Cabibbo-Kobayashi-Maskawa matrix - AbsoluteAstronomy.com

Standard Model

The Standard Model of particle physics is a theory concerning the electromagnetic, weak, and strong nuclear interactions, which mediate the dynamics of the known subatomic particles. Developed throughout the mid to late 20th century, the current formulation was finalized in the mid 1970s upon...

of particle physics

Particle physics

Particle physics is a branch of physics that studies the existence and interactions of particles that are the constituents of what is usually referred to as matter or radiation. In current understanding, particles are excitations of quantum fields and interact following their dynamics...

, the Cabibbo–Kobayashi–Maskawa matrix (CKM matrix, quark mixing matrix, sometimes also 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 quark

Quark

A quark is an elementary particle and a fundamental constituent of matter. Quarks combine to form composite particles called hadrons, the most stable of which are protons and neutrons, the components of atomic nuclei. Due to a phenomenon known as color confinement, quarks are never directly...

s when they propagate freely and when they take part in the weak interaction

Weak interaction

Weak interaction , is one of the four fundamental forces of nature, alongside the strong nuclear force, electromagnetism, and gravity. It is responsible for the radioactive decay of subatomic particles and initiates the process known as hydrogen fusion in stars...

s. It is important in the understanding of CP violation

CP violation

In particle physics, CP violation is a violation of the postulated 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 were interchanged with its antiparticle , and left and right were swapped...

. A precise mathematical definition of this matrix is given in the article on the formulation of the standard model

Standard model (basic details)

This is a detailed description of the standard model of particle physics. It describes how the leptons, quarks, gauge bosons and the Higgs particle fit together...

. This matrix was introduced for three generations of quarks by Makoto Kobayashi

Makoto Kobayashi (physicist)

is a Japanese physicist known for his work on CP-violation who was awarded one quarter of the 2008 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."- Biography :After completing his PhD at...

and Toshihide Maskawa

Toshihide Maskawa

is a Japanese theoretical physicist known for his work on CP-violation who was awarded one quarter of the 2008 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."-Biography:A native of Aichi...

, adding one generation

Generation (particle physics)

In particle physics, a generation is a division of the elementary particles. Between generations, particles differ by their quantum number and mass, but their interactions are identical....

to the matrix previously introduced by Nicola Cabibbo

Nicola Cabibbo

Nicola Cabibbo was an Italian physicist, best known for his work on the weak interaction. He was also the president of the Italian National Institute of Nuclear Physics from 1983 to 1992, and from 1993 until his death he was the president of the Pontifical Academy of Sciences...

. This matrix is also an extension of the GIM mechanism

GIM mechanism

In quantum field theory, the GIM mechanism is the mechanism by which flavour-changing neutral currents are suppressed. It also explains why weak interactions that change strangeness by 2 are suppressed while those that change strangeness by 1 are allowed...

, which only includes two of the three current families of quarks.

The matrix

In 1963, Nicola Cabibbo

Nicola Cabibbo

introduced the Cabibbo angle (θ_c) to preserve the universality of the weak interaction

Weak interaction

. Cabibbo was inspired by previous work by Murray Gell-Mann

Murray Gell-Mann

Murray Gell-Mann is an American physicist and linguist who received the 1969 Nobel Prize in physics for his work on the theory of elementary particles...

and Maurice Lévy, on the effectively rotated nonstrange and strange vector and axial weak currents, which he references.

In light of current knowledge (quarks were not yet theorized), the Cabibbo angle is related to the relative probability that down

Down quark

The down quark or d quark is the second-lightest of all quarks, a type of elementary particle, and a major constituent of matter. It, along with the up quark, forms the neutrons and protons of atomic nuclei...

and strange quark

Strange quark

The strange quark or s quark is the third-lightest of all quarks, a type of elementary particle. Strange quarks are found in hadrons, which are subatomic particles. Example of hadrons containing strange quarks include kaons , strange D mesons , Sigma baryons , and other strange particles...

s decay into up quark

Up quark

The up quark or u quark is the lightest of all quarks, a type of elementary particle, and a major constituent of matter. It, along with the down quark, forms the neutrons and protons of atomic nuclei...

s (|V_ud|² and |V_us|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′. Mathematically this is:

or using the Cabbibo angle:

Using the currently accepted values for |V_ud| and |V_us| (see below), the Cabbibo 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

Matrix (mathematics)

In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...

as:

or using the Cabibbo angle

where the various |V_ij|² represent the probability that the quark of i flavor decays into a quark of j flavor. This 2 × 2 rotation matrix is called the Cabibbo matrix. Observing that CP-violation could not be explained in a four-quark model, Kobayashi and Maskawa generalized the Cabbibo matrix into the Cabibbo–Kobayashi–Maskawa matrix (or CKM matrix) to keep track of the weak decays of three generations of quarks:

On the left is the weak interaction

Weak interaction

doublet partners of up-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 |V_ij|².

Currently, the best determination of the magnitude

Absolute value

In mathematics, the absolute value |a| of a real number a is the numerical value of a without regard to its sign. So, for example, the absolute value of 3 is 3, and the absolute value of -3 is also 3...

s of the CKM matrix elements is:

Note that the choice of usage of down-type quarks in the definition is purely arbitrary and does not represent some sort of deep physical asymmetry between up-type and down-type quarks. We could just as easily define the matrix the other way around, describing weak interaction partners of mass eigenstates of down-type quarks, u′, c′ and t′, in terms of u, c, and t. Since the CKM matrix is unitary (and therefore its inverse is the same as its conjugate transpose), we would obtain essentially the same matrix.

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 flavour

Flavour (particle physics)

In particle physics, flavour or flavor is a quantum number of elementary particles. In quantum chromodynamics, flavour is a global symmetry...

s) 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 an overall common phase is unobservable. 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, 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
  CP violation
  In particle physics, CP violation is a violation of the postulated 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 were interchanged with its antiparticle , and left and right were swapped...
  
  .

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

Standard Model

case (N = 3), there are three mixing angles and one CP-violating complex phase.

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

Kaon

In particle physics, a kaon is any one of a group of four mesons distinguished by the fact that they carry a quantum number called strangeness...

decays already in 1964, the emergence of the Standard Model

Standard model (basic details)

This is a detailed description of the standard model of particle physics. It describes how the leptons, quarks, gauge bosons and the Higgs particle fit together...

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

Bottom quark

The bottom quark, also known as the beauty quark, is a third-generation quark with a charge of − e. Although all quarks are described in a similar way by the quantum chromodynamics, the bottom quark's large bare mass , combined with low values of the CKM matrix elements Vub and Vcb, gives it a...

at Fermilab

Fermilab

Fermi National Accelerator Laboratory , located just outside Batavia, Illinois, near Chicago, is a US Department of Energy national laboratory specializing in high-energy particle physics...

(by Leon Lederman's group) in 1976 therefore immediately started off the search for the missing third-generation quark, the top quark

Top quark

The top quark, also known as the t quark or truth quark, is an elementary particle and a fundamental constituent of matter. Like all quarks, the top quark is an elementary fermion with spin-, and experiences all four fundamental interactions: gravitation, electromagnetism, weak interactions, and...

Weak universality

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

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

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 boson

Vector boson

In particle physics, a vector boson is a boson with the spin quantum number equal to 1.The vector bosons considered to be elementary particles in the Standard Model are the gauge bosons or, the force carriers of fundamental interactions: the photon of electromagnetism, the W and Z bosons of the...

s 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

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

Complex plane

In mathematics, the complex plane or z-plane is a geometric representation of the complex numbers established by the real axis and the orthogonal imaginary axis...

. There are six choices of i and j (three independent), 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

CP violation

phase. The area vanishes for the specific parameters in the Standard Model for which there would be no CP violation

CP violation

. The orientation of the triangles depend on the phases of the quark fields.

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

Belle experiment

The Belle experiment is a particle physics experiment conducted by the Belle Collaboration, an international collaboration of more than 400 physicists and engineers investigating CP-violation effects at the High Energy Accelerator Research Organisation in Tsukuba, Ibaraki Prefecture, Japan.The...

and the American BaBar

BaBar experiment

The BaBar experiment, or simply BaBar, is an international collaboration of more than 500 physicists and engineers studying the subatomic world at energies of approximately ten times the rest mass of a proton . Its design was motivated by the investigation of CP violation...

experiments, as well as at LHCb

LHCb

LHCb is one of six particle physics detector experiments collecting data at the Large Hadron Collider accelerator at CERN. LHCb is a specialized b-physics experiment, that is measuring the parameters of CP violation in the interactions of b-hadrons...

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 (δ). Cosines and sines of the angles are denoted c_i and s_i, respectively. θ₁ is the Cabibbo angle.

"Standard" parameters

A "standard" parameterization of the CKM matrix uses three Euler angles

Euler angles

The Euler angles are three angles introduced by Leonhard Euler to describe the orientation of a rigid body. To describe such an orientation in 3-dimensional Euclidean space three parameters are required...

(θ₁₂, θ₂₃, θ₁₃) and one CP-violating phase (δ₁₃). 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. θ₁₂ is the Cabibbo angle.

The currently best known values for the standard parameters are:

θ₁₂ = °, θ₁₃ = °, θ₂₃ = °, and δ₁₃ = .

Wolfenstein parameters

A third parameterization of the CKM matrix was introduced by Lincoln Wolfenstein

Lincoln Wolfenstein

Lincoln Wolfenstein is an American particle physicist who studies the weak interaction. Wolfenstein was born in 1923 and obtained his PhD in 1949 from the University of Chicago. He retired from Carnegie Mellon University in 2000 after being a faculty member for 52 years, but still lectures there...

with the four parameters λ, A, ρ, and η. The four Wolfenstein parameters have the property that all are of order 1 and are related to the "standard" parameterization:

λ = s₁₂

Aλ² = s₂₃

Aλ³(ρ − iη) = s₁₃e^−iδ

The Wolfenstein parameterization of the CKM matrix, to order λ³, 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:

λ = , A = , ρ = , and η = .

Nobel Prize

In 2008, Kobayashi and Maskawa shared one half of the Nobel Prize in Physics

Nobel Prize in Physics

The Nobel Prize in Physics is awarded once a year by the Royal Swedish Academy of Sciences. It is one of the five Nobel Prizes established by the will of Alfred Nobel in 1895 and awarded since 1901; the others are the Nobel Prize in Chemistry, Nobel Prize in Literature, Nobel Peace Prize, and...

"for the discovery of the origin of the broken symmetry which predicts the existence of at least three families of quarks in nature". Some physicists were reported to harbor bitter feelings about the fact that the Nobel Prize committee failed to reward the work of Cabibbo, on which the work of the other two was based. Asked for a reaction on the prize, Cabibbo preferred to give no comment.

The matrix

Counting

Observations and predictions

Weak universality

The unitarity triangles

Parameterizations

KM parameters

"Standard" parameters

Wolfenstein parameters

Nobel Prize

See also

Further reading