Английская Википедия:Coleman–Weinberg potential
Шаблон:Short description The Coleman–Weinberg model represents quantum electrodynamics of a scalar field in four-dimensions. The Lagrangian for the model is
- <math>L = -\frac{1}{4} (F_{\mu \nu})^2 + |D_{\mu} \phi|^2 - m^2 |\phi|^2 - \frac{\lambda}{6} |\phi|^4</math>
where the scalar field is complex, <math>F_{\mu \nu}=\partial_\mu A_\nu-\partial_\nu A_\mu </math> is the electromagnetic field tensor, and <math>D_{\mu}=\partial_\mu-\mathrm i (e/\hbar c)A_\mu </math> the covariant derivative containing the electric charge <math>e</math> of the electromagnetic field.
Assume that <math>\lambda</math> is nonnegative. Then if the mass term is tachyonic, <math>m^2<0</math> 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, <math>m^2>0</math> the vacuum expectation of the field <math>\phi</math> is zero. At the classical level the latter is true also if <math>m^2=0</math>. 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 <math>\phi</math> 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 <math>m^2</math>. 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 <math> \kappa\equiv\lambda/e^2</math>, with a tricritical point near <math> \kappa=1/\sqrt 2</math> 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 <math>\kappa</math> 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 <math>\kappa=0.76/\sqrt{2}</math>, i.e., slightly below the value <math>\kappa=1/\sqrt{2}</math> where type-I goes over into type-II superconductor. The prediction was confirmed in 2002 by Monte Carlo computer simulations.[2]
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