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.

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 - model have 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.[1] 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 tricitical point lies at roughly \kappa=0.76/\sqrt{2}, i.e., slightly below the value \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]



  1. ^ H. Kleinert (1982). "Disorder Version of the Abelian Higgs Model and the Order of the Superconductive Phase Transition" (PDF). Lett. Nuovo Cimento 35: 405–412. doi:10.1007/BF02754760. 
  2. ^ J. Hove, S. Mo, A. Sudbo (2002). "Vortex interactions and thermally induced crossover from type-I to type-II superconductivity" (PDF). Phys. Rev. B 66: 064524. arXiv:cond-mat/0202215. Bibcode:2002PhRvB..66f4524H. doi:10.1103/PhysRevB.66.064524.