CP transformation of a non-Abelian gauge connection (${\mathcal{A}}_{\mu }^{CP}=-P_{\mu }{}^{\nu }{\mathcal{A}}_{\nu }^T(x_P)$)

For vector coupling to a Dirac fermion, write the Hermitian color connection ${\mathcal{A}}_{\mu }=A_{\mu }^a{T}^a$. Charge conjugation reverses the vector current and transposes its color matrix, requiring ${\mathcal{A}}_{\mu }^C=-{\mathcal{A}}_{\mu }^T$. Combining with parity gives ${\mathcal{A}}_{\mu }^{CP}(x)=-P_{\mu }{}^{\nu }{\mathcal{A}}_{\nu }^T(x_P)$, where $P=\mathrm{diag}(1,-1,-1,-1)$. The transpose acts only on color indices; intrinsic fermion phases cancel in the bilinear.