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

Assume that λ λ {\displaystyle \lambda } is nonnegative. Then if the mass term is tachyonic, 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} the vacuum expectation of the field ϕ ϕ {\displaystyle \phi } is zero. At the classical level the latter is true also if 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 ϕ {\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}} . 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 κ ≡ ≡ λ λ / e 2 {\displaystyle \kappa \equiv \lambda /e^{2}} , with a tricritical point near κ κ = 1 / 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.cite-ref-1[1] If the Ginzburg–Landau parameter κ κ {\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 κ = 0.76 / 2 {\displaystyle \kappa =0.76/{\sqrt {2}}} , i.e., slightly below the value κ κ = 1 / 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.cite-ref-2[2]

Contents

• Literature

• See also

• References

Literature

• citerefs-coleman-and-e-weinberg1973S. 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.

• citerefl-d-landau1937L.D. Landau (1937). "On the theory of phase transitions. II". Zhurnal Eksperimental'noi i Teoreticheskoi Fiziki. 7: 627.

• citerefv-l-ginzburg-and-l-d-landau2009V.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.

• citerefm-tinkham2004M.Tinkham (2004). Introduction to Superconductivity. Dover Books on Physics (2nd ed.). Dover. ISBN 0-486-43503-2.