The CKM matrix & its parametrizations

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

4. CP Violation antiparticle • C (charge conjugation) : particle 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 very important for CP study • For quarks, • weak interaction eigenstates  mass eigenstates • mixing of flavor through CKM matrix

6. 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!

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

8. =0.1440.025 =0.342+0.016-0.015

9. UnitarityTests of Mixing Matrices The quark sector Unitarity:

10. Physics should be independent of a particular parametrization of the CKM matrix !

11. Although differentparametrizations 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.

12. Parametrizations of the CKM matrix • Exact parametrizations • -- KM parametrization (1973) • -- Maianiparametrization (1977) • -- CK (Standard) parametrization (1984) • Approximate parametrizations • -- Wolfensteinparametrization (1983) • -- Qin-Maparametrization (2011)

13. Kobayashi-Maskawaparametrization (1973) • The first parametrization of the CKM matrix by KM • From the experimental data • nearly 90o : maximal CP violation

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.

15. Maianiparametrization (1977) • Thisparametrization 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.

16. Chau-Keung parametrization (1984) • The standard parametrization for the quark mixing matrix • From the experimental data

17. Thisparametrization 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 .

18. Wolfensteinparametrization (1983) • In 1983, it was realized thatthe 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.

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

20. Qin-Ma parametrization (2011) • AWolfenstein-like parametrization • With the data on the magnitudes of the CKM matrix elements in the KM parametrization, • To a good approximation, let • “Triminimalparametrization” • with • To make the lowest order be the unit matrix, adjust the phases of • quarks with

22. Qin-Ma parametrization • maximal CP violation • Wolfensteinparametrization

23. Qin-Ma argued that “one has difficulty to arrive at the Wolfensteinparametrization from the triminimalparametrization of the KM matrix.” • However, it can be shown that both Wolfenstein & Qin-Ma parametrizations can be obtained easily from the KM & CK parametrizationsto be discussed from now on.

24. CK  Wolfensteinparametrization • Let

25. KM  Wolfensteinparametrization • Rotate the phases of the quark fields • Let • nearly 90o

26. Wolfenstein Qin-Maparametrization • Rotate the phases of the quark fields

27. Let • nearly 90o

28. The rephasing-invariant quantity: “Jarlskog invariant” • Wolfenstein • Qin-Ma • nearly 90o

29. CK  Qin-Maparametrization • Rotate the phases of the quark fields • Let

30. WolfensteinParametrization at Higher Order

31. The CKM matrix elements are the fundamental parameters in the SM, the precise determinationof 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 termsin the CKM matrix elements to be determined in the future precision experiments will become more and more important.

32. It was pointed out that as in any perturbative expansion, high order terms in l are not unique in the Wolfensteinparametrization, 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 .

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.

34. Wolfensteinparametrization (1983)

35. TheWolfensteinparametrization up to l6

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

40. Thus, the seeming discrepancies are resolved !

41. Qin-Ma parametrization (2011)

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