CKM matrix

Flavour in particle physics

Flavour quantum numbers

Y=B+S+C+B'+T
Q=I_z+Y/2
Q=T_z+Y_W/2
B-L conserved in the standard model

The Matrix

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

On the left is the CKM Matrix along with a vector of strong force eigenstates of the quarks, and on the right is the weak force eigenstates of the quarks. The CKM matrix describes the probability of a transition from one quark q to another quark q' . This transition is proportional to $\left| V_{qq'} \right| ^2$ .

Experimentally, the values in the matrix have been found to be roughly:

$\begin{bmatrix} 0.9753 & 0.221 & 0.003 \\ 0.221 & 0.9747 & 0.040 \\ 0.009 & 0.039 & 0.9991 \end{bmatrix}$

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 (ie, 2N flavours) then

A N×N complex matrix contains 2N² real numbers, ie, 2 for each entry.
The constraint of unitarity is ∑_k V_ikV^*_jk = δ_ij. Therefore, for the diagonal terms (i=j) there are N constraints, and for the remaining terms, N(N-1). The number of independent complex numbers in an unitary matrix is therefore N².
One phase can be absorbed into each quark field. An overall common phase is unobservable. Hence there are <b>2N-1 less independent numbers, giving the total number of free variables to be (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.

Observations and predictions

The idea of Cabibbo originated from a need to explain two observed phenomena:

the transitions u↔d and e↔ν_e, 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 postulaing [[#weak universality|]weak universality] to resolve issue 1, along with a mixing angle θ_c, now called the Cabibbo angle, between the d and s quarks to resolve issue 2.

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 K meson decays already in 1964, the emergence of the standard model 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 by Leon Lederman in 1976 therefore immediately started off the search for the top quark.

Weak universality

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

\sum | V i j | 2 = 1 j

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 after Nicola Cabibbo, who first pointed it out 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

$\sum_k V_{ik}V^*_{jk} = 0$

For any fixed i and j, this is a constraint on three complex numbers, one for each k, which says that these numbers form the vertices of a triangle in the complex plane. There are six choices of i and j, 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 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.

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 Japanse BELLE and the Californian BaBar experiments.

References and external links

Povh, Bogdan et al., (1995). Particles and Nuclei: An Introduction to the Physical Concepts. New York: Springer. ISBN 3540201688
CP violation, by I.I. Bigi and A.I. Sanda (Cambridge University Press, 2000) [ISBN 0521443490]
Particle Data Group on CP violation
The Babar experiment at SLAC and the BELLE experiment at KEK Japan
N. Cabibbo, Phys. Rev. Lett. 10 (1963) 531.
M. Kobayashi and K. Maskawa, Prog. Theor. Phys. 49 (1973) 652.