Spontaneous Symmetry Breaking and the Higgs Mechanism

Analyze spontaneous breaking of a global continuous symmetry in a scalar field theory and derive Goldstone's theorem.

Apply the Higgs mechanism to an Abelian U(1) gauge theory, demonstrating how the gauge boson acquires mass while the Goldstone boson is eliminated.

Construct the SU(2)xU(1) gauge structure of the electroweak theory, defining weak isospin and hypercharge.

Assign the Standard Model fermions to representations of SU(2)xU(1) and write their covariant derivatives.

Introduce the Higgs doublet and its potential, showing how its vacuum expectation value spontaneously breaks SU(2)xU(1) down to U(1)em.

Derive the masses of the W and Z bosons from the electroweak Lagrangian after spontaneous symmetry breaking.

Derive the mixing of the neutral gauge bosons (B and W³) to produce the physical photon (A) and Z boson fields, defining the Weinberg angle.

Introduce Yukawa couplings to the Higgs field and show how they generate fermion masses after symmetry breaking.