Does charge conjugation affect parity?

[silmaril89]

"Notice that these transformations do not alter the chirality of particles. A left-handed neutrino would be taken by charge conjugation into a left-handed antineutrino, which does not interact in the Standard Model." --https://en.wikipedia.org/wiki/C-symmetry

The excerpt above seems to unambiguously answer this question. But, then:

"You can easily convince yourself (exercise II.1.9) that the charge conjugate of a left handed field is right handed and vice versa." --Quantum Field Theory in a Nutshell, A. Zee

These statements appear to be contradictory. What's going on here?

Also, it does seem easy to convince myself of Zee's comment (following Zee's convention that $\psi \to \psi_c = \gamma^2 \psi^\ast$):

Suppose $\psi$ is left-handed (i.e. $P_L \psi = \psi$ and $P_R \psi = 0$), then
$$P_L \psi_c = P_L \gamma^2 \psi^\ast = \gamma^2 P_R \psi^\ast = \gamma^2 (P_R \psi)^\ast = 0$$
and
$$P_R \psi_c = P_R \gamma^2 \psi^\ast = \gamma^2 P_L \psi^\ast = \gamma^2 (P_L \psi)^\ast = \psi_c$$
Therefore, it appears that Zee's comment is correct. Can anyone help me understand why the two quotes above are or are not in contradiction?

 

[Avodyne]

Both statements are technically true, but I think Zee's is misleading. If we work in basis where $\gamma_5$ is diagonal, then a Dirac field $\Psi$ can be written as a left-handed Weyl field $\chi$ stacked on top of a right-handed Weyl field $\xi^\dagger$,
$$\Psi=\pmatrix{\chi\cr\xi^\dagger}$$
The charge conjugate field is then
$$\Psi^c=\pmatrix{\xi\cr\chi^\dagger}$$
Now if we set $\xi=0$, then we recover Zee's statement (and your algebra). But I think it is more correct to say that the charge conjugate of the left-handed field $\chi$ is the left-handed field $\xi$. Then, if we use $\Psi$ as a Dirac field for neutrinos, $\chi$ creates left-handed neutrinos, and $\xi$ creates left-handed antineutrinos, which is consistent with the wikipedia statement.

 

[silmaril89]

Ok, thanks for the reply. I think I'm still a little confused, but you've put me in a particular direction to begin investigating this further.