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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.

The Standard Model is a gauge theory with group $SU(3{)}_c\times SU(2{)}_L\times U(1{)}_Y$, matter quarks and leptons, gauge bosons and a Higgs field. It combines Quantum chromodynamics with the electroweak interaction. Its successes do not make its field content a complete account of nature. Established missing phenomena should be distinguished from theoretical questions about its parameters.

neutrino oscillation demonstrates nonzero mass-squared differences and mixing of flavor states. The minimal renormalizable Standard Model has no right-handed neutrino and no neutrino mass term. The atmospheric experiment compared zenith-angle distributions of muon-like and electron-like events in a large water Cherenkov detector. A direction-dependent muon-neutrino deficit, with different path lengths through Earth and comparison samples, discriminated against a uniform flux-normalization error.^[ref] One extension adds right-handed neutrinos and small Dirac Yukawa couplings; another has a Majorana mass term and seesaw mechanism, with light eigenvalue of order $m_D^2/M$. Oscillations alone do not decide whether neutrinos are Majorana neutrinos.

dark matter is inferred from gravitational observations such as galactic motions and gravitational lensing; Standard Model particles do not supply the required dominant cold component. Active neutrinos contribute some matter but cannot constitute all the cold dark matter. A new stable weakly interacting particle, an axion, or other hidden-sector states are possible explanations, with different scattering, production and astrophysical signals. Dark-matter evidence does not identify a particular new particle. The institutional account distinguishes gravitational evidence from proposed particle candidates.^[ref]

The matter-antimatter imbalance poses a different problem. Dynamical baryogenesis needs Sakharov conditions, including baryon-number violation, CP violation and departure from thermal equilibrium. Although the Standard Model contains relevant ingredients in principle, they do not generate the observed asymmetry in its conventional thermal history with measured parameters. Additional CP phases, a stronger electroweak transition, or leptogenesis from heavy Majorana neutrinos are possible extensions. Each needs a quantitative mechanism.

gravity is absent as a dynamical quantum interaction, motivating quantum gravity and unification. The hierarchy problem asks why the Higgs mass scale is small compared with much higher scales when scalar masses receive sensitive quantum corrections. supersymmetry can cancel particular contributions between partner fields if its breaking pattern is suitable; compositeness gives another interpretation of the Higgs sector. Neither motivation proves that the corresponding new particles exist. The Strong CP problem asks why the allowed QCD CP-violating parameter is so small; an axion mechanism is one possible answer.

A grand unified theory embeds the separate gauge factors in a larger group and can relate couplings or fermion representations, sometimes predicting proton decay. Its tests differ from dark-matter or neutrino-mass searches. A useful extension must explain a specified deficiency, preserve successful low-energy predictions and supply distinguishable experimental consequences. Attractive new symmetries alone are not evidence.

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.

The Strong CP problem asks why the physical QCD theta angle is smaller than roughly $10^{-10}$ even though no Standard Model symmetry requires this small value.