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The Higgs mechanism is described by the potential

V(\phi) = \mu^2 \phi^{\dagger} \phi + \lambda (\phi^{\dagger} \phi)^2

. If

\mu^2 < 0

and

\lambda > 0

, the potential is minimized at a non-zero vacuum expectation value (VEV),

\langle \phi \rangle = v / \sqrt{2}

. This VEV breaks the symmetry, generating mass terms for the gauge bosons

W

and

Z

through the kinetic term

\mathcal{L}_{kin} = (D_{\mu} \phi)^* (D^{\mu} \phi)

, and for fermions via Yukawa couplings

\mathcal{L}_{Yukawa} = -y_f \bar{\psi}_L \ \tilde{\phi} \ \tau^a \ \frac{\sigma^a}{2} \ \rightarrow \ \bar{\psi}_L \ \frac{m_f}{v} \ \bar{\psi}_R

Higgs Mechanism