Pseudoscalar current of Majorana fields

[Andrea M.]

Consider a Majorana spinor
$$
\Phi=\left(\begin{array}{c}\phi\\\phi^\dagger\end{array}\right)
$$
and an pseudoscalar current $\bar\Phi\gamma^5\Phi$. This term is invariant under hermitian conjugation:
$$
\bar\Phi\gamma^5\Phi\to\bar\Phi\gamma^5\Phi
$$
but if I exploit the two component structure
$$
\bar\Phi\gamma^5\Phi=-\phi\phi+\phi^\dagger\phi^\dagger
$$
the invariance under hermitian conjugation seems lost
$$
-\phi\phi+\phi^\dagger\phi^\dagger\to\phi\phi-\phi^\dagger\phi^\dagger
$$
Where is the catch?

 

[Orodruin]

Where is the catch?

You are forgetting that you are dealing with Grassmann numbers and therefore miss a minus sign (in addition to not separating $\phi^\dagger$ and $\phi^c$ ...).

 

[Andrea M.]

You are forgetting that you are dealing with Grassmann numbers and therefore miss a minus sign (in addition to not separating $\phi^\dagger$ and $\phi^c$ ...).

Writing down the spinor indices the product $\phi\phi$ becomes
$$
\phi\phi=\phi^\alpha\phi_\alpha=\phi^\alpha\epsilon_{\alpha\beta}\phi^\beta
$$
the component $\phi^\alpha$ are Grassmann numbers but the product $\phi\phi$ should commute, am I wrong?
For the difference between $\phi^\dagger$ and $\phi^c$ I'm using the conventions of this review, so I have that
$$
\Psi=\left(\begin{array}{c}\chi_\alpha\\\eta^{\dot\alpha\dagger}\end{array}\right)\quad\quad\bar\Psi=\left(\eta^\alpha,\chi_{\dot\alpha}^{\dagger}\right)\quad\quad\Psi^c=\left(\begin{array}{c}\eta_\alpha\\\chi^{\dot\alpha\dagger}\end{array}\right)
$$

 

[Andrea M.]

Ok, i found the (silly) error:
$$
\bar\Phi\gamma^5\Phi=\Phi^\dagger\gamma^0\gamma^5\Phi
$$ so under hermitian conjugation this becomes
$$
\Phi^\dagger\gamma^5\gamma^0\Phi=-\Phi^\dagger\gamma^0\gamma^5\Phi=-\bar\Phi\gamma^5\Phi
$$
that imply
$$
\bar\Phi\gamma^5\Phi+h.c.=0
$$
the same result that we found exploiting the two component structure. Correct?