= CP transformation of a charged quark current
{c}
{title2=$\mathcal L_{cc}(V)\xrightarrow{CP}\mathcal L_{cc}(V^*)$}
Under <charge conjugation>, $\bar u\gamma^\mu(1-\gamma^5)d$ becomes $-\bar d\gamma^\mu(1+\gamma^5)u$. <Parity> reverses the chirality again and applies the Lorentz parity matrix. The minus sign cancels that of charge conjugation on the charged vector field. Thus each charged-current operator is exchanged with its Hermitian conjugate under <CP symmetry>, giving the displayed transformation for the <CKM matrix> $V$. For nondegenerate quark masses, physical CP conservation means that allowed quark rephasings can make $V$ real. A nonzero <Jarlskog invariant> obstructs this. A merely complex phase convention is not proof of physical CP violation, and $V=I$ is a CP-conserving counterexample to unconditional noninvariance.
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