**Coleman–Weinberg Potential**

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

where the scalar field is complex, is the electromagnetic field tensor, and the covariant derivative containing the electric charge of the electromagnetic field. The model illustrates the generation of mass by fluctuations of the vector field. Equivalently one may say that the model possesses a first-order phase transition as a function of . 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.

Interestingly, 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, with a tricritical point near 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 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, i.e., slightly below the value where type-I goes over into type-II superconductor. The prediction was confirmed in 2002 by Monte Carlo computer simulations.

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