Chapter 1 More SSB , then Higgs

This lagrangian is, of course, invariant under SO(N), rotations of φ in the N dimensional internal space. φj(x) → Ojkφk(x), with O an orthogonal matrix, which leaves ∑ φ invariant. Notice that the mass term has the wrong sign. Had that sign been minus, with a + in the potential energy, the potential’s absolute minimum would have been at ~ φ = 0, the classical vacuum would be at ~ φ = 0, and the rotational symmetry would be intact. But with the plus in the lagrangian, ~ φ = 0 is a local maximum of the potential energy, and not a classical ground state. Instead a classical lowest energy state will have ~ φ = ~ φ0 with φ0 := |~ φ0| = μ/ √ λ. To get a state of lowest energy, we need not only that V (φ) is minimized at each point x, but also that the kinetic energy (~ ∇φ)2 is minimized, which means vanishing. So although the potential only tells us that ~ φ(x) should lie on a sphere of radius φ0, minimizing the energy means it has to be the same ~ φ throughout space. If we apply a global SO(N) transformation, we get a new state which is of the same energy as our state ~ φ0, but which has an overlap with ~ φ0(x) at every point which is less that one, so that raised to the infinite power from