Question

Is $\overline{\psi_{L}^{c}}\psi_{R}^{c}=\overline{\psi_{R}}\psi_{L}$ true for two different spin 1/2 fermions?

Answer 1

In the context of seesaw mechanism or Dirac and Majorana mass terms, one often see the following identity

ψcL¯¯¯¯¯¯ψcR=ψR¯¯¯¯¯¯ψL.ψLc¯ψRc=ψR¯ψL.

Here, I am using 4 components notation in the chiral basis. The convention for the charge conjugation is ψc=−iγ2ψ∗ψc=−iγ2ψ∗, and ψcL=(ψL)cψLc=(ψL)c. The following is my effort of proving it.

ψcL¯¯¯¯¯¯ψcR=iγ2ψ∗L¯¯¯¯¯¯¯¯¯¯¯¯¯iγ2ψ∗R=(iγ2ψ∗L)+γ0iγ2ψ∗R=ψTLiγ2γ0iγ2ψ∗RψLc¯ψRc=iγ2ψL∗¯iγ2ψR∗=(iγ2ψL∗)+γ0iγ2ψR∗=ψLTiγ2γ0iγ2ψR∗

=−ψTLγ2γ0γ2ψ∗R=ψTLγ2γ2γ0ψ∗R=−ψTLγ0ψ∗R=−ψL,iγ0ijψ∗R,j.=−ψLTγ2γ0γ2ψR∗=ψLTγ2γ2γ0ψR∗=−ψLTγ0ψR∗=−ψL,iγij0ψR,j∗.

Now if ψL,iψL,i and ψR,iψR,i are anticommuting, then one have

−ψL,iγ0ijψ∗R,j=ψ∗R,jγ0jiψL,i=ψR¯¯¯¯¯¯ψL.

Answer 2

By ψL=χLψL=χL, I mean to replace all the ψLψL by χLχLin the above derivation. So now χLχL and ψRψR are two different Weyl spinors. They are not just the projection of a single Dirac fermion. I guess your answer is that if χLχL and ψRψR are two different fields, then they commute. So we will have χcL¯¯¯¯¯¯ψcR=−ψR¯¯¯¯¯¯χLχLc¯ψRc=−ψR¯χL instead of χcL¯¯¯¯¯¯ψcR=ψR¯¯¯¯¯¯χLχLc¯ψRc=ψR¯χL. –