Coleman–Weinberg potential

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

$L = -\frac{1}{4} (F_{\mu \nu})^2 + (D_{\mu} \phi)^2 - m^2 \phi^2 - \frac{\lambda}{6} \phi^4$

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. The model illustrates the generation of mass by fluctuations of the vector field. Equivalently one may say that the model possesses a first-order phase transition as a function of m². 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. Interestingly, the three-dimensional version of the Coleman–Weinberg model has both a first and a second-order phase transition depending on the ratio of the Ginzburg–Landau parameter $\kappa\equiv\lambda/3e^2$, with a tricritical point near $\kappa=1/\sqrt 2$ which separates type I from type II superconductivity.

References

• S. Coleman and E. Weinberg (1973). "Radiative Corrections as the Origin of Spontaneous Symmetry Breaking". Physical Review D 7: 1888. Bibcode 1973PhRvD...7.1888C. doi:10.1103/PhysRevD.7.1888. 
• L.D. Landau (1937). Zhurnal Eksperimental'noi i Teoreticheskoi Fiziki 7: 627. 
• V.L. Ginzburg and L.D. Landau (1950). Zhurnal Eksperimental'noi i Teoreticheskoi Fiziki 20: 1064. 
• M.Tinkham (2004). Introduction to Superconductivity. Dover Books on Physics (2nd ed.). Dover. ISBN 0-486-43503-2.