= Solution
Write $J_V^\mu=\bar\psi\gamma^\mu\psi$ for the <Dirac current> and $J_A^\mu=\bar\psi\gamma^\mu\gamma^5\psi$ for the <axial current>. Under <parity symmetry in quantum field theory>, the <Dirac spinor> phases cancel and
$$
\hat P J_V^\mu(x)\hat P^{-1}=\mathbb P^\mu{}_{\nu}J_V^\nu(x_P),\qquad
\hat P J_A^\mu(x)\hat P^{-1}=-\mathbb P^\mu{}_{\nu}J_A^\nu(x_P).
$$
The extra sign for the <axial current> follows from $\gamma^0\gamma^5\gamma^0=-\gamma^5$. Combining these transformations with part (c), both currents transform under <CP symmetry> as $J_{V,A}^\mu(x)\mapsto-\mathbb P^\mu{}_{\nu}J_{V,A}^\nu(x_P)$. The vector <relativistic quantum field> transforms as $V^\mu(x)\mapsto\eta_C\mathbb P^\mu{}_{\nu}V^{\nu\dagger}(x_P)$. The two parity matrices cancel in the <tensor contraction>, giving
$$
\boxed{(\widehat{CP})\left[\bar\psi\gamma^\mu(g_v-g_a\gamma^5)\psi V_\mu\right](x)(\widehat{CP})^{-1}
=-\eta_C\left[\bar\psi\gamma^\mu(g_v-g_a\gamma^5)\psi\right](x_P)V_\mu^\dagger(x_P).}
$$
For a real vector <relativistic quantum field> with $\eta_C=-1$, the interaction is \b[invariant under <CP symmetry>]. A complex vector <relativistic quantum field> instead maps to the conjugate-field interaction, which must be compared with its <Hermitian conjugation> partner.
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