Higgs Mechanism:
Basically, I will only look at U(1) symmetry. Electroweak interactions need a U(1) x SU(2) symmetry, but SU(2) requires 2 by 2 matrices, and the software on this forum is inadequate to deal with matrices. But you can get the flavor just by doing U(1) symmetry and how mass is introduced in the Lagrangian of equation (5).
I will rewrite this equation as:
(11) L = ∂μΦ† ∂μΦ - ¼ Fμν Fμν – V(Φ†Φ).
(12) where V(Φ†Φ) = (m2)/(2φ2) [Φ†Φ - φ2 ] 2
Three important things to note:
(13) The field Φ is now a complex number, denoted by (Φ1, Φ2) or Φ = Φ1 + iΦ2 ( i being the imaginary number, square root of – 1), and Φ† = Φ1 – iΦ2.
(14) the minimum field energy is obtained when Φ†Φ = φ2.
(15) The number of possible vacuum states is infinite. We break this symmetry by requiring that Φ is real, we take the vacuum state to be (φ,0), and expand:
Φ = φ + (1/2½)h
Substituting 7,8,9, 12, and 15 into 11, we get
(17) L = {(∂μ - iqAμ)( φ + (½ ½)h)}{( ∂μ + iqAμ)( φ + (½ ½)h} - ¼ Fμν Fμν - (m2)/(2φ2) [2½φh + ½h2 ]2
After calculating the Lagrangian, we separate it into two parts:
(18) L = Lfree + Lint
where
(19) Lfree = ½∂μh∂μh - m2h2 - ¼ Fμν Fμν + q2φ2AμAμ
All the remaining terms are lumped into Lint, which offer no interest.
So, we can see that by breaking the symmetry, we end up with two massive particles. In equation 19, the second term refers to a scalar particle with mass equal to 2½m, associated with h (the higgs field) and the fourth term, a vector boson with mass 2½qφ, associated with Aμ( the electromagnetic field).
