In theStandard Model ofparticle physics, theCabibbo–Kobayashi–Maskawa matrix,CKM matrix,quark mixing matrix, orKM matrix is aunitary matrix that contains information on the strength of theflavour-changingweak interaction. Technically, it specifies the mismatch ofquantum states ofquarks when they propagate freely and when they take part in theweak interactions. It is important in the understanding ofCP violation. This matrix was introduced for three generations of quarks byMakoto Kobayashi andToshihide Maskawa, adding onegeneration to the matrix previously introduced byNicola Cabibbo. This matrix is also an extension of theGIM mechanism, which only includes two of the three current families of quarks.

In 1963,Nicola Cabibbo introduced theCabibbo angle (θ_c) to preserve the universality of theweak interaction.^[1] Cabibbo was inspired by previous work byMurray Gell-Mann and Maurice Lévy,^[2]on the effectively rotated nonstrange and strange vector and axial weak currents, which he references.^[3]

In light of current concepts (quarks had not yet been proposed), the Cabibbo angle is related to the relative probability thatdown andstrange quarks decay intoup quarks ( |V_ud|²   and   |V_us|² , respectively). In particle physics terminology, the object that couples to the up quark via charged-current weak interaction is a superposition of down-type quarks, here denoted byd′.^[4]Mathematically this is:

${\displaystyle d^{\prime }=V_{\mathrm{u}\mathrm{d}}d+V_{\mathrm{u}\mathrm{s}}s,}$

or using the Cabibbo angle:

${\displaystyle d^{\prime }=\cos {\theta }_{\mathrm{c}}d+\sin {\theta }_{\mathrm{c}}s.}$

Using the currently accepted values for   |V_ud|   and   |V_us|   (see below), the Cabibbo angle can be calculated using

${\displaystyle \tan {\theta }_{\mathrm{c}}=\frac{|V_{\mathrm{u}\mathrm{s}}|}{|V_{\mathrm{u}\mathrm{d}}|}=\frac{0.22534}{0.97427}\Rightarrow {\theta }_{\mathrm{c}}={13.02}^{\circ }.}$

When thecharm quark was discovered in 1974, it was noticed that the down and strange quark could transition into either the up or charm quark, leading to two sets of equations:

${\displaystyle d^{\prime }=V_{\mathrm{u}\mathrm{d}}d+V_{\mathrm{u}\mathrm{s}}s,}$

${\displaystyle s^{\prime }=V_{\mathrm{c}\mathrm{d}}d+V_{\mathrm{c}\mathrm{s}}s;}$

or using the Cabibbo angle:

${\displaystyle d^{\prime }=\cos {\theta }_{\mathrm{c}}d+\sin {\theta }_{\mathrm{c}}s,}$

${\displaystyle s^{\prime }=-\sin {\theta }_{\mathrm{c}}d+\cos {\theta }_{\mathrm{c}}s.}$

This can also be written inmatrix notation as:

or using the Cabibbo angle

where the various |V_ij|² represent the probability that the quark of flavorj decays into a quark of flavori. This 2×2 rotation matrix is called the "Cabibbo matrix", and was subsequently expanded to the 3×3 CKM matrix.

In 1973, observing thatCP-violation could not be explained in a four-quark 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 are theweak interaction doublet partners of down-type quarks, and on the right is the CKM matrix, along with a vector of mass eigenstates of down-type quarks. The CKM matrix describes the probability of a transition from one flavourj quark to another flavouri quark. These transitions are proportional to |V_ij|².

As of 2023, the best determination of the individualmagnitudes of the CKM matrix elements was:^[6]

Using those values, one can check the unitarity of the CKM matrix. In particular, we find that the first-row matrix elements give:${\displaystyle |V_{\mathrm{u}\mathrm{d}}{|}^2+|V_{\mathrm{u}\mathrm{s}}{|}^2+|V_{\mathrm{u}\mathrm{b}}{|}^2=.999997\pm .0007}$

making the experimental results in line with the theoretical value of 1.

The choice of usage of down-type quarks in the definition is a convention, and does not represent a physically preferred asymmetry between up-type and down-type quarks. Other conventions are equally valid: The mass eigenstatesu,c, andt of the up-type quarks can equivalently define the matrix in terms oftheir weak interaction partnersu′,c′, andt′. Since the CKM matrix is unitary, its inverse is the same as itsconjugate transpose, which the alternate choices use; it appears as the same matrix, in a slightly altered form.

To generalize the matrix, count the number of physically important parameters in this matrixV which appear in experiments. If there areN generations of quarks (2Nflavours) then

• AnN × N unitary matrix (that is, a matrixV such thatV^†V = I, whereV^† is theconjugate transpose ofV andI is the identity matrix) requiresN² 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 the matrix is independent of a common phase. Hence, the total number of free variables independent of the choice of the phases of basis vectors isN² − (2N − 1) = (N − 1)².
  • Of these,⁠1/2⁠N(N − 1) are rotation angles calledquark mixing angles.
  • The remaining⁠1/2⁠(N − 1)(N − 2) are complex phases, which causeCP violation.

For the caseN = 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 theCabibbo angle after its inventorNicola Cabibbo.

For theStandard Model case (N = 3), there are three mixing angles and one CP-violating complex phase.^[7]

Cabibbo's idea originated from a need to explain two observed phenomena:

1. the transitionsu ↔ d,e ↔ ν_e , andμ ↔ ν_μ had similar amplitudes.
2. the transitions with change in strangenessΔS = 1 had amplitudes equal to⁠ 1 /4⁠ of those withΔS = 0 .

Cabibbo's solution consisted of postulatingweak universality (see below) to resolve the first issue, along with a mixing angleθ_c, now called theCabibbo angle, between thed ands quarks to resolve the second.

For two generations of quarks, there can be no CP violating phases, as shown by the counting of the previous section. Since CP violationshad already been seen in 1964, in neutralkaon decays, theStandard Model that emerged soon after clearly indicated the existence of a third generation of quarks, as Kobayashi and Maskawa pointed out in 1973. The discovery of thebottom quark atFermilab (byLeon Lederman's group) in 1976 therefore immediately started off the search for thetop quark, the missing third-generation quark.

Note, however, that the specific values that the angles take on arenot a prediction of the standard model: They arefree parameters. At present, there is no generally-accepted theory that explains why the angles should have the values that are measured in experiments.

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

${\displaystyle \sum _k|V_{jk}{|}^2=\sum _k|V_{kj}{|}^2=1}$

separately for each generationj. This implies that the sum of all couplings of anyone of the up-type quarks toall the down-type quarks is the same for all generations. This relation is calledweak universality and was first pointed out byNicola Cabibbo in 1967. Theoretically it is a consequence of the fact that allSU(2) doublets couple with the same strength to thevector bosons of weak interactions. It has been subjected to continuing experimental tests.

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

${\displaystyle \sum _k{V}_{ik}{V}_{jk}^{\ast }=0.}$

For any fixed and differenti andj, this is a constraint on three complex numbers, one for eachk, which says that these numbers form the sides of a triangle in thecomplex plane. There are six choices ofi andj (three independent), and hence six such triangles, each of which is called aunitary triangle. Their shapes can be very different, but they all have the same area, which can be related to theCP violating phase. The area vanishes for the specific parameters in the Standard Model for which there would be noCP 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 theJarlskog invariant (introduced byCecilia Jarlskog in 1985),

${\displaystyle J=c_{12}{c}_{13}^2{c}_{23}{s}_{12}{s}_{13}{s}_{23}\sin \delta \approx 3\cdot {10}^{-5}.}$

For Greek indices denoting up quarks and Latin ones down quarks, the 4-tensor${\displaystyle (\alpha ,\beta ;i,j)\equiv \mathrm{Im}(V_{\alpha i}{V}_{\beta j}{V}_{\alpha j}^{\ast }{V}_{\beta i}^{\ast })}$ is doubly antisymmetric,

${\displaystyle (\beta ,\alpha ;i,j)=-(\alpha ,\beta ;i,j)=(\alpha ,\beta ;j,i).}$

Up to antisymmetry, it only has 9 = 3 × 3 non-vanishing components, which, remarkably, from the unitarity ofV, can be shown to beall identical in magnitude, that is,

so that

${\displaystyle J=(u,c;s,b)=(u,c;d,s)=(u,c;b,d)=(c,t;s,b)=(c,t;d,s)=(c,t;b,d)=(t,u;s,b)=(t,u;b,d)=(t,u;d,s).}$

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 JapaneseBELLE and the AmericanBaBar experiments, as well as atLHCb in CERN, Switzerland.

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.

The original parameterization of Kobayashi and Maskawa used three angles ( θ₁,θ₂,θ₃ ) and a CP-violating phase angle ( δ ).^[5]θ₁ is the Cabibbo angle. For brevity, the cosines and sines of the anglesθ_k are denotedc_k ands_k, fork = 1, 2, 3 respectively.

A "standard" parameterization of the CKM matrix uses threeEuler angles ( θ₁₂,θ₂₃,θ₁₃ ) and one CP-violating phase ( δ₁₃ ).^[8]θ₁₂ is the Cabibbo angle. This is the convention advocated by theParticle Data Group. Couplings between quark generationsj andk vanish ifθ_jk = 0. Cosines and sines of the angles are denotedc_jk ands_jk, respectively.

The 2008 values for the standard parameters were:^[9]

θ₁₂ =13.04°±0.05°,θ₁₃ =0.201°±0.011°,θ₂₃ =2.38°±0.06°

and

δ₁₃ =1.20±0.08 radians =68.8°±4.5°.

A third parameterization of the CKM matrix was introduced byLincoln Wolfenstein with the four real parametersλ,A,ρ, andη, which would all 'vanish' (would be zero) if there were no coupling.^[10] The four Wolfenstein parameters have the property that all are of order 1 and are related to the 'standard' parameterization:

${\displaystyle \lambda =s_{12},}$	${\displaystyle \lambda =s_{12},}$
${\displaystyle A{\lambda }^2=s_{23},}$	${\displaystyle A=\frac{s_{23}}{s_{12}^2},}$
${\displaystyle A{\lambda }^3(\rho -i\eta )=s_{13}{e}^{-i\delta },}$	${\displaystyle \rho ={\mathcal{R}}_{\mathcal{e}}\left\{\frac{s_{13}{e}^{-i\delta }}{s_{12}{s}_{23}}\right\},\eta =-{\mathcal{I}}_{\mathcal{m}}\left\{\frac{s_{13}{e}^{-i\delta }}{s_{12}{s}_{23}}\right\}.}$

Although the Wolfenstein parameterization of the CKM matrix can be as exact as desired when carried to high order, it is mainly used for generating convenient approximations to the standard parameterization. The approximation to orderλ³, good to better than 0.3% accuracy, is:

Rates ofCP violation correspond to the parametersρ andη.

Using the values of the previous section for the CKM matrix, as of 2008 the best determination of the Wolfenstein parameter values is:^[6]

λ =.22500 ± 0.0067,  A =0.826+0.018
−0.015,  ρ = 0.159±0.010,   and  η = 0.348±0.010.

In 2008, Kobayashi and Maskawa shared one half of theNobel 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".^[11] Some physicists were reported to harbor bitter feelings about the fact that the Nobel Prize committee failed to reward the work ofCabibbo, whose prior work was closely related to that of Kobayashi and Maskawa.^[12] Asked for a reaction on the prize, Cabibbo preferred to give no comment.^[13]

• Formulation of the Standard Model andCP violations
• Quantum chromodynamics,flavour andstrong CP problem
• Weinberg angle, a similar angle for Z and photon mixing
• Pontecorvo–Maki–Nakagawa–Sakata matrix, the equivalent mixing matrix forneutrinos
• Koide formula

• D.J. Griffiths (2008).Introduction to Elementary Particles (2nd ed.).John Wiley & Sons.ISBN 978-3-527-40601-2.

• B. Povh; et al. (1995).Particles and Nuclei: An Introduction to the Physical Concepts.Springer.ISBN 978-3-540-20168-7.

• I.I. Bigi, A.I. Sanda (2000).CP violation.Cambridge University Press.ISBN 978-0-521-44349-4.

• "Particle Data Group: The CKM quark-mixing matrix"(PDF).
• "Particle Data Group: CP violation in meson decays"(PDF).
• "The Babar experiment". atSLAC, California, and"the BELLE experiment". atKEK, Japan.

• Standard Model
• Electroweak theory
• Matrices

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