the maximum likelihood method used to analyse nemo-3 results interest of the method technical...
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The maximum likelihood method used to analyse NEMO-3 results
• interest of the method
• technical explanation of the method
• very preliminary results obtained
Laurent SIMARD, LAL-ORSAYILIAS Prague meeting, 20-21/04/06
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Interest of the method
not a simple counting method, use the information of all 2e- events in the spectrum above 2 MeV
Use all information from the events, notonly Etot = E1 + E2, but also E1, E2, cos
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P(event i)= xPo+ xradonPradon + x int 208Tl P int 208Tl + … +(1- x- xradon - x int 208Tl - …) P2
Method of fit of the fraction
For each signal/backgroundP is obtained from simulation
P = P(Etot)P(Emin/Etot)P(cos /Emin)
L = events
ieventP )~(Maximize L as a function of x
Fixed with channelswith higher statistics
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List of processes taken into account
• radon (in fact 214Bi) emitted from the tracking volume or deposited on the foil surface
• 208Tl in the sources• 214Bi in the sources• 208Tl in the glass of the PMTs• 214Bi in the glass of the PMTs
Signal• either from <m> • or from V+A process • or Majoron
Backgrounds
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Parametrisation of Etot
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Fit of 2 SSD
Monte Carlo used
• 500 000 000 events• 10 000 000 events between 1.8 and 2.1 MeV• 10 000 000 events between 2.1 and 2.4 MeV• 10 000 000 events between 2.4 and 2.7 MeV• 10 000 000 events between 2.7 and 2.9 MeV• 10 000 000 events between 2.9 and 3 MeV• 10 000 000 events between 3 and 3.1 MeV
weights to add these MC calculated from the theoretical formula (taken from the simulation)
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Fit between 2 and 2.7 MeV
Etot/me Etot/me
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Fit between 2.7 and 3.1 MeV
Etot/me Etot/me
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Fit between 3.1 and 3.27 MeV
Etot/me Etot/me
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Fit of
Monte Carlo used : 10 000 000 events
Etot/me
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Fit between 2 and 2.76 MeV
Etot/me
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Fit between 2.76 and 2.81 MeV
Etot/me
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Fit between 2.81 and 4.1MeV
Etot/me
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Fit of 208Tl internal
Monte Carlo used : 675 000 000 events
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Fit between 2 and 2.04 MeV
Etot/me
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Fit between 2.04 and 2.15 MeV
Etot/me
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Fit between 2.15 and 3.07 MeV
Etot/me
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Fit between 3.07 and 4.34 MeV
Etot/me
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Parametrisation of Emin/Etot
The Monte Carlo statistics above 2 MeV in Etot is divided in binsof 50 keV width
For each signal or background 2 steps :• fit Emin for each bin of Etot with some parameters• then fit the parameters as a function of Etot
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for 0.26 MeV < Emin < 0.36 MeV : threshold effect (cut at 200 keV)
Fit of Emin in bins of Etot for
for 0.36 MeV<Emin<Etot/2 analogy with Doi : P(E1,E2) = E1p1E2p2
2 parameters to fit as a function of Etot
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2 MeV<Etot<2.05 MeV
2.05 MeV<Etot<2.1 MeV
2.1 MeV<Etot<2.15 MeV
2.15 MeV<Etot<2.2 MeV
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2.4 MeV<Etot<2.45 MeV
2.45 MeV<Etot<2.5 MeV
2.5 MeV<Etot<2.55 MeV
2.55 MeV<Etot<2.6 MeV
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2.8 MeV<Etot<2.85 MeV
2.85 MeV<Etot<2.9 MeV
2.9 MeV<Etot<2.95 MeV
2.95 MeV<Etot<3MeV
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Fit of the parameters as a function of Etot for
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2 MeV<Etot<2.05 MeV
2.05 MeV<Etot<2.1 MeV
2.1 MeV<Etot<2.15 MeV
2.15 MeV<Etot<2.2 MeV
Parameterization of Emin for V+A
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2.2 MeV<Etot<2.25 MeV
2.25 MeV<Etot<2.3 MeV
2.3 MeV<Etot<2.35 MeV
2.35 MeV<Etot<2.4 MeV
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2.4 MeV<Etot<2.45 MeV
2.45 MeV<Etot<2.5 MeV
2.5 MeV<Etot<2.55 MeV
2.55 MeV<Etot<2.6 MeV
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2.6 MeV<Etot<2.65 MeV
2.65 MeV<Etot<2.7 MeV
2.7 MeV<Etot<2.75 MeV
2.75 MeV<Etot<2.8 MeV
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2.8 MeV<Etot<2.85 MeV
2.85 MeV<Etot<2.9 MeV
2.9 MeV<Etot<2.95 MeV
2.95 MeV<Etot<3 MeV
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3 MeV<Etot<3.05 MeV
3.05 MeV<Etot<3.1 MeV
3.1 MeV<Etot<3.15 MeV
3.15 MeV<Etot<3.2 MeV
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3.2 MeV<Etot<3.25 MeV
3.25 MeV<Etot<3.3 MeV
3.3 MeV<Etot<3.35 MeV
3.35 MeV<Etot<3.4 MeV
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Fits of cos /Emin
The Monte Carlo statistics above 0.25 MeV in Emin is divided in bins of 50 keV width
Same formula for all processes
try to use derive formulae from Doi for and
P(cos /Emin) = const ( 1 – coef1(cos + coef2 (cos 2 + coef3 (cos 3 + coef4 (cos 4)
For -1<cos <0.9
For –0.9<cos
P(cos /Emin) =pente (racine - cos
parameters to fit asa function of Emin
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0.25 MeV<Emin<0.3 MeV
0.3 MeV<Emin<0.35 MeV
0.35 MeV<Emin<0.4 MeV
0.4 MeV<Emin<0.45 MeV
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0.85 MeV<Emin<0.9 MeV
0.9 MeV<Emin<0.95 MeV
0.95 MeV<Emin<1 MeV
1 MeV<Emin<1.05 MeV
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1.05 MeV<Emin<1.1 MeV
1.1 MeV<Emin<1.15 MeV
1.15 MeV<Emin<1.2 MeV
1.2 MeV<Emin<1.25 MeV
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Fit of the parameters as a function of Emin for
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Fit of the parameters as a function of Emin for
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radon activity is measured in the tracking detector using the e- channel A(radon in the tracking
volume) ~0.95 Bq (high-radon period), 0.14 Bq(low-radon period)
Fraction of the backgrounds (except ) is fixed using
dedicated higher-statistics channelsExample : radon fraction which contribute to the 2e- channel above 2 MeV
Then using simulation, the expected number
of 2e - events above 2 MeV due to radon is derived
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208Tl fraction from the sources which contribute to the 2e- channel above 2 MeV
Then using simulation, the expected number of 2e -
events above 2 MeV due to 208Tl in the sources is derived
208Tl activity in the sources is measured using the e-2 and e-3 channel
A(208Tl) from the 100Mo sources ~ 100 Bq/kg
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Limits obtained for 25 MC samples after 5 years for 100Mo,
with :
• T(1/2)() = 7.7 1018 y
• A(208Tl internal) = 100 Bq/kg
• A(214Bi internal) = 300 Bq/kg
• no radon
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T ½ limits with window,1D,2D,3D likelihood
Window2900-3300 keV
In corrected energy (gas…)
1.3 1024 y
1D likelihoodEtot
1.1 1024 y
2D likelihoodEtot, Emin1.3 1024 y
3D likelihoodEtot, Emin
cos 1.3 1024 y
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Correlations between T ½ limits
gain whenadding Emin
~ same limitwith window
or 3D-lik
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T ½ V+A limits with window,1D,2D,3D likelihood
Window2900-3300 keV
In corrected energy (gas…)
0.5 1024 y
1D likelihoodEtot
0.5 1024 y
2D likelihoodEtot, Emin0.7 1024 y
3D likelihoodEtot, Emin
cos 0.8 1024 y
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Correlations between T ½ V+A limits
gain whenadding Emin
Better limitwith 3D-lik
than for window
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Very preliminary results for likelihood for 100Mo : low radon period
3D Likelihood (90% CL)T½ (<m > 3 1023 y
T½ (V+A) > 2.2 1023 y
Window (90% CL)2.9 MeV-3.3 MeV in
corrected energyNexpected = 2.6Nobserved = 2
Nexcluded = 3.7T½ (<m > 3.6 1023 y T½ (V+A) > 1.5 1023 y
2D Likelihood (90% CL)T½ (<m > 3 1023 y
T½ (V+A) > 2.3 1023 y
1D Likelihood (90% CL)T½ (<m > 3.5 1023 y
T½ (V+A) > 1.6 1023 y
6452 events above 2 MeV(dec 04 -> mar 06 : 257.1 days)
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Etot Etot
Emin cos
3
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Very preliminary results for likelihood for 82Se : low radon period
3D Likelihood (90% CL)T½ (<m > 2.2 1023 y
T½ (V+A) > 1.2 1023 y
Window (90% CL)
2.8 MeV-3.3 MeV in corrected energy
Nexpected = 1Nobserved = 0
Nexcluded = 2.3T½ (<m > 1.6 1023 y
T½ (V+A) > 0.7 1023 y
2D Likelihood (90% CL)T½ (<m > 2.1 1023 y
T½ (V+A) > 1.2 1023 y
1D Likelihood (90% CL)T½ (<m > 2.1 1023 y
T½ (V+A) > 1 1023 y
115 events above 2 MeV (dec 04 -> mar 06 : 257.1 days)
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Etot Etot
Emin cos
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Conclusion
• method to take into account all information from a tracko-calo detector (not only total energy deposited, but also individual energies, angle)
• Gain in sensitivity obtained for the V+A process, mainly by using Emin information