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The labeling of the neutrino mass eigenstates is arbitrary [PDG Neutrino Review]. Regularly it is assumed for convenience that $$\Delta m^2_{21} = m_2^2 - m_1^2 > 0.$$ Then, values for the mixing angles are given that lead to the PMNS matrix $V$ with the $V_{1,1}$ entry being the largest in both its row and column [i.e. Wikipedia]. This implies that the lighter mass eigenstate is mostly of electron flavor (even in an inverted hierarchy setting, between $\nu_e$ and $\nu_\mu$, it is always $\nu_e$ that is shown to be lighter).

The vacuum oscillation observables only depend on the cosine of $\Delta m^2$, or the squared sine, so its acutal sign is irrelevant here. In matter, there is a CP violation induced that is proportional to the $\Delta m^2$, but since we have three flavor mixing and $\Delta m_{31}^2 \gg \Delta m_{21}^2$, these effects will be used to determine the sign of $\Delta m^2_{31}$.

What is the status of the sign of $\Delta m^2_{21}$?

What implications would a negative sign of $\Delta m^2_{21}$ have?

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    $\begingroup$ I thought $\Delta m^{2}_{21}$ was positive by definition, rather than it being a convenient assumption. $\endgroup$ – dukwon Dec 19 '16 at 13:09
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    $\begingroup$ @dukwon Well, you can use it do define a numbering scheme, but the question then becomes what I wrote in the second paragraph: How do we know the lighter of the two states is mostly of electron flavor? $\endgroup$ – Neuneck Dec 19 '16 at 13:51
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This is the neutrino mass hierarchy conundrum. It can only be resolved experimentally. The Hyper Kamiokande experiment aims to resolve this.

Hyper-Kamiokande will observe large numbers of neutrinos produced by the collisions of cosmic rays with nuclei in the atmosphere. Those which are produced in the atmosphere on the opposite side of the Earth will be influenced by its matter on their way to the detector. Accordingly, the oscillations of such muon neutrinos into electron neutrinos as well as the oscillations of muon antineutrinos into electron antineutrinos will be affected. However, the extent of these effects depends upon the mass hierarchy such that for a normal hierarchy oscillations into electron neutrinos are enhanced, while for an inverted hierarchy oscillations into electron antineutrinos are enhanced. For this reason, the number of events coming from the opposite side of the Earth that oscillate into electron neutrinos will be larger if the hierarchy is normal than if it is inverted (Figure 2). On the other hand, the number of events that oscillate into electron antineutrinos will be larger for the inverted hierarchy than for the normal hierarchy. Though the change in the event rate due to the mass hierarchy is only between about 5 and 15%, because Hyper-Kamiokande is so large it will be able to detect even this small difference.

This is extended to the question of the position of the third neutrino, which is also to be resolved experimentally , probably in the next decade or so.

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