Neutrions are produced and detected as flavor eigenstates $\nu_{\alpha}$ with $\alpha=e, \mu, \tau$. These states have no fixed mass, but are the combinations of three mass eigenstates $\nu_{k}$ with $k=1, 2, 3$, with mass $m_1$, $m_2$ and $m_3$, respectively. My questions are:
a) do neutrinos travel from source to the detector as flavor eigenstates or mass eigenstates?
b) is it possible to know which mass eigenstate the neutrino is in?
Best Answer
(a) They start as a flavor eigenstate, which is a super position of mass eigenstates. The mass eigenstates have different time evolution, hence the state is, in general, a mixed state in either basis.
(b) No. As an analogy, consider polarized photons and Faraday rotation--it may start out + polarized, rotate to a mixture of + & - (with coefficients a & b) and then at your + detector you see it $a^2$ fraction of the time, and $b^2$ you don't. In either case, you can't say which state a particular photon was in.
(b') Can we detect a $\nu_e$ and know it's mass? Can it have the mass of a $\nu_{\tau}$? The $\nu_e$ doesn't have "a" mass, it has 3:
$|\nu_e\rangle=0.82|\nu_1\rangle+0.54|\nu_2\rangle-0.15|\nu_3\rangle$
while a tau-neutrino:
$|\nu_{\tau}\rangle=0.44|\nu_1\rangle-0.45|\nu_2\rangle-0.77|\nu_3\rangle$
So, "yes", if we measure it's mass, then it will have a mass that a tau neutrino mass measurement could yield.
In theory: it's not a sensible question to ask, since flavor eigenstates aren't mass eigenstates.
In practice: we do not know the masses of the mass eigenstates, and their differences are much less than an eV--so how are you going the measure that?