Fermion propagator in terms of $psi_{L,R}$ and $bar{psi}_{L,R}$?

Suppose we have a fermion lagrangian
begin{equation}
mathcal{L}=bar{psi}igamma^{mu}partial_{mu}psi-mbar{psi}psi,
end{equation}
which can be written in terms of $psi_{L,R}$ and $bar{psi}_{L,R}$
begin{equation}
mathcal{L}=bar{psi}_Ligamma^{mu}partial_{mu}psi_L+bar{psi}_Rigamma^{mu}partial_{mu}psi_R-m(bar{psi}_Lpsi_R+bar{psi}_Rpsi_L).
end{equation}

So how should I write the propagator in terms of $psi_{L,R}$ and $bar{psi}_{L,R}$?
Does the propagtor contain four pieces $langlepsi_Lbar{psi}_Lrangle, langlepsi_Rbar{psi}_Rrangle, langlepsi_Rbar{psi}_Lrangle, langlepsi_Lbar{psi}_Rrangle$?