Coleman–Weinberg potential

The Coleman–Weinberg model represents quantum electrodynamics of a scalar field in four-dimensions. The Lagrangian for the model is 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 u })^{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 u }=partial _{mu }A_{ u }-partial _{ u }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 }} the covariant derivative containing the electric charge e {displaystyle e} of the electromagnetic field. Assume that 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 {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 - model have a conformal anomaly). The same can happen in other gauge theories. In the broken phase the fluctuations of the scalar field 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 ≡ λ / e 2 {displaystyle kappa equiv lambda /e^{2}} , with a tricritical point near 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 temperatureinterval where fluctuations are large (Ginzburg interval) is extremely small.The question was finally settledin 1982. If the Ginzburg-Landau parameter 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 atroughly κ = 0.76 / 2 {displaystyle kappa =0.76/{sqrt {2}}} , i.e., slightly below the value 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.