Electroweak Unification

Why bother unifying?

The weak and electromagnetic forces look almost nothing alike. Electromagnetism has infinite range, governs every chemical bond, and is mediated by a single massless particle, the photon. The weak force has a range of 10⁻¹⁸ m, only manifests itself in beta decay and neutrino scattering, violates parity, and is mediated by three massive bosons (W+, W−, Z⁰). Yet they are the same force.

The clue was in the couplings. Although the weak force looks ~10⁻⁶ times weaker than electromagnetism at low energies, that's because of the W mass — not because the intrinsic coupling is small. Strip away the mass suppression and weak coupling g ≈ 0.65 is actually larger than electromagnetic coupling e ≈ 0.31. Above the W mass scale (~80 GeV) the two forces are comparable in strength, and the gauge structures they live in turn out to mix.

The Glashow–Salam–Weinberg model writes one combined gauge theory based on SU(2)_L × U(1)_Y. Below the electroweak scale of ~250 GeV, the Higgs field breaks the symmetry — three of the four gauge bosons acquire mass and become W+, W−, Z⁰; the fourth stays massless and is the photon. Above 250 GeV, the symmetry is restored and the four bosons act on equal footing.

The structure of the theory

Original (unbroken)	Coupling	After Higgs breaking	Mass	Role
SU(2)_L gauge bosons W¹, W², W³	g ≈ 0.653	W+ = (W¹ − iW²)/√2	80.4 GeV	Charged-current weak
		W− = (W¹ + iW²)/√2	80.4 GeV	Charged-current weak
W³ and B mix:		Z⁰ = W³ cos θ_W − B sin θ_W	91.2 GeV	Neutral-current weak
U(1)_Y gauge boson B	g' ≈ 0.358	γ = W³ sin θ_W + B cos θ_W	0	Electromagnetism

The mixing angle θ_W (the "Weinberg angle") is a measured parameter of nature. Its value at the Z mass is:

sin²θ_W ≈ 0.23121,  so  θ_W ≈ 28.74°

The electric charge — the unbroken combination of couplings — is:

e = g sin θ_W = g' cos θ_W ≈ 0.303

And the Z mass and W mass are related by:

M_W / M_Z = cos θ_W   ⇒   80.379 / 91.188 = 0.8814   ✓

That last identity is a non-trivial prediction of the theory — it ties together two independently measured masses through one angle. It works.

The Higgs mechanism in one paragraph

Start with a complex scalar doublet H (four real components) that transforms under SU(2)_L × U(1)_Y. Give it a potential V(H) = −μ²|H|² + λ|H|⁴ shaped like a Mexican hat: the origin is a maximum, the minimum is a circle at |H| = v/√2 where v = √(μ²/λ) ≈ 246 GeV. The vacuum picks one direction on that circle, breaking SU(2)_L × U(1)_Y → U(1)_EM. Three of the four scalar degrees of freedom are massless Goldstone modes (a generic consequence of spontaneous symmetry breaking); but in a gauge theory those Goldstones are "eaten" by the gauge bosons — three of the four become massive (W+, W−, Z⁰) and acquire longitudinal polarizations. The fourth scalar component survives as the physical Higgs boson, observed at 125.25 GeV in 2012. The photon stays massless because it corresponds to the unbroken U(1)_EM.

Worked example — checking sin²θ_W

1. Measured: M_W = 80.379 GeV, M_Z = 91.188 GeV.
2. Compute cos θ_W = M_W / M_Z = 0.8814.
3. So sin²θ_W = 1 − 0.8814² = 1 − 0.7768 = 0.2232 (tree level).
4. Direct measurement at LEP (effective leptonic mixing angle): sin²θ_W^eff ≈ 0.23148 ± 0.00017.
5. Tiny difference (~0.005) is explained by one-loop radiative corrections — chiefly virtual top quark contributions, themselves a precision test.
6. So sin²θ_W ≈ 0.231 is the "official" value, used throughout particle physics for cross-section and branching-fraction predictions.

JavaScript — electroweak relations

// Constants
const M_W = 80.379;          // GeV
const M_Z = 91.188;          // GeV
const G_F = 1.166e-5;        // GeV^-2
const alpha = 1 / 137.036;   // fine structure (low E)
const alpha_MZ = 1 / 127.95; // fine structure at M_Z
const sin2_theta_W = 0.23121;
const v = 246.22;            // Higgs VEV in GeV

// Weinberg angle from masses (tree level)
function sin2thW_tree() {
  return 1 - (M_W / M_Z) ** 2;
}
console.log(`sin²θ_W tree: ${sin2thW_tree().toFixed(4)}`);
// 0.2232 — slightly different from running value at Z mass

// Gauge couplings
const e = Math.sqrt(4 * Math.PI * alpha_MZ);
console.log(`e at M_Z = ${e.toFixed(4)}`);   // 0.3133

const g = e / Math.sqrt(sin2_theta_W);
console.log(`g (SU(2)) = ${g.toFixed(4)}`);  // 0.6520

const gp = e / Math.sqrt(1 - sin2_theta_W);
console.log(`g' (U(1)) = ${gp.toFixed(4)}`); // 0.3573

// Cross-check: G_F = g² / (4√2 M_W²)
const G_F_predicted = (g * g) / (4 * Math.sqrt(2) * M_W * M_W);
console.log(`Predicted G_F: ${G_F_predicted.toExponential(3)} GeV^-2`);
console.log(`Measured G_F:  ${G_F.toExponential(3)} GeV^-2`);
// agree to 1e-7 percent

// W and Z masses from Higgs VEV and couplings
const M_W_pred = g * v / 2;
const M_Z_pred = Math.sqrt(g * g + gp * gp) * v / 2;
console.log(`M_W predicted: ${M_W_pred.toFixed(3)} GeV (measured ${M_W})`);
console.log(`M_Z predicted: ${M_Z_pred.toFixed(3)} GeV (measured ${M_Z})`);

// Mixing rotation: γ and Z from W³ and B
function rotateBoson(W3, B, theta_W) {
  const gamma = W3 * Math.sin(theta_W) + B * Math.cos(theta_W);
  const Z = W3 * Math.cos(theta_W) - B * Math.sin(theta_W);
  return { gamma, Z };
}

const theta_W = Math.asin(Math.sqrt(sin2_theta_W));
console.log(`θ_W = ${(theta_W * 180 / Math.PI).toFixed(2)}°`);  // 28.74°

// Test: pure W³ rotates to (sin θ_W, cos θ_W) → (γ, Z)
console.log(rotateBoson(1, 0, theta_W));  // gamma=sin, Z=cos
console.log(rotateBoson(0, 1, theta_W));  // gamma=cos, Z=-sin

// Energy regimes
function effectiveFermi(E_GeV) {
  // weak interaction at energy E
  // schematic: 1/M_W² propagator if E ≪ M_W; otherwise comparable to EM
  if (E_GeV < M_W) {
    return `weak effective at ${E_GeV} GeV ≈ G_F (very weak, ${G_F.toExponential(2)} GeV^-2)`;
  } else {
    return `at ${E_GeV} GeV ≥ M_W: weak ≈ EM (both ~1/E²)`;
  }
}
console.log(effectiveFermi(0.001));   // nuclear energies
console.log(effectiveFermi(100));     // LHC scale — unified

// Higgs scalar mass squared from VEV
const lambda = 0.13;
const m_H = Math.sqrt(2 * lambda) * v;
console.log(`Higgs mass: ${m_H.toFixed(2)} GeV (measured 125.25)`);

Neutral currents — the smoking gun

A unique prediction of electroweak unification was the existence of weak interactions in which neither flavour nor charge changes: ν + e → ν + e mediated by the Z⁰ boson. Such processes are forbidden in the original Fermi theory, which has only charged currents (W exchange). Electroweak unification forces them because Z is a mixture of W³ and B, and the W³ couplings to leptons follow directly from SU(2).

The first observation came at CERN's Gargamelle bubble chamber in July 1973. A muon antineutrino entered the bubble chamber, produced no muon — so no W+ charged current — but kicked an atomic electron, which left a curved track. The recoil signature matched the Z exchange prediction. The full discovery paper, "Observation of neutrino-like interactions without muon or electron in the Gargamelle neutrino experiment," appeared a few months later.

Neutral currents proved that the SU(2) × U(1) structure was real, that the weak and electromagnetic forces were genuinely mixed, and that an associated Z⁰ boson must exist with a specific mass. Ten years later that boson was found, at the right mass, at CERN's proton–antiproton collider.

History — the architects

Sheldon Glashow (1961) wrote down the SU(2) × U(1) gauge structure but couldn't give the W and Z masses. Abdus Salam and Steven Weinberg (1967, independently) added the Higgs mechanism, producing a complete renormalizable theory. 't Hooft and Veltman (1971–72) proved renormalizability — an essential technical requirement that took years to settle. Neutral currents were observed in 1973. The 1979 Nobel Prize was awarded to Glashow, Salam, and Weinberg "for their contributions to the theory of the unified weak and electromagnetic interaction." The W and Z were detected at CERN in 1983 (Rubbia, van der Meer); LEP measured their properties to extraordinary precision between 1989 and 2000. Today the Standard Model's electroweak sector is the most precisely tested theory in physics — parts-per-million agreement between hundreds of independent measurements.

Common misconceptions

• "Electroweak unification means the forces become identical." Not quite — they remain distinct gauge bosons even above the electroweak scale, but they sit inside one symmetry group and have comparable couplings. The W mass goes to zero at very high energies, but its couplings to fermions remain non-trivial.
• "The forces 'fuse' above 250 GeV." The Higgs field doesn't switch off above v — the symmetry is broken in the same way, just less consequential at energies where the W and Z masses are small compared to E.
• "The Higgs causes the unification." The unification is in the gauge structure SU(2) × U(1), which is symmetric independently of the Higgs. The Higgs is what breaks the symmetry to give W and Z mass — it's a feature of the low-energy theory.
• "The Weinberg angle is fixed by theory." No — it's a free parameter of the Standard Model, measured experimentally to be sin²θ_W ≈ 0.231. Grand-unified theories predict a value at high energy (~3/8 in SU(5)), but the SM does not predict it.
• "Photon comes from B, Z comes from W³." Both arise from mixtures. The photon is the linear combination that the Higgs vacuum leaves invariant; the Z is orthogonal to it. Both involve both W³ and B with angle θ_W.
• "Electroweak unification is the same as grand unification." No. Electroweak unification is fully confirmed and built into the SM. Grand unification adds the strong force into a single group at ~10¹⁶ GeV and remains hypothetical.

Electroweak unification vs other unification programs

Program	Forces merged	Scale	Status	Key prediction
Electroweak (Glashow–Salam–Weinberg)	Weak + EM	~250 GeV (Higgs VEV)	Confirmed (Nobel 1979)	Neutral currents, W/Z masses
Maxwell EM	Electric + magnetic	Any (already done)	Confirmed (19th c.)	EM waves, c constant
Grand Unification (SU(5), SO(10))	Electroweak + strong	~10¹⁶ GeV	Hypothetical	Proton decay ~10³⁴ yr
String / M-theory	All four (incl. gravity)	~10¹⁸–10¹⁹ GeV (Planck)	Hypothetical	Extra dimensions, supersymmetry
Supersymmetry	Bosons ↔ fermions	≲ TeV (not seen)	Searched (no SUSY found)	Sparticles within LHC reach
Newton (gravity)	Celestial + terrestrial	Any	Confirmed (17th c.)	Universal inverse-square law

Where electroweak unification matters

• LHC physics. Every prediction of W, Z, and Higgs production cross-sections relies on electroweak unification; precision measurements (M_W to 10 MeV, sin²θ_W to 0.0001) probe the theory at the loop level.
• Neutrino physics. Neutral-current interactions (Z exchange) are how SNO, Super-K, and IceCube detect neutrinos of all flavours; charged-current (W exchange) gives flavour identification.
• Early universe. The electroweak phase transition at T ≈ 160 GeV (about 10⁻¹² s after the big bang) marks when the Higgs field turned on and elementary particles acquired their masses.
• Standard Model anchoring. The electroweak sector ties together α, G_F, M_W, M_Z, sin²θ_W, and m_H into a tightly constrained set of relations — the most precise consistency check in physics.
• Beyond the Standard Model. Any extension (supersymmetry, extra dimensions, composite Higgs) must reproduce electroweak unification at low energy or be ruled out by the precision data.
• Cosmology. Baryogenesis scenarios using electroweak sphalerons rely on the unified structure to convert baryon number into lepton number above the EW phase transition.
• Pedagogy. Cleanest example of how a deep symmetry, broken at one scale, produces forces that look entirely different in everyday life.