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.
