Coleman–Weinberg potential

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_{μ ν})^{2} + | D_{μ} ϕ |^{2} - m^{2} | ϕ |^{2} - \frac{λ}{6} | ϕ |^{4}

where the scalar field is complex, $F_{μ ν} = \partial_{μ} A_{ν} - \partial_{ν} A_{μ}$ is the electromagnetic field tensor, and $D_{μ} = \partial_{μ} - i (e / ℏ c) A_{μ}$ the covariant derivative containing the electric charge $e$ of the electromagnetic field.

Assume that $λ$ 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 $ϕ$ 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 - the model has a conformal anomaly).

The same can happen in other gauge theories. In the broken phase the fluctuations of the scalar field $ϕ$ 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 $κ \equiv λ / e^{2}$ , with a tricritical point near $κ = 1 / \sqrt{2}$ 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 $κ$ 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 $κ = 0.76 / \sqrt{2}$ , i.e., slightly below the value $κ = 1 / \sqrt{2}$ where type-I goes over into type-II superconductor. The prediction was confirmed in 2002 by Monte Carlo computer simulations.^[2]

Literature

S. Coleman and E. Weinberg (1973). "Radiative Corrections as the Origin of Spontaneous Symmetry Breaking". Physical Review D. 7 (6): 1888–1910. arXiv:hep-th/0507214. Bibcode:1973PhRvD...7.1888C. doi:10.1103/PhysRevD.7.1888. S2CID 6898114.
L.D. Landau (1937). "On the theory of phase transitions. II". Zhurnal Eksperimental'noi i Teoreticheskoi Fiziki. 7: 627.
V.L. Ginzburg and L.D. Landau (2009). "On the Theory of Superconductivity". On Superconductivity and Superfluidity. pp. 113–137. doi:10.1007/978-3-540-68008-6_4. ISBN 978-3-540-68004-8.
M.Tinkham (2004). Introduction to Superconductivity. Dover Books on Physics (2nd ed.). Dover. ISBN 0-486-43503-2.

References

↑ H. Kleinert (1982). "Disorder Version of the Abelian Higgs Model and the Order of the Superconductive Phase Transition" (PDF). Lettere al Nuovo Cimento. 35 (13): 405–412. doi:10.1007/BF02754760. S2CID 121012850.
↑ 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 (6) 064524. arXiv:cond-mat/0202215. Bibcode:2002PhRvB..66f4524H. doi:10.1103/PhysRevB.66.064524. S2CID 13672575.

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[1] H. Kleinert (1982). "Disorder Version of the Abelian Higgs Model and the Order of the Superconductive Phase Transition" (PDF). Lettere al Nuovo Cimento. 35 (13): 405–412. doi:10.1007/BF02754760. S2CID 121012850.

[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 (6) 064524. arXiv:cond-mat/0202215. Bibcode:2002PhRvB..66f4524H. doi:10.1103/PhysRevB.66.064524. S2CID 13672575.

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Literature

See also

References