Supergravity Resolution of the Unification to Planck Scale Hierarchy

It is demonstrated how the hierarchy between the gauge coupling unification scale of minimal supersymmetry and the Planck (or string) scale, which resembles in order of magnitude a loop factor, can actually be explained as such in supergravity-coupled supersymmetry. A gauge and global singlet field acquires a linear potential term due to its one-loop supergravity interactions and slides to the desired scale. The singlet field can then provide the seed for the breaking of the unified theory at the appropriate scale via its couplings to fields in the adjoint representation. The minimal supersymmetric extension of the Standard Model (the MSSM) of electroweask and strong interactions is well known to be consistent with the unification of the electroweak and strong couplings at a scale MG ≃ 3 × 10 16 GeV. Given the current measured values of the couplings, the unification holds at the percentile level (in the units of the unified coupling αG ∼ 0.04) with only O(1) ambiguity in the unification scale (for example, see Refs. [1, 2]). And because the apparent unification scale is well below the Planck scale, Planck-suppressed corrections are sufficiently small that one can trust the field theory calculation [1]. While an impressive result, this particular scale for unification is poorly understood. Specifically, the unification scale lies two orders of magnitude below the (reduced) Planck scale, MP , and an order of magnitude below the predicted unification scale in perturbative heterotic string theory, ∼ 5 × 10 GeV [3]. Within the context of string theory, there are a number of proposals for alleviating this discrepancy, including extra matter at intermediate scales, altered unification conditions, and nonperturbative/M-theory effects (for a review, see Ref. [4].) Alternatively, there may be a true grand-unified theory (GUT) in the decades between the Planck (or string) scale and the phenomenologically determined unification scale. However, in this latter case, one is usually forced to introduce the scale of GUTbreaking (i.e., the unification scale) as an additional fundamental scale in the problem. It would clearly be preferable to find some mechanism by which one or more of the seemingly fundamental scales in the theory (the Planck scale, the GUT scale, the supersymmetry-breaking scale) could be derived from the others. There are already well-motivated explanations of the supersymmetry-breaking scale as the strong coupling scale of some new gauge interaction, replacing it as a fundamental scale in favor of the Planck scale and an O(1) gauge coupling [6]. In this paper we will derive a mechanism by which the GUT scale can in turn be extracted as a function of the Planck scale, once supersymmetry is broken. Several models already exist in the literature for doing just this. One of the earliest is the “inverted hierarchy” model of Witten [7]. In this toy model the only fundamental scale is the scale of supersymmetry-breaking. The GUT is broken at tree-level at a scale determined by the vacuum expectation value (vev) of a gauge singlet. That singlet, however, is undetermined at tree-level and only later fixed by logarithmically divergent corrections to the potential. Because of the logarithms, the GUT scale is exponentially far from the supersymmetry-breaking scale. Other more realistic models have also relied on logarithmically divergent contributions to generate the GUT scale, but down from the Planck scale instead of up from the supersymmetry-breaking scale. Since the GUT scale is so close to the Planck scale, though, the exponential hierarchy must be arranged to be small. The model of Goldberg [8] generates the GUT scale through the vev of a singlet when its supersymmetry-breaking mass-squared is driven 1 The actual embedding of a unified theory into a string theory is not straightforward and will not be addressed here. However recent results on non-perturbative solutions to string theories (e.g., F-theory [5]) show, in principle, tremendous flexibility in the embeddings that can be arranged.