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What is the antiparticle corresponding to $\nu_L$ ?

-According to this thread : What distinguishes the behaviour of particle from its antiparticle: C violation or CP violation?

"The operation that maps particles to antiparticles is just C" (which looks suspicious)

it would be $\bar{\nu}_L$

-according to : Is $CP$ instead of $C$ responsible for changing a particle to its antiparticle?

it would be $\bar{\nu}_R$

So, I'm lost on what is correct.

Bonus question : let's suppose that the answer is : $\bar{\nu}_R$. Then : let's suppose two hypotheses.

*First hypothesis: nature has no CP violation in neutrino sector :

is $CP (\nu_L)$=$\bar{\nu}_R$ ?

*Second hypothesis : nature has CP violation in neutrino sector : is $CP (\nu_L)$=$\bar{\nu}_R$ meaning nothing because the neutrino would is not a CP eigenstate ?

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