The Higgs Mechanism

4.1 Spontaneous Symmetry Breaking

The Higgs mechanism gives mass to the $W^\pm$ and $Z^0$ bosons while preserving gauge invariance. The Key idea: a scalar field acquires a non-zero vacuum expectation value (VEV), spontaneously Breaking the electroweak symmetry.

The Higgs potential:

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

With $\mu^2 \lt 0$ and $\lambda \gt 0$ . This is the Mexican hat potential: the minimum is at $|\phi| = v/\sqrt{2}$ where $v = \sqrt{-\mu^2/\lambda}$ .

4.2 Worked Example: Mexican Hat Potential Analysis

Example 4.1: Detailed analysis of spontaneous symmetry breaking

Consider the complex scalar field $\phi = (\phi_1 + i\phi_2)/\sqrt{2}$ with the potential:

$V(\phi) = \mu^2\lvert\phi\rvert^2 + \lambda\lvert\phi\rvert^4$

With $\mu^2 \lt 0$ , $\lambda \gt 0$ .

Step 1: Find the minimum. Setting $\partial V/\partial\phi_i = 0$ :

$\frac{\partial V}{\partial\phi_1} = \phi_1(\mu^2 + \lambda(\phi_1^2 + \phi_2^2)) = 0$ $\frac{\partial V}{\partial\phi_2} = \phi_2(\mu^2 + \lambda(\phi_1^2 + \phi_2^2)) = 0$

The solutions are:

$\phi_1 = \phi_2 = 0$ : This is a local maximum since $V"' = \mu^2 \lt 0$ .
$\phi_1^2 + \phi_2^2 = -\mu^2/\lambda \equiv v^2$ : This is the circle of minima.

The VEV is $v = \sqrt{-\mu^2/\lambda}$ . The symmetry is $\mathrm{U}(1)$ (phase rotations $\phi \to e^{i\alpha}\phi$ ), which has a continuous set of degenerate ground states.

Step 2: Expand around the vacuum. Choose the vacuum $\langle\phi\rangle = v/\sqrt{2}$ by Fixing the gauge. Parameterise:

$\phi(x) = \frac{1}{\sqrt{2}}(v + h(x))$

Where $h(x)$ is a real scalar field (the “Higgs field” fluctuation).

Step 3: Compute the physical mass. Substituting into the potential:

$V = \mu^2 \cdot \frac{1}{2}(v + h)^2 + \lambda \cdot \frac{1}{4}(v + h)^4$

Expanding and keeping terms through quadratic order in $h$ :

$V = \underbrace{\frac{\mu^2 v^2}{2} + \frac{\lambda v^4}{4}}_{\mathrm{constant} + \underbrace{(\mu^2 v + \lambda v^3)}_{= 0}\,h + \underbrace{\left(\frac{\mu^2}{2} + \frac{3\lambda v^2}{2}\right)}_{m_h^2/2}\,h^2 + \cdots}$

Using $v^2 = -\mu^2/\lambda$ The coefficient of $h$ vanishes (as required at the minimum) And:

$m_h^2 = \mu^2 + 3\lambda v^2 = -\lambda v^2 + 3\lambda v^2 = 2\lambda v^2$

The physical Higgs mass is $m_h = v\sqrt{2\lambda}$ .

Step 4: Goldstone boson. The original complex field had two real degrees of freedom. After symmetry breaking, one ( $h$ ) becomes a massive particle. The other (the phase Fluctuation) is the Goldstone boson --- a massless mode corresponding to motion Along the circle of degenerate minima. In gauge theory, this Goldstone boson is “eaten” By the gauge field to become its longitudinal polarisation. $\blacksquare$

4.3 Mass Generation for Gauge Bosons

The Higgs field is an SU(2) doublet:

$\phi = \begin{pmatrix} \phi^+ \\ \phi^0 \end{pmatrix}, \quad \langle\phi\rangle_0 = \frac{1}{\sqrt{2}}\begin{pmatrix} 0 \\ v \end{pmatrix}$

Where $v \approx 246$ GeV.

Expanding around the VEV, the kinetic term $|D_\mu\phi|^2$ (with $D_\mu$ the covariant derivative) Produces mass terms:

$m_W = \frac{1}{2}gv, \quad m_Z = \frac{1}{2}\sqrt{g^2 + g'^2}\,v, \quad m_\gamma = 0$

Where $g$ and $g'$ are the SU(2) and U(1) coupling constants.

Proof of the mass relation. The covariant derivative is:

$D_\mu = \partial_\mu - ig\frac{\tau^a}{2}W^a_\mu - ig'\frac{Y}{2}B_\mu$

The kinetic term $|D_\mu\phi|^2$ evaluated at the VEV gives:

$\lvert D_\mu\langle\phi\rangle_0\rvert^2 = \frac{v^2}{8}\left[g^2(W^1_\mu)^2 + g^2(W^2_\mu)^2 + (gW^3_\mu - g'B_\mu)^2\right]$

Identifying $W^\pm_\mu = (W^1_\mu \mp iW^2_\mu)/\sqrt{2}$ The first two terms give:

$\frac{1}{2}\left(\frac{gv}{2}\right)^2 W^+_\mu W^{-\mu} \implies m_W = \frac{gv}{2}$

The last term, after the rotation to $Z^0$ and $A$ Gives:

$m_Z = \frac{v}{2}\sqrt{g^2 + g'^2}, \quad m_\gamma = 0 \quad \blacksquare$

The ratio:

$\frac{m_W}{m_Z} = \cos\theta_W$

Defines the Weinberg angle $\theta_W \approx 28.7^\circ$ .

4.4 Fermion Masses and Yukawa Couplings

Fermions acquire mass through Yukawa couplings to the Higgs field:

$\mathcal{L}_{\mathrm{Yukawa} = -y_f \bar{\psi}_L \phi \psi_R + \mathrm{h}.c.}$

After symmetry breaking, this gives $m_f = y_f v/\sqrt{2}$ . The Yukawa couplings $y_f$ are free Parameters of the Standard Model, not predicted by the theory.

Example 4.2: Yukawa coupling of the top quark

The top quark has mass $m_t \approx 173$ GeV/ $c^2$ . The Yukawa coupling is:

$y_t = \frac{\sqrt{2}\,m_t}{v} = \frac{\sqrt{2} \times 173\;\mathrm{GeV}{246\;\mathrm{GeV} \approx 0.994}}$

This is remarkably close to 1 --- the top quark has the largest Yukawa coupling of all Fermions and is the only fermion with a coupling of order unity. For comparison:

Electron: $y_e = m_e\sqrt{2}/v \approx 2.9 \times 10^{-6}$
Muon: $y_\mu \approx 6.1 \times 10^{-4}$
Bottom quark: $y_b \approx 0.024$

The enormous range of Yukawa couplings (over five orders of magnitude) is the flavour Problem and remains unexplained within the Standard Model.

4.5 Physical Higgs Boson

After gauge fixing (unitary gauge), the four degrees of freedom of the Higgs doublet become:

Three Goldstone bosons absorbed by $W^\pm$ and $Z^0$ (giving them their longitudinal polarisation states).
One physical scalar particle $H$ with mass $m_H = \sqrt{2\lambda}\,v \approx 125$ GeV/ $c^2$ .

The Higgs boson was discovered at the LHC by ATLAS and CMS on July 4, 2012, with $m_H \approx 125$ GeV/ $c^2$ . The discovery confirmed the mechanism of electroweak symmetry Breaking and earned the 2013 Nobel Prize for Englert and Higgs.

Mark this page as reviewed