Cabibbo-Kobayashi-Maskawa matrix
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In the Standard Model of particle physics, the Cabibbo-Kobayashi-Maskawa matrix (CKM matrix, quark mixing matrix, sometimes earlier 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 quarks when they propagate freely and when they take part in the weak interactions. It is important in the understanding of CP violations. A precise mathematical definition of this matrix is given in the article on the formulation of the standard model. 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.

Contents

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 mass 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, combining a large number of independent measurements, the magnitudes of the values in the matrix have been found to be[1] (only central values presented here, uncertainties are excluded):

V_{ij} = \begin{bmatrix} 0.97383 & 0.2272 & 0.00396 \\ 0.2271 & 0.97296 & 0.04221 \\ 0.00814 & 0.04161 & 0.999100 \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 (2N flavours) then

  1. An N×N complex matrix contains 2N2 real numbers.
  2. The constraint of unitarity is ∑k VikV*jk = δij. Therefore, for the diagonal terms (i=j) there are N constraints, and for the remaining terms, N(N−1). The number of independent real numbers in a unitary matrix is therefore N2.
  3. One phase can be absorbed into each quark field. An overall common phase is unobservable. Hence there are 2N−1 fewer independent numbers, giving the total number of free variables to be (N−1)2.
  4. Of these, N(N−1)/2 are rotation angles called quark mixing angles.
  5. The remaining (N−1)(N−2)/2 are complex phases, which cause CP violation.

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 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:

  1. the transitions u↔d and e↔νe, μ↔νμ had similar amplitudes.
  2. 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 issue 1, along with a mixing angle θc, 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 kaon 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 at Fermilab (by Leon Lederman's group) in 1976 therefore immediately started off the search for the missing third-generation quark, the top quark.

Weak universality

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

∑ | Vik | 2 = 1
k

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 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, 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 Japanese BELLE and the Californian BaBar experiments.

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.

The original parameterization of Kobayashi and Maskawa used three angles (θ1, θ2, θ3) and a CP-violating phase (δ).[2] Cosines and sines of the angles are denoted ci and si, respectively. θ1 is the Cabibbo angle.

\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}

A "standard" parameterization of the CKM matrix uses three Euler angles (θ12, θ23, θ13) and one CP-violating phase (δ13).[3] Couplings between quark generation i and j vanish if θij = 0. Cosines and sines of the angles are denoted cij and sij, respectively. θ12 is the Cabibbo angle.

\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}

A third parameterization of the CKM matrix was introduced by Wolfenstein with four variables (λ, A, ρ, η) all of order one.[4] The four Wolfenstein variables 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

\begin{bmatrix} 1-\lambda^2/2 & \lambda & A\lambda^3(\rho-i\eta) \\ -\lambda & 1-\lambda^2/2 & A\lambda^2 \\ A\lambda^3(1-\rho-i\eta) & -A\lambda^2 & 1 \end{bmatrix}

See also

References

  1. ^ W.-M. Yao et al., J. Phys. G 33, 1 (2006) and 2007 partial update for the 2008 edition available on the PDG WWW pages (URL: http://pdg.lbl.gov/), Chapter 11. The CKM Quark-Mixing Matrix
  2. ^ M. Kobayashi and T. Maskawa, Progress in Theoretical Physics 49 652 (1973).
  3. ^ L. L. Chau and W.-Y. Keung, Physical Review Letters 53 1802 (1984).
  4. ^ L. Wolfenstein, Physical Review Letters 51 1945 (1983).
  • Griffiths, David J. (1987). Introduction to Elementary Particles. Wiley, John & Sons, Inc. ISBN 0-471-60386-4. 
  • Povh, Bogdan et al., (1995). Particles and Nuclei: An Introduction to the Physical Concepts. New York: Springer. ISBN 3-540-20168-8

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