Soft SUSY Masses and the Dynamical Determination of the Gravitino Mass

We discuss in detail the possibility of determining dynamicaly the gravitino mass m3/2, which is related to the supersymmetry breaking scale, within the minimal supersymmetric standard model (MSSM). Using the complete MSSM spectrum, we minimize the vacuum energy including oneloop corrections and a cosmological term of O(m3/2) induced by the underlying fundamental theory. We find that both terms are necessary to determine dynamically the gravitino mass. Other useful constraints for the low energy phenomenology are also obtained. IOA 313/95 NTUA 47/95 January 1995 (∗)On leave of absence from Physics Department, University of Ioannina, 451 10 Ioannina, Greece It is widely believed that the only plausible solution to the gauge hierarchy problem is N=1 local Supersymmetry [1]. The gauge hierarchy problem arises from quadratically divergent one-loop corrections to the effective potential, those being of the form (Λ2StrM2/(32π2)), where Λ is the momentum cut-off, while StrM(z, z̄) = ∑ n (−1)(2sn + 1)mn(z, z̄) (1) The sum is over all particles with field-dependent masses squared mn and spin sn. Since M2 contains also the Higgs mass-squared, this term induces a divergent contribution destabilising the hierarchy MW << MP l, where MP l is the Planck mass. In the spontaneously broken N=1 local Supersymmetry the StrM2, which appears as a coefficient of the one-loop quadratically divergent contributions, is given in terms of the field dependent gravitino mass m3/2 by the formula [2] StrM = 2Q(z, z̄)m3/2(z, z̄) (2) where the dimensionless functionQ(z, z̄) depends on the fields z and z̄ through the Riçi tensor of the Kähler manifold and the function fab(z, z̄) which determines the kinetic terms of the vector supermultiplets as well as the gauge coupling constants. In the fundamental theory of quantum gravity the non–vanishing of Q(z, z̄) would imply corrections to the effective potential of the order O(m3/2M P l) which cannot be cancelled by any contribution of low energy physics. The gravitino mass is given by m3/2(z, z̄) =| W(z) | e (3) where W(z) is the superpotential. The value of m3/2 is related to the scale of supersymmetry breaking which should not be much larger than the electroweak breaking scale. Since m3/2(z, z̄) is field dependent, its vacuum expectation value (vev) should arise from the minimization of the potential. Then, quadratically divergent loop corrections proportional to StrM2 will induce either m3/2 → 0 (unbroken supersymmetry) or m3/2 → MP l, therefore destabilizing again the hierarchy. A possible solution to the hierarchy problem requires the vanishing of Q(z, z̄) which motivated the no-scale supergravity models[3]. A further step towards this problem has been taken the last few years by going beyond the N=1 local Supersymmetry, the Superstring theory. In the context of the latter, and in particular of their four-dimensional version [4], the effective supergravity theory is