I know the flavor and mass eigenstates are different, but are they related? What I mean is, in a process like fusion where electron neutrinos are created, do they start in the 1 mass eigenstate? My knowledge of QFT is nonexistent, so I've never really seen a Neutrino field written out mathematically, so I don't really know what's going on. I just see the mass eigenkets and I am comfortable with that, but I wasn't clear on this question. The mass and flavor eigenstates are different, but in what ways are they related?
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The mixing matrix tells you exactly the correspondence between the mass states and the flavor states. This is true in the quark sector, too, but unlike the quarks where the mass--flavor identification is pretty strong, it is very weak in the neutrino sector.
The elements of the mixing matrix are exactly the flavor content of each mass state $$ \nu_\alpha = \sum_{i=1}^3 V_{\alpha,i} \nu_i \,, $$ for $V$ the mixing matrix (in the notation used in 2).
This image (linked rather than imported because I can't find any license information) shows the flavor makeup of each mass state graphically for both sectors. I believe that there is an assumption of normal hierarchy in that figure as the results do depend a little on the hierarchy. (Image from here.)
The strongest identification in the neutrino sector is between $\nu_1$ and $\nu_e$, and even that is very rough: flavor changes could occur almost immediately, as evidenced by the recent success of the $\theta_{13}$ experiments (Daya Bay, RENO and Double Chooz) in observing electron-anti-neutrino oscillation to more than five sigma at ranges on order of 1 kilometer.
