Coleman–Weinberg Potential

The Coleman–Weinberg model represents quantum electrodynamics of a scalar field in four-dimensions. The Lagrangian for the model is

:$L = -\frac (F_)^2 + , D_ \phi, ^2 - m^2 , \phi, ^2 - \frac , \phi, ^4$

where the scalar field is complex, $F_=\partial_\mu A_\nu-\partial_\nu A_\mu$ is the electromagnetic field tensor, and $D_=\partial_\mu-\mathrm i (e/\hbar c)A_\mu$ the covariant derivative containing the electric charge $e$ of the electromagnetic field.

Assume that $\lambda$ 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 $\phi$ 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 - the model has a conformal anomaly).

The same can happen in other gauge theories. In the broken phase the fluctuations of the scalar field $\phi$ 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 $\kappa\equiv\lambda/e^2$, with a tricritical point near $\kappa=1/\sqrt 2$ 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 $\kappa$ 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 tricritical point lies at
roughly
$\kappa=0.76/\sqrt$, i.e., slightly below the value $\kappa=1/\sqrt$
where type-I goes over into type-II superconductor.
The prediction was confirmed in 2002 by Monte Carlo computer simulations.

Literature

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See also

* Quartic interaction

References

Quantum field theory
Quantum mechanical potentials
{{quantum-stub