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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 = − 1 4 ( F μ ν ) 2 + | D μ ϕ | 2 − m 2 | ϕ | 2 − λ 6 | ϕ | 4 {\displaystyle L=-{\frac {1}{4}}(F_{\mu \nu })^{2}+|D_{\mu }\phi |^{2}-m^{2}|\phi |^{2}-{\frac {\lambda }{6}}|\phi |^{4}} {\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, F μ ν = ∂ μ A ν − ∂ ν A μ {\displaystyle F_{\mu \nu }=\partial _{\mu }A_{\nu }-\partial _{\nu }A_{\mu }} {\displaystyle F_{\mu \nu }=\partial _{\mu }A_{\nu }-\partial _{\nu }A_{\mu }} is the electromagnetic field tensor, and D μ = ∂ μ − i ( e / ℏ c ) A μ {\displaystyle D_{\mu }=\partial _{\mu }-\mathrm {i} (e/\hbar c)A_{\mu }} {\displaystyle D_{\mu }=\partial _{\mu }-\mathrm {i} (e/\hbar c)A_{\mu }} the covariant derivative containing the electric charge e {\displaystyle e} {\displaystyle e} of the electromagnetic field.

Assume that λ {\displaystyle \lambda } {\displaystyle \lambda } is nonnegative. Then if the mass term is tachyonic, m 2 < 0 {\displaystyle m^{2}<0} {\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, m 2 > 0 {\displaystyle m^{2}>0} {\displaystyle m^{2}>0} the vacuum expectation of the field ϕ {\displaystyle \phi } {\displaystyle \phi } is zero. At the classical level the latter is true also if m 2 = 0 {\displaystyle m^{2}=0} {\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 - the model has a conformal anomaly).

The same can happen in other gauge theories. In the broken phase the fluctuations of the scalar field ϕ {\displaystyle \phi } {\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 m 2 {\displaystyle m^{2}} {\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 κ ≡ λ / e 2 {\displaystyle \kappa \equiv \lambda /e^{2}} {\displaystyle \kappa \equiv \lambda /e^{2}}, with a tricritical point near κ = 1 / 2 {\displaystyle \kappa =1/{\sqrt {2}}} {\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 κ {\displaystyle \kappa } {\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 κ = 0.76 / 2 {\displaystyle \kappa =0.76/{\sqrt {2}}} {\displaystyle \kappa =0.76/{\sqrt {2}}}, i.e., slightly below the value κ = 1 / 2 {\displaystyle \kappa =1/{\sqrt {2}}} {\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]

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