Quantum Field Theory Course

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

Part VIII: Standard Model • Chapter 2

The Higgs Mechanism

Spontaneous symmetry breaking and the origin of mass

The Mass Problem

The Crisis: Gauge symmetry forbids mass terms!

A mass term for gauge bosons like m²WμWμ explicitly breaks gauge invariance. But experimentally:

  • • W boson: mW = 80.4 GeV (massive!)
  • • Z boson: mZ = 91.2 GeV (massive!)
  • • Photon: mγ = 0 (massless ✓)

How can we maintain gauge invariance while giving W and Z mass?

The Solution: Spontaneous Symmetry Breaking

The Higgs mechanism (Englert-Brout-Higgs-Guralnik-Hagen-Kibble, 1964) shows that gauge bosons can acquire mass if the vacuum itself breaks the symmetry, even though the Lagrangian remains symmetric.

1. The Higgs Doublet

Introduce a complex scalar field in the SU(2) doublet:

Φ = (φ+, φ0)T ~ (1, 2, +1/2)

In component form:

Φ = 1/√2 ((φ1 + iφ2), (φ3 + iφ4))T
  • • 4 real scalar fields: φ₁, φ₂, φ₃, φ₄
  • • Hypercharge Y = +1/2: gives Q(φ+) = +1, Q(φ0) = 0
  • • Weak isospin doublet: transforms under SU(2)L

The Higgs Potential

The most general renormalizable potential:

V(Φ) = μ²Φ†Φ + λ(Φ†Φ)²

For μ² < 0 (tachyonic mass), the potential has:

  • • Unstable origin: V(0) = 0 is a maximum, not minimum
  • • Continuous minimum: at |Φ| = v where v² = -μ²/λ
  • • Mexican hat shape: minimum forms a circle in field space

The vacuum expectation value (VEV):

v = √(-μ²/λ) ≈ 246 GeV

This is the electroweak scale that sets all W, Z, and fermion masses!

2. Spontaneous Symmetry Breaking

Choosing a Vacuum

The vacuum must pick one direction in field space. By convention, choose:

⟨Φ⟩ = 1/√2 (0, v)T = 1/√2 (0, 246 GeV)T

This choice:

  • • Breaks SU(2)L × U(1)Y: vacuum is not invariant under full symmetry
  • • Preserves U(1)EM: photon remains massless (Q⟨Φ⟩ = 0)
  • • Gives mass to W±, Z: via covariant derivative term

Key principle:

The Lagrangian is symmetric, but the vacuum is not. This is spontaneous symmetry breaking—the ground state chooses a direction.

Goldstone Theorem

When continuous symmetry is spontaneously broken, Goldstone bosons appear:

Goldstone's Theorem: One massless scalar for each broken generator.

  • • SU(2)×U(1) has 4 generators
  • • U(1)EM has 1 unbroken generator
  • • Therefore: 4 - 1 = 3 Goldstone bosons

But in a gauge theory, Goldstone bosons are "eaten":

Higgs Mechanism:

The 3 would-be Goldstone bosons become the longitudinal polarizations of W+, W-, Z. This is how massless gauge bosons acquire a third polarization state (massive spin-1 needs 3 DOF).

3. Gauge Boson Masses

The kinetic term for the Higgs field contains the covariant derivative:

ℒkin = (DμΦ)†(DμΦ)

where:

Dμ = ∂μ + ig Wiμτi/2 + ig' Y Bμ/2

When Φ takes its VEV ⟨Φ⟩ = (0, v/√2)T, we get mass terms:

W Boson Mass:

mW = gv/2 ≈ 80.4 GeV

From W± = (W¹ ∓ iW²)/√2 coupling to Higgs VEV

Z Boson Mass:

mZ = v√(g² + g'²)/2 ≈ 91.2 GeV

Mixing of W³ and B creates massive Z

Photon Mass:

mγ = 0 (exactly!)

The orthogonal combination Aμ = sin θW W³ + cos θW B remains massless because U(1)EM is unbroken: Q⟨Φ⟩ = 0

Mass Relation:

mW/mZ = cos θW ≈ 0.882

Tree-level prediction, tested to 0.02% precision! Radiative corrections depend on mt, mH.

4. The Weinberg Angle

The weak mixing angle θW (Weinberg angle) relates couplings:

tan θW = g'/g

It determines how W³ and B mix to form Z and γ:

Zμ = cos θW W³μ - sin θW Bμ
Aμ = sin θW W³μ + cos θW Bμ

The electromagnetic coupling emerges as:

e = g sin θW = g' cos θW

Measured value:

sin² θW ≈ 0.231 at MZ (MS̄ scheme)

This runs with energy scale Q due to loop corrections—crucial for precision tests!

5. Fermion Masses: Yukawa Couplings

Fermions also get mass via the Higgs through Yukawa interactions:

ℒYukawa = -yf Φ̄ ψ̄L ψR + h.c.

When Φ → v/√2, this becomes a mass term:

mf = yf v/√2

Examples:

  • Top quark: yt ≈ 1.0 → mt ≈ 173 GeV (largest Yukawa!)
  • Bottom quark: yb ≈ 0.024 → mb ≈ 4.2 GeV
  • Electron: ye ≈ 3 × 10-6 → me ≈ 0.5 MeV

The Hierarchy Problem:

Why do Yukawa couplings span 6 orders of magnitude (10-6 to 1)? The SM provides no explanation—these are 9 independent parameters (up/down × 3 gen + 3 leptons).

6. The Physical Higgs Boson

After symmetry breaking, expand around the vacuum:

Φ(x) = 1/√2 (0, v + H(x))T

where H(x) is the physical Higgs field (one real scalar remains).

Higgs Boson Mass:

mH² = -2μ² = 2λv²

The Higgs mass is not predicted by symmetry alone—it's a free parameter (related to λ).

Experimental discovery:

ATLAS & CMS at LHC, July 4, 2012: mH = 125.1 ± 0.1 GeV

Higgs Couplings

The Higgs couples to particles proportionally to their mass:

  • • To fermions: gHff̄ = mf/v (Yukawa coupling)
  • • To W bosons: gHWW ∝ mW/v
  • • To Z bosons: gHZZ ∝ mZ/v
  • • To itself: Higgs self-coupling λHHH = 3mH²/v²

This means H couples most strongly to top quarks and W/Z bosons!

Higgs Decay Modes

At mH = 125 GeV, dominant decays:

H → bb̄: ~58%
H → WW*: ~21%
H → gg: ~8% (loop)
H → τ+τ-: ~6%
H → ZZ*: ~3%
H → γγ: ~0.2% (loop)

Despite being rare, H → γγ and H → ZZ* → 4ℓ were crucial for discovery due to clean signatures!

7. Vacuum Stability

Is our electroweak vacuum stable? This depends on how λ runs with energy:

Potential problem:

At high energies, the running Higgs self-coupling λ(Q) can turn negative due to top quark loops!

dλ/dt ∝ -yt⁴ (top Yukawa contribution)

Current status (mH = 125 GeV, mt = 173 GeV):

  • • Vacuum is metastable: λ turns negative around 10¹⁰ GeV
  • • True vacuum has V < 0 at Planck scale (universe would collapse!)
  • • But tunneling time τ > 10¹⁰⁰ years (universe is safe...for now)
  • • Sensitive to mt and mH—slight changes could restore absolute stability

Summary

  • ✓ Higgs doublet Φ ~ (1, 2, +1/2) with VEV v = 246 GeV
  • ✓ Spontaneous breaking: SU(2)L × U(1)Y → U(1)EM
  • ✓ 3 Goldstone bosons eaten → longitudinal W±, Z polarizations
  • ✓ Gauge boson masses: mW = gv/2, mZ = mW/cos θW, mγ = 0
  • ✓ Fermion masses: Yukawa couplings mf = yfv/√2
  • ✓ Physical Higgs: mH = 125.1 GeV, couplings ∝ mass
  • ✓ Vacuum stability: Metastable, sensitive to mt and mH

Further Resources

  • • Peskin & Schroeder - Chapter 20 (Higgs Mechanism in Electroweak Theory)
  • • Schwartz - Chapter 28 (Spontaneous Symmetry Breaking)
  • • Weinberg - Vol II, Chapter 21 (Spontaneous Breaking of Gauge Symmetries)
  • • ATLAS/CMS Papers - Phys. Lett. B 716 (2012) - Higgs discovery