= CP invariance of a neutral chiral vector interaction
{c}
{title2=$\mathcal L=(j_\mu-j_{5\mu})V^\mu$}
The <vector current> is odd and the <axial current> is even under <charge conjugation>. Under <parity symmetry in quantum field theory>, the first is a vector and the second a pseudovector. Thus their difference obeys $j_L^\mu(x)\mapsto-P^\mu{}_{\nu}j_L^\nu(x_P)$ under <CP>, with $P=\operatorname{diag}(1,-1,-1,-1)$. If a neutral vector has $V^\mu(x)\mapsto\xi P^\mu{}_{\nu}V^\nu(x_P)$ under <CP>, the interaction transforms by the factor $-\xi$. It is invariant for $\xi=-1$. Being a real field restricts its intrinsic phases but does not alone fix this sign. The conventional C-odd neutral gauge-vector assignment makes real diagonal <Z boson> vector and axial couplings CP invariant, despite their separate C and P violations.
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