= CP transformation of a non-Abelian gauge connection
{c}
{title2=$\mathcal A_\mu^{CP}=-P_\mu{}^\nu\mathcal A_\nu^T(x_P)$}
For vector coupling to a Dirac <fermion>, write the Hermitian color connection $\mathcal A_\mu=A_\mu^aT^a$. <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 $P=\operatorname{diag}(1,-1,-1,-1)$. The transpose acts only on color indices; intrinsic <fermion> phases cancel in the bilinear.
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