Cabibbo–Kobayashi–Maskawa matrix

The Cabibbo - Kobayashi - Maskawa matrix ( CKM matrix ) is a unitary 3 × 3 matrix that represents in the context of the Standard Model of particle physics, in which statistical units quarks of three flavor generations (each type quarks with charge 2 / 3 e, or -type quarks with charge -1 / 3 e) can be converted into other quarks the corresponding charge by interaction with a ( charged) W boson ( ie after normalization with respect to all other phase-space dependencies). The CKM matrix is therefore also known as quark mixing matrix.

Overview

Developed by Nicola Cabibbo in consideration of two quark generations theoretical concept of mixing of quark generations through flavor changing charged currents ( FCCC) has been extended by Makoto Kobayashi and Toshihide Masukawa with the CKM matrix for three generations. Sometimes they will only Kobayashi and Maskawa named ( Kobayashi - Maskawa matrix KM matrix).

Their definition is derived from the observation of certain transition probabilities:

If a type quark of a certain flavor, has turned into a type quark by emitting a positively charged boson, then the absolute square of the matrix element, the ( suitably normalized ) corresponds to the transition probability for a quark of the flavor.

Also by definition, the value also, conversely, the ( suitably normalized ) probability for the transition of a quark to quark; assuming the associated emission of a boson.

The CKM matrix is described by three real parameters physically unique and a complex phase ( five more phases that occur mathematically, have no physical meaning ). The transition probabilities of the quarks are therefore not completely independent, but obey certain relations - according to the requirement of the standard model, which is experimentally verifiable and tests so far has withstood. Therefore, the experimentally determined matrix of values ​​, the amount of squares the experimentally determined quark transition probabilities represent total, also called the CKM matrix.

The physical meaning of the complex phase is the CP violation of the weak interaction. It is noteworthy that only a physical complex phase can occur from a dimension of three, ie CP violation (at least) requires three quark generations. For her based on this consideration prediction of a third generation of quarks Kobayashi and Maskawa were given together with Yoichiro Nambu 2008 Nobel Prize in Physics.

From neutrino experiments it is known that there is also a leptonic mixing matrix in analogy to the CKM matrix. This is referred to as Maki Nakagawa Sakata matrix ( MNS matrix).

The CKM matrix and transformations between eigenstate systems

As already outlined above, the CKM matrix describes the relationship between the quark flavor content of a given initial state and a corresponding final state, the transition was complete by flavor - changing charged currents ( ie by - boson interaction of the first order ) effects.

Corresponding matrix equations (in which the nine CKM matrix elements and the flavor eigenstates of the six quarks are explicitly named) be

For an initial state containing only type quarks; and by definition also

For an initial state containing only type quarks.

Now there is an obvious theoretical possibility (and also a secure experimental findings) that the corresponding CKM matrix differs from a unit matrix:

In other words, it is said that the electro- weak interaction mixes the three considered quark generations, the assignment of the quark flavor content of certain initial and final states is carried out in the three generations experimentally due to their significantly different quark masses.

(In a common notation eigenstates of the electroweak interaction are also called, and so on referred to. )

The comparison with the first matrix equation shows that the CKM matrix can be understood as a product of two unitary transformation matrices, respectively, isolated for - or -type quarks, the relationship between the system of eigenstates of the electroweak interaction and the system of quark mass eigenstates (that is also eigenstates of the flavor ) represent:

In a more compact form is this matrix product:

The CKM matrix can itself be regarded also as a transformation matrix that mediates between the reference system of the type quarks and another suitable reference system, are the three independent elements in another, likewise common notation, and referred (but conceptually different from the must be distinguished above eigenstates of the weak interaction ). It is just around the states that couple under - boson interaction first order each accurate and complete to the type quarks or respectively. Correspondingly, one writes:

Unitarity of the CKM matrix as a requirement of the standard model and the subject of current research

As noted at the outset, the term " CKM matrix " is used both for the matrix, the Kobayashi and Maskawa defined in the framework of the theory of electroweak interactions to construct a mechanism of CP violation, as well as in the context of experimental physics to be determined matrix of values ​​whose absolute squares represent measured quark transition probabilities.

The CKM matrix in the theoretical sense on the one hand is defined as unitary and especially exactly representable as a product of two unitary transformation (which, in each case for the quarks of the same charge, the context or the mixture of mass eigenstates and eigenstates of the weak interaction describe).

The CKM matrix in the experimental sense, on the other hand does not necessarily fulfilled and from the outset the Unitaritäts condition. Instead, only experimentally to answer by obtaining measured values ​​whether or within which accuracy this matrix is unitary or not.

The prediction that the experimental matrix is indeed unitary, and consequently the theory of electroweak interactions (GWS - theory ) is suitable with three generations of quark flavors and sufficient to describe quantitatively correct all findable changes of quark flavor content also and in summarize the form of values ​​of the elements of an exactly unitary 3 × 3 matrix, an essential (that is by no means trivial ) aspect of the Standard Model.

In the mathematical condition of unitarity of the 3 × 3 matrix partial conditions can be distinguished, which in turn correspond to specific aspects of the standard model. In particular, the following so-called diagonal condition can be considered separately:

For each quark flavor or. This corresponds to the experimental expectation of weak universality that any interaction strength, which leads to changes of the quark flavor content, for all quarks is equal to total (and thus does not have to be explicitly taken into account in the normalization ). Also thus combines the model expectation and the previous experimental finding that any changes of the quark flavor content ( that is, apart from pair creation or destruction ) exclusively by the electroweak interactions take place (ie coupling to bosons ) within three quark generations.

The remaining part unitarity conditions for a 3 × 3 matrix ( off-diagonal terms ) can be represented by so-called unitary triangles. The corresponding standard model corresponding experimental expectations or predictions explicitly refer among other things to measured values ​​for CP violation.

The values ​​of the coefficients of the CKM matrix are:

Counting of free parameters

To count the free parameters of the CKM matrix, proceed as follows:

Of these, the rotation angle, which are referred to as curd mixture angle. The remaining parameters are complex phases which cause the CP- injury. Especially remains so for only a mixing angle for the quarks, the Cabibbo angle, whereas the position in the case for the standard model of three quark mixing angles and one CP- violating complex phase.

Observations and predictions

One recognizes that quark transitions occur within a generation with the highest probability ( diagonal elements close to one ), while transitions between different generations are suppressed ( for example, the decay of an s-quark in the lighter, more stable u-quark ). This explains the relatively long life for some mesons containing quarks of higher generations.

From the unitarity condition we obtain the following relations:

Since the products of the matrix elements are again complex, this can be represented as vectors in the complex plane. Since the sum of these vectors is zero, one can these vectors together to form a triangle and thus acquires the so-called Unitaritätsdreieck. Many research groups are working currently with determining the angle of this triangle on the decays of - and - mesons.

The unitarity of the CKM matrix is the subject of current research. One example tried to measure on the electroweak top quark production matrix element or to find inconsistencies in Unitaritätsdreieck. If the unitarity of the CKM matrix be violated, this would be an indication of physics beyond the Standard Model.

So far, the unitarity of the CKM matrix, however, has withstood the experimental verifications. In December 2006, such direct measurement is the first time succeeded. It was found that the unitarity remains with a significance of safeguarded. The task now is to increase the accuracy of the measurement.