Consequences of the complex character of the internal symmetry in supersymmetric theories

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_z^cG_z^c is always degenerate with a vacuum associated to a pointz, whereG_z^c=G_z^c; 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_z^c=G_z^c. 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.Partially supported by the Swiss National Science Foundation and INFN, Sezione di PadovaOn leave of absence from the Department of Physics of the University of Padova, Italy