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CP conjugates the charged-current coupling and replaces $V_{\mathrm{CKM}}$ by $V_{\mathrm{CKM}}^{\ast }$. CP is conserved only if quark-field rephasings can make the entire matrix real. For three generations an irreducible phase prevents this. Equivalently, the rephasing-invariant Jarlskog invariantis nonzero for distinct rows and columns. Since $J$ changes sign under complex conjugation, $J\ne 0$ proves CP violation in electroweak interactions.

The Standard Model has two independent sources of CP violation. Weak CP violation comes from the irreducible phase of the Cabibbo-Kobayashi-Maskawa matrix; its basis-independent measure is the Jarlskog invariant. Strong CP violation is governed bywhich multiplies the Yang-Mills theta term $G_{\mu \nu }^a{\tilde{G}}^{a\mu \nu }$. Quark chiral rephasings shift ${\theta }_{\mathrm{QCD}}$ and the mass-matrix phase oppositely, leaving $\bar{\theta }$ invariant. The CKM phase and $\bar{\theta }$ are otherwise independent parameters: observed weak CP violation does not explain why neutron-electric-dipole bounds require $|\bar{\theta }|\ll 1$. This unexplained smallness is the Strong CP problem.