Abstract
The consequences of the invariance of the superpotential under the complexificationG c of the internal symmetry group on the determination of the possible patterns of symmetry and supersymmetry breaking are established in a globally supersymmetric theory. In particular, in the case of global internal symmetry we show that a vacuum associaated to a pointz, whereG cz ≠G cz is always degenerate with a vacuum associated to a pointz′, whereG cz′ =G cz′ ; all the other degeneracies of the minimum of the potential on an orbit ofG c are also determined and shown to be completely removed when the internal symmetry is gauged. The zeroes of theD-term of a supersymmetric gauge theory are characterized as the points of the closed orbits ofG c which are at minimum distance from the origin; at these pointsG cz =G cz . It is rigorously proved that the minimum of the potential is zero if the gradient of the superpotential vanishes somewhere. It is also shown that theD-term necessarily vanishes at the minimum of the potential if the direction of spontaneous supersymmetry breaking is invariant byG.
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We refer to two recent books: Wess, J., Bagger, J.: Supersymmetry and supergravity. Princeton, NJ: Princeton University Press 1983
Gates, G., Jr., Grisaru, M.T., Roček, M., Siegel, W.: Superspace. Reading, MA: Benjamin Cummings 1983
See, for instance, Zumino, B.: Int. Conf. on high energy physics (Lisbon, July 1981), CERN preprint 3167, LBL preprint 13691 and references therein
Ovrut, B., Wess, J.: SupersymmetryR ξ gauge and radiative symmetry breaking. Phys. Rev.D 25, 409 (1982)
Bagger, J., Wess, J.: Op cit.
Gatto, R., Sartori, G.: Gauge symmetry breaking in supersymmetric gauge theories: Necessary and sufficient condition. Phys. Lett.118, 79 (1982), and Zeros of theD-term and complexification of the gauge group in supersymmetric theories.157 B, 389 (1985)
Girardi, G., Sorba, P., Stora, R.: Comments on the spontaneous symmetry breaking in supersymmetric theories. Phys. Lett.144, 212 (1984)
Procesi, C., Schwarz, G.W.: The geometry of orbit spaces and gauge symmetry breaking in supersymmetric gauge theories. Phys. Lett.161, 117 (1985)
Borel, A.: Linear algebraic groups. New York: Benjamin 1969
Humphreys, J.G.: Linear algebraic groups. Graduate texts in mathematics, Vol. 21. Berlin, Heidelberg, New York: Springer 1975
Fogarty, J.: Invariant theory. New York: Benjamin 1969
Mumford, D.: Geometric invariant theory. Erg. Math., Bd. 34. Berlin, Heidelberg, New York: Springer 1965
Buchmüller, W., Peccei, R.D., Yanagida, T.: Quasi Nambu-Goldstone fermions. Nucl. Phys.B 227, 503 (1983), and references therein
Barbieri, R., Masiero, A., Veneziano, G.: Hierarchy of fermion masses in supersymmetric composite models. Phys. Lett.128, 179 (1983)
Lerche, W.: On Goldstone fields in supersymmetric theories. Nucl. Phys.B 238, 582 (1984)
Sharatchandra, H.S.: Max-Planck-Institute preprint MPI-PAE/Pth 2/84
Kugo, T., Ojima, I., Yanagida, T.: Superpotential symmetries and pseudo Nambu-Goldstone supermultiplets. Phys. Lett.135, 402 (1984)
Bando, M., Kuramoto, T., Maskawa, T., Uehara, S.: Structure of non-linear realization in supersymmetric theories. Phys. Lett.138, 94 (1984)
Lee, C., Sharatchandra, H.S.: Max-Planck-Institute preprint PAE/Pth 54/83
Lerche, W.: See [10],
Bando, M., Kuramoto, T., Maskawa, T., Uehara, S.: See [10]
Shore, G.: Generalizations of Dashen's formula in supersymmetric QCD. Nucl. Phys.B231, 139 (1984), and Supersymmetric Higgs mechanism with non-doubled Goldstone bosons.B 248, 123 (1984)
Goiti, J.L.: Mass generation in theories with quasi-Goldstone fermions. Nucl. Phys.B 261, 66 (1985)
Kempf, G., Ness, L.: In: Algebraic geometry. Lecture Notes in Mathematics, Vol.732. Berlin, Heidelberg, New York: Springer 1978
Matsushima, Y.: Nagoya Math. J.16, 215 (1960)
Luna, D.: Slices étales. Bull. Soc. Math. France, Mémoire33, 81 (1973)
Abud, M., Sartori, G.: The geometry of orbit-space and natural minima of Higgs potentials. Phys. Lett.104 B, 147 (1981), and The geometry of spontaneous symmetry breaking. Ann. Phys.150, 307 (1983)
Schwarz, G. W.: Lifting smooth homotopies of orbit spaces. Inst. Hautes Etudes Sci. Publ. Math. (1980)
Gatto, R., Sartori, G.: Relies of supersymmetry in the mass spectrum after spontaneous breaking. Int. J. Mod. Phys. A1, 683 (1986)
Zumino, B.: In: Unified theories of elementary particles. Breitenlohner, P., Dürr, H.P. (eds.). Berlin, Heidelberg, New York: Springer 1982
Fayet, P., Iliopoulos, J.: Spontaneously broken supergauge symmetries and Goldstone spinors. Phys. Lett.51 B, 461 (1974)
Buccella, F., Derendinger, J.P., Ferrara, S., Savoy, C.A.: Patterns of symmetry breaking in supersymmetric gauge theories. Phys. Lett.115 B, 375 (1982)
Luna, D.: Invent. Math.29, 231 (1975)
Sartori, G.: A theorem on orbit structures (strata) of compact linear Lie groups. J. Math. Phys.24, 765 (1983)
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Communicated by L. Alvarez-Gaumé
Partially supported by the Swiss National Science Foundation and INFN, Sezione di Padova
On leave of absence from the Department of Physics of the University of Padova, Italy
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Gatto, R., Sartori, G. Consequences of the complex character of the internal symmetry in supersymmetric theories. Commun.Math. Phys. 109, 327–352 (1987). https://doi.org/10.1007/BF01215226
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DOI: https://doi.org/10.1007/BF01215226