Quantum Field Theory Course

Total: 440+ pages covering classical field theory through the Standard Model

Part V: Gauge Field Theories - Chapter 7

Electroweak Theory

SU(2)×U(1) unification of electromagnetic and weak interactions

Video Lectures

For comprehensive video lectures on Electroweak Theory:

  • Tobias Osborne QFT 2016:YouTube Playlist

    Covers spontaneous symmetry breaking and Higgs mechanism

  • David Tong Gauge Theory:Lecture Notes (Cambridge)

    Chapter 5 on Electroweak Theory

  • Peskin & Schroeder:Chapters 20-21 on Standard Model

The Electroweak Gauge Group

The electroweak theory is based on the gauge group:

SU(2)L × U(1)Y

L denotes weak isospin (left-handed fermions), Y is weak hypercharge. The electromagnetic U(1)EM emerges after spontaneous symmetry breaking.

Fermion Representations

Left-handed fermions form SU(2) doublets, right-handed are singlets:

Leptons

LL = (νe, e)L   Y = -1/2
eR                      Y = -1

Quarks

QL = (u, d)L       Y = 1/6
uR                    Y = 2/3
dR                    Y = -1/3

Electric Charge Formula

Q = T3 + Y (Gell-Mann–Nishijima formula)

Gauge Bosons

Before symmetry breaking, there are four massless gauge bosons:

SU(2)L Bosons

W1μ, W2μ, W3μ

Coupling g

U(1)Y Boson

Bμ

Coupling g'

The Higgs Mechanism

A complex scalar doublet acquires a vacuum expectation value (VEV):

Φ = (1/√2) (0, v + h)T,    v ≈ 246 GeV

The Higgs potential with spontaneous symmetry breaking:

V(Φ) = μ²|Φ|² + λ|Φ|&sup4;,    μ² < 0

Nobel Prize 2013

Higgs and Englert received the Nobel Prize for the theoretical discovery of the mechanism. The Higgs boson was discovered at CERN in 2012 (mH = 125 GeV).

Symmetry Breaking Pattern

SU(2)L × U(1)Y → U(1)EM

After symmetry breaking, the gauge bosons mix:

Charged Bosons

W± = (W1 &mp; iW2)/√2
MW = gv/2 ≈ 80.4 GeV

Neutral Bosons

Z0 = W3cosθW - B sinθW
A = W3sinθW + B cosθW

Weinberg angle: tanθW = g'/g,   sin²θW ≈ 0.231

Mass Relations

W Mass

MW = gv/2 = 80.4 GeV

Z Mass

MZ = MW/cosθW = 91.2 GeV

Photon Mass

Mγ = 0 (massless)

ρ Parameter

ρ = MW²/(MZ²cos²θW) = 1

Tree-Level Prediction

The ratio MW/MZ = cosθW is a key prediction of electroweak theory, verified to high precision.

Fermion Masses

Fermion masses arise from Yukawa couplings to the Higgs:

LYukawa = -ye L̄LΦ eR - yd Q̄LΦ dR - yu Q̄LΦ̃ uR + h.c.

After SSB: mf = yfv/√2

CKM Matrix

The quark mass eigenstates differ from weak eigenstates, leading to the 3×3 CKM mixing matrix with 4 independent parameters (3 angles + 1 CP phase).

Weak Interactions

Charged Current

W± couples to left-handed fermion doublets
jμCC = (g/√2) ν̄LγμeL

Neutral Current

Z couples to both L and R with different strengths
jμNC = (g/cosθW) f̄γμ(gV - gAγ5)f

At low energies (Q << MW), the Fermi theory is recovered: GF/√2 = g²/(8MW²)

Key Experimental Tests

W and Z Discovery (1983)

UA1 and UA2 at CERN discovered W± and Z&sup0; at predicted masses. Nobel Prize to Rubbia and van der Meer.

LEP Precision Tests

Z pole measurements confirmed electroweak theory to 0.1% precision. Predicted top quark mass before discovery.

Higgs Discovery (2012)

ATLAS and CMS discovered the Higgs boson at 125 GeV, completing the Standard Model.

Historical Development

1961 - Glashow:

Proposed SU(2)×U(1) structure for electroweak interactions

1964 - Higgs, Englert, Brout:

Discovered spontaneous symmetry breaking mechanism for gauge theories

1967-68 - Weinberg, Salam:

Combined Higgs mechanism with electroweak gauge theory

1971 - 't Hooft:

Proved renormalizability of spontaneously broken gauge theories

1973 - Neutral currents:

Gargamelle experiment discovered Z-mediated neutral currents

Summary

  • ✓ Gauge group: SU(2)L×U(1)Y → U(1)EM
  • ✓ Higgs mechanism: Scalar doublet VEV breaks symmetry, generates masses
  • ✓ Gauge bosons: W± (80 GeV), Z&sup0; (91 GeV), γ (massless)
  • ✓ Weinberg angle: sin²θW ≈ 0.231, relates couplings
  • ✓ Fermion masses: Yukawa couplings to Higgs doublet
  • ✓ CKM matrix: Quark mixing with CP violation
  • ✓ Precision tests: Verified to <0.1% at LEP and LHC