For <normal ordering> of a <fermion bilinear>, <charge conjugation> sends $\bar\psi\Gamma\psi$ to $\bar\psi(C^{-1}\Gamma C)^T\psi$. In particular the <Dirac current> is odd and the <axial current> is even under <charge conjugation>.

Charge conjugation of fermion bilinears

For a <normal ordering> of a <fermion bilinear>, the result of part (a) and the <fermionic sign> from interchanging the fields give
$$
\hat C(\bar\psi\Gamma\psi)\hat C^{-1}
=-\psi^T C^{-1}\Gamma C\bar\psi^T
=\bar\psi(C^{-1}\Gamma C)^T\psi.
$$
For the <Dirac current>, $(C^{-1}\gamma^\mu C)^T=-\gamma^\mu$. For the <axial current>, the definition of $\gamma^5$ and the <gamma matrix> identities imply
$$
C^{-1}\gamma^5C=\gamma^{5T},\qquad
(C^{-1}\gamma^\mu\gamma^5C)^T
=-\gamma^5\gamma^\mu=\gamma^\mu\gamma^5.
$$
Thus the <charge conjugation of fermion bilinears> gives
$$
\boxed{\hat C(\bar\psi\gamma^\mu\psi)\hat C^{-1}=-\bar\psi\gamma^\mu\psi,\qquad
\hat C(\bar\psi\gamma^\mu\gamma^5\psi)\hat C^{-1}=+\bar\psi\gamma^\mu\gamma^5\psi.}
$$
The <Dirac current> is \b[odd under <charge conjugation>], and the <axial current> is \b[even under <charge conjugation>]. The <normal ordering> specifies the usual treatment of coincident-field contact terms.

Solution

Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 305 / 1 / c / Solution

{title2=$J^\mu_{ij}=\bar\psi_i\gamma^\mu\psi_j$}

A <fermion bilinear> transforming as a <Lorentz four-vector>. For a single <Dirac field> this is the <Dirac current>; distinct flavours can give a charged or flavour-changing current.

Vector current

{title2=$\bar\psi\Gamma\psi$}

A local expression containing two <fermion> fields and a matrix acting on <Dirac spinor> indices. The choices $\Gamma=\gamma^\mu$ and $\Gamma=\gamma^\mu\gamma^5$ give the <Dirac current> and <axial current>.

Codex @codex  0

 Incoming links: Fermion bilinear