Abstract
In this paper we consider the bosonic sector of the electroweak theory. It has been shown in the work of Ambjorn and Olesen that when the Higgs mass equals to the mass of theZ boson, the model in two dimensions subject to the 't Hooft periodic boundary condition may be reduced to a Bogomol'nyi system and that the solutions of the system are vortices in a “dual superconductor”. We shall prove using a constrained variational reformulation of the problem the existence of such vortices. Our conditions for the existence of solutions are necessary and sufficient when the vortex numberN=1,2.
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References
Abrikosov, A. A.: On the magnetic properties of superconductors of the second group. Sov. Phys. JETP5, 1174–1182 (1957)
Ambjorn, J., Olesen, P.: Anti-screening of large magnetic fields by vector bosons. Phys. Lett.B214, 565–569 (1988)
Ambjorn, J., Olesen, P.: A magnetic condensate solution of the classical electroweak theory. Phys. Lett.B218, 67–71 (1989)
Ambjorn, J., Olesen, P.: On electroweak magnetism. Nucl. Phys.B315, 606–614 (1989)
Ambjorn, J., Olesen, P.: A condensate solution of the electroweak theory which interpolates between the broken and the symmetry phase. Nucl. Phys.B330, 193–204 (1990)
Aubin, T.: Meilleures constantes dans le théorème d'inclusion de Sobolev et un théoréme de Fredholm non linéaire pour la transformation conforme de la courbure scalaire. J. Funct. Anal.32, 148–174 (1979)
Aubin, T.: Nonlinear Analysis on Manifolds: Monge-Ampére Equations. Berlin, Heidelberg, New York: Springer 1982.
Jaffe, A., Taubes, C. H.: Vortices and Monopoles. Boston: Birkhäuser 1980
Kazdan, J. L., Warner, F. W.: Curvature functions for compact 2-manifolds. Ann. Math.99, 14–47 (1974)
Kibble, T. W. B.: Some implications of a cosmological phase transition. Phys. Rep.67, 183–199 (1980)
Skalozub, V. V.: Abrikosov lattice in the theory of electroweak interactions. Sov. J. Nucl. Phys.43, 665–669 (1986)
Skalozub, V. V.: The structure of vacuum in the Weinberg-Salan theory. Sov. J. Nucl. Phys.45, 1058–1064 (1987)
t'Hooft, G.: A property of electric and magnetic flux in nonabelian gauge theories. Nucl. Phys.B153, 141–160 (1979)
Wang, S., Yang, Y.: Abrikosov's vortices in the critical coupling, preprint, 1990
Yang, Y.: Existence of the massiveSO(3) vortices. J. Math. Phys.32, 1395–1399 (1991)
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Communicated by A. Jaffe
Research supported in part by NSF grant DMS-88-02858 and DOE grant DE-FG02-86ER250125
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Spruck, J., Yang, Y. On multivortices in the electroweak theory I: Existence of periodic solutions. Commun.Math. Phys. 144, 1–16 (1992). https://doi.org/10.1007/BF02099188
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DOI: https://doi.org/10.1007/BF02099188