Spontaneous Symmetry Breaking

Explains how a symmetric potential selects a nonsymmetric vacuum. Theory, formulas, worked examples and interactive graphs.

Conceptual context

Spontaneous symmetry breaking occurs when the laws describing a system are symmetric but the lowest-energy state is not. In this leaf, a real $\phi$ is represented along a symmetric $V$. The equations do not lose their symmetry; instead, the system selects one among several equivalent vacua. The key quantity is the $mu2$, which controls the curvature near the origin, while the $\lambda$ keeps the energy bounded at large field values. This explains how the vacuum may acquire a nonzero $v$ and why small fluctuations around the chosen vacuum behave as excitations with a definite $mh$.

🟢 Essential level

Picture a ball on a perfectly symmetric surface. If the lowest point lies at the center, the ball can rest there and the symmetry remains visible.

If the center becomes a hilltop and two equal valleys appear, the surface is still symmetric, but the ball must choose one valley. That choice represents spontaneous symmetry breaking. The $mu2$ decides whether the origin of the $\phi$ is stable or unstable. When the origin is stable, the vacuum $v$ is zero. When it becomes unstable, two opposite minima appear with the same $V$. The $\lambda$ keeps the energy bounded and controls how rapidly it rises away from the minima. Near the selected vacuum, small oscillations of the $\phi$ behave as particles whose $mh$ depends on the local curvature of the $V$.

This mechanism was proposed nearly simultaneously by three groups in 1964: Robert Brout and François Englert in August, Peter Higgs shortly afterward, and Gerald Guralnik, Carl Hagen, and Tom Kibble in November. All three papers independently showed how a spontaneously broken gauge symmetry can give mass to field bosons.

🔵 Formal level

Basic formulas

The minimal model uses a real $\phi$ with reflection symmetry. Its potential energy density is described by:

Symmetric quartic potential

The sign of the $mu2$ controls the local geometry of the $V$, while the $\lambda$ determines its growth at large amplitude. Candidate homogeneous vacuum configurations are found by requiring the $dV_dphi$ to vanish:

Vacuum stationarity condition

This condition always gives the central solution and, when the parameters allow it, two nonzero solutions. When the $mu2$ is positive and the $\lambda$ is positive, the origin is the stable minimum and the $v$ is zero. When the $mu2$ is negative, the origin becomes a local maximum and two degenerate minima appear.

Real student questions

Does the symmetry disappear? No. The dynamical law and the $V$ remain symmetric, but the selected vacuum is not.

Why are there two vacua? Because the $V$ is invariant under a sign change of the $\phi$, so both minima have equal energy.

Does negative $mu2$ mean an imaginary particle mass? It signals that the original expansion point is unstable. Around the correct vacuum, the physical $mh$ is real and positive.

What selects one minimum? A fluctuation, an initial condition, an environment, or an arbitrarily small bias.

Does vanishing $dV_dphi$ guarantee a vacuum? No. It may also identify a maximum or an inflection point.

Final synthesis

The $mu2$ determines whether the origin is stable or whether the $\phi$ settles at a nonzero vacuum. The $\lambda$ stabilizes the $V$, the $v$ measures the broken phase, and the local curvature fixes the $mh$. The theory remains symmetric; the selected vacuum does not.