Quark rephasing and CP conservation

For nondegenerate quark masses, independent phase changes of the up- and down-quark mass eigenfields alter the numerical CKM matrix by row and column phases without changing physics. Thus CP symmetry requires that some allowed phase convention makes the matrix real, rather than requiring real entries in every convention. With $N$ generic families, removing $2N-1$ phases from a unitary matrix leaves $(N-1)(N-2)/2$ physical CP-violating phases and $N(N-1)/2$ mixing angles.

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.