Coleman-Weinberg potential

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

$L\; =\; -frac\{1\}\{4\}\; (F\_\{mu\; u\})^2\; +\; (D\_\{mu\}\; phi)^2\; -\; m^2\; phi^2\; -\; frac\{lambda\}\{6\}\; phi^4$

where the scalar field is complex, $F\_\{mu\; u\}=partial\_mu\; A\_\; u-partial\_\; u\; A\_mu$ 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 $m^2$. 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. 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 $kappaequivlambda/3e^2$, with a
tricritical point near $kappa=1/sqrt\; 2$ which separates type I from type II superconductivity.

References

*S. Coleman and E. Weinberg, Phys. Rev. D7, 1888 (1973). [http://prola.aps.org/abstract/PRD/v7/i6/p1888_1]

*L.D. Landau, Zh. Elsp. Teor. Fiz. 7, 627 (1937).

*V.L. Ginzburg and L.D. Landau, ibid. 20, 1064 (1950).

*cite book | author=Michael Tinkham | title=Introduction to Superconductivity | edition = 2^nd ed. | publisher=Dover Books on Physics | year=2004 | id=ISBN 0-486-43503-2 (Paperback)
*Hagen Kleinert, Lett. Nuovo Cimento {f 35}, 405 (1982) (http://www.physik.fu-berlin.de/~kleinert/97/97.pdf).

*Hagen Kleinert, "Gauge Fields in Condensed Matter", Vol. I, " SUPERFLOW AND VORTEX LINES", pp. 1--742, Vol. II, "STRESSES AND DEFECTS", pp. 743-1456, [http://www.worldscibooks.com/physics/0356.htm World Scientific (Singapore, 1989)] ; Paperback ISBN 9971-5-0210-0 " (also available online: [http://www.physik.fu-berlin.de/~kleinert/kleiner_reb1/contents1.html Vol. I] and [http://www.physik.fu-berlin.de/~kleinert/kleiner_reb1/contents2.html Vol. II] )"

*Hagen Kleinert, "Limitations on Coleman-Weinberg Mechanism", Phys. Lett. B 128, 69 (1983) (http://www.physik.fu-berlin.de/~kleinert/106/106.pdf).