I have evaluated the MACRO combined data as presented in the plot produced
by Francesco for its statistical interpretation under different oscillation
signatures. I find that I mostly agree with Francesco and disagree with Nat
in the interpretation of the data. This may be due to a simple calculational
error in Nat's more complicated statistical calculation... I doubt it is due
to a fundamental difference due to the statistical treatment. I suspect that
the main difference is due to a difference in the expected suppresion
factors in oscillation models. Hence, I explicitly provide my calculations
of the suppresion factors for various oscillation models below.
My calculation follows the one done by Francesco (I think). Unlike
Francesco however, I find that oscillation models allow for better chi-squared
than without oscillations and that the overall normalization factor for
minimum chi-squared is well outside of the 17% error estimate on the absolute
flux. I find that the change in the zenith distribution due to oscillations
does have some affect on the chi-squared but the improvement is generally
small enough that the chi-squared per DOF is not improved at all. I find that
no oscillation model fits the data very well. Hence, we are forced to assume
that there is some unknown systematic error, either in the data or MC
calculation or other unknown physics. Since the chi-squared is *always*
dominated by either bins 1 and 2 or bin 4 (depending on the overall
normalization), if we assume a correction of this systematic error then
the chi-squared will be good for either an oscillation model or no
oscillations.
The calculation that I have done is very simple. I add the statistical and
experimental systematic errors for each data point in quadrature as the
total error to be used in calculation of chi-squared. I then allow the
MC predicted value to fluctuate up to 17% in the direction which minimizes
chi-squared. In some cases, I allow the normalization to vary more just to
see what normalization is required to minimize chi-squared. I do not include
the bin nearest to horizontal in the calculation (same as Francesco) due to
concerns about punch-through from Teramo.
The data which I have used is shown below. The bins run from vertical
to horizontal from left to right:
MACRO number of events /26.8,21.8,40.3,49.5,22.0,28.0,24.0,25.0,11.0,6.0/
Statistical Error /5.2,4.7,6.5,6.9,4.7,5.3,4.9,5.0,3.3,2.4/
Systematic Error /1.2,1.2,2.0,2.8,1.2,1.1,2.5,2.4,3.1,2.0/
Bartol MC prediction /54.0,49.2,45.4,40.8,39.0,32.1,25.5,19.7,11.0,2.3/
The factors calculated for various delta m**2 (dm2 below) for the suppresion
of upgoing muons is shown below. The calculation has been done assuming a
minimum detectable muon energy of 1 GeV. Oscfac is the reduction factor which
each bin of the MC prediction above should be multiplied by if neutrino
oscillations exist with sin**2 2theta=1. and dm2 as shown:
ccc dm2=.002
ccc data oscfac/0.57,0.52,0.59,0.61,0.63,0.65,0.69,0.74,0.78,0.91/
ccc dm2=.001
ccc data oscfac/0.69,0.70,0.71,0.73,0.75,0.78,0.81,0.84,0.88,0.96/
ccc dm2=.0004
ccc data oscfac/0.79,0.80,0.81,0.83,0.85,0.88,0.91,0.95,0.98,0.99/
ccc dm2=.02
ccc data oscfac/0.51,0.52,0.52,0.53,0.54,0.55,0.56,0.58,0.61,0.67/
ccc dm2=.004
ccc data oscfac/0.57,0.59,0.60,0.61,0.63,0.65,0.67,0.70,0.76,0.83/
Using the above numbers, I have calculated the following chi-squared values:
delta m**2 Scale Factor chi-squared DOF chi-squared/DOF Note #
No oscillations 1.0 73.6 9 8.2
No oscillations 0.83 38.9 9 4.3 1
No oscillations 0.67 28.0 8 3.5 1,2
.0004 1.0 33.2 8 4.2 3
.0004 0.83 23.8 8 3.0 4
.001 1.0 24.3 8 3.0
.001 0.95 23.4 8 2.9 4
.002 1.0 23.2 8 2.9
.002 1.15 19.8 8 2.5 4
.004 1.0 25.4 8 3.2
.004 1.10 23.4 8 2.9 4
.02 1.0 34.4 8 4.3 5
.02 1.17 26.6 8 3.3
.02 1.25 25.3 7 3.6 3
no osc (no bins 1,2) 1.0 15.8 6 2.6 6
no osc (no bins 1,2) 0.86 10.6 6 1.8 6
.002 (no bin 4) 1.0 12.3 7 1.8 6
.002 (no bin 4) 1.10 11.4 7 1.6 6
Notes:
1. I keep DOF=9 until the scale factor drops below 0.83.
2. I find the best normalization with no oscillations to be 0.67. This is
quite different than Francesco's 0.81.
3. Setting the scale factor (equivalent to sin**2 2theta) to 1. keeps
one more DOF. Varying the scale factor within the MC error should
therefor retain 1 DOF with respect to any scaling outside of that
error. Clearly, the mixing strength (should it exist) would not be
well constrained.
4. Optimized scale factor
5. Although larger delta m**2 appears to be relatively disfavored by
the chi-squared, the fact that no chi-squared is good forces us to
assume that an unknown systematic error is at work which could make
the larger delta m**2 just as legitimate if oscillations are in fact
at work. An example of such an error could be extra background coming
from backscattering. Of course, this could also work the other way
and make everything consistent with no oscillations. Bigger delta m**2
are just equivalent to scale factors and will give similar chi-squared
to the 0.67 normalization with no oscillations.
6. These are calculated to show the effect of "fixing" the worst bins
by just leaving them out of the chi-squared calculation.