The Higgs Mechanism

When a local (gauge) symmetry is spontaneously broken, Goldstone's theorem is evaded in the most elegant way: the would-be massless Goldstone bosons are "eaten" by the gauge bosons, providing the longitudinal polarization that massive spin-1 particles require. The gauge bosons acquire mass, the Goldstone bosons disappear from the physical spectrum, and one massive neutral scalar — the Higgs boson — remains. Discovered at CERN in 2012, it is the last piece of the Standard Model.

Key Concepts

Degree of freedom count: N_scalar + N_gauge (massless) = N_scalar−1 + N_gauge (massive)
Abelian Higgs model: U(1) + charged scalar → massive photon + Higgs; Mγ = ev
Gauge-fixing (unitary gauge): sets Goldstone π(x) = 0, making the absorption explicit
Rξ gauges: keep Goldstone as a ghost-like field; ξ→∞ gives unitary gauge, ξ=1 gives Feynman gauge
In SM: SU(2)_L×U(1)_Y → U(1)_EM: W±, Z get mass; photon stays massless; Higgs doublet gives 1 physical Higgs
Fermion masses from Yukawa: ℒ_Y = −y_f L̄φf_R+h.c. → m_f = y_f v/√2 after SSB

Key Equations

Gauge boson mass (Abelian Higgs)

|D_\mu\phi|^2\supset e^2v^2A_\mu A^\mu,\quad M_A=ev

W and Z masses (SM)

M_W=\tfrac{1}{2}gv\approx80.4\text{ GeV},\quad M_Z=\frac{gv}{2\cos\theta_W}\approx91.2\text{ GeV}

Higgs boson mass

m_H=\sqrt{2\lambda}\,v\approx125.1\text{ GeV}\quad(v=246\text{ GeV})

Top quark mass

m_t=\frac{y_t\,v}{\sqrt{2}}\approx173\text{ GeV}\quad(y_t\approx1)

Worked Example

Example Problem

Problem

Using v = 246 GeV, g = 0.653, and sin²θW = 0.231, compute MW, MZ, and the Higgs self-coupling λ from mH = 125.1 GeV.

Solution

MW = gv/2 = 0.653×246/2 = 80.3 GeV ✓. cosθW = √(1−0.231) = 0.877, MZ = MW/cosθW = 80.3/0.877 = 91.6 GeV ✓. λ = m²H/(2v²) = (125.1)²/(2×246²) = 15650/121032 = 0.1293.

Key Takeaways

The Higgs mechanism: local gauge symmetry + SSB → gauge bosons absorb Goldstone bosons and become massive; degrees of freedom are conserved
In the SM, the Higgs doublet breaks SU(2)_L×U(1)_Y → U(1)_EM, giving mass to W± and Z while leaving the photon massless
Fermion masses arise from Yukawa couplings y_f to the Higgs: mf = yf v/√2; the Higgs boson couples to fermions proportional to their mass
The Higgs boson (mH = 125.1 GeV, discovered at CERN July 2012) is the smoking gun of SSB and completes the Standard Model