The maximum likelihood method used to analyse NEMO-3 results

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

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

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

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

Parametrisation of Etot

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)

Fit between 2 and 2.7 MeV

Etot/me Etot/me

Fit between 2.7 and 3.1 MeV

Etot/me Etot/me

Fit between 3.1 and 3.27 MeV

Etot/me Etot/me

Fit of

Monte Carlo used : 10 000 000 events

Etot/me

Fit between 2 and 2.76 MeV

Etot/me

Fit between 2.76 and 2.81 MeV

Etot/me

Fit between 2.81 and 4.1MeV

Etot/me

Fit of 208Tl internal

Monte Carlo used : 675 000 000 events

Fit between 2 and 2.04 MeV

Etot/me

Fit between 2.04 and 2.15 MeV

Etot/me

Fit between 2.15 and 3.07 MeV

Etot/me

Fit between 3.07 and 4.34 MeV

Etot/me

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

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

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

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

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

Fit of the parameters as a function of Etot for

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

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

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

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

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

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

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

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

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

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

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

Fit of the parameters as a function of Emin for

Fit of the parameters as a function of Emin for

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

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

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

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

Correlations between T ½ limits

gain whenadding Emin

~ same limitwith window

or 3D-lik

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

Correlations between T ½ V+A limits

gain whenadding Emin

Better limitwith 3D-lik

than for window

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)

Etot Etot

Emin cos

3

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)

Etot Etot

Emin cos

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