Coleman–Weinberg potential

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

${\displaystyle L=-{\frac {1}{4}}(F_{\mu \nu })^{2}+|D_{\mu }\phi |^{2}-m^{2}|\phi |^{2}-{\frac {\lambda }{6}}|\phi |^{4}}$

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

Assume that ${\displaystyle \lambda }$ is nonnegative. Then if the mass term is tachyonic, ${\displaystyle m^{2}<0}$ 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, ${\displaystyle m^{2}>0}$ the vacuum expectation of the field ${\displaystyle \phi }$ is zero. At the classical level the latter is true also if ${\displaystyle m^{2}=0}$. 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 ${\displaystyle \phi }$ 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 ${\displaystyle m^{2}}$. 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 ${\displaystyle \kappa \equiv \lambda /e^{2}}$, with a tricritical point near ${\displaystyle \kappa =1/{\sqrt {2}}}$ 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.[1] If the Ginzburg-Landau parameter ${\displaystyle \kappa }$ 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 ${\displaystyle \kappa =0.76/{\sqrt {2}}}$, i.e., slightly below the value ${\displaystyle \kappa =1/{\sqrt {2}}}$ where type-I goes over into type-II superconductor. The prediction was confirmed in 2002 by Monte Carlo computer simulations.[2]