CKM matrix

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In the Standard Model of particle physics, 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 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. This matrix is also an extension of the GIM mechanism, which only includes 2 of the 3 current families of quarks.

1 The matrix
2 Counting
3 Observations and predictions
4 Weak universality
5 The unitarity triangles
6 Parameterizations
7 Nobel Prize
8 See also
9 Notes
10 References
11 External links

The matrix

A pictorial representation of the six quarks' decay modes, with mass increasing from left to right.

In 1963, Nicola Cabibbo introduced the Cabibbo angle (θ_c) to preserve the universality of the weak interaction.¹ In light of current knowledge (quarks were not yet theorized), the Cabibbo angle is related to the relative probability that down and strange quarks decays into up quarks (|V_ud|² and |V_us|² respectively). In particle physics parlance, the d and s quarks were said to form a weak interaction eigenstate, here denoted by d′.² Mathematically this is:

$|d^\prime \rangle = V_{ud} | d \rangle + V_{us} | s \rangle \ ,$

or using the Cabbibo angle:

$|d^\prime \rangle = \cos \theta_c | d \rangle + \sin \theta_c | s \rangle \ .$

Using the currently accepted values for |V_ud| and |V_us| (see below), the Cabbibo angle can be calculated using

$\tan\theta_c=\frac{|V_{us}|}{|V_{ud}|}=\frac{0.2257}{0.97419} \rarr \theta_c= ~13.04^\circ \ .$

If this were a valid treatment, the probabilities |V_ud|² and |V_us|² should add up to 1, but instead add up to 0.9999866461, indicative of the missing top quark (the missing term is |V_ut|² = 0.0000128881). However, determining |V_ud|² and |V_us|² with a high-enough precision to predict the existence of the t quark was not possible at the time.

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:

The Cabibbo angle represents the rotation of the strong eigenstate vector space formed by the strong eigenstates $\scriptstyle{| d \rangle , \ | s \rangle}$ into the weak eigensate vector space formed by the weak eigenstates $\scriptstyle{| d^\prime \rangle , \ | s^\prime \rangle}$ . θ_C = 13.04°.

$| d^\prime \rangle = V_{ud} | d \rangle + V_{us} | s \rangle \ ;$

$| s^\prime \rangle = V_{cd} | d \rangle + V_{cs} | s \rangle \ ,$

or using the Cabibbo angle:

$| d^\prime \rangle = \cos{\theta_C} | d \rangle + \sin{\theta_C} | s \rangle \ ;$

$| s^\prime \rangle = -\sin{\theta_C} | d \rangle + \cos{\theta_C} | s \rangle \ .$

This can also be written in matrix notation (a type of mathematical table) as:

$\begin{bmatrix} \left| d^\prime \right \rangle \\ \left| s^\prime \right \rangle \end{bmatrix} = \begin{bmatrix} V_{ud} & V_{us} \\ V_{cd} & V_{cs}\\ \end{bmatrix} \begin{bmatrix} \left| d \right \rangle \\ \left| s \right \rangle \end{bmatrix} \ ,$

or using the Cabibbo angle

$\begin{bmatrix} \left| d^\prime \right \rangle \\ \left| s^\prime \right \rangle \end{bmatrix} = \begin{bmatrix} \cos{\theta_C} & \sin{\theta_C} \\ -\sin{\theta_C} & \cos{\theta_C}\\ \end{bmatrix} \begin{bmatrix} \left| d \right \rangle \\ \left| s \right \rangle \end{bmatrix} \ ,$

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 matrix is called the Cabibbo matrix.^{citation needed} 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:³

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

On the left is the weak interaction eigenstates of the quarks, and on the right is the CKM Matrix along with a vector of strong interaction eigenstates of the 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 magnitudes of the CKM matrix elements is:⁴

$\begin{bmatrix} |V_{ud}| & |V_{us}| & |V_{ub}| \\ |V_{cd}| & |V_{cs}| & |V_{cb}| \\ |V_{td}| & |V_{ts}| & |V_{tb}| \end{bmatrix} = \begin{bmatrix} 0.97419 \pm 0.00022 & 0.2257 \pm 0.010 & 0.00359 \pm 0.0016 \\ 0.2256 \pm 0.0010 & 0.97334 \pm 0.00023 & 0.0415^{+0.0010}_{-0.0011} \\ 0.00874^{+0.00026}_{-0.00037} & 0.0407 \pm 0.0010 & 0.999133^{+0.000044}_{-0.000043} \end{bmatrix}.$

Counting

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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

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 (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.

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.⁵

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 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

∑	\| V_ik \| 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.

KM parameters

The original parameterization of Kobayashi and Maskawa used three angles (θ₁, θ₂, θ₃) and a CP-violating phase (δ).⁶ Cosines and sines of the angles are denoted c_i and s_i, respectively. θ₁ is the Cabibbo angle.

$\begin{bmatrix} c_1 & -s_1 c_3 & -s_1 s_3 \\ s_1 c_2 & c_1 c_2 c_3 - s_2 s_3 e^{i\delta} & c_1 c_2 s_3 + s_2 c_3 e^{i\delta}\\ s_1 s_2 & c_1 s_2 c_3 + c_2 s_3 e^{i\delta} & c_1 s_2 s_3 - c_2 c_3 e^{i\delta} \end{bmatrix}.$

"Standard" parameters

A "standard" parameterization of the CKM matrix uses three Euler angles (θ₁₂, θ₂₃, θ₁₃) and one CP-violating phase (δ₁₃).⁷ 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.

$\begin{bmatrix} c_{12}c_{13} & s_{12} c_{13} & s_{13}e^{-i\delta_{13}} \\ -s_{12}c_{23} - c_{12}s_{23}s_{13}e^{i\delta_{13}} & c_{12}c_{23} - s_{12}s_{23}s_{13}e^{i\delta_{13}} & s_{23}c_{13}\\ s_{12}s_{23} - c_{12}c_{23}s_{13}e^{i\delta_{13}} & -c_{12}s_{23} - s_{12}c_{23}s_{13}e^{i\delta_{13}} & c_{23}c_{13} \end{bmatrix}.$

The currently best known values for the standard parameters are θ₁₂ = 13.04° ± 0.05°, θ₁₃ = 0.201° ± 0.011°, θ₂₃ = 2.38° ± 0.06° and δ₁₃ = 1.20 ± 0.08.⁸

Wolfenstein parameters

A third parameterization of the CKM matrix was introduced by Lincoln Wolfenstein with the four parameters λ, A, ρ, and η.⁹ 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

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

Using the values of the previous section for the CKM matrix, the best determination of the Wolfenstein variables is λ = 0.2257+0.0009−0.0010, A = 0.814+0.021−0.022, ρ = 0.135+0.031−0.016 and η = 0.349+0.015−0.017.⁴

Nobel Prize

In 2008, Kobayashi and Maskawa shared one half of the 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".¹⁰ Some physicists, especially Italian, had bitter feelings that the Nobel Prize committee failed to reward the work of Cabibbo, on which the work of the other two was based.¹¹ Asked for a reaction on the prize, Cabibbo preferred to give no comment. According to sources close to him, he was very embittered.¹²

Notes

^ N. Cabibbo (1963). "Unitary Symmetry and Leptonic Decays". Physical Review Letters 10 (12): 531–533. doi:10.1103/PhysRevLett.10.531.
^ I.S. Hughes (1991). "Chapter 11.1 – Cabibbo Mixing". Elementary Particles (3rd ed.). Cambridge University Press. pp. 242–243. ISBN 0-521-40402-9. http://books.google.ca/books?id=JN6qlZlGUG4C&pg=PA242&lpg=PA242&dq=cabbibo+angle&source=bl&ots=O-9mv4enjH&sig=2WKAm3JFiK9m9nXaFX6lectLojs&hl=en&sa=X&oi=book_result&resnum=11&ct=result#PPR11,M1.
^ M. Kobayashi, T. Maskawa (1973). "CP-Violation in the Renormalizable Theory of Weak Interaction". Progress of Theoretical Physics 49 (2): 652–657. doi:10.1143/PTP.49.652. http://ptp.ipap.jp/link?PTP/49/652/pdf.
^ ^a ^b C. Amsler et al. (2008). "Review of Particles Physics" (PDF). Physics Letters B667: 1–1340. http://pdg.lbl.gov/2008/reviews/kmmixrpp.pdf.
^ J.C. Baez (March 6, 2005). "Neutrinos and the Mysterious Maki-Nakagawa-Sakata Matrix". Retrieved on January 4, 2009. "In fact, the Maki-Nakagawa-Sakata matrix actually affects the behavior of all leptons, not just neutrinos. Furthermore, a similar trick works for quarks - but then the matrix U is called the Cabibbo-Kobayashi-Maskawa matrix. This one was actually discovered first."
^ M. Kobayashi and T. Maskawa, Progress in Theoretical Physics 49 652 (1973).
^ L. L. Chau and W.-Y. Keung, Physical Review Letters 53 1802 (1984).
^ Values obtained from values of Wolfenstein parameters in the 2008 Particle Data Group review.
^ L. Wolfenstein, Physical Review Letters 51 1945 (1983).
^ "The Nobel Prize in Physics 2008". nobelprize.org (7 October 2008).
^ "Physics Nobel snubs key researcher". New Scientist (7 October 2008).
^ "Nobel, l'amarezza dei fisici italiani". Corriere della Sera (7 October 2008).

References

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

External links

CP violation, by I.I. Bigi and A.I. Sanda (Cambridge University Press, 2000) ISBN 0-521-44349-0
Particle Data Group: the CKM matrix
Particle Data Group: CP violation in meson decays
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.

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