# How do we know that the neutrino flavour states are the linear combination of mass states?

I am studying the neutrino oscillation phenomenom in which the flavour of a neutrino can change when it evolves in space-time.
What I understand is that this means that the 3 neutrino flavour states can be built in a common base of states. That is, if $$\langle\nu_\mu|\nu_e\rangle \neq 0 \xrightarrow{} |\nu_\alpha\rangle = \sum_iU_{\alpha i}|\nu_i\rangle$$ With $$\alpha$$, and $$i$$ denoting the 3 possible flavours and base states respectively.
My question is, if this interpretation is correct, how do we know that this $$|\nu_i\rangle$$ states are mass states?

$$\textbf{Edit:}$$ Maybe what I'm trying to ask, is: how do we mathematically come up with the energy/mass eigenstates? Where do they come from?

• How do you know that the $| \nu_i \rangle$ are the mass eigenstates? You diagonalize the neutrino mass matrix and determine the corresponding eigenvectors. Mar 8 at 13:51
• @Hyperon And by what magic do you know the neutrino mass matrix? Mar 10 at 17:32
• @John Doty In the minimal version of the SM, the neutrinos are exactly massless and, as a consequence, there are no neutrino oscillations, clearly in conflict with the experimental evidence of the measured neutrino mass differences and mixing angles. Therefore, the SM must be extended by adding additional leptonic and/or scalar degrees of freedom. Presently, the ultimate form of this necessary extension is not known (there are numerous competing models), further experimental input (e.g. Majorana/Dirac nature of $\nu$, CP-violation in the lepton sector) will lead to further restrictions. / ctd Mar 10 at 21:37
• @John Doty But independently of the actual form of the extension of the standard model, you will have a mass matrix in the neutral lepton sector and the diagonalization of this mass matrix allows to identify the (neutrino) mass-eigenfields as linear combinations of the weak eigenfields. This allows a largely model-independendent discussion of neutrino oscillations in terms of neutrino mass differences, mixing angles and $CP$-violating phases. Mar 10 at 21:45
• @Hyperon OK, show me the values of the mass matrix elements. If you can't, then "You diagonalize the neutrino mass matrix and determine the corresponding eigenvectors." is empty of physical content. Mar 10 at 22:25