Neutrino oscillations are sensitive to $\Delta m^2$, but not to the absolute mass scale, therefore we have the problem of the mass hierarchy.

Which experiment could resolve this problem? Is there any proposal?


The mass hierarchy problem is distinct from the problem of measuring the absolute neutrino mass scale. The neutrino mass hierarchy problem refers to the fact that we currently only have measured the absolute value of $\Delta m_{32}^2$ which is defined as:

$$\Delta m_{32}^2 \equiv m_3^2 - m_2^2$$

and therefore we don't know whether $m_3$ is greater than or less than $m_2$.

Currently (2018) the experiment NOVA has already disfavored the "inverted" hierarchy (where $m_3 < m_2$) at 95% confidence (see https://arxiv.org/abs/1806.00096). However, in high energy physics the threshold for claiming a discovery is usually set at $5\sigma$ which corresponds to a confidence level of 99.9999%.

The T2K experiment in Japan also favors the normal hierarchy with a posterior probability of 70% (see https://arxiv.org/abs/1701.00432).

The DUNE experiment which is currently under construction will also have some sensitivity to the mass hierarchy. For an in-depth review see https://arxiv.org/abs/1506.07917.

As for the other problem of determining the absolute neutrino mass scale, the KATRIN experiment in Germany is expected to have a limit on the mass of the electron neutrino on the order of $0.2~\mathrm{eV}$.

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