Hadronic production of a Higgs boson in association with two jets at ...

1.2. Electroweak Symmetry Breaking 14example we have spontaneously broken the O(N) continuous symmetry by choosingto express the ground state in terms of a particular direction in φ space. Theremaining symmetry is O(N − 1) so we would describe this breaking as O(N) →O(N − 1).The appearance of massless fields as a result of spontaneous symmetry breakingis a general result of theorem proven by Goldstone [46, 47]. Goldstone’s theoremstates that for every spontaneously broken continuous symmetry the theory mustcontain a massless particle. In the above example the original symmetry O(N)had (N(N − 1))/2 symmetries, when it was broken to O(N − 1) this changed to(N − 2)(N − 1)/2. This resulted in a loss of N − 1 symmetries, hence we observedN − 1 Goldstone bosons.1.2.2 Spontaneous breaking of scalar QEDNext we consider the following Lagrangian which couples a complex scalar to itselfand to an electromagnetic field,L = − 1 4 (F µν) 2 + |D µ φ| 2 − V (φ) (1.34)where D µ = ∂ µ + ieA µ . As can be seen from the discussion of section 1.1.1 theLagrangian is invariant under the following U(1) transformations (provided V (φ) isa function of φ ∗ φ),φ(x) → e iα(x) φ(x), A µ (x) → A µ (x) − 1 e ∂ µα(x). (1.35)An interesting, and relevant choice of potential is the followingV (φ) = −µ 2 φ ∗ φ + λ 2 (φ∗ φ) 2 . (1.36)In exactly the same manner as the previous section when µ 2 > 0 there is a non-zerovacuum expectation value (vev),( ) µ2 1/2〈φ〉 = φ 0 = . (1.37)λWhen we expand φ(x) areound the vacuum state,φ(x) = φ 0 + √ 1 (φ 1 (x) + φ 2 (x)) (1.38)2