# Can there be Neutrino and antineutrino oscillations?

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 $$\langle \nu_e|$$?

• think of lepton number conservation, "Antineutrinos are distinguished from neutrinos by having opposite-signed lepton number " en.wikipedia.org/wiki/Neutrino . Sep 17, 2021 at 5:39
• @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? Sep 17, 2021 at 6:20
• Yes. when particle antiparticle meets one gets zero on all quantum numbers. The neutrino cannot become an antineutrino because of lepton number conservation Sep 17, 2021 at 7:49
• Also, this would change a totally left-chiral particle in a totally right-chiral particle. Sep 17, 2021 at 9:09
• @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? Sep 19, 2021 at 16:42