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At the KIT they have been measuring the mass of the electron neutrino with a huge spectrometer (i.e. they make an enormous effort) and already published limits on the highest possible electron neutrino mass (it is already less than $1$eV). How is this compatible with neutrino oscillations? The electron neutrino flavor state is a mixture of at least 2 mass eigenstates:

$$| \overline{\nu_e} \rangle = \cos\theta | \nu_1\rangle + \sin \theta |\nu_2\rangle $$

where $| \nu_1\rangle$ is the first mass eigenstate and $|\nu_2\rangle $ is the second mass eigenstate. I assume that the mass of both is significantly different. $|\overline{\nu_e}\rangle$ is the electron neutrino flavor eigenstate. As far as I understand the mass of the flavor eigenstate is not known and upon measurement it would project on one of the 2 possible mass eigenstates. So what are they measuring actually?

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If you are asking about the KATRIN tritium beta decay experiment, they are measuring the endpoint of the product electron spectrum in $$ ^3_1H \to ^3_2{He^+} + {e^-} + \bar{\nu}_e, $$ whose half-life is 12ish years.

But, as you observe, $\bar{\nu}_e$ has no well-defined mass, since it is not a mass eigenstate, only a convenient shorthand for the superposition of states coupling to the electron. So there are three possible decays, to $\bar\nu_1,\bar\nu_2,\bar\nu_3$, not observed directly, but limited in the maximum energy/momentum they may fly away with, and thus constraining the maximum energy of the electron momentum measured. (There is an additional complication that $\bar{\nu}_e$ overlaps with the heaviest $\bar{\nu}_3$ only very little, but no mind.)

The actual composite quantity measured is an "effective" incoherent average quantity, $$ (m^{eff}_{\nu_e})^2= \sum_{i=1}^3 m_i^2|U_{ei}|^2, $$ pulled up by the $\bar{\nu}_3$ in the normal hierarchy, but whose influence is diminished by the smallness of the corresponding PMNS U mixing element ($|U_{e3}|^2$~2%). After elaborate analysis, they conclude what an upper bound for it might be.

There are no oscillations involved here!

enter image description here

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  • $\begingroup$ Thank you for the answer. So they measure $m^{eff}_{\overline{\nu}_e}$ ? If this is the case, although they claim that the measurement is important for cosmology (& particle physics), does it help to know this very special quantity if actually the real mass $m_{\overline{\nu}_e}$ is needed for this? $\endgroup$ Oct 22, 2023 at 18:25
  • $\begingroup$ The electron anti neutrino has no mass! But the mass of the 3rd (heaviest) eigenstate is important for cosmology… $\endgroup$ Oct 22, 2023 at 18:37
  • $\begingroup$ Sorry, I meant $m_{\nu_1}$. If $m_{\nu_3}$ is the most important quantity to know, how this is related to $m^{eff}_{\nu_e}$ ? And how many parameters are involved in this which also have to be known with a sufficient precision? $\endgroup$ Oct 22, 2023 at 19:20
  • $\begingroup$ I was thinking of the normal hierarchy, by now all but established, so the heaviest is $\nu_3$. I think the incoherent average measured limits the precision quite a bit, and there are lots of parameters and other experimental input (from oscillation experiments) involved. But I really haven't looked into their papers; maybe you have or can. I am a mere theorist... $\endgroup$ Oct 22, 2023 at 20:34
  • $\begingroup$ Cosmology bounds another, complementary, quantity, $\sum_i m_{\nu_i}$, dominated by the heaviest $\nu_3$... $\endgroup$ Oct 22, 2023 at 20:45

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