CKM matrix

The Cabibbo-Kobayashi-Maskawa matrix ( CKM matrix ) is a unitary 3 × 3 matrix which, within the framework of the standard model of particle physics, represents the statistical proportions in which quarks of three flavor generations (each -type quarks with charge ² ⁄ ₃ e ; or -type quarks with charge - ¹ ⁄ ₃ e ) through interaction with a (charged) W boson into other quarks with the corresponding charge (i.e. after normalization with regard to all other phase space dependencies). The CKM matrix is ​​therefore also referred to as the quark mixture matrix . ${\displaystyle V_{UD}}$ ${\displaystyle u}$   ${\displaystyle d}$ 

overview

The theoretical concept of mixing quark generations by flavor-changing charged currents (FCCC) developed by Nicola Cabibbo considering two quark generations was extended to three generations by Makoto Kobayashi and Toshihide Masukawa (Maskawa). The mixture of the flavor states is described by the so-called CKM matrix (named after the initials of the three physicists). Since the Nobel Prize was awarded to Kobayashi and Masukawa, but not to Cabibbo, it has sometimes been called the Kobayashi-Maskawa matrix (KM matrix).

Their definition results from the consideration of certain transition probabilities :

If a -type quark of a certain flavor ,, has converted into a -type quark with the emission of a positively charged -Boson , then the square of the amount of the matrix element ,, corresponds to the (suitably normalized) transition probability to a quark of the flavor . ${\displaystyle u}$ ${\displaystyle u_{i}}$ ${\displaystyle W^{+}}$ ${\displaystyle d}$ ${\displaystyle |V_{ij}|^{2}}$ ${\displaystyle d_{j}}$

Likewise by definition, the value also corresponds conversely to the (suitably normalized) probability for the transition from a quark to a quark ; assuming the associated emission of a -Boson. ${\displaystyle |V_{ij}|^{2}}$ ${\displaystyle d_{j}}$ ${\displaystyle u_{i}}$ ${\displaystyle W^{-}}$

The CKM matrix is ​​clearly described physically by three real parameters and a complex phase (another five phases that occur mathematically have no physical meaning). The transition probabilities of the quarks are therefore not completely independent of each other, but obey certain relationships - in accordance with the requirements of the standard model, which can be verified experimentally and has withstood previous tests. Therefore, the matrix of values ​​to be determined experimentally, the squares of which represent the quark transition probabilities determined experimentally, is also called the CKM matrix.

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

It is known from neutrino experiments that there is also a leptonic mixture matrix, analogous to the CKM matrix. This is referred to as the Pontecorvo-Maki-Nakagawa-Sakata matrix (PMNS matrix).

The CKM matrix and transformations between eigenstate systems

As already outlined above, the CKM matrix describes the relationship between the quark flavor contents of a given initial state and a corresponding final state, the transition of which was completely caused by flavor-changing charged currents ( i.e. first-order boson interaction). ${\displaystyle W}$

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

${\displaystyle {\begin{bmatrix}V_{ud}&V_{us}&V_{ub}\\V_{cd}&V_{cs}&V_{cb}\\V_{td}&V_{ts}&V_{tb}\end{bmatrix}}{\begin{bmatrix}\langle d|D_{\rm {init}}\rangle \\\langle s|D_{\rm {init}}\rangle \\\langle b|D_{\rm {init}}\rangle \end{bmatrix}}={\begin{bmatrix}\langle u|U_{\rm {end}}\rangle \\\langle c|U_{\rm {end}}\rangle \\\langle t|U_{\rm {end}}\rangle \\\end{bmatrix}}}$

for an initial state that contained only -type quarks; and by definition as well ${\displaystyle |D_{\rm {init}}\rangle }$ ${\displaystyle d}$

${\displaystyle {\begin{bmatrix}\langle u|U_{\rm {init}}\rangle \\\langle c|U_{\rm {init}}\rangle \\\langle t|U_{\rm {init}}\rangle \\\end{bmatrix}}={\begin{bmatrix}V_{ud}&V_{us}&V_{ub}\\V_{cd}&V_{cs}&V_{cb}\\V_{td}&V_{ts}&V_{tb}\end{bmatrix}}{\begin{bmatrix}\langle d|D_{\rm {end}}\rangle \\\langle s|D_{\rm {end}}\rangle \\\langle b|D_{\rm {end}}\rangle \\\end{bmatrix}}}$

for an initial state that only contained -type quarks. ${\displaystyle |U_{init}\rangle }$ ${\displaystyle u}$

Now there is a natural theoretical possibility (and also a secure experimental findings) therein, is that the corresponding CKM matrix of a unit matrix is different:

${\displaystyle V_{UD}\neq 1^{3\times 3}}$

In other words, it is said that the electroweak interaction mixes the three quark generations under consideration, with the quark flavor content of certain initial and final states being assigned experimentally to the three generations based on their significantly different quark masses.

A reference system for the representation of the described initial and final states can, however, instead also be selected in such a way that their relationship caused by flavor-changing charged currents is represented by an identity matrix. Such diagonalization is by reference to the system of so-called eigenstates of the electro-weak interaction ( , , and so on) is achieved: ${\displaystyle |d_{w}\rangle }$ ${\displaystyle |u_{w}\rangle }$ ${\displaystyle |s_{w}\rangle }$

${\displaystyle {\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\\\end{bmatrix}}{\begin{bmatrix}\langle d_{w}|D_{\rm {init}}\rangle \\\langle s_{w}|D_{\rm {init}}\rangle \\\langle b_{w}|D_{\rm {init}}\rangle \end{bmatrix}}={\begin{bmatrix}\langle u_{w}|U_{\rm {end}}\rangle \\\langle c_{w}|U_{\rm {end}}\rangle \\\langle t_{w}|U_{\rm {end}}\rangle \\\end{bmatrix}}}$

(In one common notation eigenstates of the electro-weak interaction are also called , , and designated so on.) ${\displaystyle |d'\rangle }$ ${\displaystyle |u'\rangle }$ ${\displaystyle |s'\rangle }$

The comparison with the first matrix equation shows that the CKM matrix can be understood as the product of two unitary transformation matrices, respectively , which, separately for - or - type quarks, show the relationship between the system of eigenstates of the electroweak interaction and the system of quark mass. Representing eigenstates (i.e. also eigenstates of the flavor): ${\displaystyle A_{{U_{w}}U}}$ ${\displaystyle A_{{D_{w}}D}}$ ${\displaystyle u}$ ${\displaystyle d}$

${\displaystyle {\begin{bmatrix}V_{ud}&V_{us}&V_{ub}\\V_{cd}&V_{cs}&V_{cb}\\V_{td}&V_{ts}&V_{tb}\end{bmatrix}}\equiv {\begin{bmatrix}\langle u|u_{w}\rangle &\langle u|c_{w}\rangle &\langle u|t_{w}\rangle \\\langle c|u_{w}\rangle &\langle c|c_{w}\rangle &\langle c|t_{w}\rangle \\\langle t|u_{w}\rangle &\langle t|c_{w}\rangle &\langle t|t_{w}\rangle \end{bmatrix}}{\begin{bmatrix}\langle d_{w}|d\rangle &\langle d_{w}|s\rangle &\langle d_{w}|b\rangle \\\langle s_{w}|d\rangle &\langle s_{w}|s\rangle &\langle s_{w}|b\rangle \\\langle b_{w}|d\rangle &\langle b_{w}|s\rangle &\langle b_{w}|b\rangle \end{bmatrix}}}$

In a more compact form, this matrix product reads :

${\displaystyle V_{UD}\equiv A_{{U_{w}}U}^{\dagger }A_{{D_{w}}D}}$

The CKM matrix itself can also be understood as a transformation matrix that mediates between the reference system of the -type quarks and a suitable other reference system, the three independent elements of which are designated in another, also common notation , and (which, however, conceptually from the above-mentioned eigenstates of the weak interaction must be distinguished). These are precisely the states that couple exactly and completely to the -type quarks or respectively under the first order boson interaction . Accordingly one writes: ${\displaystyle V_{UD}}$ ${\displaystyle d}$ ${\displaystyle |d'\rangle }$ ${\displaystyle |s'\rangle }$ ${\displaystyle |b'\rangle }$ ${\displaystyle W}$ ${\displaystyle u}$ ${\displaystyle |u\rangle }$ ${\displaystyle |c\rangle }$ ${\displaystyle |t\rangle }$

${\displaystyle {\begin{bmatrix}V_{ud}&V_{us}&V_{ub}\\V_{cd}&V_{cs}&V_{cb}\\V_{td}&V_{ts}&V_{tb}\end{bmatrix}}{\begin{bmatrix}\left|d\right\rangle \\\left|s\right\rangle \\\left|b\right\rangle \end{bmatrix}}={\begin{bmatrix}\left|d'\right\rangle \\\left|s'\right\rangle \\\left|b'\right\rangle \end{bmatrix}}}$

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

As mentioned at the beginning, the term “CKM matrix” is used both for the matrix that Kobayashi and Maskawa defined in the context of the theory of the electroweak interaction in order to construct a mechanism of CP violation, as well as in the context of experimental physics Matrix of values ​​to be determined, the squares of which represent measured Quark transition probabilities.

The CKM matrix in the theoretical sense on the one hand is defined as unitary and, in particular, can be exactly represented as a product of two unitary transformation matrices (which describe the relationship or the mixture of mass eigenstates and eigenstates of the weak interaction, each for the quarks with the same charge).

On the other hand, the CKM matrix in the experimental sense does not necessarily and from the outset fulfill the unitarity condition. Instead, it is only possible to answer experimentally by obtaining measured values ​​as to whether or within what accuracy this matrix is ​​unitary or not.

The prediction that the experimental matrix is ​​actually unitary and that consequently the theory of the electroweak interaction (GWS theory) with three generations of quark flavors is suitable and sufficient to quantitatively correctly describe all detectable changes in quark flavor contents and in Summarizing the form of values ​​of the elements of an exactly unitary 3 × 3 matrix is ​​an essential (i.e. by no means trivial) aspect of the Standard Model.

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

${\displaystyle \sum _{k=1}^{3}|V_{ik}|^{2}=\sum _{n=1}^{3}|V_{nj}|^{2}=1,}$

for every single quark flavor or . This corresponds to the experimental expectation of weak universality that any strength of interaction that leads to changes in the quark flavor content is the same for all quarks (and therefore does not have to be explicitly taken into account in the standardization). This also connects the model expectation and the previous experimental finding that any changes in the quark flavor content (i.e. apart from pair creation or annihilation) occur exclusively through the electroweak interaction ( i.e. coupling to bosons) within three quark generations. ${\displaystyle u_{i}}$ ${\displaystyle d_{j}}$ ${\displaystyle W}$

The remaining partial conditions for the unitarity of a 3 × 3 matrix (secondary diagonal conditions) can be represented by so-called unitary triangles . The corresponding experimental expectations or predictions belonging to the standard model relate, among other things, expressly to measured values ​​for CP violation.

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

${\displaystyle (|V_{ij}|)={\begin{bmatrix}0{,}97427&0{,}22534&0{,}00351\\0{,}22520&0{,}97344&0{,}0412\\0{,}00867&0{,}0404&0{,}999146\end{bmatrix}}}$

Counting of the free parameters

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

1. A complex N × N matrix has real parameters. ${\displaystyle 2N^{2}}$
2. The CKM matrix is ​​unitary, so it holds . Conditions arise that reduce the number of free parameters to . ${\displaystyle \sum _{k}V_{ik}V_{jk}^{*}=\delta _{ij}}$ ${\displaystyle N^{2}}$ ${\displaystyle N^{2}}$
3. Each quark field can absorb one phase. A global phase is unobservable. The free parameters are therefore reduced by further and free parameters remain . ${\displaystyle 2N-1}$ ${\displaystyle (N-1)^{2}}$

Of these, angles of rotation are called quark mixing angles . The remaining parameters are complex phases that cause the CP violation. In particular, there remains only one mixing angle for the quarks, the Cabibbo angle , while in the case of the standard model there are three quark mixing angles and one CP-violating complex phase. ${\displaystyle N(N-1)/2}$ ${\displaystyle (N-1)(N-2)/2}$ ${\displaystyle N=2}$ ${\displaystyle N=3}$

Standard parameterization

In terms of the three mixing angles and the phase, follows for the CKM matrix

${\displaystyle V={\begin{pmatrix}c_{12}c_{13}&s_{12}c_{13}&s_{13}e^{-\mathrm {i} \delta }\\-s_{12}c_{23}-c_{12}s_{23}s_{13}e^{\mathrm {i} \delta }&c_{12}c_{23}-s_{12}s_{23}s_{13}e^{\mathrm {i} \delta }&s_{23}c_{13}\\s_{12}s_{23}-c_{12}c_{23}s_{12}e^{\mathrm {i} \delta }&-s_{23}c_{23}s_{13}e^{\mathrm {i} \delta }&c_{23}c_{13}\end{pmatrix}},}$

where the abbreviations and were introduced. The relation between the entries in the matrix and the angles is found: ${\displaystyle s_{ij}=\sin \theta _{ij}}$ ${\displaystyle c_{ij}=\cos \theta _{ij}}$

${\displaystyle s_{12}=|V_{us}|\qquad s_{13}=|V_{ub}|\qquad s_{23}=|V_{cb}|}$

Wolfenstein parameterization

There is an approximate parameterization of the CKM matrix according to Lincoln Wolfenstein in terms of the sine of the Cabibbo angle . This parameterization is only unitary up to the order , but clearly shows the order of magnitude of the individual matrix elements. The Wolfenstein parameterization is ${\displaystyle \lambda =s_{12}}$ ${\displaystyle \lambda ^{4}}$

${\displaystyle V={\begin{pmatrix}1-{\frac {1}{2}}\lambda ^{2}&\lambda &A\lambda ^{3}(\rho -\mathrm {i} \eta )\\-\lambda -\mathrm {i} A\lambda ^{5}\eta &1-{\frac {1}{2}}\lambda ^{2}&A\lambda ^{2}\\A\lambda ^{3}(1-\rho -\mathrm {i} \eta )&-A\lambda ^{2}-\mathrm {i} A\lambda ^{4}\eta &1\end{pmatrix}}+{\mathcal {O}}(\lambda ^{4}).}$

The three remaining parameters are related to the angles of the standard parameterization as follows: ${\displaystyle A,\eta ,\rho }$

${\displaystyle A={\frac {s_{23}}{s_{12}^{2}}}\qquad \eta ={\frac {s_{13}}{s_{12}s_{23}}}\sin \delta \qquad \rho ={\frac {s_{13}}{s_{12}s_{23}}}\cos \delta }$

Observations and predictions

It can be seen that quark transitions occur within a generation with the greatest probability (diagonal elements close to one), while transitions between different generations (e.g. the decay of an s-quark into the lighter, stable u-quark) are suppressed. This explains the relatively long lifespan for some mesons that contain higher generation quarks.

From the unitarity condition we get the following relationships: ${\displaystyle AA^{\dagger }=1^{3\times 3}}$

${\displaystyle V_{\text{ud}}{\bar {V}}_{\text{us}}+V_{\text{cd}}{\bar {V}}_{\text{cs}}+V_{\text{td}}{\bar {V}}_{\text{ts}}=0}$

${\displaystyle V_{\text{ud}}{\bar {V}}_{\text{ub}}+V_{\text{cd}}{\bar {V}}_{\text{cb}}+V_{\text{td}}{\bar {V}}_{\text{tb}}=0}$

${\displaystyle V_{\text{us}}{\bar {V}}_{\text{ub}}+V_{\text{cs}}{\bar {V}}_{\text{cb}}+V_{\text{ts}}{\bar {V}}_{\text{tb}}=0}$

Since the products of the matrix elements are complex, they can be represented as vectors in the complex number plane. Since the sum of these vectors results in zero, these vectors can be combined to form a triangle and thus the so-called unitarity triangle is obtained. Many research groups are currently working on determining the angle of this triangle using the decays of - and - mesons . ${\displaystyle B}$ ${\displaystyle K}$

The unitarity of the CKM matrix is ​​the subject of current research. One tries, for example, to measure the matrix element via the electroweak top quark production or to find inconsistencies in the unitarity triangle. Should the unitarity of the CKM matrix be violated, this would be an indication of physics beyond the standard model. ${\displaystyle V_{tb}}$

Individual evidence

1. ↑ The name Masukawa is spelled correctly, but in literature the form Maskawa is almost always chosen.
2. ↑ J. Beringer et al. ( Particle Data Group ): Phys. Rev. D 86 , 010001 (2012). For details on measurement inaccuracies, please refer to the data from the Particle Data Group.
3. ^ Organization Européenne pour la Recherche Nucléaire (CERN): The CKM Matrix and the Unitarity Triangle . Ed .: M. Battaglia et al. Geneva 2003, p. 3 , arxiv : hep-ph / 0304132 .
4. ↑ Lincoln Wolfenstein: Parametrization of the Kobayashi-Maskawa Matrix . In: Physical Review Letters . tape 51 , no. 21 , 1983.

literature

• Makoto Kobayashi, Toshihide Maskawa: CP-Violation in the Renormalizable Theory of Weak Interaction . In: Progress of Theoretical Physics . tape 49 , no. 2 , 1973, p. 652-657 , doi : 10.1143 / PTP.49.652 .