The CKM matrix & its parametrizations

Sechul Oh Yonsei University (Int’l Campus)

with Y.H. Ahn and H.Y. ChengPhys. Lett. B701, 614 (2011) Phys. Lett. B703, 571 (2011)

Particle Phys., Yonsei, December 1, 2011

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Outline

Introdution

Parametrizations of the CKM matrix

Wolfenstein & Wolfenstein-like parametrizations at high order

Summary

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• C (charge conjugation) : particle

P (parity) : right-handed left-handed

antiparticle

• 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).

CP Violation

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For quarks,weak interaction eigenstates mass eigenstates mixing of flavor through CKM matrix

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very importantfor CP study

The Quark Mixing & Lepton Mixing Matrices

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Good approximation for neutrino mixing: The tri-bimaximal matrix

Good approximation for quark mixing: The unit matrix

Very different mixing patterns for quarks and neutri-nos!

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Cabibbo-Kobayashi-Maskawa (CKM) matrix

Unitarity:

Unitarity triangle:(g)

(a)

(b)

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=0.1440.025=0.342+0.016

-0.015

Unitarity Tests of Mixing Matrices

The quark sector

Unitarity:

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Physics should be independent of a partic-ular parametrization of the CKM matrix !

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Although different parametrizations of the quark mixing matrix are mathematically equivalent, the consequences of experi-mental analysis may be distinct.

The magnitude of the elements Vij are physical quantities which do not depend on parametrization. However, the CP-vio-lating phase does.As a result, the understanding of the origin of CP violation is as-sociated with the parametrization.

e.g., the prediction based on the maximal CP violation hypothe-sis is related with the parametrization, or in other words, phase convention.

i.e., with the original KM parametrization, one can get success-ful predictions on the unitarity triangle from the maximal CP vi-olation hypothesis.

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Parametrizations of the CKM matrix

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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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Kobayashi-Maskawa parametrization (1973)

The first parametrization of the CKM matrix by KM

From the experimental data

nearly 90o : maximal CP viola-tion

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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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Maiani parametrization (1977)

This parametrization has a nice feature that its imag-inary part is proportional to s23 sin f , which is of or-der 10-2 .It was once proposed by PDG (1986 eidtion) to be the

standard parametrization for the quark mixing ma-trix.

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Chau-Keung parametrization (1984)

The standard parametrization for the quark mixing matrix

From the experimental data

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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 .

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Wolfenstein parametrization (1983)

In 1983, it was realized that the bottom quark de-cays predominantly to the charm quark:

Wolfenstein then noticed that and intro-duced

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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-- The parameter is small and serves as an expan-sion parameter.

-- The parameter , because .-- Since , the parameters and should

be smaller than one. From the experimental data

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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 parametriza-tion”

To make the lowest order be the unit matrix, adjust the phases of

quarks with

with

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Qin-Ma parametrization

Wolfenstein parametrization maximal CP viola-tion

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Qin-Ma argued that “one has difficulty to arrive at the Wolfenstein parametrization from the trimini-mal 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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CK Wolfenstein parametrization

Let

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KM Wolfenstein parametrization

Rotate the phases of the quark fields

Let

nearly 90o

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Wolfenstein Qin-Ma parametrization

Rotate the phases of the quark fields

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Let

nearly 90o

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The rephasing-invariant quantity: “Jarlskog in-variant”

nearly 90o

Wolfenstein

Qin-Ma

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CK Qin-Ma parametrization

Rotate the phases of the quark fields

Let

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Wolfenstein Parametrization

at Higher Order

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The CKM matrix elements are the fundamental pa-rameters in the SM, the precise determination of which is highly crucial and will be performed in fu-ture experiments such as LHCb and Super B factory ones.Apparently, if the CKM matrix is expressed in a par-ticular 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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It was pointed out that as in any perturbative ex-pansion, 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 cer-tain 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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Since the CKM matrix can be parametrized in infin-itely many ways with three rotation angles and one CP-odd phase, it is desirable to find a certain sys-tematic way to resolve these complications and to keep consistency between the CKM matrix ele-ments expressed in different parametrizations.

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Wolfenstein parametrization (1983)

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The Wolfenstein parametrization up to l6

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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

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Thus, the seeming discrepancies are re-solved !

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Qin-Ma parametrization (2011)

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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!