Last updated

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 flavour-changing 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.

## The matrix

### Predecessor: Cabibbo matrix

In 1963, Nicola Cabibbo introduced the Cabibbo anglec) to preserve the universality of the weak interaction. [1] Cabibbo was inspired by previous work by Murray 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 knowledge (quarks were not yet theorized), the Cabibbo angle is related to the relative probability that down and strange quarks decay into up quarks (|Vud|2 and |Vus|2 respectively). In particle physics parlance, the object that couples to the up quark via charged-current weak interaction is a superposition of down-type quarks, here denoted by d′. [4] Mathematically this is:

${\displaystyle d^{\prime }=V_{ud}d+V_{us}s,}$

or using the Cabibbo angle:

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

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

${\displaystyle \tan \theta _{\mathrm {c} }={\frac {|V_{us}|}{|V_{ud}|}}={\frac {0.22534}{0.97427}}\Rightarrow \theta _{\mathrm {c} }=~13.02^{\circ }.}$

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:

${\displaystyle d^{\prime }=V_{ud}d+V_{us}s;}$
${\displaystyle s^{\prime }=V_{cd}d+V_{cs}s,}$

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 in matrix notation as:

${\displaystyle {\begin{bmatrix}d^{\prime }\\s^{\prime }\end{bmatrix}}={\begin{bmatrix}V_{ud}&V_{us}\\V_{cd}&V_{cs}\\\end{bmatrix}}{\begin{bmatrix}d\\s\end{bmatrix}},}$

or using the Cabibbo angle

${\displaystyle {\begin{bmatrix}d^{\prime }\\s^{\prime }\end{bmatrix}}={\begin{bmatrix}\cos {\theta _{\mathrm {c} }}&\sin {\theta _{\mathrm {c} }}\\-\sin {\theta _{\mathrm {c} }}&\cos {\theta _{\mathrm {c} }}\\\end{bmatrix}}{\begin{bmatrix}d\\s\end{bmatrix}},}$

where the various |Vij|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 CP-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]

${\displaystyle {\begin{bmatrix}d^{\prime }\\s^{\prime }\\b^{\prime }\end{bmatrix}}={\begin{bmatrix}V_{ud}&V_{us}&V_{ub}\\V_{cd}&V_{cs}&V_{cb}\\V_{td}&V_{ts}&V_{tb}\end{bmatrix}}{\begin{bmatrix}d\\s\\b\end{bmatrix}}.}$

On the left is the weak 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 quark i to another quark j. These transitions are proportional to |Vij|2.

As of 2020, the best determination of the magnitudes of the CKM matrix elements was: [6]

${\displaystyle {\begin{bmatrix}|V_{ud}|&|V_{us}|&|V_{ub}|\\|V_{cd}|&|V_{cs}|&|V_{cb}|\\|V_{td}|&|V_{ts}|&|V_{tb}|\end{bmatrix}}={\begin{bmatrix}0.97370\pm 0.00014&0.2245\pm 0.0008&0.00382\pm 0.00024\\0.221\pm 0.004&0.987\pm 0.011&0.0410\pm 0.0014\\0.0080\pm 0.0003&0.0388\pm 0.0011&1.013\pm 0.030\end{bmatrix}}.}$

Using those values, one can check the unitarity of the CKM matrix. In particular, we find that the first-row matrix elements give: ${\displaystyle |V_{ud}|^{2}+|V_{us}|^{2}+|V_{ub}|^{2}=0.9985\pm 0.0005}$,

which exhibits an apparent violation of the unitarity condition at 3 standard deviations, and provides interesting hints to physics beyond the Standard Model.

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, such as defining the matrix in terms of weak interaction partners of mass eigenstates of up-type 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.

## General case construction

To generalize the matrix, count the number of physically important parameters in this matrix, V which appear in experiments. 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 N2 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 is N2  (2N  1) = (N  1)2.
• Of these, 1/2N(N  1) are rotation angles called quark mixing angles.
• The remaining 1/2(N  1)(N  2) are complex phases, which cause CP violation.

### N = 2

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.

### N = 3

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

## Observations and predictions

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

1. the transitions u ↔ d, e ↔ νe , and μ ↔ νμ had similar amplitudes.
2. the transitions with change in strangeness ΔS = 1 had amplitudes equal to ¼ of those with ΔS = 0 .

Cabibbo's solution consisted of postulating weak universality to resolve the first issue, along with a mixing angle θc, 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 had already been seen in 1964, in neutral kaon decays, the Standard 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 the bottom quark at Fermilab (by Leon Lederman's group) in 1976 therefore immediately started off the search for the top quark, the missing third-generation quark.

Note, however, that the specific values that the angles take on are not a prediction of the standard model: They are free parameters. At present, there is no generally-accepted theory that explains why the angles should have the values that are measured in experiments.

## Weak universality

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

${\displaystyle \sum _{k}|V_{ik}|^{2}=\sum _{i}|V_{ik}|^{2}=1}$

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 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 CKM-matrix can be written in the form

${\displaystyle \sum _{k}V_{ik}V_{jk}^{*}=0.}$

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,

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

For Greek indices denoting up quarks and Latin ones down quarks, the 4-tensor ${\displaystyle (\alpha ,\beta ;i,j)\equiv \operatorname {Im} (V_{\alpha i}V_{\beta j}V_{\alpha j}^{*}V_{\beta i}^{*})}$ 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 of V, can be shown to be all identical in magnitude, that is,

${\displaystyle (\alpha ,\beta ;i,j)=J~{\begin{bmatrix}0&1&-1\\-1&0&1\\1&-1&0\end{bmatrix}}_{\alpha \beta }\otimes {\begin{bmatrix}0&1&-1\\-1&0&1\\1&-1&0\end{bmatrix}}_{ij},}$

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 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 (θ1, θ2, θ3) and a CP-violating phase angle (δ). [5] θ1 is the Cabibbo angle. Cosines and sines of the angles θk are denoted ck and sk, for k = 1,2,3 respectively.

${\displaystyle {\begin{bmatrix}c_{1}&-s_{1}c_{3}&-s_{1}s_{3}\\s_{1}c_{2}&c_{1}c_{2}c_{3}-s_{2}s_{3}e^{i\delta }&c_{1}c_{2}s_{3}+s_{2}c_{3}e^{i\delta }\\s_{1}s_{2}&c_{1}s_{2}c_{3}+c_{2}s_{3}e^{i\delta }&c_{1}s_{2}s_{3}-c_{2}c_{3}e^{i\delta }\end{bmatrix}}.}$

### "Standard" parameters

A "standard" parameterization of the CKM matrix uses three Euler angles (θ12, θ23, θ13) and one CP-violating phase (δ13). [8] θ12 is the Cabibbo angle. Couplings between quark generations j and k vanish if θjk = 0 . Cosines and sines of the angles are denoted cjk and sjk, respectively.

{\displaystyle {\begin{aligned}&{\begin{bmatrix}1&0&0\\0&c_{23}&s_{23}\\0&-s_{23}&c_{23}\end{bmatrix}}{\begin{bmatrix}c_{13}&0&s_{13}e^{-i\delta _{13}}\\0&1&0\\-s_{13}e^{i\delta _{13}}&0&c_{13}\end{bmatrix}}{\begin{bmatrix}c_{12}&s_{12}&0\\-s_{12}&c_{12}&0\\0&0&1\end{bmatrix}}\\&={\begin{bmatrix}c_{12}c_{13}&s_{12}c_{13}&s_{13}e^{-i\delta _{13}}\\-s_{12}c_{23}-c_{12}s_{23}s_{13}e^{i\delta _{13}}&c_{12}c_{23}-s_{12}s_{23}s_{13}e^{i\delta _{13}}&s_{23}c_{13}\\s_{12}s_{23}-c_{12}c_{23}s_{13}e^{i\delta _{13}}&-c_{12}s_{23}-s_{12}c_{23}s_{13}e^{i\delta _{13}}&c_{23}c_{13}\end{bmatrix}}.\end{aligned}}}

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°.

### Wolfenstein parameters

A third parameterization of the CKM matrix was introduced by Lincoln Wolfenstein with the four parameters λ, A, ρ, and η, 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:

λ = s12
A λ2 = s23
A λ3 ( ρiη ) = s13eiδ

The Wolfenstein parameterization of the CKM matrix, is an approximation of the standard parameterization. To order λ3, it is:

${\displaystyle {\begin{bmatrix}1-{\tfrac {1}{2}}\lambda ^{2}&\lambda &A\lambda ^{3}(\rho -i\eta )\\-\lambda &1-{\tfrac {1}{2}}\lambda ^{2}&A\lambda ^{2}\\A\lambda ^{3}(1-\rho -i\eta )&-A\lambda ^{2}&1\end{bmatrix}}+O(\lambda ^{4}).}$

The CP violation can be determined by measuring ρiη.

Using the values of the previous section for the CKM matrix, the best determination of the Wolfenstein parameters is: [11]

λ = 0.2257+0.0009
−0.0010
,   A = 0.814+0.021
−0.022
,   ρ = 0.135+0.031
−0.016
,   and  η = 0.349+0.015
−0.017
.

## 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]

## Related Research Articles

Angular displacement of a body is the angle in radians through which a point revolves around a centre or line has been rotated in a specified sense about a specified axis. When a body rotates about its axis, the motion cannot simply be analyzed as a particle, as in circular motion it undergoes a changing velocity and acceleration at any time (t). When dealing with the rotation of a body, it becomes simpler to consider the body itself rigid. A body is generally considered rigid when the separations between all the particles remains constant throughout the body's motion, so for example parts of its mass are not flying off. In a realistic sense, all things can be deformable, however this impact is minimal and negligible. Thus the rotation of a rigid body over a fixed axis is referred to as rotational motion.

In linear algebra, an orthogonal matrix, or orthonormal matrix, is a real square matrix whose columns and rows are orthonormal vectors.

In linear algebra, a complex square matrix U is unitary if its conjugate transpose U* is also its inverse, that is, if

An ellipsoid is a surface that may be obtained from a sphere by deforming it by means of directional scalings, or more generally, of an affine transformation.

Ray transfer matrix analysis is a mathematical form for performing ray tracing calculations in sufficiently simple problems which can be solved considering only paraxial rays. Each optical element is described by a 2×2 ray transfer matrix which operates on a vector describing an incoming light ray to calculate the outgoing ray. Multiplication of the successive matrices thus yields a concise ray transfer matrix describing the entire optical system. The same mathematics is also used in accelerator physics to track particles through the magnet installations of a particle accelerator, see electron optics.

In continuum mechanics, the infinitesimal strain theory is a mathematical approach to the description of the deformation of a solid body in which the displacements of the material particles are assumed to be much smaller than any relevant dimension of the body; so that its geometry and the constitutive properties of the material at each point of space can be assumed to be unchanged by the deformation.

In the mathematical field of differential geometry, one definition of a metric tensor is a type of function which takes as input a pair of tangent vectors v and w at a point of a surface and produces a real number scalar g(v, w) in a way that generalizes many of the familiar properties of the dot product of vectors in Euclidean space. In the same way as a dot product, metric tensors are used to define the length of and angle between tangent vectors. Through integration, the metric tensor allows one to define and compute the length of curves on the manifold.

Linear elasticity is a mathematical model of how solid objects deform and become internally stressed due to prescribed loading conditions. It is a simplification of the more general nonlinear theory of elasticity and a branch of continuum mechanics.

In linear algebra, linear transformations can be represented by matrices. If is a linear transformation mapping to and is a column vector with entries, then

Neutrino oscillation is a quantum mechanical phenomenon in which a neutrino created with a specific lepton family number can later be measured to have a different lepton family number. The probability of measuring a particular flavor for a neutrino varies between three known states, as it propagates through space.

A transversely isotropic material is one with physical properties that are symmetric about an axis that is normal to a plane of isotropy. This transverse plane has infinite planes of symmetry and thus, within this plane, the material properties are the same in all directions. Hence, such materials are also known as "polar anisotropic" materials. In geophysics, vertically transverse isotropy (VTI) is also known as radial anisotropy.

In cartography, a Tissot's indicatrix is a mathematical contrivance presented by French mathematician Nicolas Auguste Tissot in 1859 and 1871 in order to characterize local distortions due to map projection. It is the geometry that results from projecting a circle of infinitesimal radius from a curved geometric model, such as a globe, onto a map. Tissot proved that the resulting diagram is an ellipse whose axes indicate the two principal directions along which scale is maximal and minimal at that point on the map.

In Hamiltonian mechanics, the linear canonical transformation (LCT) is a family of integral transforms that generalizes many classical transforms. It has 4 parameters and 1 constraint, so it is a 3-dimensional family, and can be visualized as the action of the special linear group SL2(R) on the time–frequency plane (domain).

In particle physics, the Pontecorvo–Maki–Nakagawa–Sakata matrix, Maki–Nakagawa–Sakata matrix, lepton mixing matrix, or neutrino mixing matrix is a unitary mixing matrix which contains information on the mismatch of quantum states of neutrinos when they propagate freely and when they take part in the weak interactions. It is a model of neutrino oscillation. This matrix was introduced in 1962 by Ziro Maki, Masami Nakagawa and Shoichi Sakata, to explain the neutrino oscillations predicted by Bruno Pontecorvo.

In geometry, various formalisms exist to express a rotation in three dimensions as a mathematical transformation. In physics, this concept is applied to classical mechanics where rotational kinematics is the science of quantitative description of a purely rotational motion. The orientation of an object at a given instant is described with the same tools, as it is defined as an imaginary rotation from a reference placement in space, rather than an actually observed rotation from a previous placement in space.

In mathematics, the axis–angle representation of a rotation parameterizes a rotation in a three-dimensional Euclidean space by two quantities: a unit vector e indicating the direction of an axis of rotation, and an angle θ describing the magnitude of the rotation about the axis. Only two numbers, not three, are needed to define the direction of a unit vector e rooted at the origin because the magnitude of e is constrained. For example, the elevation and azimuth angles of e suffice to locate it in any particular Cartesian coordinate frame.

The quark–lepton complementarity (QLC) is a possible fundamental symmetry between quarks and leptons. First proposed in 1990 by Foot and Lew, it assumes that leptons as well as quarks come in three "colors". Such theory may reproduce the Standard Model at low energies, and hence quark–lepton symmetry may be realized in nature.

In particle physics, CP violation is a violation of CP-symmetry : the combination of C-symmetry and P-symmetry. CP-symmetry states that the laws of physics should be the same if a particle is interchanged with its antiparticle while its spatial coordinates are inverted. The discovery of CP violation in 1964 in the decays of neutral kaons resulted in the Nobel Prize in Physics in 1980 for its discoverers James Cronin and Val Fitch.

In mathematical physics, the Berezin integral, named after Felix Berezin,, is a way to define integration for functions of Grassmann variables. It is not an integral in the Lebesgue sense; the word "integral" is used because the Berezin integral has properties analogous to the Lebesgue integral and because it extends the path integral in physics, where it is used as a sum over histories for fermions.

In theoretical physics, the composition of two non-collinear Lorentz boosts results in a Lorentz transformation that is not a pure boost but is the composition of a boost and a rotation. This rotation is called Thomas rotation, Thomas–Wigner rotation or Wigner rotation. The rotation was discovered by Llewellyn Thomas in 1926, and derived by Wigner in 1939. If a sequence of non-collinear boosts returns an object to its initial velocity, then the sequence of Wigner rotations can combine to produce a net rotation called the Thomas precession.

## References

1. Cabibbo, N. (1963). "Unitary Symmetry and Leptonic Decays". Physical Review Letters . 10 (12): 531–533. Bibcode:1963PhRvL..10..531C. doi:.
2. Gell-Mann, M.; Lévy, M. (1960). "The Axial Vector Current in Beta Decay". Il Nuovo Cimento . 16 (4): 705–726. Bibcode:1960NCim...16..705G. doi:10.1007/BF02859738. S2CID   122945049.
3. Maiani, L. (2009). "Sul Premio Nobel Per La Fisica 2008" (PDF). Il Nuovo Saggiatore. 25 (1–2): 78. Archived from the original (PDF) on 22 July 2011. Retrieved 30 November 2010.
4. Hughes, I.S. (1991). "Chapter 11.1 – Cabibbo Mixing". Elementary Particles (3rd ed.). Cambridge University Press. pp. 242–243. ISBN   978-0-521-40402-0.
5. Kobayashi, M.; Maskawa, T. (1973). "CP-Violation in the Renormalizable Theory of Weak Interaction". Progress of Theoretical Physics . 49 (2): 652–657. Bibcode:1973PThPh..49..652K. doi:.
6. Zyla, P.A.; et al. (2020). "Review of Particle Physics: CKM Quark-Mixing Matrix" (PDF). PETP . 2020 (8): 083C01. doi:.
7. Baez, J.C. (4 April 2011). "Neutrinos and the Mysterious Pontecorvo-Maki-Nakagawa-Sakata Matrix" . Retrieved 13 February 2016. 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.
8. Chau, L.L.; Keung, W.-Y. (1984). "Comments on the Parametrization of the Kobayashi-Maskawa Matrix". Physical Review Letters . 53 (19): 1802–1805. Bibcode:1984PhRvL..53.1802C. doi:10.1103/PhysRevLett.53.1802.
9. Values obtained from values of Wolfenstein parameters in the 2008 Review of Particle Physics .
10. Wolfenstein, L. (1983). "Parametrization of the Kobayashi-Maskawa Matrix". Physical Review Letters . 51 (21): 1945–1947. Bibcode:1983PhRvL..51.1945W. doi:10.1103/PhysRevLett.51.1945.
11. Amsler, C.; Doser, M.; Antonelli, M.; Asner, D.M.; Babu, K.S.; Baer, H.; et al. (Particle Data Group) (2008). "The CKM Quark-Mixing Matrix" (PDF). Physics Letters B . Review of Particles Physics. 667 (1): 1–1340. Bibcode:2008PhLB..667....1A. doi:10.1016/j.physletb.2008.07.018.
12. "The Nobel Prize in Physics 2008" (Press release). The Nobel Foundation. 7 October 2008. Retrieved 24 November 2009.
13. Jamieson, V. (7 October 2008). "Physics Nobel Snubs key Researcher". New Scientist . Retrieved 24 November 2009.
14. "Nobel, l'amarezza dei fisici italiani". Corriere della Sera (in Italian). 7 October 2008. Retrieved 24 November 2009.