It seemed intended for the macro-upmu mailing list. Apologies in advance
for duplicates.
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---------- Forwarded message ----------
Date: Wed, 21 Apr 1999 16:19:29 +0200
From: "AXPBO::SPURIO - TEL. 630 5226" <Maurizio.Spurio@bo.infn.it>
To: macro@BUDOE.BU.EDU, baldini@pisa.infn.it, barish@hep.caltech.edu,
bernardini@lecce.infn.it, cei@pisa.infn.it, devincenzi@roma1.infn.it,
dicredico@lngs.infn.it, ediehl@hep.uchicago.edu, giacomelli@bo.infn.it
Subject: Statistical consideration on the low energy neutrino samples.
Statistical consideration on the low energy neutrino samples.
=======================================================================
Because at the incoming summer conferences we must present a statystical
interpretation of our low-energy neutrino events, I produced the following
result. PLEASE, READ AND SEND COMMENTS before to show the figures
outside the collaboration. I include also the data/MC to produce alternative
results. In the following mail, there is the postscript of the figures.
Maurizio Spurio
***************************************************************************
I tested the two alternative hypoteses (with the chi^2 probability):
- H0 = no neutrino oscillation, but reduced cross section and nu flux;
- H1 = two-flavors neutrino oscillations;
for the IU events (Lecce sample), the ID+UGS (Bologna sample) and the
combination of the IU and ID+UGS. I recall that for the two samples,
the measured ratios are:
(Data/MC)_IU = 0.57 +- 0.06(stat) +-0.06 (sys) + 0.14 (theo)
(Data/MC)_(ID+UGS)= 0.72 +- 0.06(stat) +-0.07 (sys) + 0.17 (theo)
(see Calthech meeting).
*** NO-OSCILLATION TEST *******
Mimic SK, the chi^2 funcion used to test H0 is:
chi^2 = \sum ((Data(i) - a*MC(i))/s_data(i) )**2 + (1-a/s_a)**2
where: data(i) = IU (4 bin), or ID+UGS (4 bin), or IU+(ID+UGS) (8 bin)
s_data(i) = stat error + 10% syst error on the data bin (in quadrature)
MC(i) = MC value (Bartol flux, Lipari cross section,GRVL0)
a = absolute flux normalization (0.5<a<=1)
s_a = 25% (theor. error on the absolute flux)
The chi^2 probability that the Bartol flux and Lipari cross section, with
an overall reduction factor (1-a), represents the true parent
distribution (TPD), is presented in the following figure (left).
Dashed lines are for the two data set, full line for the combination of
all data. Note:
1. an absolute reduction of 28% (flux/cross section) has a 13% (maximum)
probability to be the TPD for the ID+UGS events (such a reduction
is more than one sigma away from the theoretical prediction);
2. a 40% (flux/cross section) reduction has a 0.3% probability for the
IU events (which required a larger reduction);
3. evaluating the chi^2 for both the ID+UGS and the IU
***** the maximum of the probability is 0.06% ****
(only one absolute flux normalization constant cannot explain
the two data sets).
*** NEUTRINO-OSCILLATION TEST *******
To test H1, I assumed full mixing and the reduction factors for nu_mu
disappearence evaluated by myself (ID+UGS) and Antonio (IU). The tables
are for 10 bins of cos(zenith) and 25 intervals of dm^2. In the chi^2
function, the statistical error and a 10% sys. error are added in quadrature.
(The result does not change significatively if sys=8% or 12%).
The chi^2 probability vs. dm^2 for the two data sets, and for the combined
data (full line) is in the figure (rigth). Note:
1. The probability for the IU reach a constant value for dm^2> 5*10^-4 eV^2
(nu_mu fully oscillated, and zero sensitivity for downgoing events);
2. The probability for the ID+UGS decreases for dm^2> 2*10^-2 eV^2;
starting from this value, also ID events should be reduced.
3. Using the combined data set, the maximum of the chi^2 probability
(72%) is reached at dm^2= 6*10^-3 eV^2. The probability is >10%
for 2*10-4< dm^2 <6*10-2 eV^2.
*** CONCLUSIONS ********
The maximum of the probability that the (Bartol nu flux* Lipari c.s.) represents
the low energy MACRO neutrino data is 0.06%. Using the neutrino oscillation
hypotesis, the maximum of the probability (72%) is reached for max. mixing
and dm^2= 6*10^-3 eV^2. The probability region for 10% of the best-fit value
is 2*10-4< dm^2 <6*10-2 eV^2.
*** HOMEWORK ****
You do not agree? You suspect an error? Help us producing an alternative/correct
result. You can use for the following data:
* Corrected IU values:
data le/58.86,30.86,16.86, 7.86,1.43/
* MC predictions (no-osci)
data lem/86.04,62.67,37.60,14.22,1.43/
* Corrected ID+UGS values:
data bo/ 74.6,69.7,37.8, 9.7,0.97/
* MC predictions (no-osci)
data bom/105.7,92.6,57.6,17.3,0.65/
* Averaged reduction for nu_mu oscillations (25 bins from log10(dm^2) = -5
up to -0.2) for lecce and Bologna events:
data rile/.995,.988,.972,.935,0.870,0.774,0.680,0.621,0.594,0.590,
+ .587,.586,.583,.573,0.572,0.567,0.562,0.561,0.567,0.561,
+ 0.561,0.567,0.566,0.567,0.567/
data ribo/1.00,0.99,0.99,0.97,0.93,0.88,0.82,0.78,0.765,0.75
+ ,0.74,0.73,0.72,0.71,0.70,0.69,0.65,0.61,0.55,0.52
+ ,0.51,0.50,0.495,0.49,0.48/
(With a coffee charge, I can send you the bin by bin reduction).
maurizio