The Coleman–Weinberg model represents quantum electrodynamics of a scalar field in four-dimensions. The Lagrangian for the model is
L = − 1 4 ( F μ ν ) 2 + ( D μ ϕ ) 2 − m 2 ϕ 2 − λ 6 ϕ 4
where the scalar field is complex, F μ ν = ∂ μ A ν − ∂ ν A μ is the electromagnetic field tensor, and D μ = ∂ μ − i ( e / ℏ c ) A μ the covariant derivative containing the electric charge e of the electromagnetic field.
Assume that λ is nonnegative. Then if the mass term is tachyonic, m 2 < 0 there is a spontaneous breaking of the gauge symmetry at low energies, a variant of the Higgs mechanism. On the other hand, if the squared mass is positive, m 2 > 0 the vacuum expectation of the field ϕ is zero. At the classical level the latter is true also if m 2 = 0 However, as was shown by Sidney Coleman and Erick Weinberg even if the renormalized mass is zero spontaneous symmetry breaking still happens due to the radiative corrections (this introduces a mass scale into a classically conformal theory - model have a conformal anomaly).
The same can happen in other gauge theories. In the broken phase the fluctuations of the scalar field ϕ will manifest themselves as a naturally light Higgs boson, as a matter of fact even too light to explain the electroweak symmetry breaking in the minimal model - much lighter than vector bosons. There are non-minimal models that give a more realistic scenarios. Also the variations of this mechanism were proposed for the hypothetical spontaneously broken symmetries including supersymmetry.
Equivalently one may say that the model possesses a first-order phase transition as a function of m 2 . The model is the four-dimensional analog of the three-dimensional Ginzburg–Landau theory used to explain the properties of superconductors near the phase transition.
The three-dimensional version of the Coleman–Weinberg model governs the superconducting phase transition which can be both first- and second-order, depending on the ratio of the Ginzburg–Landau parameter κ ≡ λ / e 2 , with a tricritical point near κ = 1 / 2 which separates type I from type II superconductivity. Historically, the order of the superconducting phase transition was debated for a long time since the temperature interval where fluctuations are large (Ginzburg interval) is extremely small. The question was finally settled in 1982. If the Ginzburg-Landau parameter κ that distinguishes type-I and type-II superconductors (see also here) is large enough, vortex fluctuations becomes important which drive the transition to second order. The tricitical point lies at roughly κ = 0.76 / 2 , i.e., slightly below the value κ = 1 / 2 where type-I goes over into type-II superconductor. The prediction was confirmed in 2002 by Monte Carlo computer simulations.
