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]

References

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.