| Cabibbo–Kobayashi–Maskawa matrix | |
|---|---|
| Name | Cabibbo–Kobayashi–Maskawa matrix |
| Type | Mixing matrix |
| Introduced | 1963 (Cabibbo), 1973 (Kobayashi–Maskawa) |
| Associated | Standard Model, Weak interaction |
Cabibbo–Kobayashi–Maskawa matrix
The Cabibbo–Kobayashi–Maskawa matrix (commonly abbreviated as CKM matrix) is a unitary matrix in the Standard Model of particle physics that describes the mixing between the three generations of quark flavor eigenstates under the weak interaction. It encodes the probabilities for transitions among up-type quarks and down-type quarks in charged-current processes, and its complex phase(s) provide a source of CP violation. The CKM matrix is central to understanding weak decays, precision tests at facilities such as the Large Hadron Collider and KEK, and implications for cosmological matter–antimatter asymmetry.
The CKM matrix arises from the misalignment between the mass eigenstates produced by the Higgs mechanism and the weak interaction eigenstates defined by the SU(2) gauge symmetry of the electroweak interaction. In quantum field theory terms, diagonalizing the quark mass matrices introduces a unitary transformation that is the CKM matrix. Its elements are complex numbers constrained by unitarity and are experimentally determined from processes studied at institutions such as CERN, Fermilab, SLAC, and KEK. The CKM framework interlinks with precision tests of the Standard Model and searches for new physics.
Mathematically, the CKM matrix V is a 3×3 unitary matrix: V = (V_{ud}, V_{us}, V_{ub}; V_{cd}, V_{cs}, V_{cb}; V_{td}, V_{ts}, V_{tb}). Unitarity implies orthogonality relations such as the unitarity triangles, which are graphical representations used in analyses by collaborations like Belle, BaBar, and LHCb. Common parameterizations include the original Cabibbo angle for two generations extended by Kobayashi and Maskawa to three generations, and the Wolfenstein parameterization expressed in parameters (λ, A, ρ, η). The Jarlskog invariant J quantifies the magnitude of CP violation in a parameterization-independent way. Constraints on matrix elements come from theoretical frameworks such as quantum chromodynamics (QCD) and techniques like lattice QCD for hadronic matrix elements.
In charged-current weak interactions mediated by the W boson, the CKM matrix multiplies the left-handed quark fields to connect up-type and down-type currents. This mixing explains observed phenomena including strangeness-changing decays, weak decays of B mesons, kaons, and D mesons. The pattern of hierarchical magnitudes (e.g., |V_{ud}| ≈ 0.974, |V_{us}| ≈ 0.224) reflects the smallness of intergenerational mixing except for nearby generations, a feature often discussed in flavor model building and in proposals such as Froggatt–Nielsen mechanism. Collider experiments measuring branching fractions and angular distributions provide inputs for global fits performed by groups like the CKMfitter Group and UTfit.
The complex phase in the CKM matrix is a source of CP violation within the Standard Model, first used to explain observed CP-violating processes in kaon decays and later in B meson systems. Quantification through the unitarity triangle angles (α, β, γ) is a major program of experiments including Belle II and LHCb. While the CKM mechanism produces CP violation consistent with many laboratory measurements, its magnitude appears insufficient to account for the observed cosmic baryon asymmetry of the Universe under standard electroweak baryogenesis scenarios, motivating searches for additional sources of CP violation in models such as supersymmetry, left–right symmetric model, or theories with extended Higgs sectors.
Determination of CKM elements relies on a wide array of experimental inputs: superallowed nuclear beta decays constraining |V_{ud}|, kaon decays and semileptonic decays measuring |V_{us}|, charm and bottom decay studies constraining |V_{cd}|, |V_{cs}|, |V_{cb}|, and rare processes and B oscillations informing |V_{td}| and |V_{ts}|. Key experimental facilities and collaborations include SuperKEKB, LHCb, ATLAS, CMS, BaBar, Belle, and fixed-target experiments. Precision theoretical inputs from lattice gauge theory and global fits by the CKMfitter and UTfit collaborations synthesize results into recommended values used by the Particle Data Group.
The CKM matrix is embedded in flavor physics models and guides extensions of the Standard Model that attempt to explain the pattern of masses and mixings. Proposed frameworks include flavor symmetries (e.g., U(1), S_3), grand unified theories such as SU(5) and SO(10), and mechanisms linking quark and lepton mixing via the PMNS matrix for neutrinos. Discrepancies between CKM-based predictions and experimental results are sought as evidence of new physics — for example via flavor-changing neutral current anomalies, lepton flavor universality tests, or CP-violating observables beyond CKM expectations.
The two-generation mixing concept originated with Nicola Cabibbo (1963) introducing the Cabibbo angle to explain weak decays involving strangeness. The three-generation extension with a CP-violating phase was formulated by Makoto Kobayashi and Toshihide Maskawa (1973), anticipating the existence of the bottom quark and top quark; their work contributed to the award of the Nobel Prize in Physics (2008) for discoveries related to the origin of the broken symmetry that predicts such phenomena. Subsequent experimental verification and refinement involved collaborations and institutions such as CERN, Fermilab, SLAC, KEK, the Belle and BaBar experiments, and theoretical developments by researchers including C. Jarlskog and proponents of parametrizations like Lincoln Wolfenstein.
Category:Quantum physics Category:Particle physics Category:Standard Model