In the abelian Higgs model (U(1) gauge field coupled to a complex scalar with a Mexican hat potential), the photon acquires a mass m_A = ev. Where does the longitudinal degree of freedom come from?
AFrom the timelike component of the gauge field
BFrom the Goldstone boson — the angular mode of the scalar field is absorbed into the gauge field, giving it the third polarization (longitudinal) needed for a massive vector boson
CFrom the ghost fields introduced during gauge fixing
DFrom vacuum fluctuations of the electromagnetic field
Before symmetry breaking, the theory has a massless gauge boson with 2 polarizations and a complex scalar with 2 degrees of freedom (4 total). After symmetry breaking, the gauge boson is massive with 3 polarizations and one real scalar (the Higgs boson) remains (4 total). The Goldstone boson has been absorbed: it provides the longitudinal polarization of the now-massive gauge boson. In unitary gauge, this is manifest — the Goldstone field disappears from the Lagrangian entirely, and the gauge field mass term appears explicitly.
Question 2 True / False
The Higgs mechanism violates gauge invariance because a massive gauge boson is not gauge invariant.
TTrue
FFalse
Answer: False
The Lagrangian of the Higgs mechanism is fully gauge invariant — the symmetry is spontaneously broken, not explicitly broken. The mass term arises from the gauge-covariant coupling of the gauge field to the scalar field with a nonzero vacuum expectation value: |D_mu phi|^2 evaluated at <phi> = v generates (ev)^2 A_mu A^mu/2. In the unitary gauge, this looks like an explicit mass term, but it originated from a gauge-invariant expression. This is crucial: explicit breaking of gauge invariance would destroy renormalizability, but spontaneous breaking preserves it. The proof of renormalizability of the Higgs mechanism (by 't Hooft and Veltman, Nobel Prize 1999) was one of the most important results in 20th century physics.
Question 3 Multiple Choice
The Higgs boson was discovered at the LHC in 2012 with a mass of approximately 125 GeV. Why does the Standard Model not predict the Higgs mass, even though it predicts the W and Z masses?
ABecause the LHC was not precise enough to test the prediction
BBecause the Higgs mass depends on the self-coupling lambda in the Higgs potential, which is a free parameter of the Standard Model — unlike the W and Z masses (which are determined by the gauge couplings and the Higgs vacuum expectation value v = 246 GeV), the Higgs mass m_H = sqrt(2 lambda) v requires knowing lambda independently
CBecause the Higgs boson is a composite particle
DBecause quantum corrections make the Higgs mass incalculable
The W mass is m_W = gv/2 and the Z mass is m_Z = sqrt(g^2 + g'^2) v/2, where g and g' are the SU(2) and U(1) gauge couplings (measured independently) and v = 246 GeV is fixed by the Fermi constant. These are predictions. The Higgs mass m_H = sqrt(2 lambda) v depends on the quartic coupling lambda, which is a free parameter. Measuring m_H = 125 GeV determines lambda = m_H^2/(2v^2) approximately 0.13. The Standard Model has no mechanism to predict lambda from other parameters.
Question 4 Short Answer
Explain how fermion masses are generated in the Standard Model through the Higgs mechanism, and why this is necessary given the chiral structure of the electroweak interaction.
Think about your answer, then reveal below.
Model answer: In the Standard Model, left-handed fermions form SU(2) doublets while right-handed fermions are SU(2) singlets. A Dirac mass term m psi-bar psi = m(psi-bar_L psi_R + psi-bar_R psi_L) couples left- and right-handed components, but since they transform differently under SU(2), this term is not gauge invariant. The Higgs mechanism solves this: a Yukawa coupling y psi-bar_L phi psi_R (where phi is the Higgs doublet) is gauge invariant. When phi acquires a vacuum expectation value <phi> = (0, v/sqrt(2))^T, this term becomes y v/sqrt(2) psi-bar_L psi_R, which is a mass term with m = yv/sqrt(2). Each fermion's mass is proportional to its Yukawa coupling y, which is a free parameter. This is why fermion masses span such a huge range (electron: 0.5 MeV; top quark: 173 GeV).
The Yukawa couplings are among the most puzzling free parameters of the Standard Model. Why the top quark's Yukawa coupling is close to 1 while the electron's is 10^{-6} is the fermion mass hierarchy problem — one of the major open questions in particle physics.