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Unformatted text preview: 11. CKM quark-mixing matrix 1 11. THE CKM QUARK-MIXING MATRIX Revised February 2010 by A. Ceccucci (CERN), Z. Ligeti (LBNL), and Y. Sakai (KEK). 11.1. Introduction The masses and mixings of quarks have a common origin in the Standard Model (SM). They arise from the Yukawa interactions with the Higgs condensate, L Y = Y d ij Q I Li d I Rj Y u ij Q I Li u I Rj + h . c ., (11 . 1) where Y u,d are 3 3 complex matrices, is the Higgs field, i, j are generation labels, and is the 2 2 antisymmetric tensor. Q I L are left-handed quark doublets, and d I R and u I R are right-handed down- and up-type quark singlets, respectively, in the weak-eigenstate basis. When acquires a vacuum expectation value, = (0 , v/ 2), Eq. (11 . 1) yields mass terms for the quarks. The physical states are obtained by diagonalizing Y u,d by four unitary matrices, V u,d L,R , as M f diag = V f L Y f V f R ( v/ 2), f = u, d . As a result, the charged-current W interactions couple to the physical u Lj and d Lk quarks with couplings given by V CKM V u L V d L = V ud V us V ub V cd V cs V cb V td V ts V tb . (11 . 2) This Cabibbo-Kobayashi-Maskawa (CKM) matrix [1,2] is a 3 3 unitary matrix. It can be parameterized by three mixing angles and the CP-violating KM phase . Of the many possible conventions, a standard choice has become  V = c 12 c 13 s 12 c 13 s 13 e i s 12 c 23 c 12 s 23 s 13 e i c 12 c 23 s 12 s 23 s 13 e i s 23 c 13 s 12 s 23 c 12 c 23 s 13 e i c 12 s 23 s 12 c 23 s 13 e i c 23 c 13 , (11 . 3) where s ij = sin ij , c ij = cos ij , and is the phase responsible for all CP-violating phenomena in avor-changing processes in the SM. The angles ij can be chosen to lie in the first quadrant, so s ij , c ij 0. It is known experimentally that s 13 s 23 s 12 1, and it is convenient to exhibit this hierarchy using the Wolfenstein parameterization. We define  s 12 = = | V us | p | V ud | 2 + | V us | 2 , s 23 = A 2 = V cb V us , s 13 e i = V ub = A 3 ( + i ) = A 3 ( + i ) 1 A 2 4 1 2 [1 A 2 4 ( + i )] . (11 . 4) These relations ensure that + i = ( V ud V ub ) / ( V cd V cb ) is phase-convention-independent, and the CKM matrix written in terms of , A, , and is unitary to all orders in . The definitions of , reproduce all approximate results in the literature. For example, = (1 2 / 2 + . . . ) and we can write V CKM to O ( 4 ) either in terms of , or, traditionally, V = 1 2 / 2 A 3 ( i ) 1 2 / 2 A 2 A 3 (1 i ) A 2 1 + O ( 4 ) . (11 . 5) K. Nakamura et al. , JPG 37 , 075021 (2010) (http://pdg.lbl.gov) July 30, 2010 14:36 2 11. CKM quark-mixing matrix Figure 11.1: Sketch of the unitarity triangle.Sketch of the unitarity triangle....
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