1. The CKM matrix & its parametrizations
Sechul Oh
Yonsei University (Int’l Campus)
with Y.H. Ahn and H.Y. Cheng
Phys. Lett. B701, 614 (2011)
Phys. Lett. B703, 571 (2011)
Particle Phys., Yonsei, December 1, 2011

2. Outline Introdution Parametrizations of the CKM matrix
Wolfenstein & Wolfenstein-like parametrizations
at high order
Summary
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4. CP Violation • C (charge conjugation) : particle antiparticle
P (parity) : right-handed
left-handed
• Matter-antimatter asymmetry in universe requires
CP-violating interactions (Sakharov 1967)
• CP violation has been experimentally observed:
in K meson system (1963)
in B meson system (1998)
• The Standard Model:
the origin of CP violation is a complex phase
of the “CKM matrix” (1973).

5. The Quark Mixing & Lepton Mixing Matrices
For quarks,
weak interaction eigenstates  mass eigenstates
mixing of flavor through CKM matrix
very important
for CP study
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6. Very different mixing patterns for quarks and neutrinos!
Good approximation for
neutrino mixing:
The tri-bimaximal matrix
Good approximation for
quark mixing: The unit matrix
Very different mixing patterns for quarks and neutrinos!
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7. Cabibbo-Kobayashi-Maskawa (CKM) matrix
Unitarity:
(a)
Unitarity triangle:
(g)
(b)

8. =0.1440.025 =

9. Unitarity Tests of Mixing Matrices
The quark sector
Unitarity:

10. Physics should be independent of a particular parametrization of the CKM matrix !
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11. Although different parametrizations of the quark mixing matrix are mathematically equivalent, the consequences of experimental analysis may be distinct.
The magnitude of the elements Vij are physical quantities which do not depend on parametrization. However, the CP-violating phase does.
As a result, the understanding of the origin of CP violation is associated with the parametrization.
 e.g., the prediction based on the maximal CP violation hypothesis is related with the parametrization, or in other words, phase convention.
i.e., with the original KM parametrization, one can get successful predictions on the unitarity triangle from the maximal CP violation hypothesis.
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12. Parametrizations of the CKM matrix
Exact parametrizations
-- KM parametrization (1973)
-- Maiani parametrization (1977)
-- CK (Standard) parametrization (1984)
Approximate parametrizations
-- Wolfenstein parametrization (1983)
-- Qin-Ma parametrization (2011)
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13. Kobayashi-Maskawa parametrization (1973)
The first parametrization of the CKM matrix by KM
From the experimental data
nearly 90o : maximal CP violation
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14. There is one disadvantage in this parametrization:
the matrix element Vtb (of order 1) has a large imaginary
part.
 Since CP-violating effects are known to be small, it is
desirable to parameterize the mixing matrix in such a
way that the imaginary part appears with a smaller
coefficient.
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15. Maiani parametrization (1977)
This parametrization has a nice feature that its imaginary part is proportional to s23 sin f , which is of order
It was once proposed by PDG (1986 eidtion) to be the
standard parametrization for the quark mixing matrix.
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16. Chau-Keung parametrization (1984)
The standard parametrization for the quark mixing matrix
From the experimental data
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17. This parametrization is equivalent to the Maiani one, after
the quark field redefinition:
The imaginary part is proportional to s13 sin f , which is
of order
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18. Wolfenstein parametrization (1983)
In 1983, it was realized that the bottom quark decays predominantly to the charm quark:
Wolfenstein then noticed that and introduced
an approximate parametrization of the CKM matrix
-- a parametrization in which unitarity only holds
approximately.
This parametrization is practically very useful and has since become very popular.
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19. -- The parameter is small and serves as an expansion parameter.
-- The parameter , because
-- Since , the parameters and should be
smaller than one.
From the experimental data
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20. Qin-Ma parametrization (2011)
A Wolfenstein-like parametrization
With the data on the magnitudes of the CKM matrix elements in the KM parametrization,
To a good approximation, let
“Triminimal parametrization”
with
To make the lowest order be the unit matrix, adjust the phases of
quarks with
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21. Sechul Oh

22. Qin-Ma parametrization
maximal CP violation
Wolfenstein parametrization
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23. Qin-Ma argued that “one has difficulty to arrive at the Wolfenstein parametrization from the triminimal parametrization of the KM matrix.”
However, it can be shown that both Wolfenstein & Qin-Ma parametrizations can be obtained easily from the KM & CK parametrizations to be discussed from now on.
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24. CK  Wolfenstein parametrization
Let
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25. KM  Wolfenstein parametrization
Rotate the phases of the quark fields
Let
nearly 90o
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26. Wolfenstein  Qin-Ma parametrization
Rotate the phases of the quark fields
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27. Let
nearly 90o
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28. The rephasing-invariant quantity: “Jarlskog invariant”
Wolfenstein
Qin-Ma
nearly 90o
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29. CK  Qin-Ma parametrization
Rotate the phases of the quark fields
Let
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30. Wolfenstein Parametrization
at Higher Order
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31. The CKM matrix elements are the fundamental parameters in the SM, the precise determination of which is highly crucial and will be performed in future experiments such as LHCb and Super B factory ones.
Apparently, if the CKM matrix is expressed in a particular parametrization, such as the Wolfenstein one, having an approximated form in terms of a small expansion parameter l , then high order l terms in the CKM matrix elements to be determined in the future precision experiments will become more and more important.
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32. It was pointed out that as in any perturbative expansion, high order terms in l are not unique in the Wolfenstein parametrization, though the nonuniqueness of the high order terms does not change the physics.
Thus, if one keeps using only one parametrization, there would not be any problem.
However, if one tries to compare the values of certain parameters, such as l , used in one parametrization with those used in another parametrization, certain complications can occur, because of the nonuniqueness of the high order terms in l .
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33. Since the CKM matrix can be parametrized in infinitely many ways with three rotation angles and one CP-odd phase, it is desirable to find a certain systematic way to resolve these complications and to keep consistency between the CKM matrix elements expressed in different parametrizations.
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34. Wolfenstein parametrization (1983)
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35. The Wolfenstein parametrization up to l6
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37. In comparison with the data
which Wolfenstein used for his original parametrization, the current data indicates
Thus, propose to define the parameters and of order unity by scaling the numerically small (of order l ) parameters and as
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40. Thus, the seeming discrepancies are resolved !
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41. Qin-Ma parametrization (2011)
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42. Sechul Oh

43. Summary
We have discussed several different parametrizations of the quark
mixing matrix.
The approximated parametrizations, such as the Wolfenstein &
Qin-Ma ones, can be obtained easily from the exact parametrizations,
such as the KM & CK ones.
Seeming discrepancies appearing at high order in the Wolfenstein
& Wolfenstein-like parametrization can be systematically resolved.
Thank you!
Sechul Oh