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An experimentally observed example of CP violation is the long-lived neutral kaon decay $K_L\to {\pi }^{+}{\pi }^{-}$. The two-pion spin-zero final state is CP even, whereas $K_L$ would be the CP-odd neutral-kaon state in the CP-conserving limit. Its two-pion decay reveals CP violation. The historical experimental account is cern-courier.web.cern.ch/a/cp-violations-early-days/ .

For the CKM matrix, diagonalize the up- and down-quark mass matrices independently by biunitary transformations. Writewith $L_u^{†}{M}_u^{\prime }{R}_u$ and $L_d^{†}{M}_d^{\prime }{R}_d$ positive diagonal matrices. Only the left transformations enter the weak charged current, because its chiral projector removes the right-handed fields. Substitution givesThe matrix is unitary but need not be diagonal: the two left mass diagonalizations generally differ.

Using the field transformations in the subparts and reordering the anticommuting Dirac fields, the charged bilinear transforms aswhere the fields on the right are both evaluated at $x_P$. The W boson contributes its second minus sign and Lorentz parity factor, so the product maps into its Hermitian-conjugate product. Crucially CP symmetry acts unitarily: it does not separately complex-conjugate the numerical coefficient $V_{ij}$. Thus the transformed interaction isComparing with the original interaction requires $V_{ij}=V_{ij}^{\ast }$ in this canonical CP convention. The physical condition is quark rephasing and CP conservation:A merely complex-looking matrix in an arbitrary mass-eigenstate phase convention is not itself proof of CP violation.

For CKM parameter counting, a unitary $N\times N$ matrix has $N^2$ real parameters: $N(N-1)/2$ mixing angles and $N(N+1)/2$ phases. Rephasing the $N$ up and $N$ down mass eigenfields removes $2N-1$ phases, since their common phase cancels out of $V$. For generic nondegenerate masses this leavesThere is one physical phase for three families and none for one or two. Degenerate masses would permit additional transformations, so the generic count need not apply unchanged.

For the QCD part of the gauge covariant derivative, define the Hermitian colour matrix ${\mathcal{A}}_{\mu }=A_{\mu }^a{T}^a$ and chooseThis sign convention agrees with the field-strength definition ${\mathcal{F}}_{\mu \nu }=-i[D_{\mu },D_{\nu }]/g_s$ in the source. The kinetic term contains $-g_s\overline{u}{\gamma }^{\mu }{\mathcal{A}}_{\mu }u$. A vector bilinear under CP changes sign, gains the Lorentz parity matrix, and transposes its colour matrix upon exchanging quark and antiquark. Invariance of this interaction therefore requires the CP transformation of a non-Abelian gauge connection:Both the time/spatial distinction and the colour transpose are essential; individual colour components need not all share one charge-conjugation sign.

The same gauge covariant derivative givesApply the transformation above and the chain rule ${\partial }_{\mu }f(x_P)={\Pi }_{\mu }{}^{\alpha }({\partial }_{\alpha }f)(x_P)$. The transpose reverses commutator order, $[X^T,Y^T]=-[X,Y{]}^T$, so the nonlinear term transforms with the same overall minus sign as the derivatives. This proves the CP transformation of non-Abelian field strength:Finally $\mathrm{tr}(T^a{T}^b)={\delta }^{ab}/2$ rewrites the theta density as $2\theta {ϵ}^{\mu \nu \rho \sigma }\mathrm{tr}({\mathcal{F}}_{\mu \nu }{\mathcal{F}}_{\rho \sigma })$. The two minus signs cancel, and transposition followed by cyclicity leaves the colour trace unchanged. But the four parity matrices acting on the Levi-Civita symbol contribute $\det \Pi =-1$. HenceThe density is C even and P odd, therefore CP odd. A generic fixed nonzero theta coefficient violates CP symmetry. This is the source of the Strong CP problem, separate from the charged-current phase in the CKM matrix.

An anomalous chiral redefinition moves phase between the bare QCD angle and the quark mass matrix, so the measurable parameter isThe neutron electric dipole moment requires $|\bar{\theta }|$ to be extremely small, although no generic Standard Model symmetry enforces this. This is the Strong CP problem.

In Peccei–Quinn theory, an anomalous spontaneously broken global symmetry replaces $\bar{\theta }$ by a dynamical axion field. Nonperturbative QCD generates an axion potential whose minimum lies at the CP-conserving value, dynamically relaxing the effective angle to zero.

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.

For nondegenerate quark masses, changing the constant quantum phases of quark fields preserves their diagonal mass terms. Adopt the conventionSuch quantum field rephasings alter individual entries of the CKM matrix without altering physics. Physical CP symmetry in this mixing sector means that some allowed convention makes the CKM matrix real. Toshihide Maskawa emphasized the distinction between complex entries and an irremovable CP-violating phase.^[ref]

Here is a constructive two-generation argument. For a generic $2\times 2$ unitary matrix, use the three effective row/column phases to make $V_{11},V_{12},V_{22}$ real and positive. Column orthogonality then forces $V_{21}$ to be real and negative. Row normalization leaves just the real rotation specified by the Cabibbo angle. Zero entries are limiting cases and introduce no weak phase. Thus two-generation quark mixing cannot violate CP symmetry through this mechanism, even when its entries initially look complex.

For three quark generations, the Jarlskog invariant provides an obstruction to repeating that construction: all four phase factors cancel from its defining quartet. If its imaginary part is nonzero, no choice of quantum field rephasings can make every entry real. With nondegenerate quark masses, this proves physical CP violation in the mixing sector. This argument concerns the weak interaction; it does not dispose of the separate Strong CP problem.