Physic Labs

Particle physics

The Higgs boson and Higgs mechanism

The Higgs field has a nonzero vacuum expectation value; coupling to it gives W and Z masses after electroweak symmetry breaking, while the Higgs boson is a field excitation.

For μ²<0, the potential has a Mexican-hat shape: its minima lie on a ring rather than at Φ=0. Choosing a vacuum spontaneously selects a direction; in a gauge theory the would-be Goldstone modes are absorbed, giving gauge bosons mass.

V(Φ)=μ2Φ†Φ+λ(Φ†Φ)2,v=−μ2/λV(\Phi)=\mu^2\Phi^\dagger\Phi+\lambda(\Phi^\dagger\Phi)^2,\quad v=\sqrt{-\mu^2/\lambda}

Definition: Definition and physical meaning

In unitary gauge the Higgs doublet has ⟨Φ⟩=(0,v/√2)ᵀ, with v≈246 GeV. Then m_W=gv/2 and m_Z=v√(g²+g′²)/2, while the photon remains massless.

Adjust the parameters and drag the figure to explore the model; readouts are quantitative illustrations under the stated assumptions.

Structure and interpretation

After choosing the vacuum, three Goldstone degrees of freedom in the doublet are absorbed by W± and Z⁰; one physical degree remains, the Higgs boson h. This is the Higgs realization of spontaneous symmetry breaking in a gauge theory.

Example: Quantitative example

With λ≈0.13 and v≈246 GeV, m_h≈√(2λ)v≈125 GeV. This is the observed Higgs-boson mass, not the W/Z mass; precise values depend on running parameters and conventions.

Solution

√(2×0.13)×246 GeV≈125.5 GeV. This estimate uses the classical quartic potential and effective parameters; precision predictions include quantum corrections.

The Higgs field also couples to fermions through Yukawa interactions. After symmetry breaking this coupling becomes a fermion mass, mf=yfv/2m_f=y_fv/\sqrt2, so different Yukawa values produce different masses. The model does not explain why the coefficients yfy_f span many orders of magnitude; this remains a flavor-structure problem. Nonzero neutrino masses require an additional mechanism.

Spontaneous symmetry breaking does not mean the fundamental laws lose symmetry: the equations remain symmetric, but the vacuum selects one configuration. Oscillations around the potential minimum produce the Higgs boson; W and Z masses arise from the covariant derivative of the doublet. Fermions gain mass through Yukawa couplings, while the photon stays massless because the residual U(1)emU(1)_{\mathrm{em}} symmetry remains unbroken.

Quick check

Approximately what is the Standard Model Higgs vacuum value v?

Which boson remains massless in the Higgs mechanism?

References

  1. Michael E. Peskin and Daniel V. Schroeder (1995). An Introduction to Quantum Field Theory
  2. Steven Weinberg (1995). The Quantum Theory of Fields, Volume I