← Back to Particle Physics

Part IV: The Higgs Mechanism

Introduction

The Higgs mechanism explains how fundamental particles acquire mass through spontaneous electroweak symmetry breaking. Proposed independently by several theorists in 1964 (Higgs, Englert, Brout, Guralnik, Hagen, Kibble), it predicts the existence of the Higgs boson, discovered at the LHC in 2012.

Nobel Prize 2013: François Englert and Peter Higgs

Spontaneous Symmetry Breaking

The Higgs Potential

The Higgs field $\phi$ is a complex $SU(2)_L$ doublet with potential:

$$ V(\phi) = -\mu^2|\phi|^2 + \lambda|\phi|^4 $$

For $\mu^2 > 0$, minimum at $|\phi| = v = \sqrt{\mu^2/(2\lambda)} \approx 246$ GeV

The "Mexican hat" potential has a circle of degenerate minima. Choosing one minimum spontaneously breaks $SU(2)_L \times U(1)_Y \to U(1)_{\text{EM}}$.

Vacuum Expectation Value

In unitary gauge, the Higgs doublet acquires a vacuum expectation value (VEV):

$$ \langle\phi\rangle = \frac{1}{\sqrt{2}}\begin{pmatrix} 0 \\ v \end{pmatrix}, \quad v \approx 246 \text{ GeV} $$

Expanding around minimum: $\phi = \frac{1}{\sqrt{2}}\begin{pmatrix} 0 \\ v + h \end{pmatrix}$

Mass Generation

Gauge Boson Masses

The Higgs VEV gives mass to $W^\pm$ and $Z^0$ through their couplings:

$$ m_W = \frac{gv}{2} \approx 80.4 \text{ GeV}, \quad m_Z = \frac{v}{2}\sqrt{g^2 + g'^2} \approx 91.2 \text{ GeV} $$$$ m_\gamma = 0 \quad \text{(photon remains massless)} $$

Fermion Masses (Yukawa Couplings)

Fermions acquire mass through Yukawa interactions with the Higgs:

$$ \mathcal{L}_{\text{Yukawa}} = -y_f \bar{f}_L \phi f_R + \text{h.c.} $$

After SSB:

$$ m_f = \frac{y_f v}{\sqrt{2}} $$

Top Quark:

$m_t \approx 173$ GeV

$y_t \approx 1$

Bottom Quark:

$m_b \approx 4.2$ GeV

$y_b \approx 0.024$

Electron:

$m_e \approx 0.511$ MeV

$y_e \approx 3 \times 10^{-6}$

The Higgs Boson

Properties

Higgs Boson Characteristics:

  • Mass: $m_h = 125.25 \pm 0.17$ GeV (measured at LHC)
  • Spin: $J = 0$ (scalar boson)
  • Parity: $P = +1$ (even parity)
  • Charge: neutral
  • Width: $\Gamma_h \approx 4.1$ MeV (lifetime $\sim 10^{-22}$ s)

Theoretical Prediction:

$$ m_h^2 = 2\mu^2 = 2\lambda v^2 $$

Self-coupling: $\lambda \approx 0.13$

Production and Decay

At the LHC, Higgs bosons are primarily produced via:

Gluon Fusion (ggF):

$gg \to h$ (via top loop)

~90% of production

Vector Boson Fusion (VBF):

$qq \to qqh$

~7% of production

Associated Production:

$Wh, Zh$

~3% combined

$t\bar{t}h$ Production:

Direct top Yukawa

~1% of production

Main Decay Channels:

  • $h \to b\bar{b}$: 58% (largest branching ratio)
  • $h \to WW^*$: 21%
  • $h \to \tau^+\tau^-$: 6.3%
  • $h \to ZZ^*$: 2.6%
  • $h \to \gamma\gamma$: 0.23% (golden channel for discovery)
  • $h \to Z\gamma$: 0.15%

Discovery at the LHC

July 4, 2012: Historic Announcement

ATLAS and CMS experiments at CERN announced the discovery of a new particle consistent with the Higgs boson:

ATLAS Result:

  • $m_h = 126.0 \pm 0.4$ GeV
  • Significance: $5.9\sigma$
  • Channels: $\gamma\gamma, ZZ^*$

CMS Result:

  • $m_h = 125.3 \pm 0.4$ GeV
  • Significance: $5.0\sigma$
  • Channels: $\gamma\gamma, ZZ^*$

Key Discovery Channels

$h \to \gamma\gamma$ (Diphoton Channel):

Clean signature: two high-energy photons

Excellent mass resolution: $\Delta m/m \sim 1-2\%$

Loop process via $W$ bosons and top quarks

$h \to ZZ^* \to 4\ell$ (Four-Lepton Channel):

"Golden channel": four charged leptons

Very low background, excellent mass resolution

Clear peak in $m_{4\ell}$ distribution

Higgs Coupling Measurements

Testing the Standard Model

Coupling strengths should be proportional to particle mass: $g_{hff} \propto m_f$, $g_{hVV} \propto m_V^2$

Measured Couplings (relative to SM prediction):

  • $\mu_{WW} = 1.15 \pm 0.10$ (within 1.5σ of SM)
  • $\mu_{ZZ} = 1.06 \pm 0.09$ (excellent agreement)
  • $\mu_{b\bar{b}} = 1.04 \pm 0.13$ (confirmed 2018)
  • $\mu_{t\bar{t}} = 1.10 \pm 0.15$ (confirmed 2018)
  • $\mu_{\tau\tau} = 1.09 \pm 0.11$ (good agreement)
  • $\mu_{\gamma\gamma} = 1.10 \pm 0.07$ (loop process)

All measurements consistent with Standard Model predictions at $\sim 10\%$ level.

Open Questions

Remaining Mysteries

  • Hierarchy Problem: Why is $m_h \ll M_{\text{Planck}}$? Quantum corrections should drive Higgs mass to very high scales.
  • Yukawa Hierarchy: Why do fermion masses span 6 orders of magnitude? ($m_e/m_t \sim 10^{-6}$)
  • Triviality Problem: Is the Higgs self-coupling fundamental or emergent?
  • Higgs Self-Coupling: Not yet directly measured. Need $hh$ production (HL-LHC goal)
  • CP Properties: Is the Higgs purely CP-even? Searches for CP violation ongoing
  • Invisible Decays: Could Higgs decay to dark matter? Current limit $< 11\%$

Key Takeaways

  • •Higgs mechanism explains mass generation via spontaneous $SU(2)_L \times U(1)_Y$ breaking
  • •Higgs VEV: $v = 246$ GeV sets electroweak scale
  • •Discovered July 4, 2012 at LHC with mass $m_h = 125.25$ GeV
  • •Nobel Prize 2013: Englert and Higgs for theoretical prediction
  • •All measured couplings consistent with SM predictions
  • •Discovery channels: $\gamma\gamma$ and $ZZ^* \to 4\ell$
  • •Many open questions remain: hierarchy problem, Yukawa hierarchy, self-coupling
← Part III: Electroweak TheoryPart V: Experimental Methods →