# Why neutrino mass hierarchy scenario do not include the case $m_2<m_1<m_3$ and the case $m_3<m_2<m_1$?

The neutrino mass hierarchies are made of two hypotheses/scenarios:

• the normal hierarchy: $$m1
• the inverted hierarchy: $$m3

Why neutrino mass hierarchy scenario do not include also the cases:

• $$m2?

• $$m3?

I have seen the post: Neutrino mass hierarchy

but two people are in contradiction and there was no conclusion on who is right. @SRS is saying in this thread that it is a convention to have $$m_1. Is he right? wrong? Is there a proof of the convention from a statement in a document?

We know that $$m_1$$ is less than $$m_2$$ from solar neutrino measurements of the MSW effect. Here's a quote from a paper that addresses this

Solar neutrinos allow to fix the sign of $$\Delta m^2_{21}$$for the standard value of $$V_e$$. The sign determines the resonance channel (neutrino or antineutrino) and the mixing in matter. The facts that due to smallness of the 1-3 mixing the problem is reduced approximately to the $$2\nu$$-problem and that suppression of signal averaged over the oscillations at high energies is stronger than 1/2, selects $$\Delta m^{2}_{21} > 0$$.

Put simply we measure the survival probability of solar neutrinos to be (more or less) like this... Image from Wikipedia.

Where the probability at low energies is ~60% and at high energies is about 30%. If $$\Delta m^{2}_{21}$$ were negative you would see at high energies the survival probability dip to some value of ~50% at ~1MeV then go back up to ~60% at higher energies. (Caveat, most of the numbers above are made up, take them with a grain of salt).

• Is m1 < m3 < m2 ruled out? – JollyJoker Feb 25 at 9:12
• Yes, that possibility is ruled out by the fact that $\Delta m^{2}_{21}$ is ~$10^{-5}$ eV$^2$ and $\Delta m^{2}_{31}$ is around $10^{-3}$ eV$^2$. I.e the mass difference between the $m_{1}$ state and the $m_{3}$ state is very large compared to the difference between the $m_{1}$ state and the $m_{2}$ state. So the only remaining possibilities are $m_{1} < m_{2} < m_{3}$ or $m_{3} < m_{1} < m_{2}$. – mze Feb 26 at 18:04

I was also trying to find a concrete answer to this question and just came across this quote from https://arxiv.org/abs/1801.04946;

Note that the observation of matter effects in the Sun constrains the product $$\Delta m_{21}^2 \cos 2θ_{12}$$ to be positive. Therefore, depending on the convention chosen to describe solar neutrino oscillations, matter effects either fix the sign of the solar mass splitting $$\Delta m_{21}^2$$ or the octant of the solar angle $$θ_{12}$$, with $$\Delta m_{21}^2$$ positive by definition.

So the answer appears to be both. Though I have seen more documents fixing the octant of the angle than the sign of the splitting.