Wing Sheung Chan

12 The Standard Model and lepton flavour violation The interaction terms in the electroweak sector of the SM Lagrangian are L EW , int = − ¯Ψ L γ µ g T · W µ + 1 2 g 0 Y B µ Ψ L − 1 2 ¯ ψ R γ µ ( g 0 Y B µ ) ψ R . (1.10) The capitalised notation for the left-handed fermion fields is to emphasise that they are indeed doublets of up-type and down-type quarks, or neutrinos and charged leptons: Ψ L = u L d 0 L ! or ν 0 L ` L ! . (1.11) Here, we denoted any up-type and down-type quarks collectively by u and d respectively. The primed symbols indicate that they are the weak interaction eigenstates instead of the mass eigenstates, the meaning of which will be elaborated later in Section 1.2.5. W ± bosons and the charged-current interaction We choose a basis such that the weak isospin operator can be expressed in terms of the Pauli matrices τ : T i = τ i / 2 . Since the first two Pauli matrices, τ 1 = 0 1 1 0 ! and τ 2 = 0 − i i 0 ! , (1.12) mix the components of the fermion field doublets, the corresponding W 1 and W 2 fields cannot be physical. However, if we define τ ± = ( τ 1 ± iτ 2 ) / 2 and W ± = ( W 1 ∓ iW 2 ) / √ 2 , we can rewrite the relevant terms in the Lagrangian into: L charged current = − g ¯Ψ L γ µ T 1 W 1 µ + T 2 W 2 µ Ψ L = − g √ 2 ¯Ψ L γ µ τ + W + µ + τ − W − µ Ψ L = − g 2 √ 2 ¯Ψ γ µ 1 − γ 5 τ + W + µ + τ − W − µ Ψ . (1.13) In our representation, τ + = 1 2 0 1 0 0 ! and τ − = 1 2 0 0 1 0 ! . (1.14) In this basis, components of the fermion field doublets are not mixed anymore. It becomes immediately apparent that interactions with the W + field changes a down-type quark into an up-type quark, or a charged lepton into a neutrino, raising the electric charge by one in the process. Vice versa, interactions with the W − field lower the electric charge of a an up-type quark or a neutrino, turning them into a down-type quark or a charged lepton respectively.