G. PANOTOPOULOS

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378

fermion. All the couplings in the model are given in

terms of the electric charge. It is interesting that there is a

quartic self-interaction coupling for the scalar fields even

in the absence of a coupling

. Within the one-loop

renormalization program we give the expression for the

wave-function renormalization, and according to the

standard prescription we compute the critical exponents

,

from the beta function and the anomalous dimen-

sions. Finally, we have discussed spontaneous supersym-

metry breaking a la Fayet-Iliopoulos mechanism. There

is a case in which both supersymmetry and gauge sym-

metry can be broken at the same time. The photon ac-

quires a non-vanishing mass, and the Landau-Ginzburg

parameter is computed. We find that its value corre-

sponds to type II superconductors. Our values of the

Ginzburg parameter and of the critical exponents are

similar to the ones obtained in [4], without many fer-

mions and without the introduction of a second coupling

constant for the scalar quartic self-interaction.

5. Acknowledgments

The author acknowledges financial support from FPA

2008-02878 and Generalitat Valenciana under the grant

PROMETEO/2008/004.

6. References

[1] V. L. Ginzburg and L. D. Landau, “On the Theory of

Superconductivity,” Zhurnal Eksperimental’noi i Teo-

reticheskoi Fiziki, Vol. 20, 1950, pp. 1064-1082.

[2] B. I. Halperin, T. C. Lubensky and S. K. Ma, “First-

Order Phase Transitions in Superconductors and

Smectic-A Liquid Crystals,” Physical Review Letters,

Vol. 32, 1974, pp. 292-295. J. H. Chen, T. C. Lubensky

and D. R. Nelson, “Crossover near Fluctuation-Induced

First-Order Phase Transitions in Superconductors,”

Physical Review B, Vol. 17, No. 11, 1978, pp. 4274-4286.

doi:10.1103/PhysRevB.17.4274

[3] I. D. Lawrie, “On the Phase Transitions in Abelian Higgs

Models,” Nuclear Physics B, Vol. 200, No. 1, 1982, pp.

1-19. doi:10.1016/0550-3213(82)90055-4

I. F. Herbut and Z. Tesanovic, “Critical Fluctuations in

Superconductors and the Magnetic Field Penetration

Depth,” Physical Review Letters, Vol. 76, No. 24, 1996,

pp. 4588-4591. doi:10.1103/PhysRevLett.76.4588

[4] H. Kleinert and F. S. Nogueira, “Critical Behavior of the

Ginzburg-Landau Model Coupled to Massless Dirac Fer-

mions,” Physical Review B, Vol. 66, No. 1, 2002, p.

012504. doi:10.1103/PhysRevB.66.012504

[5] J. B. Marston, “U(1) Gauge Theory of the Heisenberg

Antiferromagnet,” Physical Review Letters, Vol. 61, No.

17, 1988, pp. 1914-1917.

doi:10.1103/PhysRevLett.61.1914

J. B. Marston and I. Affleck, “Large-n Limit of the Hub-

bard-Heisenberg Model,” Physical Review B, Vol. 39, No.

16, 1989, pp. 11538-11558.

doi:10.1103/PhysRevB.39.11538

D. H. Kim and P. A. Lee, “Theory of Spin Excitations in

Undoped and Underdoped Cuprates,” Annals of Physics,

Vol. 272, No. 1, 1999, pp. 130-164.

doi:10.1006/aphy.1998.5888

[6] H. Kleinert, “Disorder Version of the Abelian Higgs

Model and the Order of the Superconductive Phase Tran-

sition,” Lette re Al Nuov o Ciment o, Vol. 35, No. 13, 1982,

pp. 405-412. doi:10.1007/BF02754760

[7] S. Mo, J. Hove and A. Sudbo, “Order of the

Metal-to-Superconductor Transition,” Physical Review B,

Vol. 65, No. 10, 2002, p. 104501.

doi:10.1103/PhysRevB.65.104501

[8] M. Le Bellac, “Quantum and Statistical Field Theory,”

Oxford University Press, Oxford, 1992.

[9] H. Kleinert and V. Schulte-Frohlinde, “Critical Phenom-

ena in 4

-Theory,” World Scientific, Singapore, 2001.

http://www.physik.fu-berlin.de/kleinert/b8

[10] W. Hollik, E. Kraus and D. Stockinger, “Renormalization

and Symmetry Conditions in Supersymmetric QED,”

European Physical Journal C, Vol. 11, No. 2, 1999, pp.

365-381. doi:10.1007/s100529900216

[11] S. Ferrara and O. Piguet, “Perturbation Theory and Re-

normalization of Supersymmetric Yang-Mills Theories,”

Nuclear Physics B, Vol. 93, No. 2, 1975, pp. 261-302.

doi:10.1016/0550-3213(75)90573-8

[12] J. D. Bjorken and S. D. Drell, “Relativistic Quantum

Mechanics,” McGraw-Hill, New York, 1998.

[13] F. S. Nogueira and H. Kleinert, “Field Theoretical Ap-

proaches to the Superconducting Phase Transition,” The

Smithsonian/NASA Astrophysics D ata System.

[14] M. E. Peskin and D. V. Schroeder, “An Introduction to

Quantum Field Theory (Frontiers in Physics),” Westview

Press, New York, 1995.

M. Srednicki, “Quantum Field Theory,” Cambridge Uni-

versity Press, Cambridge, 2007.

[15] G. ‘t Hooft and M. J. G. Veltman, “Regularization and

Renormalization of Gauge Fields,” Nuclear Physics B,

Vol. 44, No. 1, 1972, pp. 189-213.

doi:10.1016/0550-3213(72)90279-9

[16] M. E. Machacek and M. T. Vaughn, “Two-Loop Renor-

malization Group Equations in a General Quantum Field

Theory: (I). Wave Function Renormalization,” Nuclear

Physics B, Vol. 222, No. 1, 1983, pp. 83-103.

doi:10.1016/0550-3213(83)90610-7

[17] J. Wess and J. Bagger, “Supersymmetry and Supergrav-

ity,” Princeton University Press, New Jersey, 1992.

[18] P. Fayet and J. Iliopoulos, “Spontaneously Broken Super-

gauge Symmetries and Goldstone Spinors,” Physics Let-

ters B, Vol. 51, No. 5, 1974, pp. 461-464.

doi:10.1016/0370-2693(74)90310-4