PhD thesis - CP3-Origins

Introduction to Elementary Particle Physics 15Now we can write the Higgs Lagrangian asL H = Tr (D µ Φ) † D µ Φ + µ 2 TrΦ † Φ − λ ( TrΦ † Φ ) 2, (2.42)where the covariant derivative isThe electroweak symmetry acts as followsD µ φ = ∂ µ Φ + i g 2 σ · W µΦ − i g′2 B µΦσ 3 . (2.43)SU(2) L :Φ → LΦ,U(1) Y : Φ → Φe − i 2 σ3θ .(2.44)We can make the global symmetry manifest by taking the hypercharge interactions tovanish, g ′ → 0. In this limit the Higgs Lagrangian has a global SU(2) R symmetrySU(2) R : Φ → ΦR † . (2.45)Therefore the Higgs Lagrangian has SU(2) L × SU(2) R symmetry which breaks down toSU(2) L+R when the Higgs field aquires a vev⎛ ⎞〈Φ〉 = √ 1 ⎝ v 0 ⎠ . (2.46)2 0 vThis breaking pattern yields three Goldstone bosons which are eaten by the Higgsmechanism providing masses to the weak gauge bosons.M 2 W = 1 4 g2 v 2M 2 Z = 1 4 (g2 + g ′2 )v 2 .(2.47)Thus, at tree levelρ =M 2 WM 2 Z cos2 θ W= 1, (2.48)where the θ W is the Weinberg angle. In the limit g ′ → 0 the W + , W − and Z bosonsform a triplet under the SU(2) L+R explaining why the masses are degenerate in this