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)²For 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 GeVThis 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)TThis 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/2When Φ takes its VEV ⟨Φ⟩ = (0, v/√2)T, we get mass terms:
W Boson Mass:
mW = gv/2 ≈ 80.4 GeVFrom W± = (W¹ ∓ iW²)/√2 coupling to Higgs VEV
Z Boson Mass:
mZ = v√(g² + g'²)/2 ≈ 91.2 GeVMixing 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 ≈ 0.882Tree-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'/gIt determines how W³ and B mix to form Z and γ:
Zμ = cos θW W³μ - sin θW BμAμ = sin θW W³μ + cos θW BμThe electromagnetic coupling emerges as:
e = g sin θW = g' cos θWMeasured 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/√2Examples:
- 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))Twhere H(x) is the physical Higgs field (one real scalar remains).
Higgs Boson Mass:
mH² = -2μ² = 2λv²The 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:
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