Neutrino mass hierarchy I am a beginner in particle physics. I just want to know why there is a sign ambiguity in the neutrino mass square difference $\Delta m_{31}^2$ and not in $\Delta m_{21}^2$?
 A: How we label the eigenstates of the neutrino mass matrix is not relevant for the answer. The important point is that vacuum oscillations are only affected by the absolute value of the squared mass differences $\Delta m^2_{ij} = m^2_i - m^2_j$ (the indices may be in the opposite order depending on the notation).
What allows to determine the sign of the so-called solar squared mass difference, $\Delta m^2_{21}$, is the fact that for the electron neutrinos emitted in the center of the sun and travelling towards earth, neutrino oscillations start to take place in the very dense interior of the sun, before they propagate in the outer space. If one takes into account the effect of matter, the oscillation pattern changes (it is called MSW effect after the names of Mikheyev-Smirnov-Wolfenstein) and the sign of the squared mass difference plays a role, so $\Delta m^2_{21}$ becomes measurable.
To measure also the sign of $\Delta m^2_{31}$ we should measure the effect of neutrino-matter interactions for neutrinos that cross the interior of earth, but in that case the density is much smaller and consequently the effect is smaller.
For more details, see for example
C. Giunti, C. W. Kim, "Fundamentals of Neutrino Physics and Astrophysics", New York, Oxford University Press, 2007, p. 369, ISBN 978-0-19-850871-7
A: I disagree with the answer provided by @StefanoGariazzo. 
Oscillation probabilities are insensitive to the signs of $\Delta m_{ij}^2$ for all $i$ and $j$.
However, the numbering of massive neutrinos, $\nu_{1}$ $\nu_2$ and $\nu_3$, is arbitrary. The usual convention is to number the neutrinos such that $m_2>m_1$ which makes $\Delta m^2_{21}>0$ by convention. With this choice made, there can be two possibilities: either $m_1<m_2<m_3$ or $m_3<m_1<m_2$. This is why only absolute value of the atmospheric mass squared difference can be inferred from oscillation experiments. This is explained in PDG and Lecture-2 by F. Feruglio.
