Upper bounds on supersymmetry breaking from gauge coupling unification

I derive conservative upper bounds on the supersymmetry breaking parameter m1/2 as a function of the strong coupling in the Standard Supersymmetric Model (SSM) using gauge coupling unification. I find that over more than 99% of the parameter space, α3 > 0.120 implies that m1/2 is below 10TeV and α3 > 0.129 implies that m1/2 is below 1TeV. I express the variation of these bounds over the SSM parameter space with a numerical coefficient, c. I also find that in the SSM, a reasonable value of 50GeV < m1/2 < 1TeV requires α3 > 0.119 over the whole parameter space. These bounds are particularly sensitive to the value of sin θW = 0.2317 ± 0.0005 used in the calculation. In more realistic models, heavy thresholds and gravitational effects will modify this result. Although these effects are theoretically calculable in specific models, more realistic models contain many unknown parameters in practice. I illustrate this point with minimal supersymmetric SU(5) where the combined constraints of gauge coupling unification and proton decay require α3 > 0.119 for m1/2 < 1TeV and the upper bound on the supersymmetry breaking scale is greatly relaxed. MIU-THP-71/95 February 1995 Gauge coupling unification [1] applied to precision LEP measurements has provided strong motivation for supersymmetry, [2] and inspired numerous attempts to extract the supersymmetry breaking scale and constrain the parameter space of specific models [3]. However, definite conclusions only result from very specific models: threshold effects and gravitational corrections make model-independent statements difficult [4]. In this paper, I find interesting constraints from gauge coupling unification within the specific framework of the Standard Supersymmetric Model (SSM) as defined in reference [5] and the minimal supersymmetric SU(5) model. Although these models are an excellent beginning, they can hardly be considered ultimate theories as neither include gravity and the fine-tuning problem in minimal supersymmetric SU(5) requires some modification of the GUT structure. However, there is hope that similar constraints could be derived for specific realistic string models where the additional threshold and gravitational effects are in principle calculable. Direct searches for supersymmetric particles have continually increased the lower bounds on their masses. However, the only upper bounds on supersymmetric masses come from naturalness arguments [6] or cosmological constraints on the LSP relic density [7]. Although compelling, the naturalness bounds are not rigorous, and whether the upper bound on supersymmetric masses is 1 TeV, 10TeV or even 100TeV is not clear and somewhat a matter of taste. The cosmological bounds can be evaded, for example by breaking R-parity. It would be extremely useful to have some other method of bounding the supersymmetric masses. In this paper, I focus on a first step in this direction by deriving an upper bound on the soft supersymmetry breaking parameter m1/2 as a function of the strong coupling in the Standard Supersymmetric Model (SSM). Our approach attempts a general and analytic analysis to reveal the underlying physics, and is meant to complement the many numerical searches of the SSM parameter space in the literature [3]. Unless otherwise indicated, our notation assumes all gauge couplings are renormalized at mZ in the MS scheme. Ignoring gravitational effects, gauge coupling unification in the SSM gives a simple prediction for the soft supersymmetry breaking parameter m1/2. Restricting attention to values of m1/2 for which all the supersymmetric thresholds are above mZ , this prediction is [8]: ln( m1/2 mZ ) = −X + 7π α3 + 7 ln [