Wing Sheung Chan

16 The Standard Model and lepton flavour violation different generations. These superposition states are the weak interaction eigenstates. By convention, a basis is chosen such that the mass eigenstates of up-type quarks are also the weak interaction eigenstates. This forces the weak interaction eigenstates of down-type quarks to be superpositions of the mass eigenstates. The quark doublets can be written as Ψ L = u L d 0 L ! , c L s 0 L ! and t L b 0 L ! , (1.22) where the primed fields are the interaction eigenstates and are related to the mass eigenstates by the Cabibbo-Kobayashi-Maskawa (CKM) matrix V CKM :   d 0 s 0 b 0   = V CKM   d s b   . (1.23) The CKM matrix is a unitary matrix and its elements are fundamental parameters of the SM. The magnitudes of the CKM matrix elements have been measured by experiments [6] to be V ij CKM =   0 . 97446 ± 0 . 00010 0 . 22452 ± 0 . 00044 0 . 00365 ± 0 . 00012 0 . 22438 ± 0 . 00044 0 . 97359 ± 0 . 00011 0 . 04214 ± 0 . 00076 0 . 00896 ± 0 . 00024 0 . 04133 ± 0 . 00074 0 . 999105 ± 0 . 000032   . (1.24) The off-diagonal elements of the matrix are relatively small, implying that the probability of transition of a quark from one generation to another is relatively low. Neutrino mixing The observation of neutrino oscillations [20, 21] has indisputably proven that neutrinos have finite rest masses, however small they are. And similar to the quarks, there is a mismatch between the neutrino mass eigenstates and the neutrino weak interaction eigenstates. In the conventional basis where the charged lepton mass eigenstates are also the interaction eigenstates, the neutrino eigenstates are related by the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix U PMNS , analogous to the CKM matrix for quark mixing:   ν e ν µ ν τ   = U PMNS   ν 1 ν 2 ν 3   , (1.25)