Higgs Mechanism

The Higgs Mechanism

The Higgs mechanism is the process by which the W and Z bosons — and all fundamental fermions — acquire mass without violating the gauge symmetries of the Standard Model. The discovery of the Higgs boson at CERN in 2012 (Nobel Prize 2013) confirmed this picture.

Spontaneous Symmetry Breaking and the VEV

The Higgs field $\phi$ acquires a non-zero vacuum expectation value (VEV) $v$ at the minimum of its potential:

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

The VEV is fixed by the Fermi constant $G_F$:

$v = \frac{1}{\sqrt{\sqrt{2}\, G_F}} \approx 246.22 \text{ GeV}$

where $G_F = 1.1663788 \times 10^{-5}$ GeV$^{-2}$.

Gauge Boson Masses

Expanding around the VEV gives mass terms for the W and Z bosons. The W boson trilinear coupling to the Higgs is:

$g_{HWW} = \frac{2 m_W^2}{v}$

Similarly for the Z: $g_{HZZ} = 2m_Z^2/v$.

Yukawa Couplings and Fermion Masses

Fermions couple to the Higgs field through Yukawa interactions:

$\mathcal{L}_Y = -y_f \bar{\psi}_f \phi \psi_f$

After symmetry breaking, the fermion mass is $m_f = y_f v / \sqrt{2}$, so:

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

The top quark ($m_t \approx 173$ GeV) has $y_t \approx 1$ — it is nearly as strongly coupled to the Higgs as theoretically natural. The electron ($m_e \approx 0.511$ MeV) has $y_e \approx 3 \times 10^{-6}$, an unexplained hierarchy.

Your Task

Implement:

• higgs_vev() — the Higgs VEV in GeV from $G_F$
• yukawa_coupling(m_fermion_GeV) — Yukawa coupling $y_f = \sqrt{2}\,m_f/v$ (compute $v$ inline)
• higgs_to_WW_coupling(m_W_GeV) — trilinear Higgs–WW coupling $2m_W^2/v$ (compute $v$ inline)

All constants must be defined inside each function body.