Physics:Coleman–Weinberg potential

Short description: Potential arising from loop effects

The Coleman–Weinberg model represents quantum electrodynamics of a scalar field in four-dimensions. The Lagrangian for the model is

[math]\displaystyle{ L = -\frac{1}{4} (F_{\mu \nu})^2 + |D_{\mu} \phi|^2 - m^2 |\phi|^2 - \frac{\lambda}{6} |\phi|^4 }[/math]

where the scalar field is complex, [math]\displaystyle{ F_{\mu \nu}=\partial_\mu A_\nu-\partial_\nu A_\mu }[/math] is the electromagnetic field tensor, and [math]\displaystyle{ D_{\mu}=\partial_\mu-\mathrm i (e/\hbar c)A_\mu }[/math] the covariant derivative containing the electric charge [math]\displaystyle{ e }[/math] of the electromagnetic field.

Assume that [math]\displaystyle{ \lambda }[/math] is nonnegative. Then if the mass term is tachyonic, [math]\displaystyle{ m^2\lt 0 }[/math] 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, [math]\displaystyle{ m^2\gt 0 }[/math] the vacuum expectation of the field [math]\displaystyle{ \phi }[/math] is zero. At the classical level the latter is true also if [math]\displaystyle{ m^2=0 }[/math]. 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 [math]\displaystyle{ \phi }[/math] 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 [math]\displaystyle{ m^2 }[/math]. 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 [math]\displaystyle{ \kappa\equiv\lambda/e^2 }[/math], with a tricritical point near [math]\displaystyle{ \kappa=1/\sqrt 2 }[/math] 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. If the Ginzburg–Landau parameter [math]\displaystyle{ \kappa }[/math] 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 [math]\displaystyle{ \kappa=0.76/\sqrt{2} }[/math], i.e., slightly below the value [math]\displaystyle{ \kappa=1/\sqrt{2} }[/math] where type-I goes over into type-II superconductor. The prediction was confirmed in 2002 by Monte Carlo computer simulations.

Literature

• S. Coleman and E. Weinberg (1973). "Radiative Corrections as the Origin of Spontaneous Symmetry Breaking". Physical Review D 7 (6): 1888–1910. doi:10.1103/PhysRevD.7.1888. Bibcode: 1973PhRvD...7.1888C. 
• 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 (1950). "On the theory of superconductivity". Zhurnal Eksperimental'noi i Teoreticheskoi Fiziki 20: 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. 

See also

• Quartic interaction