Fitting OPERA’s Result
Yesterday I tried to point out that the results from OPERA are not necessarily contradicted by the SN1987a results.
Today I have tried to take this simple analysis one step further, by fitting a simple quadratic ansatz for the OPERA results and comparing to the SN1987a and also the MINOS measurement (Phys. Rev. D76 072005 2007).
I assumed that
δv/c = (E/M)2
where E is the energy of the neutrino and M is a phenomenological mass parameter to be determined by measurements. A simple χ2 linear regression analysis indicates that
M = 7.9 ± 1.4 TeV.
I ignored the systematic uncertainties which are fully correlated between the two OPERA measurements. Continuing to take only the statistical uncertainties into account, I find χ2 = 5.9 for one degree of freedom, which is rather poor, but including the systematic uncertainty would improve this value somewhat.
Here is a plot showing the fit:
The fit looks incorrect (it passes close to the second point but is far from the first point) because it is constrained to come from the origin (ie, zero neutrino energy means zero deviation). This is not a parabolic fit — it is a linear fit to 1/M2. The MINOS data were not included in the fit.
Taking the fit as it is, we can compare to SN1987a again:
The SN1987a constraint is indicated by the dashed line – the region above the line is disallowed. Clearly the red line is not excluded by the SN1987a data.
The MINOS data point appears to be inconsistent with the red line, but one has to be quite careful with a double-log plot like this one. The solid error bars are one-sigma limits as reported by MINOS. The lower two-sigma limit, however, encompasses zero and is indicated by the dotted line. As you can see in the linear plot above, the MINOS data is consistent with the red curve at better than the two-sigma level.
This little analysis does not indicate that the OPERA results are correct, of course. The experimental work done is very impressive and I admire the Lyon group for their achievement. But I agree with everyone else, including the OPERA Collaboration, that confirmation by another experiment is needed before the result can be taken to be true. Meanwhile, common statements that SN1987a rules out the OPERA result should be couched in more tentative language, in my opinion.
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