Dynamical determination of the unification scale by gauge-mediated supersymmetry breaking

We propose a mechanism for generating the grand-unification (GUT) scale dynamically from the Planck scale. The idea is that the GUT scale is fixed by the vacuum expectation value of a “GUT modulus” field whose potential is exactly flat in the supersymmetric limit. If supersymmetry is broken by gauge mediation, a potential for the GUT modulus is generated at 2 loops, and slopes away from the origin for a wide range of parameters. This potential is stabilized by Planck-suppressed operators in the Kähler potential, and the GUT scale is fixed to be of order $M_{*}/\left(4\mathrm{}{\pi }^2\right)$ (where $M_{*}\sim {10}^{18}\mathrm{GeV}$ is the reduced Planck scale) independently of the supersymmetry breaking scale. The cosmology of this scenario is acceptable if there is an epoch of inflation where the inflationary energy density is small compared to the supersymmetry-breaking scale. We construct a realistic GUT that realizes these ideas. The model is based on the gauge group SU(6), and solves the doublet-triplet splitting problem by a sliding singlet mechanism. The GUT sector contains no dimensionful couplings or tuned parameters, and all mass scales other than the Planck scale are generated dynamically. This model can be viewed as a realistic implementation of the inverted hierarchy mechanism.