Coleman-Weinberg potential

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

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, F_{\mu \nu}=\partial_\mu A_\nu-\partial_\nu A_\mu is the electromagnetic field tensor, and D_{\mu}=\partial_\mu-(e/\hbar c)A_\mu the covariant derivative containing the electric charge e 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 m2. 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 has both a first and a second-order phase transition depending on the ratio of the Ginzburg-Landau parameter \kappa\equiv\lambda/3e^2, with a tricritial point near \kappa=1/\sqrt 2 which separates type I from type II superconductivity.

[edit] References

  • S. Coleman and E. Weinberg, Phys. Rev. D7, 1888 (1973). [1]