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I don't know have rigor knowledge in neutrino oscillations. I just know that there is not a definite mass state for a definite flavor Just like with spin where you might have eigenstate $|+\rangle_z$ but that would not have definite spin angular momentum in $x$ or $y$ direction.


For two-level system, We can see the mixing of the state with $$\begin{pmatrix} \nu_e\\ \nu_\mu \end{pmatrix}=\begin{pmatrix} \cos\theta& \sin\theta\\ -\sin\theta& \cos\theta \end{pmatrix}\begin{pmatrix} \nu_1\\ \nu_2 \end{pmatrix}$$

If we start with $\psi(0)=|\nu_e\rangle $ then at later instant of time $$|\psi(t)\rangle =U(t)|\psi(0)\rangle =U(t)|\nu_e\rangle=U(t)[|\nu_1\rangle \cos \theta+|\nu_2\rangle \sin\theta]=e^{-iE_1 t/\hbar}\cos\theta|\nu_1\rangle +e^{-iE_2t/\hbar}\sin\theta|\nu_2\rangle $$

One can find the transition probability $P(\nu_e\rightarrow \nu_\mu)$.


Is there oscillation between neutrino and antineutrino states? I don't how to work with neutrino and antineutrino. How to represent an antineutrino state? According to Wikipedia, It says, they are denoted by the complex conjugate. Is that mean bra vector $\langle \nu_e|$?

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  • $\begingroup$ think of lepton number conservation, "Antineutrinos are distinguished from neutrinos by having opposite-signed lepton number " en.wikipedia.org/wiki/Neutrino . $\endgroup$
    – anna v
    Sep 17, 2021 at 5:39
  • $\begingroup$ @annav Lepton number should be conserved if the neutrino and antineutrino have opposite lepton then a neutrino can't be converted to anti-neutrino. Is this what you mean? $\endgroup$ Sep 17, 2021 at 6:20
  • $\begingroup$ Yes. when particle antiparticle meets one gets zero on all quantum numbers. The neutrino cannot become an antineutrino because of lepton number conservation $\endgroup$
    – anna v
    Sep 17, 2021 at 7:49
  • $\begingroup$ Also, this would change a totally left-chiral particle in a totally right-chiral particle. $\endgroup$
    – JulianDeV
    Sep 17, 2021 at 9:09
  • $\begingroup$ @annav There can't be oscillation between neutrino and anti-neutrino states. Is it possible that to have oscillations between two mass eigenstates of antineutrino? How do we generally work with anti-neutrino states? $\endgroup$ Sep 19, 2021 at 16:42

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There is an additional intricacy involved when discussing antineutrinos. Roughly speaking there are two possible ways of introducing mass terms for Fermions in QFT: We can have either Majorana or Dirac Fermions.

All charged Fermions need to be Dirac Fermions since for Majorana Fermions particle and antiparticle coincide i.e. cannot have opposite electric charges unless they are neutrally charged.

For neutrally charged particles such as Neutrinos, Majorana mass terms are a plausible option. There are actually several good reasons why Neutrinos should indeed be Majorana Fermions. On the one hand, the seesaw mechanism could then be used to produce extremely light left handed neutrinos and (yet unobserved) very heavy right handed neutrinos. This naturally explains the small neutrino masses that we observe. On the other hand, if the neutrinos were Majorana Fermions, then they would provide Lepton number violating processes. These can be relevant for Lepto- and Baryogenesis as is explored for example in Common origin of baryon asymmetry, dark matter and neutrino mass.

The difference between Majorana and Dirac Fermions can often lead to confusion - especially if it is not explicitly stressed in the literature that you are using. So maybe this is also the source of your problem.

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