken in 2009–2011. The blank box at the center shows the blind box and the inner box with blue lines shows the analysis region. Left and Right boxes with dashed black lines show the sideband data and two boxes with solid magenta lines show the time sidebands which are used for the maximum likelihood fit. A bottom center box shows the energy sideband used to evaluate the expected number of RMD events and theteγ resolution. For the illustration purpose, following loose cuts are applied;40≤Ee≤60 MeV;40≤Eγ ≤60 MeV;|θeγ| ≤200 mrad;
|φeγ| ≤200 mrad;|teγ| ≤4 ns.
6.1.4 Sideband Data
We define several sideband datasets in order to perform the calibration, evaluate the parameters required to construct the PDFs and perform the maximum likelihood analysis in the same way as used for the analysis region. Figure6.1shows distribution of the data taken with MEG trigger in 2009–2011. As shown in this figure, accidental background is distributed flat in the relative time. Therefore the parameters which are required to construct the accidental background PDFs can be extracted from the data in off-time region called time sidebands. In the same plot, there is the event concentration at the center of the relative time below the blind box (called energy sideband). These events should be the radiative muon decay (RMD) and we can use these events to extract the normalization factor and to calculate the number of RMD events in the analysis region by using the energy sideband. Figure 6.2 shows the two dimensional distribution of the events in relative angles in time sidebands. The asymmetric shape is caused by the trigger direction match algorithm with lower momentum positrons than that of the signal positron.
0 20 40 60 80
(mrad)
γ
θe
-200 -100 0 100 200
-200 -150 -100 -50 0 50
Figure 6.2: Two dimensional event distribution in θeγ vs φeγ taken in 2009–2011 in time sidebands. The center box with blue lines shows the analysis region in relative angles.
Neighboring four boxes with dashed lines show the angle sidebands which we perform the maximum likelihood fitting. For the illustration purpose, following loose cuts are applied;40≤Ee≤60 MeV;43≤Eγ ≤60 MeV;|θeγ| ≤200 mrad;|φeγ| ≤200 mrad;|teγ| ≤ 4 ns and |teγ|>1 ns.
6.1.5 Time Sidebands
The regions of the time sidebands are defined as similar to that of signal, but with a negative or a positive side of the time difference between a gamma ray and a positron.
In order to perform the maximum likelihood fitting in the time sidebands, two regions are defined as follows:
Negative time: 48 ≤ Eγ ≤ 58 MeV,
50 ≤ Ee ≤ 56 MeV,
−2.7 ≤ teγ ≤ −1.3 nsec,
−50 ≤ θeγ ≤ 50 mrad,
−50 ≤ φeγ ≤ 50 mrad Positive time: 48 ≤ Eγ ≤ 58 MeV,
50 ≤ Ee ≤ 56 MeV,
1.3 ≤ teγ ≤ 2.7 nsec,
−50 ≤ θeγ ≤ 50 mrad,
−50 ≤ φeγ ≤ 50 mrad
A pair of a positron and a gamma is selected byPositronSelectionandGammaSelection.
Time sidebands are used to extract the parameters for the accidental background as well since the accidental background events distribute uniformly in the relative time distribution. For this purpose, loose selection criteria are applied according to the ob-servable, for example, for the backgroundEe PDF, criteria associated to the gamma ray measurement can be loosen.
≤ γ ≤
50 ≤ Ee ≤ 56 MeV,
−0.7 ≤ teγ ≤ 0.7 nsec,
−50 ≤ θeγ ≤ 50 mrad,
−150 ≤ φeγ ≤ −50 mrad Positive φ: 48 ≤ Eγ ≤ 58 MeV,
50 ≤ Ee ≤ 56 MeV,
−0.7 ≤ teγ ≤ 0.7 nsec,
−50 ≤ θeγ ≤ 50 mrad, 50 ≤ φeγ ≤ 150 mrad Negative θ: 48 ≤ Eγ ≤ 58 MeV,
50 ≤ Ee ≤ 56 MeV,
−0.7 ≤ teγ ≤ 0.7 nsec,
−150 ≤ θeγ ≤ −50 mrad,
−50 ≤ φeγ ≤ 50 mrad Positive θ: 48 ≤ Eγ ≤ 58 MeV,
50 ≤ Ee ≤ 56 MeV,
−0.7 ≤ teγ ≤ 0.7 nsec, 50 ≤ θeγ ≤ 150 mrad,
−50 ≤ φeγ ≤ 50 mrad
A pair of a positron and a gamma is selected byPositronSelectionandGammaSelection.