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An experimentally observed example is $K_L\to {\pi }^{+}{\pi }^{-}$, a decay of the long-lived neutral kaon into a CP-even two-pion state. Without CP violation, the long-lived state would be CP-odd and this decay would be forbidden.

For the CKM matrix mechanism of CP violation, three quark generations are the minimum. Rephasings remove every complex phase for one or two generations, while three generations leave one physical CP-violating phase. This count concerns weak quark mixing. The Yang-Mills theta term in part (c) is a separate possible source of CP violation and does not require three generations.

Starting from a matter-antimatter symmetric ensemble, exact charge conjugation would pair every baryon-producing process with an equally probable antibaryon-producing process. Their contributions to the net baryon number cancel. Thus C violation is necessary. Violating charge conjugation alone is insufficient: if CP symmetry is exact, the parity-reflected conjugate channels still have equal probabilities and cancel in a CP symmetry invariant ensemble. One needs CP violation as well.

The remaining Sakharov conditions are baryon number violation and departure from thermal equilibrium. Conserving baryon number cannot turn an initially zero value into a nonzero one. In thermal equilibrium, under the usual CPT theorem hypotheses and with no imposed chemical potentials for charges odd under CPT symmetry, states and their conjugates have equal thermal weights, so the equilibrium expectation of net baryon number vanishes and thermal stationarity prevents its sustained production. A nonequilibrium history allows different forward and reverse populations to generate an asymmetry. These are the standard necessary conditions for dynamical baryogenesis from symmetric initial conditions, not a guarantee of a sufficiently large asymmetry.

The operator $ia\bar{\psi }{\sigma }^{\mu \nu }{\gamma }^5\psi F_{\mu \nu }$ is the relativistic fermion electric dipole moment operator. Its nonrelativistic limit contains $\mathbf{S}\cdot \mathbf{E}$: spin angular momentum is an axial vector, whereas the electric field is a polar vector. It is consequently odd under parity symmetry in quantum field theory. Both the pseudotensor fermion bilinear and the electromagnetic field tensor are odd under charge conjugation, so their product is even. ThereforeThe CPT theorem then makes it odd under time-reversal symmetry. Such an interaction can arise only from CP violation. The Cabibbo-Kobayashi-Maskawa matrix therefore induces a nonzero Standard Model electric dipole moment at sufficiently high loop order, but its flavor structure and loop suppressions make the result extraordinarily small.

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.

Two complex $N\times N$ quark Yukawa matrices contain $4N^2$ real parameters. Flavour-basis transformations $U(N{)}_Q\times U(N{)}_u\times U(N{)}_d$ remove $3N^2-1$, because the common baryon-number phase remains unbroken. Thus there arephysical parameters: $2N$ quark masses, $N(N-1)/2$ mixing angles, and $(N-1)(N-2)/2$ complex phases. A physical phase exists only for $N\ge 3$ and produces CP violation, equivalently time-reversal violation when CPT is conserved.

The standard conditions for baryogenesis from a symmetric initial ensemble are baryon number violation, both C violation and CP violation, and departure from thermal equilibrium. Conserved baryon number cannot develop a net value from zero; an exact conjugation symmetry pairs production with equal antiproduction. In thermal equilibrium, CPT symmetry makes the equilibrium expectation of net baryon number vanish for zero imposed chemical potentials for charges odd under CPT symmetry. These conditions assume the usual local quantum-field framework and do not alone guarantee a large asymmetry.

For real $a$, the Hermitian effective interaction $ia\bar{\psi }{\sigma }^{\mu \nu }{\gamma }^5\psi F_{\mu \nu }$ is odd under time-reversal symmetry. The spinor tensor and gauge field strength both acquire $-t_{\mu }{t}_{\nu }$, while the chirality matrix is even. Their signs cancel, leaving the minus sign from the explicit $i$ under antiunitarity. By the CPT theorem this is also CP violation. It is not a renormalizable tree-level Standard Model term, but a radiatively generated low-energy dipole is allowed because of the CP-violating phase in the CKM matrix.

A theory violates charge conjugation when conjugating particles to antiparticles changes a physical interaction or its transition probabilities. This is different from CP violation: an interaction can violate charge conjugation and parity symmetry in quantum field theory separately while preserving their combination.

A weak phase is a phase in a coupling or decay amplitude that reverses sign in the CP symmetry conjugate process. A physically relevant difference must survive allowed field rephasings; a complex coupling alone does not prove CP violation.