Cabibbo–Kobayashi–Maskawa matrix
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Flavour mixing 
In the Standard Model of particle physics, the Cabibbo–Kobayashi–Maskawa matrix, CKM matrix, quark mixing matrix, or KM matrix is a unitary matrix which contains information on the strength of the flavourchanging weak interaction. 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 violation. 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. This matrix is also an extension of the GIM mechanism, which only includes two of the three current families of quarks.
Contents
The matrix
Predecessor: Cabibbo matrix
In 1963, Nicola Cabibbo introduced the Cabibbo angle (θ_{c}) to preserve the universality of the weak interaction.^{[1]} Cabibbo was inspired by previous work by Murray GellMann and Maurice Lévy,^{[2]} on the effectively rotated nonstrange and strange vector and axial weak currents, which he references.^{[3]}
In light of current knowledge (quarks were not yet theorized), the Cabibbo angle is related to the relative probability that down and strange quarks decay into up quarks (V_{ud}^{2} and V_{us}^{2} respectively). In particle physics parlance, the object that couples to the up quark via chargedcurrent weak interaction is a superposition of downtype quarks, here denoted by d′.^{[4]} Mathematically this is:
or using the Cabibbo angle:
Using the currently accepted values for V_{ud} and V_{us} (see below), the Cabibbo angle can be calculated using
When the charm quark was discovered in 1974, it was noticed that the down and strange quark could decay into either the up or charm quark, leading to two sets of equations:
or using the Cabibbo angle:
This can also be written in matrix notation as:
or using the Cabibbo angle
where the various V_{ij}^{2} represent the probability that the quark of j flavor decays into a quark of i flavor. This 2 × 2 rotation matrix is called the Cabibbo matrix.
CKM matrix
In 1973, Observing that CPviolation could not be explained in a fourquark model, Kobayashi and Maskawa generalized the Cabibbo matrix into the Cabibbo–Kobayashi–Maskawa matrix (or CKM matrix) to keep track of the weak decays of three generations of quarks:^{[5]}
On the left is the weak interaction doublet partners of uptype quarks, and on the right is the CKM matrix along with a vector of mass eigenstates of downtype quarks. The CKM matrix describes the probability of a transition from one quark i to another quark j. These transitions are proportional to V_{ij}^{2}.
As of 2010, the best determination of the magnitudes of the CKM matrix elements was:^{[6]}
The choice of usage of downtype quarks in the definition is a convention, and does not represent a physically preferred asymmetry between uptype and downtype quarks. Other conventions are equally valid, such as defining the matrix in terms of weak interaction partners of mass eigenstates of uptype quarks, u′, c′ and t′, in terms of u, c, and t. Since the CKM matrix is unitary, its inverse is the same as its conjugate transpose.
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
 An N × N unitary matrix (that is, a matrix V such that VV^{†} = I, where V^{†} is the conjugate transpose of V and I is the identity matrix) requires N^{2} real parameters to be specified.
 2N − 1 of these parameters are not physically significant, because one phase can be absorbed into each quark field (both of the mass eigenstates, and of the weak eigenstates), but an overall common phase is unobservable. Hence, the total number of free variables independent of the choice of the phases of basis vectors is N^{2} − (2N − 1) = (N − 1)^{2}.
 Of these, N(N − 1)/2 are rotation angles called quark mixing angles.
 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.
For the Standard Model case (N = 3), there are three mixing angles and one CPviolating complex phase.^{[7]}
Observations and predictions
Cabibbo's idea originated from a need to explain two observed phenomena:
 the transitions u ↔ d, e ↔ ν_{e}, and μ ↔ ν_{μ} had similar amplitudes.
 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 the first issue, along with a mixing angle θ_{c}, now called the Cabibbo angle, between the d and s quarks to resolve the second.
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 thirdgeneration quark, the top quark.
Note, however, that the specific values of the angles are not a prediction of the standard model: they are open, unfixed parameters. At this time, there is no generally accepted theory that explains why the measured values are what they are.
Weak universality
The constraints of unitarity of the CKMmatrix on the diagonal terms can be written as
for all generations i. This implies that the sum of all couplings of any of the uptype quarks to all the downtype quarks is the same for all generations. This relation is called weak universality and was first pointed out by Nicola Cabibbo 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.
The unitarity triangles
The remaining constraints of unitarity of the CKMmatrix can be written in the form
For any fixed and different i and j, this is a constraint on three complex numbers, one for each k, which says that these numbers form the sides of a triangle in the complex plane. There are six choices of i and j (three independent), and hence six such triangles, each of which is called a 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.
A popular quantity amounting to twice the area of the unitarity triangle is the Jarlskog invariant, . For Greek indices denoting up quarks and Latin ones down quarks, the 4tensor is doubly antisymmetric,
Up to antisymmetry, it only has 9=3×3 nonvanishing components, which, remarkably, from the unitarity of V, can be shown to be all identical in magnitude, that is,
so that
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 American BaBar experiments, as well as at LHCb in CERN, Switzerland.
Parameterizations
Four independent parameters are required to fully define the CKM matrix. Many parameterizations have been proposed, and three of the most common ones are shown below.
KM parameters
The original parameterization of Kobayashi and Maskawa used three angles (θ_{1}, θ_{2}, θ_{3}) and a CPviolating phase (δ).^{[5]} Cosines and sines of the angles are denoted c_{i} and s_{i}, respectively. θ_{1} is the Cabibbo angle.
"Standard" parameters
A "standard" parameterization of the CKM matrix uses three Euler angles (θ_{12}, θ_{23}, θ_{13}) and one CPviolating phase (δ_{13}).^{[8]} Couplings between quark generation i and j vanish if θ_{ij} = 0. Cosines and sines of the angles are denoted c_{ij} and s_{ij}, respectively. θ_{12} is the Cabibbo angle.
The currently best known values for the standard parameters are:^{[9]}
 θ_{12} = ±0.05°, θ_{13} = 13.04±0.011°, θ_{23} = 0.201±0.06°, and δ_{13} = 2.38±0.08 rad. 1.20
Wolfenstein parameters
A third parameterization of the CKM matrix was introduced by Lincoln Wolfenstein with the four parameters λ, A, ρ, and η.^{[10]} The four Wolfenstein parameters have the property that all are of order 1 and are related to the "standard" parameterization:
 λ = s_{12}
 Aλ^{2} = s_{23}
 Aλ^{3}(ρ − iη) = s_{13}e^{−iδ}
The Wolfenstein parameterization of the CKM matrix, is an approximation of the standard parameterization. To order λ^{3}, it is:
The CP violation can be determined by measuring ρ − iη.
Using the values of the previous section for the CKM matrix, the best determination of the Wolfenstein parameters is:^{[11]}

λ = +0.0009
−0.0010, A = 0.2257+0.021
−0.022, ρ = 0.814+0.031
−0.016, and η = 0.135+0.015
−0.017. 0.349
Nobel Prize
In 2008, Kobayashi and Maskawa shared one half of the Nobel Prize in Physics "for the discovery of the origin of the broken symmetry which predicts the existence of at least three families of quarks in nature".^{[12]} Some physicists were reported to harbor bitter feelings about the fact that the Nobel Prize committee failed to reward the work of Cabibbo, whose prior work was closely related to that of Kobayashi and Maskawa.^{[13]} Asked for a reaction on the prize, Cabibbo preferred to give no comment.^{[14]}
See also
 Formulation of the Standard Model and CP violations.
 Quantum chromodynamics, flavour and strong CP problem.
 Weinberg angle, a similar angle for Z and photon mixing.
 Pontecorvo–Maki–Nakagawa–Sakata matrix, the equivalent mixing matrix for neutrinos.
 Koide formula
References
 ^ N. Cabibbo (1963). "Unitary Symmetry and Leptonic Decays". Physical Review Letters. 10 (12): 531–533. Bibcode:1963PhRvL..10..531C. doi:10.1103/PhysRevLett.10.531.
 ^ M. GellMann, M. Lévy (1960). "The Axial Vector Current in Beta Decay". Il Nuovo Cimento. 16 (4): 705–726. Bibcode:1960NCim...16..705G. doi:10.1007/BF02859738.
 ^ L. Maiani (2009). "Sul Premio Nobel Per La Fisica 2008" (PDF). Il Nuovo Saggiatore. 25 (1–2): 78.
 ^ I.S. Hughes (1991). "Chapter 11.1 – Cabibbo Mixing". Elementary Particles (3rd ed.). Cambridge University Press. pp. 242–243. ISBN 0521404029.
 ^ ^{a} ^{b} M. Kobayashi, T. Maskawa; Maskawa (1973). "CPViolation in the Renormalizable Theory of Weak Interaction". Progress of Theoretical Physics. 49 (2): 652–657. Bibcode:1973PThPh..49..652K. doi:10.1143/PTP.49.652.
 ^ Beringer, J.; Arguin, J. F.; Barnett, R. M.; Copic, K.; Dahl, O.; Groom, D. E.; Lin, C. J.; Lys, J.; Murayama, H.; Wohl, C. G.; Yao, W. M.; Zyla, P. A.; Amsler, C.; Antonelli, M.; Asner, D. M.; Baer, H.; Band, H. R.; Basaglia, T.; Bauer, C. W.; Beatty, J. J.; Belousov, V. I.; Bergren, E.; Bernardi, G.; Bertl, W.; Bethke, S.; Bichsel, H.; Biebel, O.; Blucher, E.; Blusk, S.; Brooijmans, G.; et al. (2012). "Review of Particles Physics: The CKM QuarkMixing Matrix" (PDF). Physical Review D. 80 (1): 1–1526 [162]. Bibcode:2012PhRvD..86a0001B. doi:10.1103/PhysRevD.86.010001.

^
J.C. Baez (4 April 2011). "Neutrinos and the Mysterious PontecorvoMakiNakagawaSakata Matrix". Retrieved 20160213.
In fact, the Pontecorvo–Maki–Nakagawa–Sakata matrix actually affects the behavior of all leptons, not just neutrinos. Furthermore, a similar trick works for quarks – but then the matrix U is called the Cabibbo–Kobayashi–Maskawa matrix.
 ^ L.L. Chau and W.Y. Keung (1984). "Comments on the Parametrization of the KobayashiMaskawa Matrix". Physical Review Letters. 53 (19): 1802. Bibcode:1984PhRvL..53.1802C. doi:10.1103/PhysRevLett.53.1802.
 ^ Values obtained from values of Wolfenstein parameters in the 2008 Review of Particle Physics.
 ^ L. Wolfenstein (1983). "Parametrization of the KobayashiMaskawa Matrix". Physical Review Letters. 51 (21): 1945. Bibcode:1983PhRvL..51.1945W. doi:10.1103/PhysRevLett.51.1945.
 ^ Amsler, C.; Doser, M.; Antonelli, M.; Asner, D.M.; Babu, K.S.; Baer, H.; Band, H.R.; Barnett, R.M.; Bergren, E.; Beringer, J.; Bernardi, G.; Bertl, W.; Bichsel, H.; Biebel, O.; Bloch, P.; Blucher, E.; Blusk, S.; Cahn, R.N.; Carena, M.; Caso, C.; Ceccucci, A.; Chakraborty, D.; Chen, M.C.; Chivukula, R.S.; Cowan, G.; Dahl, O.; d'Ambrosio, G.; Damour, T.; De Gouvêa, A.; Degrand, T.; et al. (2008). "Review of Particles Physics: The CKM QuarkMixing Matrix" (PDF). Physics Letters B. 667: 1–1340. Bibcode:2008PhLB..667....1A. doi:10.1016/j.physletb.2008.07.018.
 ^ "The Nobel Prize in Physics 2008" (Press release). The Nobel Foundation. 7 October 2008. Retrieved 20091124.
 ^ V. Jamieson (7 October 2008). "Physics Nobel Snubs key Researcher". New Scientist. Retrieved 20091124.
 ^ "Nobel, l'amarezza dei fisici italiani". Corriere della Sera (in Italian). 7 October 2008. Retrieved 20091124.
Further reading
 D.J Griffiths (2008). Introduction to Elementary Particles (2nd ed.). John Wiley & Sons. ISBN 9783527406012.
 B. Povh; et al. (1995). Particles and Nuclei: An Introduction to the Physical Concepts. Springer. ISBN 3540201688.
 I.I. Bigi, A.I. Sanda (2000). CP violation. Cambridge University Press. ISBN 0521443490.
 Particle Data Group: The CKM quarkmixing matrix
 Particle Data Group: CP violation in meson decays
 The Babar experiment at SLAC and the BELLE experiment at KEK Japan