Confusion about Dirac mass term In chiral basis, $\psi=\begin{pmatrix}
\psi_L\\
\psi_R
\end{pmatrix}$ and therefore, $\overline\psi=\psi^\dagger\gamma^0=\begin{pmatrix}
\psi^\dagger_L & \psi^\dagger_R
\end{pmatrix}\gamma^0=\begin{pmatrix}
\psi^\dagger_R & \psi^\dagger_L
\end{pmatrix}$. Hence, by matrix multiplication, $\overline \psi \psi=\psi^\dagger_R \psi_L+\psi^\dagger_L \psi_R$.
Again using chiral projection operators it can be shown that, $\overline\psi\psi=\bar\psi_R \psi_L+\bar\psi_L \psi_R$. 
Therefore, these two relations suggest that, we can also write, $\overline\psi$ as $\overline\psi=\begin{pmatrix}
\bar\psi_R & \bar\psi_L
\end{pmatrix}$. 


*

*Am I right? If yes, how can I show $\overline\psi=\begin{pmatrix}
\bar\psi_R & \bar\psi_L
\end{pmatrix}$starting from the $\overline\psi=\psi^\dagger\gamma^0=\begin{pmatrix}
\psi^\dagger_R & \psi^\dagger_L
\end{pmatrix}$? Moreover, this implies:
$$\psi^\dagger_R \psi_L+\psi^\dagger_L \psi_R=\bar\psi_R \psi_L+\bar\psi_L \psi_R$$

*Am I right? If yes, how can I prove the last relation starting from either side of it?
 A: I think you're confusion is with notation since, unfortunately, there are two common ways to denote projected spinors. One form is to write:
\begin{equation}
\psi \equiv \left( \begin{array}{c} 
\psi _L  \\  
\psi _R  
\end{array} \right) 
\end{equation} 
In this notation $ \psi _L $ and $ \psi _R $ are two component Weyl spinors. However, a second notation is also used where 
\begin{equation} 
\psi _L \equiv P _L \psi , \quad \psi _R \equiv P _R \psi 
\end{equation} 
and $ P _{L/R } $ are the projection operators. Now $ \psi _L $ and $ \psi _R $ are four component spinors with a zero value for two the components. 
In the first notation (where $ \psi _{ L/ R } $ are Weyl spinors) the Dirac term takes the form,
\begin{equation} 
m \bar{\psi} \psi = m \left( \psi _R ^\dagger \psi _L + h.c. \right) 
\end{equation} 
and in the second (where $ \psi _{ L/R} $ are four component objects) it takes the form,
\begin{equation} 
m \bar{\psi} \psi = m \left( \overline{ \psi _R}  \psi _L + h.c. \right) 
\end{equation} 
Its just a matter of convention and the end result is the same. 
