## Archive for September 27, 2011

### Trying to Reconcile OPERA, MINOS and SN1987a

I’ve written twice about the contradiction of the OPERA results on high-energy neutrino velocities by the low-energy SN1987a results. I tried to show that this contradiction should not be taken for granted and certainly should not be used to dismiss the OPERA results.

A much better analysis was put forth by Giacomo Cacciapaglia, Aldo Deandrea and Luca Panizzi (arXiv:1109.4980, 23-Sep-2011). The authors made some interesting theoretical remarks regarding possible dispersion relations. I had considered only a quadratic relation (δv/c) = (E_{ν}/M)^{2} but there are good arguments for other relations, and the authors allowed the exponent to vary (in other words, to be non-integer).

When considering the SN1987a and MINOS data, there are two things to keep in mind:

- the arrival times (or times of flight) of the neutrinos
- the
*spread*of arrival times, or the lack thereof

The authors emphasized the importance of the latter.

For SN1987a, there are a couple dozen events recorded with energies in the 5 – 40 MeV range, and times spreading across 10 or 12 seconds. The range in energies is quite large and should result in a much larger spread of recorded times, if the dispersion relation depends quadratically on the energy. This fact holds quite independently of the near-coincidence of the neutrino signals with the photons coming from SN1987a.

For MINOS, the story is similar. The neutrino energy spectrum peaks around 3 GeV but there is a long tail up to 120 GeV. So if the dispersion is a strong function of the energy, the time structure will be broader than the length of the spill, but this is not observed. If we write δv/c = (1/2)(E_{ν}/M)^{α}, then α should not be large. A fit to the MINOS data favors α < 0.5 which means that the actual observed δv/c from MINOS is in tension with SN1987a. The OPERA results make this tension only worse, even without an analysis of the time structure of the OPERA data. (The authors intend to do such an analysis in the near future.)

So the OPERA, MINOS and SN1987a data don’t really allow an interpretation along the lines of δv/c = (1/2)(E_{ν}/M)^{α}. (My own analysis did not incorporate the constraints from the lack of spread in time of the SN1987a and MINOS data, and hence was much weaker than this analysis.)

This means that the data require a stronger – perhaps bizarre – dispersion relation. An exponential curve does not work, as it turns out, but a smooth step curve, parametrized by a hyperbolic tangent, can be made to work. The position of the threshold would have to be around 1 GeV, and the rise would have to be quite fast, occurring within 0.1 GeV or so. Here is an example of the allowed parameter space, where δ is δv/c, μ controls the rise at the step, which is given by *m*.

No one claims that a step function parametrized by a hyperbolic tangent is well-motivated or pleasing. The point is that simpler functions don’t work so well. A second point is that the data are still sparse and poorly understood, so ad hoc treatments meant to give relatively soft answers to the question “Does this all make sense?” are justified.

Certainly an actual quantitative analysis like the one by Cacciapaglia, Deandrea and Panizzi is much better than simple dismissive statements one sees in the blogosphere…