In the standard model of particle physics the Cabibbo Kobayashi Maskawa matrix (CKM matrix, sometimes earlier called KM matrix) is a unitary matrix which contains information on the strength of flavour changing weak decays. Technically, it specifies the mismatch of quantum states of quarks when they propagate freely and when they take part in the weak interactions. It is important in the understanding of CP violations. A precise mathematical definition of this matrix is given in the article on the formulation of the standard model. This matrix was introduced for three generations of quarks by Makoto Kobayashi and Toshihide Maskawa, adding one generation to the matrix previously introduced by Nicola Cabibbo. CPT-symmetry is a fundamental symmetry of physical laws under transformations that involve the inversions of charge, parity and time simultaneously. ... CP is the product of two symmetries: C for charge conjugation, which transforms a particle into its antiparticle, and P for parity, which creates the mirror image of a physical system. ... A phenomenon is said to be chiral if it is not identical to its mirror image (see Chirality (mathematics)). The spin of a particle may be used to define a handedness for that particle. ... This is a detailed description of the standard model (SM) of particle physics. ... Particles erupt from the collision point of two relativistic (100 GeV per nucleon) gold ions in the STAR detector of the Relativistic Heavy Ion Collider. ... Quite literally, quantum state describes the state of a quantum system. ... Quarks are one of the two basic constituents of matter in the Standard Model of particle physics. ... The weak nuclear force or weak interaction is one of the four fundamental forces of nature. ... In physics, and specifically particle physics, CP violation is a violation of the postulated CP symmetry of the laws of physics. ... This is a detailed description of the standard model (SM) of particle physics. ... According to the standard model of particle physics, all the elementary particles seen in particle collision experiments can be divided into three generations. ...

The matrix

On the left is the CKM Matrix along with a vector of strong force eigenstates of the quarks, and on the right is the weak force eigenstates of the quarks. The CKM matrix describes the probability of a transition from one quark q to another quark q' . This transition is proportional to . The strong nuclear force or strong interaction (also called color force or colour force) is a fundamental force of nature which affects only quarks and antiquarks, and is mediated by gluons in a similar fashion to how the electromagnetic force is mediated by photons. ... The weak nuclear force or weak interaction is one of the four fundamental forces of nature. ...

Experimentally, the magnitudes of the values in the matrix have been found to be roughly:

Counting

To proceed further, it is necessary to count the number of parameters in this matrix, V which appear in experiments, and therefore are physically important. If there are N generations of quarks (2N flavours) then Flavour (or flavor) is a quantum number of elementary particles related to their weak interactions. ...

1. An N×N complex matrix contains 2N² real numbers.
2. The constraint of unitarity is ∑_k V_ikV^*_jk = δ_ij. Therefore, for the diagonal terms (i=j) there are N constraints, and for the remaining terms, N(N−1). The number of independent real numbers in an unitary matrix is therefore N².
3. One phase can be absorbed into each quark field. An overall common phase is unobservable. Hence there are 2N−1 less independent numbers, giving the total number of free variables to be (N−1)².
4. Of these, N(N−1)/2 are rotation angles called quark mixing angles.
5. The remaining (N−1)(N−2)/2 are complex phases, which cause CP violation.

For the case N=2, there is only one parameter which is a mixing angle between two generations of quarks. Historically, this was the first version of CKM matrix when only two generations were known. It is called the Cabibbo angle after its inventor Nicola Cabibbo. In physics, and specifically particle physics, CP violation is a violation of the postulated CP symmetry of the laws of physics. ...

For the Standard Model case N=3, there are three mixing angles and one CP-violating complex phase. The Standard Model of Fundamental Particles and Interactions The Standard Model of particle physics is a theory which describes the strong, weak, and electromagnetic fundamental forces, as well as the fundamental particles that make up all matter. ...

Observations and predictions

The idea of Cabibbo originated from a need to explain two observed phenomena:

1. the transitions u↔d and e↔ν_e, μ↔ν_μ had similar amplitudes.
2. the transitions with change in strangeness ΔS=1 had amplitudes equal to 1/4 of those with ΔS=0.

Cabibbo's solution consisted of postulating weak universality to resolve issue 1, along with a mixing angle θ_c, now called the Cabibbo angle, between the d and s quarks to resolve issue 2.

For two generations of quarks, there are no CP violating phases, as shown by the counting of the previous section. Since CP violations were seen in neutral kaon decays already in 1964, the emergence of the standard model soon after was a clear signal of the existence of a third generation of quarks, as pointed out in 1973 by Kobayashi and Maskawa. The discovery of the bottom quark at Fermilab (by Leon Lederman's group) in 1976 therefore immediately started off the search for the missing third-generation quark, the top quark. In particle physics, Kaons (also called K-mesons and denoted K) are a group of four mesons distinguished by the fact that they carry a quantum number called strangeness. ... For the Nintendo 64 emulator, see 1964 (Emulator). ... This is a detailed description of the standard model (SM) of particle physics. ... 1973 (MCMLXXIII) was a common year starting on Monday. ... The bottom quark is a third-generation quark with a charge of -(1/3)e. ... Fermilab Robert Rathbun Wilson Hall Fermi National Accelerator Laboratory (Fermilab), located in Batavia near Chicago, Illinois, is a U.S. Department of Energy national laboratory specializing in high-energy particle physics, operated for the Department of Energy by the Universities Research Association (URA). ... Leon Max Lederman (born July 15, 1922) is an American experimental physicist who was awarded the Nobel Prize in Physics in 1988 for his work on neutrinos. ... 1976 (MCMLXXVI) was a leap year starting on Thursday (the link is to a full 1976 calendar). ... The top quark is a third-generation quark with a charge of +(2/3)e. ...

Weak universality

The constraints of unitarity of the CKM-matrix on the diagonal terms can be written as

∑	| Vij | 2 = 1
j

for all generations i. This implies that the sum of all couplings of any of the up-type quarks to all the down-type quarks is the same for all generations. This relation is called weak universality after Nicola Cabibbo, who first pointed it out in 1967. Theoretically it is a consequence of the fact that all SU(2) doublets couple with the same strength to the vector bosons of weak interactions. It has been subjected to continuing experimental tests. A vector boson is a boson with spin equal to one. ...

The unitarity triangles

The remaining constraints of unitarity of the CKM-matrix can be written in the form

For any fixed i and j, this is a constraint on three complex numbers, one for each k, which says that these numbers form the vertices of a triangle in the complex plane. There are six choices of i and j, and hence six such triangles, each of which is called an unitary triangle. Their shapes can be very different, but they all have the same area, which can be related to the CP violating phase. The area vanishes for the specific parameters in the standard model for which there would be no CP violation. The orientation of the triangles depend on the phases of the quark fields. In mathematics, the complex plane is a way of visualising the space of the complex numbers. ...

Since the three sides of the triangles are open to direct experiment, as are the three angles, a class of tests of the standard model is to check that the triangle closes. This is the purpose of a modern series of experiments under way at the Japanese BELLE and the Californian BaBar experiments. Babar can refer to: Babar the Elephant The BaBar Experiment at the Stanford Linear Accelerator Babur (also spelled Baber or Babar), 16th-century ruler of India and founder of the Mogul Empire Babar Island, an island of Indonesia Safir A Babar, This is a disambiguation page — a navigational aid which...

See also

• Formulation of the standard model and CP violations.
• Quantum chromodynamics, flavour and strong CP problem.
• MNS matrix, the equivalent mixing matrix for neutrinos.

This is a detailed description of the standard model (SM) of particle physics. ... In physics, and specifically particle physics, CP violation is a violation of the postulated CP symmetry of the laws of physics. ... Quantum chromodynamics (QCD) is the theory of the strong interaction, a fundamental force describing the interactions of the quarks and gluons found in nucleons (such as the proton and neutron). ... Flavour (or flavor) is a quantum number of elementary particles related to their weak interactions. ... In particle physics, the strong CP problem is the puzzling question why Quantum Chromodynamics (QCD) does not seem to break the CP-symmetry. ... In the standard model of particle physics the Maki-Nakagawa-Sakata matrix (MNS matrix) is a unitary matrix which contains information on the mismatch between mass and weak flavour quantum states of neutrinos. ... The neutrino is an elementary particle. ...

References

• Griffiths, David J. (1987). Introduction to Elementary Particles. Wiley, John & Sons, Inc. ISBN 0471603864.
• Povh, Bogdan et al., (1995). Particles and Nuclei: An Introduction to the Physical Concepts. New York: Springer. ISBN 3540201688

External links

• CP violation, by I.I. Bigi and A.I. Sanda (Cambridge University Press, 2000) [ISBN 0521443490]
• Particle Data Group on CP violation
• The Babar experiment at SLAC and the BELLE experiment at KEK Japan
• N. Cabibbo, Phys. Rev. Lett. 10 (1963) 531.
• M. Kobayashi and K. Maskawa, Prog. Theor. Phys. 49 (1973) 652.