For two majorana field $\psi$ and $\chi$, is it true that $\bar{\psi} \chi = - \bar{\chi} \psi$?

Question

For two majorana field $\psi$ and $\chi$, which satisfy $\psi_{c}=\psi$ and $\chi_{c} = \chi$, where the charge-conjugation operation we define as $$ \Psi_{c} = C \Psi^{\ast} $$ where we work in the Weyl basis so that the matrix $C$ is given by $C = -i\gamma^2$.

It seems to me that the above implies $\bar{\psi}_{c} = ( C \psi^{\ast} )^{\dagger} \gamma^0 = \psi^{T} C^{\dagger} \gamma^0 = - \psi^{T} \gamma^0 C$, and so from this it follows that: $$ \bar{\psi} \chi = \psi^{\dagger} \gamma^0 \chi = \chi^{T} \gamma^0 \psi^{\ast} = \chi^{T} \gamma^0 C C \psi^{\ast} = - \left( - \chi^{T} \gamma^0 C \right) ( C \psi^{\ast} ) = - \bar{\chi}_c \psi_c = - \bar{\chi}\psi $$ where we have used $CC=I$, and the last equality uses the fact that these fields are Majorana fermions.

This implies that $\bar{\psi} \chi + \bar{\chi} \psi =0$ which seems wrong....where am I making a mistake?

Answer 1

I'm not adept in catching other people's mistakes, but I'll stick instead to the Bjorken & Drell v1 section 5.2 discussion and eliminate distractions. You are right in your instinct to look at the simplest bilinear, the scalar, and reject your answer, as it is odd, and $\psi=\chi$ would exclude a Dirac mass for Majorana spinors, which cannot be! So here is the trail map.

The conventions of that text, which are independent of the particular realizations and details all but guaranteed to bamboozle you, are $$ C=-C^{-1}= -C^T= -C^\dagger, $$ and imposing the Majorana condition yields $$\chi_c =C \bar{\chi}^{~T}=\chi, \qquad \psi_c= C \bar{\psi} ^{~T}=\psi ~~~~ \leadsto ~~~~\psi^T C =\bar{\psi}. $$

It then follows that $$ \bar{\psi} \chi = \psi^T C C \bar{\chi}^{~T} = -\psi^T \bar{\chi}^{~T}= \bar{\chi}\psi, $$ being a mere dot product of two vectors, and picking up a minus sign by anticommuting two Grassmann numbers at the very end.

From this point on, you may work out the evenness of the pseudoscalar and axial vector, as well, and the oddness of the vector and the tensor. It's all in the C properties of minus transposing gamma matrices.