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I was reading some articles on the analysis of electron neutrino mass limit from the SN1987A data: Bachcall & Glashow, Arnett. They both used the principle that if the electron neutrino has a non-zero mass, higher energy neutrinos will arrive before lower energy ones, with the time difference given by $\Delta t=\frac { { m }^{ 2 } }{ { E }^{ 2 } } D\frac { \Delta E }{ E } $.

However, as I understand it, for the formula to work, the neutrino has to have one definite mass m throughout the propagation. But neutrino is mixed, an electron neutrino is a superposition of three mass states $v_1$, $v_2$, and $v_3$, and oscillates as it travels. So an electron neutrino has no definite mass, how can we derive a limit on electron neutrino mass from this formula? And what does this type of analysis tell us about the mass limit on $m_1$, $m_2$, and $m_3$ separately? Thank you in advance.

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  • $\begingroup$ A guess: The upper limit on the neutrino mass should be no larger than the upper limit on the mass of the heaviest flavor of neutrino. If there were a powerful and very brief pulse of neutrinos emitted by a cosmological event, there should be a temporal distribution of detection counts depending on distance to the event, the neutrino flavor masses, and their energies. Separating the detection counts out by flavor should reveal temporal distributions determined by the event, and a good model of the event should predict those distributions as a function of mass ratios of the three flavors. $\endgroup$ – S. McGrew May 16 '18 at 4:06
  • $\begingroup$ Related: physics.stackexchange.com/questions/21351/… $\endgroup$ – dmckee May 16 '18 at 17:13
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At the time the paper was written neutrino oscillations had not been discovered so it was assumed that if the electron neutrino had a mass then it would be well defined and constant. The analysis is based on this assumption.

We now know that neutrino oscillations exist, but the analysis still tells us something useful. The analysis is based on the travel time of the neutrinos, i.e. their velocities, which is linked to their kinetic energy and therefore mass. So it still calculates a mass limit, but it is an average mass limit for the three neutrino flavours not (as originally believed) a limit on the electron neutrino mass.

The analysis cannot tell us anything about the individual values of the three masses, only their average.

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The links are within a paywall, so I cannot check if they discuss neutrino oscillations. Look at the date of the publication, 1987.

The solar neutrino problem concerned a large discrepancy between the flux of solar neutrinos as predicted from the Sun's luminosity and measured directly. The discrepancy was first observed in the mid-1960s and finally resolved around 2002.

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When neutrino detectors became sensitive enough to measure the flow of electron neutrinos from the Sun, the number detected was much lower than predicted. In various experiments, the number deficit was between one half and two thirds.

Neutrino oscillations became mainstream physics after solving this discrepancy.

The present view is that cosmology can give a limit to the sum of the three neutrino masses.

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