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I am working from the book Massive Neutrinos in Physics and Astrophysics by Mohapatra and Pal (which is available here). On page 66, the authors claim that $\psi_{L}$ is the $\mathcal{CPT}$ conjugate of $\hat{\psi}_R$ (where the hat means antiparticle).

I have attempted to verify this using the Weyl basis and equation 4.60 on page 64 by the following method: Take the Dirac spinor to be $$ \psi=\left(\begin{array}{c} \chi\\ -i\sigma_{2}\phi^{*} \end{array}\right) $$ and the charge conjugation operator (except for phase factors) $$ \mathcal{C}=\left(\begin{array}{cc} 0 &i\sigma_{2} \\ -i\sigma_{2} & 0 \end{array}\right) $$ Using this I find $$ \psi_{L}=\left(\begin{array}{cc} \chi\\ 0 \end{array}\right), \hat{\psi}_{R}=R\hat{\psi}=\left(\begin{array}{cc} 0\\ -i \sigma_{2} \chi^{*} \end{array}\right) $$ and the $\mathcal{CPT}$ conjugate of $\psi_{L}$ is (up to a phase)$$ -\gamma_{5}\psi^*_L = \left(\begin{array}{cc} 1 & 0\\ 0 & -1 \end{array}\right)\left(\begin{array}{cc} \chi^*\\ 0 \end{array}\right)=\left(\begin{array}{cc} \chi^*\\ 0 \end{array}\right) $$ But this is nothing like my expression for $\hat{\psi}_R$.

Where am I going wrong?

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