Spontaneous Symmetry Breaking and the Higgs

Takeaway

When the lowest-energy state does not share the symmetry of the laws, massless Nambu–Goldstone modes appear; with gauge fields, the Higgs mechanism gives masses to gauge bosons while preserving gauge invariance.

The problem (before → after)

Before: Symmetric equations predict massless fields or degeneracies that conflict with observations.
After: The system chooses a specific vacuum within a symmetric valley; excitations split into massless phase modes and massive amplitude modes—gauge fields can eat the phases to gain mass.

Mental model first

Picture a marble in a Mexican-hat bowl. The bowl has rotational symmetry, but the resting marble picks one direction. Small moves around the rim (phase) cost almost no energy; moving radially (amplitude) costs energy.

Just-in-time concepts

Order parameter: Quantity whose nonzero expectation signals broken symmetry.
Goldstone theorem: One massless mode per broken continuous generator (global case).
Higgs mechanism: In gauge theories, NG modes are gauged away; gauge bosons acquire mass m ∝ g v.

First-pass solution

Consider V(ϕ) = λ(|ϕ|^2 − v^2)^2. Choose a vacuum |⟨ϕ⟩| = v; expand fields around it to see one massive Higgs (radial) and massless NG (angular) mode; couple to gauge field to see gauge mass.

Explicit vs spontaneous breaking: Small symmetry-breaking terms tilt the hat, lifting degeneracy.
Finite temperature: Symmetry can restore at high T (phase transitions).
Constraints: Coleman–Mermin–Wagner forbids continuous SSB in 1–2D at finite T for short-range interactions.

Principles, not prescriptions

Vacua can break symmetries the Lagrangian has; observables remain gauge invariant.
Mass generation via symmetry structure avoids ad hoc mass terms.

Common pitfalls

Confusing gauge choice with physical breaking; only gauge-invariant statements are physical.
Assuming every broken symmetry yields a particle—count carefully with constraints.

Connections and contrasts

See also: [/blog/yang-mills-gauge-theory], [/blog/renormalization-group] (critical phenomena), [/blog/topological-order] (order without local order parameters).

Quick checks

What signals SSB? — Nonzero order parameter and degenerate vacua.
Where do masses come from in the Higgs mechanism? — Gauge fields interacting with the vacuum expectation value.
Why are NG modes massless? — Restoring symmetry costs vanishing energy for long wavelengths.