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Unformatted text preview: 11. CKM quarkmixing matrix 1 11. THE CKM QUARKMIXING 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 φ d I Rj − Y u ij Q I Li φ ∗ u I Rj + h . c ., (11 . 1) where Y u,d are 3 × 3 complex matrices, φ is the Higgs field, i, j are generation labels, and is the 2 × 2 antisymmetric tensor. Q I L are lefthanded quark doublets, and d I R and u I R are righthanded down and uptype quark singlets, respectively, in the weakeigenstate basis. When φ acquires a vacuum expectation value, φ = (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 chargedcurrent 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 CabibboKobayashiMaskawa (CKM) matrix [1,2] is a 3 × 3 unitary matrix. It can be parameterized by three mixing angles and the CPviolating KM phase [2]. Of the many possible conventions, a standard choice has become [3] V = ⎛ ⎝ c 12 c 13 s 12 c 13 s 13 e − iδ − s 12 c 23 − c 12 s 23 s 13 e iδ c 12 c 23 − s 12 s 23 s 13 e iδ s 23 c 13 s 12 s 23 − c 12 c 23 s 13 e iδ − c 12 s 23 − s 12 c 23 s 13 e iδ c 23 c 13 ⎞ ⎠ , (11 . 3) where s ij = sin θ ij , c ij = cos θ ij , and δ is the phase responsible for all CPviolating phenomena in ﬂavorchanging processes in the SM. The angles θ 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 [4–6] s 12 = λ =  V us  p  V ud  2 +  V us  2 , s 23 = Aλ 2 = λ ¯ ¯ ¯ ¯ V cb V us ¯ ¯ ¯ ¯ , s 13 e iδ = V ∗ ub = Aλ 3 ( ρ + iη ) = Aλ 3 (¯ ρ + i ¯ η ) √ 1 − A 2 λ 4 √ 1 − λ 2 [1 − A 2 λ 4 (¯ ρ + i ¯ η )] . (11 . 4) These relations ensure that ¯ ρ + i ¯ η = − ( V ud V ∗ ub ) / ( V cd V ∗ cb ) is phaseconventionindependent, and the CKM matrix written in terms of λ, A, ¯ ρ , and ¯ η is unitary to all orders in λ . The definitions of ¯ ρ, ¯ η reproduce all approximate results in the literature. For example, ¯ ρ = ρ (1 − λ 2 / 2 + . . . ) and we can write V CKM to O ( λ 4 ) either in terms of ¯ ρ, ¯ η or, traditionally, V = ⎛ ⎝ 1 − λ 2 / 2 λ A λ 3 ( ρ − iη ) − λ 1 − λ 2 / 2 Aλ 2 Aλ 3 (1 − ρ − iη ) − Aλ 2 1 ⎞ ⎠ + O ( λ 4 ) . (11 . 5) K. Nakamura et al. , JPG 37 , 075021 (2010) (http://pdg.lbl.gov) July 30, 2010 14:36 2 11. CKM quarkmixing matrix Figure 11.1: Sketch of the unitarity triangle.Sketch of the unitarity triangle....
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 Mass, The Land

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