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

1.2. Electroweak Symmetry Breaking 13mass. We follow [6] and begin by describing spontaneous symmetry of a continuoussymmetry first before moving onto discuss the breaking of gauge symmetries. Weconsider the following Lagrangian consisting of a set of N real scalar fields φ i (x),L LS = 1 2 (∂ µφ i ) 2 + 1 2 µ2 (φ i ) 2 − λ 4 [(φi ) 2 ] 2 (1.27)which is known as the linear sigma model. Here we choose λ, µ 2 > 0. The aboveLagrangian is invariant under the group of orthogonal rotations O(N),φ i → R ij φ j (1.28)The lowest-energy classical configuration is a constant field φ i 0 whose value minimisesthe potential,V (φ i ) = − 1 2 µ2 (φ i ) 2 + λ 4 [(φi ) 2 ] 2 (1.29)φ i 0 satisfies, (φ i 0) 2 = µ2λ(1.30)We observe that this constraint merely fixes the length of the vector φ i 0, its directionis arbitrary. We choose coordinates such that φ i 0 points in the N-th direction,φ i 0 = (0, 0, . . ., 0, v), (1.31)where v = µ/ √ λ. We now choose to expand the fields around the lowest energysolution,φ i (x) = (π k (x), v + σ(x)), k = 1, . . .N − 1 (1.32)Written in terms of these fields the Lagrangian takes the following form,L LS = 1 2 (∂ µπ k ) 2 + 1 2 (∂ µσ) 2 − 1 2 (2µ2 )σ 2 − √ λµσ 3 − √ λµ(π k ) 2− λ 4 σ4 − λ 2 (πk ) 2 σ 2 − λ 4 [(πk ) 2 ] 2 . (1.33)We note the appearance of one massive field σ and N − 1 massless fields π k . Theoriginal symmetry group of the Lagrangian O(N) is no longer apparent, there isonly an O(N − 1) symmetry which rotates π k fields amongst themselves. In this