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

Outline Introdution Parametrizations of the CKM matrix Wolfenstein & Wolfenstein-like parametrizations at high order Summary Sechul Oh

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).

The Quark Mixing & Lepton Mixing Matrices For quarks, weak interaction eigenstates  mass eigenstates mixing of flavor through CKM matrix very important for CP study Sechul Oh

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! Sechul Oh

Cabibbo-Kobayashi-Maskawa (CKM) matrix Unitarity: (a) Unitarity triangle: (g) (b)

=0.1440.025 =0.342+0.016-0.015

Unitarity Tests of Mixing Matrices The quark sector Unitarity:

Physics should be independent of a particular parametrization of the CKM matrix ! Sechul Oh

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. Sechul Oh

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) Sechul Oh

Kobayashi-Maskawa parametrization (1973) The first parametrization of the CKM matrix by KM From the experimental data nearly 90o : maximal CP violation Sechul Oh

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. Sechul Oh

Maiani parametrization (1977) This parametrization has a nice feature that its imaginary part is proportional to s23 sin f , which is of order 10-2 . It was once proposed by PDG (1986 eidtion) to be the standard parametrization for the quark mixing matrix. Sechul Oh

Chau-Keung parametrization (1984) The standard parametrization for the quark mixing matrix From the experimental data Sechul Oh

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 10-3 . Sechul Oh

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. Sechul Oh

-- 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 Sechul Oh

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 Sechul Oh

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Qin-Ma parametrization maximal CP violation Wolfenstein parametrization Sechul Oh

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. Sechul Oh

CK  Wolfenstein parametrization Let Sechul Oh

KM  Wolfenstein parametrization Rotate the phases of the quark fields Let nearly 90o Sechul Oh

Wolfenstein  Qin-Ma parametrization Rotate the phases of the quark fields Sechul Oh

Let nearly 90o Sechul Oh

The rephasing-invariant quantity: “Jarlskog invariant” Wolfenstein Qin-Ma nearly 90o Sechul Oh

CK  Qin-Ma parametrization Rotate the phases of the quark fields Let Sechul Oh

Wolfenstein Parametrization at Higher Order Sechul Oh

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. Sechul Oh

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 . Sechul Oh

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. Sechul Oh

Wolfenstein parametrization (1983) Sechul Oh

The Wolfenstein parametrization up to l6 Sechul Oh

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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 Sechul Oh

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Thus, the seeming discrepancies are resolved ! Sechul Oh

Qin-Ma parametrization (2011) Sechul Oh

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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