Electroweak Unification

In 1967-68, Glashow, Weinberg, and Salam unified electromagnetism and the weak force into a single SU(2)_L×U(1)_Y gauge theory. The Higgs mechanism breaks this to U(1)_EM, giving W± and Z masses while keeping the photon massless. The theory predicted neutral-current weak interactions (observed at CERN in 1973) and the W and Z bosons (discovered at CERN in 1983), earning Glashow, Salam, and Weinberg the 1979 Nobel Prize.

Key Concepts

Gauge group SU(2)_L×U(1)_Y: 3+1 = 4 generators → 4 gauge bosons W^1,W^2,W^3,B
Left-handed doublets: L=(νL,eL)^T, Q=(uL,dL)^T with Y = −½, +⅙
Right-handed singlets: eR, uR, dR with Y = −1, +⅔, −⅓ (no right-handed neutrino)
After SSB: W± = (W^1∓iW^2)/√2, Z = W^3cosθW−BsinθW, A = W^3sinθW+BcosθW
Weinberg angle θW: sinθW = g′/√(g²+g′²), sin²θW ≈ 0.231
Neutral current: couples to J^μ_3 − sin²θW J^μ_EM (predicted before observed)

Key Equations

Mass relations

M_W=\tfrac{1}{2}gv,\quad M_Z=\frac{M_W}{\cos\theta_W},\quad M_A=0

Electric charge

Q=T_3+\frac{Y}{2}

Weinberg angle

\cos\theta_W=\frac{g}{\sqrt{g^2+g^{\prime2}}}=\frac{M_W}{M_Z},\quad\sin^2\theta_W\approx0.231

Charged-current

\mathcal{L}_{CC}=\frac{g}{\sqrt{2}}W^+_\mu\bar\nu_L\gamma^\mu e_L+\text{h.c.}

Worked Example

Example Problem

Problem

With sin²θW = 0.231 and MW = 80.4 GeV, find MZ and cosθW.

Solution

cosθW = √(1−sin²θW) = √0.769 = 0.8774. MZ = MW/cosθW = 80.4/0.8774 = 91.6 GeV. (PDG: 91.188 GeV ✓.)

Key Takeaways

The electroweak gauge group SU(2)_L×U(1)_Y has 4 bosons; after SSB only 3 (W±, Z) get mass, leaving the photon massless
The Weinberg angle θW mixes SU(2) and U(1) to produce the physical mass eigenstates, with sin²θW ≈ 0.231
Weak charged currents are purely left-handed (V−A structure); the neutral current couples with both V and A with specific sin²θW-dependent coefficients
Neutral current processes (νμe → νμe) were predicted by Weinberg-Salam before being observed at CERN in 1973 — one of the great predictive triumphs of gauge theory