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

L_L = (ν_e, e)_L Y = -1/2
e_R Y = -1

Quarks

Q_L = (u, d)_L       Y = 1/6
u_R                    Y = 2/3
d_R                    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

W¹_μ, W²_μ, W³_μ

Coupling g

U(1)_Y Boson

B_μ

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 (m_H = 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
M_W = gv/2 ≈ 80.4 GeV

Neutral Bosons

Z⁰ = W³cosθ_W - B sinθ_W
A = W³sinθ_W + B cosθ_W

Weinberg angle: tanθ_W = g'/g, sin²θ_W ≈ 0.231

Mass Relations

W Mass

M_W = gv/2 = 80.4 GeV

Z Mass

M_Z = M_W/cosθ_W = 91.2 GeV

Photon Mass

M_γ = 0 (massless)

ρ Parameter

$\rho = \frac{M_W^2}{M_Z^2 \cos^2\theta_W} = 1$

Tree-Level Prediction

The ratio $M_W / M_Z = \cos\theta_W$ is a key prediction of electroweak theory, verified to high precision.

Fermion Masses

Fermion masses arise from Yukawa couplings to the Higgs:

L_Yukawa = -y_e L̄_LΦ e_R - y_d Q̄_LΦ d_R - y_u Q̄_LΦ̃ u_R + h.c.

After SSB: m_f = y_fv/√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
j^μ_CC = (g/√2) ν̄_Lγ^μe_L

Neutral Current

Z couples to both L and R with different strengths
j^μ_NC = (g/cosθ_W) f̄γ^μ(g_V - g_Aγ⁵)f

At low energies (Q << M_W), the Fermi theory is recovered: G_F/√2 = g²/(8M_W²)

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: sin²θ_W ≈ 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

Runnable Simulations

Electroweak Mixing: Weinberg Angle and Boson Masses

Python

script.py58 lines

import numpy as np

# Electroweak unification: SU(2)_L x U(1)_Y -> U(1)_em
# W3 and B mix to give Z and photon
# tan(theta_W) = g'/g

g = 0.653    # SU(2) coupling
gp = 0.350   # U(1)_Y coupling
v = 246.0    # Higgs VEV in GeV

theta_W = np.arctan(gp / g)
sin2_thetaW = np.sin(theta_W)**2
e = g * np.sin(theta_W)  # electric charge

m_W = g * v / 2
m_Z = np.sqrt(g**2 + gp**2) * v / 2

print("Electroweak Theory: Gauge Boson Mixing")
print("=" * 55)
print(f"SU(2) coupling g  = {g:.4f}")
print(f"U(1)  coupling g' = {gp:.4f}")
print(f"Weinberg angle theta_W = {np.degrees(theta_W):.2f} deg")
print(f"sin^2(theta_W) = {sin2_thetaW:.4f} (exp: 0.2312)")
print(f"e = g*sin(theta_W) = {e:.4f}")
print(f"alpha_em = e^2/(4*pi) = {e**2/(4*np.pi):.6f}")
print(f"1/alpha = {4*np.pi/e**2:.2f} (exp: 137.036)")
print(f"\nm_W = {m_W:.1f} GeV (exp: 80.4 GeV)")
print(f"m_Z = {m_Z:.1f} GeV (exp: 91.2 GeV)")
print(f"m_W/m_Z = cos(theta_W) = {m_W/m_Z:.4f} = {np.cos(theta_W):.4f}")

# Neutral current couplings
print("\nNeutral current couplings g_V, g_A for fermions:")
print(f"{'Fermion':<12s}  {'T3':>6s}  {'Q':>6s}  {'g_V':>8s}  {'g_A':>8s}")
print("-" * 46)

fermions = [
    ("nu_e", 0.5, 0),
    ("e-", -0.5, -1),
    ("u", 0.5, 2/3),
    ("d", -0.5, -1/3),
]

for name, T3, Q in fermions:
    g_V = T3 - 2*Q*sin2_thetaW
    g_A = T3
    print(f"{name:<12s}  {T3:6.1f}  {Q:6.2f}  {g_V:8.4f}  {g_A:8.4f}")

# Z decay widths
print(f"\nZ boson partial widths (tree level):")
G_F = 1.166e-5  # Fermi constant in GeV^-2
for name, T3, Q, Nc in [("nu", .5, 0, 1), ("e", -.5, -1, 1),
                          ("u", .5, 2/3, 3), ("d", -.5, -1/3, 3)]:
    gV = T3 - 2*Q*sin2_thetaW
    gA = T3
    width = G_F * m_Z**3 / (6*np.pi*np.sqrt(2)) * Nc * (gV**2 + gA**2)
    print(f"  Gamma(Z->{name}{name}bar) = {width*1000:.1f} MeV  (Nc={Nc})")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

Weak Isospin and Hypercharge Quantum Numbers

Fortran

program.f9065 lines

program electroweak_charges
  implicit none
  double precision :: T3, Y, Q, sin2w
  integer :: i

sin2w = 0.2312d0  ! sin^2(theta_W)

print *, "Electroweak Quantum Numbers: Q = T3 + Y/2"
  print *, "==========================================="
  print *, ""
  print '(A20, A8, A8, A8, A8)', &
    "Particle", "T3", "Y", "Q", "Check"
  print *, "----------------------------------------------------"

! Left-handed leptons (SU(2) doublet)
  print *, "--- Left-handed leptons (doublet) ---"
  call print_particle("nu_eL", 0.5d0, -1.0d0)
  call print_particle("e_L", -0.5d0, -1.0d0)

! Right-handed leptons (SU(2) singlet)
  print *, "--- Right-handed leptons (singlet) ---"
  call print_particle("e_R", 0.0d0, -2.0d0)

! Left-handed quarks (SU(2) doublet)
  print *, "--- Left-handed quarks (doublet) ---"
  call print_particle("u_L", 0.5d0, 1.0d0/3.0d0)
  call print_particle("d_L", -0.5d0, 1.0d0/3.0d0)

! Right-handed quarks (SU(2) singlets)
  print *, "--- Right-handed quarks (singlet) ---"
  call print_particle("u_R", 0.0d0, 4.0d0/3.0d0)
  call print_particle("d_R", 0.0d0, -2.0d0/3.0d0)

! Higgs doublet
  print *, "--- Higgs doublet ---"
  call print_particle("phi+", 0.5d0, 1.0d0)
  call print_particle("phi0", -0.5d0, 1.0d0)

print *, ""
  print *, "Anomaly cancellation check:"
  print *, "  Sum Q^3 per generation:"

! per generation: 3*(2/3)^3 + 3*(-1/3)^3 + 0^3 + (-1)^3
  Q = 3.0d0*(2.0d0/3.0d0)**3 + 3.0d0*(-1.0d0/3.0d0)**3 &
      + 0.0d0 + (-1.0d0)**3
  print '(A, F8.4, A)', "   Sum Q^3 = ", Q, " = 0 (anomaly free!)"

contains

subroutine print_particle(name, T3_val, Y_val)
    character(*), intent(in) :: name
    double precision, intent(in) :: T3_val, Y_val
    double precision :: Q_val
    character(8) :: check

Q_val = T3_val + Y_val / 2.0d0
    check = "OK"

print '(A20, F8.2, F8.2, F8.2, A8)', &
      name, T3_val, Y_val, Q_val, check
  end subroutine

end program electroweak_charges

Click Run to execute the Fortran code

Code will be compiled with gfortran and executed on the server